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classes ::: Cybernetics, Cybernetics, or Control and Communication in the Animal and the Machine, Norbert Wiener, chapterThere is a large class of phenomena in which what is observed is,
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object:1.03 - Time Series, Information, and Communication
subject class:Cybernetics
book class:Cybernetics, or Control and Communication in the Animal and the Machine
author class:Norbert Wiener
class:chapterThere is a large class of phenomena in which what is observed is
a numerical quantity, or a sequence of numerical quantities, dis-
tributed in time. The temperature as recorded by a continuous
recording thermometer, or the closing quotations of a stock in
the stock market, taken day by day, or the complete set of meteo-
rological data published from day to day by the Weather Bureau
are all time series, continuous or discrete, simple or multiple.
These time series are relatively slowly changing, and are well
suited to a treatment employing hand computation or ordinary
numerical tools such as slide rules and computing machines.
Their study belongs to the more conventional parts of statistical
theory.
What is not generally realized is that the rapidly changing
sequences of voltages in a telephone line or a television circuit
or a piece of radar apparatus belong just as truly to the field
of statistics and time series, although the apparatus by means
of which they are combined and modified must in general be
very rapid in its action, and in fact must be able to put out
results pari passu with the very rapid alterations of input. These
pieces of apparatus—­telephone receivers, wave filters, automatic
sound-­coding devices like the Vocoder of the Bell Telephone86
Chapter III
Laboratories, frequency-­modulating networks and their corre-
sponding receivers—­are all in essence quick-­acting arithmetical
devices, corresponding to the whole apparatus of computing
machines and schedules, and the staff of computers, of the sta-
tistical laboratory. The ingenuity needed in their use has been
built into them in advance, just as it has into the automatic
range finders and gun pointers of an anti-­aircraft fire-­control
system, and for the same reasons. The chain of operation has to
work too fast to admit of any human links.
One and all, time series and the apparatus to deal with them,
whether in the computing laboratory or in the telephone circuit,
have to deal with the recording, preservation, transmission, and
use of information. What is this information, and how is it mea-
sured? One of the simplest, most unitary forms of information is
the recording of a choice between two equally probable simple
alternatives, one or the other of which is bound to happen—­a
choice, for example, between heads and tails in the tossing of a
coin. We shall call a single choice of this sort a decision. If then
we ask for the amount of information in the perfectly precise
measurement of a quantity known to lie between A and B, which
may with uniform a priori probability lie anywhere in this range,
we shall see that if we put A = 0 and B = 1, and represent the
quantity in the binary scale by the infinite binary number. a 1 a 2
a 3 ... a n ..., where a 1 , a 2 , ..., each has the value 0 or 1, then the
number of choices made and the consequent amount of infor-
mation is infinite. Here
.a 1 a 2 a 3 a n =
1
1
1
1
a 1 + 2 + 2 a 2 + + n a n +
2
2
2
2
(3.01)
However, no measurement which we actually make is
performed with perfect precision. If the measurement has aTime Series, Information, and Communication
87
uniformly distributed error lying over a range of length · b 1 b 2 ...
b n ..., where b k is the first digit not equal to 0, we shall see that all
the decisions from a 1 to a k−1 , and possibly to a k , are significant,
while all the later decisions are not. The number of decisions
made is certainly not far from
− log 2 . b 1 b 2 b n
(3.02)
and we shall take this quantity as the precise formula for the
amount of information and its definition.
We may conceive this in the following way: we know a priori
that a variable lies between 0 and 1, and a posteriori that it lies on
the interval (a, b) inside (0, 1). Then the amount of information
we have from our a posteriori knowledge is
− log 2
measure of ( a , b )
measure of ( 0 , 1 )
(3.03)
However, let us now consider a case where our a priori knowledge
is that the probability that a certain quantity should lie between
x and x + dx is f 1 (x) dx, and the a posteriori probability is f 2 (x)
dx. How much new information does our a posteriori probability
give us?
This problem is essentially that of attaching a width to the
regions under the curves y = f 1 (x) and y = f 2 (x). It will be noted
that we are here assuming the variable to have a fundamental
equipartition; that is, our results will not in general be the same
if we replace x by x 3 or any other function of x. Since f 1 (x) is a
probability density, we shall have
∫
∞
−∞
f 1 ( x ) dx = 1
(3.04)
so that the average logarithm of the breadth of the region under
f 1 (x) may be considered as some sort of average of the height
of the logarithm of the reciprocal of f 1 (x). Thus a reasonable88
Chapter III
measure 1 of the amount of information associated with the
curve f 1 (x) is
∞
∫ [ log
−∞
2
f 1 ( x ) ] f 1 ( x ) dx
(3.05)
The quantity we here define as amount of information is the
negative of the quantity usually defined as entropy in similar
situations. The definition here given is not the one given by R. A.
Fisher for statistical problems, although it is a statistical defini-
tion; and can be used to replace Fisher’s definition in the tech-
nique of statistics.
In particular, if f 1 (x) is a constant over (a, b) and is zero
elsewhere,
∞
∫ [ log
−∞
2
f 1 ( x ) ] f 1 ( x ) dx =
b − a
1
1
log 2
= log 2
b − a
b − a
b − a
(3.06)
Using this to compare the information that a point is in the
region (0, 1) with the information that it is in the region (a, b),
we obtain for the measure of the difference
log 2
1
1
− log 2 1 = log 2
b − a
b − a
(3.07)
The definition which we have given for the amount of infor-
mation is applicable when the variable x is replaced by a variable
ranging over two or more dimensions. In the two-­dimensional
case, f(x, y) is a function such that
∫
∞
−∞
∞
dx ∫ dyf 1 ( x , y ) = 1
−∞
(3.08)
and the amount of information is
∫
∞
−∞
∞
dx ∫ dyf 1 ( x , y ) log 2 f 1 ( x , y )
−∞
(3.081)Time Series, Information, and Communication
89
Note that f 1 (x, y) is of the form φ(x)ψ(y) and
∫
∞
∞
−∞
φ ( x ) dx = ∫ ψ ( y ) dy = 1
(3.082)
−∞
then
∫
∞
∞
−∞
dx ∫ dy φ ( x ) ψ ( y ) = 1
(3.083
−∞
and
∫
∞
∞
−∞
dx ∫ dyf 1 ( x , y ) log 2 f 1 ( x , y )
−∞
∞ ∞
−∞ −∞
(3.084)
= ∫ dx φ ( x ) log 2 φ ( x ) + ∫ dy ψ ( y ) lo g 2 ψ ( y )
and the amount of information from independent sources
is additive. An interesting problem is that of determining the
information gained by fixing one or more variables in a prob-
lem. For example, let us suppose that a variable u lies between
x and x + dx with the probability exp ( − x 2 2 a ) dx
2 π a , while a
variable v lies between the same two limits with a probability
exp ( − x 2 2 b ) dx
2 π b . How much information do we gain con-
cerning u if we know that u + v = w? In this case, it is clear that
u = w − v, where w is fixed. We assume the a priori distributions
of u and v to be independent. Then the a posteriori distribution
of u is proportional to
 ( w − x ) 2 
 x 2 
2  a + b  

exp  −  exp  −

 = c 1 exp  − ( x − c 2 )  
 2 a 
2
2 ab  
b



(3.09)
where c 1 and c 2 are constants. They both disappear in the for-
mula for the gain in information given by the fixing of w.
The excess of information concerning x when we know w to
be that which we have in advance is90
Chapter III
1
∫
2 π [ ab ( a + b ) ]
∞
−∞
{
2 a + b  
exp   − ( x − c 2 )  
 2 ab    

}
1
a b  
a + b  
2

×   − log 2 2 π  
− ( x − c 2 )    
 log 2 e  dx
 a + b    
 2
  2 ab   

∞ 
1
x 2
 x 2    1

e xp  −    − log 2 2 π a −
log 2 e  dx
−
 2 a    2

2 a
2 π a ∫ −∞  
=
(3.091)
1
a + b 
lo g 2  


2
b 
Note that this expression (Eq. 3.091) is positive, and that it is
independent of w. It is one-­half the logarithm of the ratio of the
sum of the mean squares of u and v to the mean square of v. If v
has only a small range of variation, the amount of information
concerning u which a knowledge of u + v gives is large, and it
becomes infinite as b goes to 0.
We can consider this result in the following light: let us treat
u as a message and v as a noise. Then the information carried
by a precise message in the absence of a noise is infinite. In the
presence of a noise, however, this amount of information is
finite, and it approaches 0 very rapidly as the noise increases in
intensity.
We have said that amount of information, being the negative
logarithm of a quantity which we may consider as a probability,
is essentially a negative entropy. It is interesting to show that, on
the average, it has the properties we associate with an entropy.
Let φ(x) and ψ(x) be two probability densities; then [φ(x) +
ψ(x)]/2 is also a probability density. Then
∫
∞
φ ( x ) + ψ ( x )
φ ( x ) + ψ ( x )
dx
log
2
2
∞ ψ ( x )
∞ φ ( x )
log φ ( x ) dx + ∫
 ∫
log ψ ( x ) dx
−∞
−∞
2
2
−∞
This follows from the fact that
(3.10)Time Series, Information, and Communication
a + b
a + b 1
log
 ( a log a + b log b )
2
2
2
91
(3.11)
In other words, the overlap of the regions under φ(x) and ψ(x)
reduces the maximum information belonging to φ(x) + ψ(x). On
the other hand, if φ(x) is a probability density vanishing outside
(a, b),
∫
∞
−∞
φ ( x ) log φ ( x ) dx
(3.12)
is a minimum when φ(x) = 1/(b − a) over (a, b) and is zero else-
where. This follows from the fact that the logarithm curve is
convex upward.
It will be seen that the processes which lose information are,
as we should expect, closely analogous to the processes which
gain entropy. They consist in the fusion of regions of probability
which were originally distinct. For example, if we replace the
distribution of a certain variable by the distribution of a function
of that variable which takes the same value for different argu-
ments, or if in a function of several variables we allow some of
them to range unimpeded over their natural range of variability,
we lose information. No operation on a message can gain infor-
mation on the average. Here we have a precise application of the
second law of thermodynamics in communication engineering.
Conversely, the greater specification of an ambiguous situation,
on the average, will, as we have seen, generally gain information
and never lose it.
An interesting case is when we have a probability distribu-
tion with n-­fold density f(x 1 , ..., x n ) over the variables (x 1 , ...,
x n ), and where we have m dependent variables y 1 , ..., y m . How
much information do we get by fixing these m variables? First
let them be fixed between the limits y 1 *, y 1 * + dy 1 *; ...; y m *, y m +92
Chapter III
dy m *. Let us take as a new set of variables x 1 , x 2 , ..., x n − m , y 1 , y 2 ,
..., y m . Then over the new set of variables, our distribution func-
tion will be proportional to f(x 1 , ..., x n ) over the region R given
by y 1 *  y 1  y 1 * + dy 1 * , , y m *  y m  y m * + dy m * and 0 outside.
Thus the amount of information obtained by the specification
of the y’s will be
dx ∫ dx f ( x , , x
∫


1
R
1
n
n
) log 2 f ( x 1 , , x n )
, x
dx ∫ dx f ( x ,
∫


1
n
1
n
)
R
 − ∞ dx 1 ∞ dx n f ( x 1 , , x n ) log 2 f ( x 1 , , x n ) 
∫ −∞
 ∫ −∞

− 1


∞
∞
y 1 * , , y m * 



dx 1 ∫ dx n − m J 

∫
−∞
−∞
 x n − m + 1 , , x n 




× f ( x 1 , , x n ) log 2 f ( x 1 , , x n )


= 

− 1
∞
∞
y
y
*
,

,
*
m
1



 dx dx
 ∫ −∞ 1 ∫ − ∞ n − m J   x n − m + 1 , , x n   f ( x 1 , , x n ) 


 − ∞ dx 1 ∞ dx n f ( x 1 , , x n ) log 2 f ( x 1 , , x n ) 
∫
∫
−∞
−∞


 
 
(3.13)
Closely related to this problem is the generalization of that
which we discussed in Eq. 3.13; in the case just discussed,
how much information do we have concerning the variables
x 1 , ..., x n−m alone? Here the a priori probability density of these
variables is
∫
∞
−∞
∞
dx n − m + 1 ∫ dx n f ( x 1 , , x n )
−∞
(3.14)
and the un-­normalized probability density after fixing the y*’s is
 y 1 * , , y m * 

n − m + 1 , , x n 
∑ J  x
− 1
f ( x 1 , , x n )
(3.141)Time Series, Information, and Communication
93
where the ∑ is taken over all sets of points (x n−m+1 , ..., x n ) cor-
responding to a given set of y*’s. On this basis, we may easily
write down the solution to our problem, though it will be a
bit lengthy. If we take the set x 1 , ..., x n−m ) to be a generalized
message, the set (x n−m+1 , ..., x m ) to be a generalized noise, and
the y*’s to be a generalized corrupted message, we see that we
have given the solution of a generalization of the problem of
Expression 3.141.
We have thus at least a formal solution of a generalization
of the message-­noise problem which we have already stated. A
set of observations depends in an arbitrary way on a set of mes-
sages and noises with a known combined distribution. We wish
to ascertain how much information these observations give con-
cerning the messages alone. This is a central problem of commu-
nication engineering. It enables us to evaluate different systems,
such as amplitude modulation or frequency modulation or
phase modulation, as far as their efficiency in transmitting infor-
mation is concerned. This is a technical problem and not suit-
able to a detailed discussion here; however, certain remarks are
in order. In the first place, it can be shown that with the defini-
tion of information given here, with a random “static" on the
ether equidistributed in frequency as far as power is concerned,
and with a message restricted to a definite frequency range and
a definite power output for this range, no means of transmission
of information is more efficient than amplitude modulation,
although other means may be as efficient. On the other hand,
the information transmitted by this means is not necessarily in
the form most suitable for reception by the ear or by any other
given receptor. Here the specific characteristics of the ear and
of other receptors must be considered by employing a theory
very similar to the one just developed. In general, the efficient94
Chapter III
use of amplitude modulation or any other form of modulation
must be supplemented by the use of decoding devices adequate
to transforming the received information into a form suitable for
reception by human receptors or use by mechanical receptors.
Similarly, the original message must be coded for the greatest
compression in transmission. This problem has been attacked,
at least in part, in the design of the “Vocoder" system by the
Bell Telephone Laboratories, and the relevant general theory has
been presented in a very satisfactory form by Dr. C. Shannon of
those laboratories.
So much for the definition and technique of measuring
information. We shall now discuss the way in which informa-
tion may be presented in a form homogeneous in time. Let it
be noted that most of the telephone and other communication
devices are actually not attached to a particular origin in time.
There is indeed one operation which seems to contradict this,
but which really does not. This is the operation of modulation.
This, in its simplest form, converts a message f(t) into one of the
form f(t) sin (at + b). If, however, we regard the factor sin (at +
b) as an extra message which is put into the apparatus, it will be
seen that the situation will come under our general statement.
The extra message, which we call the carrier, adds nothing to the
rate at which the system is carrying information. All the infor-
mation it contains is conveyed in an arbitrarily short interval of
time, and thereafter nothing new is said.
A message homogeneous in time, or, as the statisticians call
it, a time series which is in statistical equilibrium, is thus a single
function or a set of functions of the time, which forms one of an
ensemble of such sets with a well-­defined probability distribu-
tion, not altered by the change of t to t + τ throughout. That is,
the transformation group consisting of the operators T λ whichTime Series, Information, and Communication
95
change f(t) into f(t + λ) leaves the probability of the ensemble
invariant. The group satisfies the properties that
T λ [ T μ f ( t ) ] = T μ + λ f ( t )
 ( −∞ < λ < ∞ )

 ( −∞ < μ < ∞ )
(3.15)
It follows from this that if Φ[f(t)] is a “functional" of f(t)—­that is,
a number depending upon the whole history of f(t)—­and if the
average of f(t) over the whole ensemble is finite, we are in a posi-
tion to use the Birkhoff ergodic theorem quoted in the previous
chapter, and to come to the conclusion that, except for a set of
values of f(t) of zero probability, the time-­average of Φ[f(t)], or in
symbols,
lim
A →∞
1 A
1 0
Φ [ f ( t + τ ) ] d τ = lim ∫ Φ [ f ( t + τ ) ] d τ
A →∞ A − A
A ∫ 0
(3.16)
exists.
There is even more here than this. We have stated in the pre-
vious chapter another theorem of ergodic character, due to von
Neumann, which states that, except for a set of elements of zero
probability, any element belonging to a system which goes into
itself under a group of measure-­preserving transformations such
as Eq. 3.15 belongs to a sub-­set (which may be the whole set)
which goes into itself under the same transformation, which has
a measure defined over itself and also invariant under the trans-
formation, and which has the further property that any portion
of this sub-­set with measure preserved under the group of trans-
formations either has the maximum measure of the sub-­set, or
measure 0. If we discard all elements except those of such a sub-­
set, and use its appropriate measure, we shall find that the time
average (Eq. 3.16) is in almost all cases the average of Φ[f(t)] over
all the space of functions f(t); the so-­called phase average. Thus in
the case of such an ensemble of functions f(t), except in a set of96
Chapter III
cases of zero probability, we can deduce the average of any statis-
tical parameter of the ensemble—­indeed we can simultaneously
deduce any countable set of such parameters of the ensemble—­
from the record of any one of the component time series, by
using a time average instead of a phase average. Moreover, we
need to know only the past of almost any one time series of the
class. In other words, given the entire history up to the present
of a time series known to belong to an ensemble in statistical
equilibrium, we can compute with probable error zero the entire
set of statistical parameters of an ensemble in statistical equilib-
rium to which that time series belongs. Up to here, we have for-
mulated this for single time series; it is equally true, however, for
multiple time series in which we have several quantities varying
simultaneously, rather than a single varying quantity.
We are now in a position to discuss various problems belong-
ing to time series. We shall confine our attention to those cases
where the entire past of a time series can be given in terms of a
countable set of quantities. For example, for quite a wide class of
functions f(t) (−∞ < t < ∞), we have fully determined f when we
know the set of quantities
0
a n = ∫ e t t n f ( t ) dt
−∞
( n = 0 , 1 , 2 , )
(3.17)
Now let A be some function of the values of t in the future, that
is, for arguments greater than 0. Then we can determine the
simultaneous distribution of (a 0 , a 1 , ..., a n , A) from the past of
almost any single time series if the set of f’s is taken in its narrow-
est possible sense. In particular, if a 0 , ..., a n are all given, we may
determine the distribution of A. Here we appeal to the known
theorem of Nikodym on conditional probabilities. The same the-
orem will assure us that this distribution, under very general cir-
cumstances, will tend to a limit as n → ∞ and this limit will giveTime Series, Information, and Communication
97
us all the knowledge there is concerning the distribution of any
future quantity. We may similarly determine the simultaneous
distribution of values of any set of future quantities, or any set
of quantities depending both on the past and the future, when
the past is known. If then we have given any adequate interpre-
tation to the “best value" of any of these statistical parameters or
sets of statistical parameters—­in the sense, perhaps, of a mean
or a median or a mode—­we can compute it from the known dis-
tribution, and obtain a prediction to meet any desired criterion
of goodness of prediction. We can compute the merit of the pre-
diction, using any desired statistical basis of this merit—­mean
square error or maximum error or mean absolute error, and so
on. We can compute the amount of information concerning any
statistical parameter or set of statistical parameters, which fixing
of the past will give us. We can even compute the whole amount
of information which a knowledge of the past will give us of
the whole future beyond a certain point; although when this
point is the present, we shall in general know the latter from the
past, and our knowledge of the present will contain an infinite
amount of information.
Another interesting situation is that of a multiple time series,
in which we know precisely only the pasts of some of the com-
ponents. The distribution of any quantity involving more than
these pasts can be studied by means very similar to those already
suggested. In particular, we may wish to know the distribution of
a value of another component, or a set of values of other compo-
nents, at some point of time, past, present, or future. The general
problem of the wave filter belongs to this class. We have a mes-
sage, together with a noise, combined in some way into a cor-
rupted message, of which we know the past. We also know the
statistical joint distribution of the message and the noise as time98
Chapter III
series. We ask for the distribution of the values of the message at
some given time, past, present, and future. We then ask for an
operator on the past of the corrupted message which will best
give this true message, in some given statistical sense. We may
ask for a statistical estimate of some measure of the error of our
knowledge of the message. Finally, we may ask for the amount of
information which we possess concerning the message.
There is one ensemble of time series which is particularly sim-
ple and central. This is the ensemble associated with the Brown-
ian motion. The Brownian motion is the motion of a particle in
a gas, impelled by the random impacts of the other particles in
a state of thermal agitation. The theory has been developed by
many writers, among them Einstein, Smoluchowski, Perrin, and
the author. 2 Unless we go down in the time scale to intervals so
small that the individual impacts of the particles on one another
are discernible, the motion shows a curious kind of undifferen-
tiability. The mean square motion in a given direction over a
given time is proportional to the length of that time, and the
motions over successive times are completely uncorrelated. This
conforms closely to the physical observations. If we normalize
the scale of the Brownian motion to fit the time scale, and con-
sider only one coordinate x of the motion, and if we let x(t) equal
0 for t = 0, then the probability that if 0  t 1  t 2   t n the
particles lie between x 1 and x 1 + dx 1 at time t 1 , ..., between x n and
x n + dx n at time t n , is
 x 2 ( x − x 1 ) 2
( x − x n − 1 ) 2 
exp  − 1 − 2
− − n
2 ( t n − t n − 1 )  
 2 t 1 2 ( t 2 − t 1 )
dx 1 dx n
( 2 π ) n t 1 ( t 2 − t 1 ) ( t n − t n − 1 )
(3.18)
On the basis of the probability system corresponding to
this, which is unambiguous, we can make the set of pathsTime Series, Information, and Communication
99
corresponding to the different possible Brownian motions
depend on a parameter α lying between 0 and 1, in such a way
that each path is a function x(t, α), where x depends on the time
t and the parameter of distribution α, and where the probability
that a path lies in a certain set S is the same as the measure of
the set of values of α corresponding to paths in S. On this basis,
almost all paths will be continuous and non-­differentiable.
A very interesting question is that of determining the average
with respect to α of x(t 1 , α) ... x(t n , α). This will be
∫
1
0
d α x ( t 1 , α ) x ( t 2 , α ) x ( t n , α )
= ( 2 π )
− n 2
[ t 1 ( t 2 − t 1 ) ( t n − t n − 1 ) ] −
1
2
∞
∞
 ξ 2 ( ξ − ξ 1 ) 2
× ∫ d ξ 1 ∫ d ξ n ξ 1 ξ 2 ξ n exp  − 1 − 2
−
−∞
−∞
 2 t 1 2 ( t 2 − t 1 )
( ξ − ξ n − 1 ) 2 
− n

2 ( t n − t n − 1 ) 
(3.19)
under the assumption that 0  t 1   t n , Let us put
ξ 1 ξ n = ∑ A k ξ 1 λ k , 1 ( ξ 2 − ξ 1 ) λ k , 2 ( ξ n − ξ n − 1 ) λ k , n
(3.20)
where λ k,1 + λ k,2 + ··· + λ k,n = n. The value of the expression in Eq.
3.19 will become
∑ A
k
( 2 π ) − 2   t 1 λ k , 1 ( t 2 − t 1 ) λ k , 2 ( t n − t n − 1 ) λ k , n 
n
− 12
∞


ξ 2
× ∏ ∫ d ξ ξ λ k , j ex p  −

−∞
−
2
t
t
(
)
j
j
−
1
j


1 ∞ λ k , j
− 1
 ξ 2 
ξ exp  −  d ξ ( t j − t j − 1 ) 2
= ∑ A k ∏
∫
−∞
2
2
π


j
 0 if any λ k , j is odd

− 1
= 
A k ∏ ( λ k , j − 1 ) ( λ k , j − 3 ) 5 �
3 �
( t j − t j − 1 ) 2
  ∑
k
j
(3.21)100
Chapter III
if every λ k,j is even,
= ∑ A k ∏ ( number of ways of dividing λ k , j terms into pairs )
k
j
× ( t j − t j − 1 )
1
2
= ∑ A k ( numbers of ways of dividing n terms into pairs
k
whose eleements both belong in the same group of
λ k , j terms into whi c h λ is separated ) × ( t j − t j − 1 )
1
2
1
= ∑ A j ∑ ∏ ∫ d α [ x ( t k , α ) − x ( t k − 1 , α ) ] [ x ( t q , α ) − x ( t q − 1 , α ) ]
j
0
Here the first ∑ sums over j; the second, over all the ways of
dividing n terms in blocks, respectively, of λ k,1 , ..., λ k,n numbers
into pairs; and the ∏ is taken over those pairs of values k and q,
where λ k,1 of the elements to be selected from t k and t q are t 1 , λ k,2
are t 2 , and so on. It immediately results that
∫
1
0
1
d α x ( t 1 , α ) x ( t 2 , α ) x ( t n , α ) = ∑ ∏ ∫ d α x ( t j , α ) x ( t k , α )
0
(3.22)
where the ∑ is taken over all partitions of t 1 , ..., t n into distinct
pairs, and the ∏ over all the pairs in each partition. In other
words, when we know the averages of the products of x(t j , α) by
pairs, we know the averages of all polynomials in these quanti-
ties, and thus their entire statistical distribution.
Up to the present, we have considered Brownian motions x(t,
α) where t is positive. If we put
ξ ( t , α , β ) = x ( t , α ) ( t �
0 )
ξ ( t , α , β ) = x ( − t , β ) ( t < 0 )
(3.23)
where α and β have independent uniform distributions over (0,
1), we shall obtain a distribution of ξ(t, α, β) where t runs overTime Series, Information, and Communication
101
the whole real infinite line. There is a well-­known mathematical
device to map a square on a line segment in such a way that area
goes into length. All we need to do is to write our coordinates in
the square in the decimal form:
α = . α 1 α 2 α n 

β = . β 1 β 2 β n 
(3.24)
and to put
γ = . α 1 β 1 α 2 β 2 α n β n
and we obtain a mapping of this sort which is one-­one for almost
all points both in the line segment and the square. Using this
substitution, we define
ξ ( t , γ ) = ξ ( t , α , β )
(3.25)
We now wish to define
∫
∞
−∞
K ( t ) d ξ ( t , γ )
(3.26)
The obvious thing would be to define this as a Stieltjes 3 integral,
but ξ is a very irregular function of t and does not make such
a definition possible. If, however, K runs sufficiently rapidly to
0 at ± ∞ and is a sufficiently smooth function, it is reasonable
to put
∫
∞
∞
−∞
K ( t ) d ξ ( t , γ ) = − ∫ K ′ ( t ) ξ ( t , γ ) dt
−∞
(3.27)
Under these circumstances, we have formally
∫
1
0
∞ ∞
−∞ −∞
d γ ∫ K 1 ( t ) d ξ ( t , γ ) ∫ K 2 ( t ) d ξ ( t , γ )
1 ∞ ∞
0 − ∞ −∞
= ∫ d γ ∫ K 1 ′ ( t ) ξ ( t , γ ) dt ∫ K 2 ′ ( t ) ξ ( t , γ ) dt
∞ ∞ 1
−∞ −∞ 0
= ∫ K 1 ′ ( s ) ds ∫ K 2 ′ ( t ) dt ∫ ξ ( s , γ ) ξ ( t , γ ) d γ
Now, if s and t are of opposite signs,
(3.28)102
1
∫
0
Chapter III
ξ ( s , γ ) ξ ( t , γ ) d γ = 0
(3.29)
while if they are of the same sign, and |s| < |t|,
1
∫
0
1
ξ ( s , γ ) ξ ( t , γ ) d γ = ∫ x ( s , α ) x ( t , α ) d α
0
∞
 u 2
( v − u ) 2 
−
du ∫ dv uv exp  −

−∞
−∞
 2 s 2 ( t − s ) 
∞
1
 u 2 
u 2 exp  −
=
du
∫
−∞
 2 s  
2 π s
=
1
2 π s ( t − s )
= s
1
2 π
∫
∞
−∞
∫
∞
(3.30)
 u 2 
u 2 exp  −  du = s
 2 
Thus:
1
∫
0
∞ ∞
−∞ −∞
d γ ∫ K 1 ( t ) d ξ ( t , γ ) ∫ K 2 ( t ) d ξ ( t , γ )
∞ s ∞ s
0 0 0 0
= − ∫ K 1 ′ ( s ) ds ∫ tK 2 ′ ( t ) d t − ∫ K 2 ′ ( s ) ds ∫ tK 1 ′ ( t ) dt
0 0 0 0
−∞ s −∞ s
+ ∫ K 1 ′ ( s ) ds ∫ tK 2 ′ ( t ) dt + ∫ K 2 ′ ( s ) ds ∫ tK 1 ′ ( t ) dt
∞
s
= − ∫ K 1 ′ ( s ) ds   sK 2 ( s ) − ∫ K 2 ( t ) dt  
0
0


∞
s
− ∫ K 2 ′ ( s ) ds   sK 1 ( s ) − ∫ K 1 ( t ) dt  
0
0


0
0

+ ∫ K 1 ′ ( s ) ds  − sK 2 ( s ) − ∫ K 2 ( t ) dt  
s
−∞


(3.31)
0
0
+ ∫ K 2 ′ ( s ) ds   − sK 1 ( s ) − ∫ K 1 ( t ) dt  
−∞
s


∞ ∞
−∞ −∞
= − ∫ sd [ K 1 ( s ) K 2 ( s ) ] = ∫ K 1 ( s ) K 2 ( s ) ds
In particular,
∫
1
0
∞ ∞
−∞ −∞
d γ ∫ K ( t + τ 1 ) d ξ ( t , γ ) ∫ K ( t + τ 2 ) d ξ ( t , γ )
∞
= ∫ K ( s ) K ( s + τ 2 − τ 1 ) ds
− ∞
Moreover,
(3.32)Time Series, Information, and Communication
∫
1
0
n
103
∞
d γ ∏ ∫ K ( t + τ k ) d ξ ( t , γ )
k = 1
−∞
(3.33)
∞
= ∑ ∏ ∫ K ( s ) K ( s + τ j − τ k ) ds
−∞
where the sum is over all partitions of τ l , ..., τ n into pairs, and the
product is over the pairs in each partition.
The expression
∫
∞
−∞
K ( t + τ ) d ξ ( τ , γ ) = f ( t , γ )
(3.34)
represents a very important ensemble of time series in the vari-
able t, depending on a parameter of distribution γ. We have just
shown what amounts to the statement that all the moments and
hence all the statistical parameters of this distribution depend
on the function
∞
Φ ( τ ) = ∫ K ( s ) K ( s + τ ) ds
−∞
(3.35)
∞
= ∫ K ( s + t ) K ( s + t + τ ) ds
−∞
which is the statisticians’ autocorrelation function with lag τ.
Thus the statistics of distribution of f(t, γ) are the same as the
statistics of f(t + t 1 , γ); and it can be shown in fact that, if
f ( t + t 1 , γ ) = f ( t , Γ )
(3.36)
then the transformation of γ into Γ preserves measure. In other
words, our time series f(t, γ) is in statistical equilibrium.
Moreover, if we consider the average of
m
 ∞ K ( t − τ ) d ξ ( t , γ )   ∞ K ( t + σ − τ ) d ξ ( t , γ ) 
 ∫ −∞
  ∫ −∞

n
(3.37)
it will consist of precisely the terms in
∫
1
0
∞
d γ   ∫ K ( t − τ ) d ξ ( t , γ )  
 −∞

m
∫
1
0
∞
d γ   ∫ K ( t + σ − τ ) d ξ ( t , γ )  
 −∞

n
(3.38)104
Chapter III
together with a finite number of terms involving as factors
powers of
∫
∞
−∞
K ( σ + τ ) K ( τ ) d τ
(3.39)
and if this approaches 0 when σ → ∞, Expression 3.38 will
be the limit of Expression ‘3.37 under these circumstances.
In other words, f(t, γ) and f(t + σ, γ) are asymptotically indepen-
dent in their distributions as σ → ∞. By a more generally phrased
but entirely similar argument, it may be shown that the simul-
taneous distribution of f(t 1 , γ), ..., f(t n , γ) and of f(σ + s 1 , γ), ...,
f(σ + s m , γ) tends to the joint distribution of the first and the
second set as σ → ∞. In other words, any bounded measurable
functional or quantity depending on the entire distribution of
the values of the function of t, f(t, γ), which we may write in the
form  [ f ( t, γ ) ] , must have the property that
{ ∫  [ f ( t , γ ) ] d γ }
1
1
lim ∫  [ f ( t , γ ) ]  [ f ( t + σ , γ ) ] d γ =
σ →∞
0
2
0
(3.40)
If now  [ f ( t, γ ) ] is invariant under a translation of t, and only
takes on the values 0 or 1, we shall have
∫
1
0
1
 [ f ( t , γ ) ] d γ = ∫ {  [ f ( t , γ ) ] d γ }
2
0
(3.41)
so that the transformation group of f(t, γ) into f(t + σ, γ) is metri-
cally transitive. It follows that if  [ f ( t, γ ) ] is any integrable func-
tional of f as a function of t, then by the ergodic theorem
∫
1
0
1
T
1
= lim
T →∞ T
 [ f ( t , γ ) ] d γ = lim
T →∞
∫ T
∫ 0
0
 [ f ( t , γ ) ] dt
− T
 [ f ( t , γ ) ] dt
(3.42)
for all values of γ except for a set of zero measure. That is, we can
almost always read off any statistical parameter of such a timeTime Series, Information, and Communication
105
series, and indeed any denumerable set of statistical parameters,
from the past history of a single example. Actually, for such a
time series,
when we know
lim
T →∞
1
T
∫
0
− T
f ( t , γ ) f ( t − τ , γ ) dt
(3.43)
we know Φ(t) in almost every case, and we have a complete sta-
tistical knowledge of the time series.
There are certain quantities dependent on a time series of this
sort which have quite interesting properties. In particular, it is
interesting to know the average of
∞
exp   i ∫ K ( t ) d ξ ( t , γ )  
 −∞

(3.44)
Formally, this may be written
∫
1
0
∞
n
i n  ∞
K ( t ) d ξ ( t , γ )  
∫

−∞


n = 0 n !
d γ ∑
= ∑
m
= ∑
m
{ ∫
m ! { ∫
( − 1 ) m
( 2 m ) !
( − 1 ) m −∞
2 m −∞
{
= exp −
∞
∞
}
[ K ( t ) ] dt }
[ K ( t ) ] 2 dt
( 2 m − 1 ) ( 2 m − 3 ) 5 �
3 �
1
m
2
1 ∞
[ K ( t ) ] 2 dt
2 ∫ −∞
m
(3.45)
}
It is a very interesting problem to try to build up a time series
as general as possible from the simple Brownian motion series.
In such constructions, the example of the Fourier developments
suggests that such expansions as Expression 3.44 are convenient
building blocks for this purpose. In particular, let us investigate
time series of the special form106
∫
b
a
Chapter III
∞
d λ exp   i ∫ K ( t + τ , λ ) d ξ ( τ , γ )  
 −∞

(3.46)
Let us suppose that we know ξ(τ, γ) as well as Expression 3.46.
Then, as in Eq. 3.45, if t 1 > t 2 ,
∫
1
0
d γ exp { is [ ξ ( t 1 , γ ) − ξ ( t 2 , γ ) ] }
b
∞
× ∫ d λ exp   i ∫ K ( t + τ , λ ) d ξ ( t , γ )  
a
 −∞

b
∞
t 1
s 2

 1
2
= ∫ d λ exp  − ∫ [ K ( t + τ , λ ) ] dt − ( t 2 − t 1 ) − s ∫ K ( t , λ ) dt 
a
−∞
t
2
2
 2

(3.47)
If we now multiply by exp[s 2 (t 2 − t 1 )/2], let s(t 2 − t 1 ) = iσ, and
let t 2 → t 1 , we obtain
∫
b
a
{
d λ exp −
}
1 ∞
[ K ( t + τ , λ ) ] 2 dt − i σ K ( t 1 , λ )
2 ∫ −∞
(3.48)
Let us take K(t 1 , λ) and a new independent variable μ and
solve for λ, obtaining
λ = Q ( t 1 , μ )
(3.49)
Then Expression 3.48 becomes
∫
K ( t 1 , b )
K ( t 1 , a )
e i μσ d μ
∂ Q ( t 1 , μ )
1 ∞
2
exp   − ∫ { K [ t + τ , Q ( t 1 , μ ) ] } dt  


2 −∞
d μ
(3.50)
From this, by a Fourier transformation, we can determine
∂ Q ( t 1 , μ )
1 ∞
2
exp   − ∫ { K [ t + τ , Q ( t 1 , μ ) ] } dt  
 2 −∞

d μ
(3.51)
as a function of μ, when μ, lies between K(t 1 , a) and K(t 1 , b). If we
integrate this function with respect to μ, we determine
∫
λ
a
{
d λ exp −
1 ∞
[ K ( t + τ , λ ) ] 2 dt
2 ∫ −∞
}
(3.52)Time Series, Information, and Communication
107
as a function of K(t 1 , λ) and t 1 . That is, there is a known function
F(u, v), such that
∫
λ
a
{
d λ exp −
}
1 ∞
[ K ( t + τ , λ ) ] 2 dt = F [ K ( t 1 , λ ) , t 1 ]
2 ∫ −∞
(3.53)
Since the left-­hand side of this equation does not depend on t 1 ,
we may write it G(λ), and put
F [ K ( t 1 , λ ) , t 1 ] = G ( λ )
(3.54)
Here, F is a known function, and we can invert it with respect to
the first argument, and put
K ( t 1 , λ ) = H [ G ( λ ) , t 1 ]
(3.55)
where it is also a known function. Then
λ
1 ∞
2
G ( λ ) = ∫ d λ exp   − ∫ { H [ G ( λ ) , t + τ ] } dt  
a
 2 −∞

(3.56)
Then the function
{
exp −
}
1 ∞
[ H ( u , t ) ] 2 dt = R ( u )
2 ∫ −∞ i
(3.57)
will be a known function, and
dG
= R ( G )
d λ
(3.58)
That is,
dG
= d λ
R ( G )
(3.59)
or
λ= ∫
dG
+ const = S ( G ) + const
R ( G )
(3.60)108
Chapter III
This constant will be given by
G ( a ) = 0
(3.61)
or
a = S ( 0 ) + const
(3.62)
It is easy to see that if a is finite, it does not matter what value we
give it; for our operator is not changed if we add a constant to all
values of λ. We can hence make it 0. We have thus determined
λ as a function of G, and thus G as a function of λ. Thus, by Eq.
3.55, we have determined K(t, λ). To finish the determination of
Expression 3.46, we need only know b. This can be determined,
however, by a comparison of
∫
b
a
{
d λ exp −
1 ∞
[ K ( t , λ ) ] 2 dt
2 ∫ −∞
}
(3.63)
with
∫
1
0
b
∞
d γ ∫ d λ exp   i ∫ K ( t , λ ) d ξ ( t , γ )  
a
 −∞

(3.64)
Thus, under certain circumstances which remain to be definitely
formulated, if a time series may be written in the form of Expres-
sion 3.46 and we know ξ(t, γ) as well, we can determine the func-
tion K(t, λ) in Expression 3.46 and the numbers a and b, except
for an undetermined constant added to a, λ, and b. There is no
extra difficulty if b = +∞, and it is not hard to extend the reason-
ing to the case where a = −∞. Of course, a good deal of work
remains to be done to discuss the problem of the inversion of
the functions inverted when the results are not single-­valued,
and the general conditions of validity of the expansions con-
cerned. Still, we have at least taken a first step toward the solu-
tion of the problem of reducing a large class of time series toTime Series, Information, and Communication
109
a canonical form, and this is most important for the concrete
formal application of the theories of prediction and of the mea-
surement of information, as we have sketched them earlier in
this chapter.
There is still one obvious limitation which we should remove
from this approach to the theory of time series: the necessity
which we are under of knowing ξ(t, γ) as well as the time series
which we are expanding in the form of Expression 3.46. The
question is: under what circumstances can we represent a time
series of known statistical parameters as determined by a Brown-
ian motion; or at least as the limit in some sense or other of
time series determined by Brownian motions? We shall confine
ourselves to time series with the property of metrical transitiv-
ity, and with the even stronger property that if we take inter-
vals of fixed length but remote in time, the distributions of any
functionals of the segments of the time series in these intervals
approach independence as the intervals recede from eachother. 4
The theory to be developed here has already been sketched by
the author.
If K(t) is a sufficiently continuous function, it is possible to
show that the zeros of
∫
∞
−∞
K ( t + τ ) d ξ ( τ , γ )
(3.65)
almost always have a definite density, by a theorem of M. Kac,
and that this density can be made as great as we wish by a proper
choice of K. Let K D be so selected that this density is D. Then the
sequence of zeros of
∫
∞
−∞
K D ( t + τ ) d ξ ( τ , γ ) from −∞ to ∞ will be
called Z n (D, γ), −∞ < n < ∞. Of course, in the numeration of these
zeros, n is determined except for an additive constant integer.110
Chapter III
Now, let T(t, μ) be any time series in the continuous variable t,
while μ is a parameter of distribution of the time series, varying
uniformly over (0, 1). Then let
T D ( t , μ γ ) = T [ t − Z n ( D , γ ) , μ ]
(3.66)
where the Z n taken is the one just preceding t. It will be seen that
for any finite set of values t 1 , t 2 , ..., t v of x the simultaneous distri-
bution of T D (t κ , μ, γ) (κ = 1, 2, ..., v) will approach the simultane-
ous distribution of T(t κ , μ) for the same t κ ’s as D → ∞, for almost
every value of μ. However, T D (t, μ, γ) is completely determined
by t, μ, D, and ξ(τ, γ). It is therefore not inappropriate to try to
express T D (t, μ, γ), for a given D and a given μ, either directly
in the form of Expression 3.46 or in some way or another as a
time series which has a distribution which is a limit (in the loose
sense just given) of distributions of this form.
It must be admitted that this is a program to be carried through
in the future, rather than one which we can consider as already
accomplished. Nevertheless, it is the program which, in the
opinion of the author, offers the best hope for a rational, consis-
tent treatment of the many problems associated with non-­linear
prediction, non-­linear filtering, the evaluation of the transmis-
sion of information in non-­linear situations, and the theory of
the dense gas and turbulence. Among these problems are per-
haps the most pressing facing communication engineering.
Let us now come to the prediction problem for time series
of the form of Eq. 3.34. We see that the only independent sta-
tistical parameter of the time series is Φ(t), as given by Eq. 3.35;
which means that the only significant quantity connected with
K(t) is
∫
∞
−∞
K ( s ) K ( s + t ) ds
(3.67)Time Series, Information, and Communication
111
Here of course K is real.
Let us put
∞
K ( s ) = ∫ k ( ω ) e i ω s d ω
(3.68)
−∞
employing a Fourier transformation. To know K(s) is to know
k(ω), and vice versa. Then
1
2 π
∫
∞
−∞
∞
K ( s ) K ( s + τ ) ds = ∫ k ( ω ) k ( − ω ) e i ωτ d ω
−∞
(3.69)
Thus a knowledge of Φ(τ) is tantamount to a knowledge of
k(ω)k(−ω). Since, however, K(s) is real,
∞
K ( s ) = ∫ k ( ω ) e − i ω s d ω
−∞
(3.70)
whence k(ω) = k(−ω). Thus |k(ω)| 2 is a known function, which
means that the real part of log |k(ω)| is a known function.
If we write
F ( ω ) =  { log [ k ( ω ) ] }
(3.71)
then the determination of K(s) is equivalent to the determina-
tion of the imaginary part of log k(ω). This problem is not deter-
minate unless we put some further restriction on k(ω). The type
of restriction which we shall put is that log k(s) shall be analytic
and of a sufficiently small rate of growth for ω in the upper half-­
plane. In order to make this restriction, k(ω) and [k(ω) −1 will be
assumed to be of algebraic growth on the real axis. Then [F(ω)] 2
will be even and at most logarithmically infinite, and the Cau-
chy principal value of
G ( ω ) =
1 ∞ F ( u )
du
π ∫ −∞ u − ω
(3.72)112
Chapter III
will exist. The transformation indicated by Eq. 3.72, known as
the Hilbert transformation, changes cos λω into sin λω and sin
λω into −cos λω. Thus F(ω) + iG(ω) is a function of the form
∫
∞
0
e i λω d [ M ( λ ) ]
(3.73)
and satisfies the required conditions for log |k(ω)| in the lower
half-­plane. If we now put
k ( ω ) = exp [ F ( ω ) + iG ( ω ) ]
(3.74)
it can be shown that k(ω) is a function which, under very general
conditions, is such that K(s), as defined in Eq. 3.68, vanishes for
all negative arguments. Thus
∞
f ( t , γ ) = ∫ K ( t + τ ) d ξ ( τ , γ )
− t
(3.75)
On the other hand, it can be shown that we may write 1/k(ω) in
the form
∞
lim ∫ e i λω dN n ( λ )
n →∞
(3.76)
0
where the N n ’s are properly determined; and that this can be
done in such a way that
τ ∞
0 − t
ξ ( τ , γ ) = lim ∫ dt ∫ Q n ( t + σ ) f ( σ , γ ) d σ
n →∞
(3.77)
Here the Q n ’s must have the formal property that
∞ ∞
− t − t
f ( t , γ ) = lim ∫ K ( t + τ ) d τ ∫ Q n ( τ + σ ) f ( σ , γ ) d σ
n →∞
(3.78)
In general, we shall have
∞ ∞
− t − t
ψ ( t ) = lim ∫ K ( t + τ ) d τ ∫ Q n ( τ + σ ) ψ ( σ ) d σ
n →∞
or if we write (as in Eq. 3.68)
(3.79)Time Series, Information, and Communication
113
∞
K ( s ) = ∫ k ( ω ) e i ω s d ω
−∞
∞
Q n ( s ) = ∫ q n ( ω ) e i ω s d ω
(3.80)
−∞
∞
ψ ( s ) = ∫ Ψ ( ω ) e i ω s d ω
−∞
then
Ψ ( ω ) = lim ( 2 π ) 2 Ψ ( ω ) q n ( − ω ) k ( ω )
3
n →∞
(3.81)
Thus
lim q n ( − ω ) =
n →∞
1
(3.82)
( 2 π ) 2 k ( ω )
3
We shall find this result useful in getting the operator of predic-
tion into a form concerning frequency rather than time.
Thus the past and present of ξ(t, γ), or properly of the “dif-
ferential" dξ(t, ξ), determine the past and present of f(t, γ), and
vice versa.
Now, if A > 0,
f ( t + A , γ ) = ∫
= ∫
∞
− t − A
− t
− t − A
K ( t + A + τ ) d ξ ( τ , γ )
K ( t + A + τ ) d ξ ( τ , γ )
(3.83)
∞
+ ∫ K ( t + A + τ ) d ξ ( τ , γ )
− t
Here the first term of the last expression depends on a range of
dξ(τ, λ) of which a knowledge of f(σ, γ) for σ  t tells us nothing,
and is entirely independent of the second term. Its mean square
value is
∫
t
− t − A
A
[ K ( t + A + τ ) ] 2 d τ = ∫ 0 [ K ( τ ) ] 2 d τ
(3.84)114
Chapter III
and this tells us all there is to know about it statistically. It may
be shown to have a Gaussian distribution with this mean square
value. It is the error of the best possible prediction of f(t + A, γ).
The best possible prediction itself is the last term of Eq. 3.83,
∫
∞
− t
k ( t + A + τ ) d ξ ( τ , γ )
∞ ∞
− t − τ
= lim ∫ k ( t + A + τ ) d τ ∫ Q n ( τ + σ ) f ( σ , γ ) d σ
n →∞
(3.85)
If we now put
k A ( ω ) =
1
2 π
∫
∞
0
K ( t + A ) e − i ω t dt
(3.86)
and if we apply the operator of Eq. 3.85 to e tωt , obtaining
∞ ∞
− t − τ
lim ∫ K ( t + A + τ ) d τ ∫ Q n ( τ + σ ) e i ωσ d σ = A ( ω ) e i γ wt
n →∞
(3.87)
we shall find out (somewhat as in Eq. 3.81) that
A ( ω ) = lim ( 2 π ) 2 q n ( − ω ) k A ( ω )
3
n →∞
= k A ( ω ) k ( ω )
∞
∞
1
=
e − i ω ( t − A ) dt ∫ k ( u ) e iut du
∫
−∞
A
2 π k ( ω )
(3.88)
This is then the frequency form of the best prediction operator.
The problem of filtering in the case of time series such as Eq.
3.34 is very closely allied to the prediction problem. Let our mes-
sage plus noise be of the form
∞
m ( t ) + n ( t ) = ∫ K ( τ ) d ξ ( t − τ , γ )
0
(3.89)
and let the message be of the form
∞ ∞
−∞ −∞
m ( t ) = ∫ Q ( τ ) d ξ ( t − τ , γ ) + ∫ R ( τ ) d ξ ( t − τ , δ )
(3.90)Time Series, Information, and Communication
115
where γ and δ are distributed independently over (0, 1). Then the
predictable part of the m(t + a) is clearly
∫
∞
0
Q ( τ + a ) d ξ ( t − τ , γ )
(3.901)
and the mean square error of prediction is
∫
a
−∞
∞
[ Q ( τ ) ] 2 d τ + ∫ −∞ [ R ( τ ) ] 2 d τ
(3.902)
Furthermore, let us suppose that we know the following
quantities:
1 1
0 0
φ 22 ( t ) = ∫ d γ ∫ d δ n ( t + τ ) n ( τ )
∞
= ∫ [ K ( t + τ ) − Q ( t + τ ) ] [ K ( τ ) − Q ( τ ) ] d τ
−∞
∞
= ∫ [ K ( t + τ ) − Q ( t + τ ) ] [ K ( τ ) − Q ( τ ) ] d τ
0
0
+ ∫ [ K ( t + τ ) − Q ( t + τ ) ] [ − Q ( τ ) ] d τ
− t
+ ∫
− t
−∞
(3.903)
∞
Q ( t + τ ) Q ( τ ) d τ + ∫ R ( t + τ ) R ( τ ) d τ
−∞
∞ ∞
0 − t
= ∫ K ( t + τ ) K ( τ ) d τ − ∫ K ( t + τ ) Q ( τ ) d τ
∞ ∞
−∞ −∞
+ ∫ Q ( t + τ ) Q ( τ ) d τ + ∫ R ( t + τ ) R ( τ ) d τ
1 1
0 0
φ 11 ( τ ) = ∫ d γ ∫ d δ m ( t + τ ) m ( τ )
∞ ∞
−∞ − ∞
= ∫ Q ( t + τ ) Q ( τ ) d τ + ∫ R ( t + τ ) R ( τ ) d τ
1 1
0 0
(3.904)
φ 12 ( τ ) = ∫ d γ ∫ d δ m ( t + τ ) n ( τ )
1 1
0 0
= ∫ d γ ∫ d δ m ( t + τ ) [ m ( τ ) + n ( τ ) ] − φ 1 1 ( τ )
1 ∞ ∞
0 − t − t
= ∫ d γ ∫ K ( σ + t ) d ξ ( τ − σ , γ ) ∫ Q ( τ ) d ξ ( τ − σ , γ ) − φ 11 ( τ )
∞
= ∫ K ( t + τ ) Q ( τ ) d τ − φ 11 ( τ )
− t
(3.905)116
Chapter III
The Fourier transforms of these three quantities are, respec-
tively,
Φ 22 ( ω ) = k ( ω ) 2 + q ( ω ) − q ( ω ) k ( ω ) − k ( ω ) q ( ω ) + r ( ω ) 2 

2
Φ 11 ( ω ) = q ( ω ) + r ( ω ) 2
 (3.906)

2
Φ 12 ( ω ) = k ( ω ) q ( ω ) − q ( ω ) − r ( ω )
 
2
where
1
2 π
1
q ( ω ) =
2 π
1
r ( ω ) =
2 π
k ( ω ) =
K ( s ) e − i ω s ds  

∞

− i ω s
Q
s
e
ds
(
)

∫ −∞

∞

− i ω s
R
s
e
ds
(
)
∫ −∞
 
∫
∞
0
(3.907)
That is,
Φ 11 ( ω ) + Φ 12 ( ω ) + Φ 12 ( ω ) + Φ 22 ( ω ) = k ( ω ) 2
(3.908)
and
q ( ω ) k ( ω ) = Φ 11 ( ω ) + Φ 21 ( ω )
(3.909)
where for symmetry we write Φ 21 ( ω ) = Φ 12 ( ω ) . We can now deter-
mine k(ω) from Eq. 3.908, as we have defined k(ω) before on the
basis of Eq. 3.74. Here we put Φ(t) for Φ 11 ( t ) + Φ 22 ( t ) + 2  [ Φ 12 ( t ) ] .
This will give us
q ( ω ) =
Φ 11 ( ω ) + Φ 21 ( ω )
k ( ω )
(3.910)
Hence
Q ( t ) = ∫
∞
−∞
Φ 11 ( ω ) + Φ 21 ( ω ) i ω t
e d ω
k ( ω )
(3.911)
and thus the best determination of m(t), with the least mean
square error, isTime Series, Information, and Communication
∫
∞
0
d ξ ( t − τ , γ ) ∫
∞
−∞
117
Φ 11 ( ω ) + Φ 21 ( ω ) i ω ( t + a )
e
d ω
k ( ω )
(3.912)
Combining this with Eq. 3.89, and using an argument similar to
the one by which we obtained Eq. 3.88, we see that the operator
on m(t) + n(t) by which we obtain the “best" representation of
m(t + a), if we write it on the frequency scale, is
∞
∞ Φ 11 ( u ) + Φ 21 ( u )
1
e iut du
e − i ω ( t − a ) dt ∫
∫
−∞
a
2 π k ( ω )
k ( u )
(3.913)
This operator constitutes a characteristic operator of what
electrical engineers know as a wave filter. The quantity a is the
lag of the filter. It may be either positive or negative; when it is
negative, −a is known as the lead. The apparatus corresponding
to Expression 3.913 may always be constructed with as much
accuracy as we like. The details of its construction are more for
the specialist in electrical engineering than for the reader of this
book. They may be found elsewhere. 5
The mean square filtering error (Expression 3.902) may be
represented as the sum of the mean square filtering error for
infinite lag:
∫
∞
∞
−∞
[ R ( τ ) ] 2 d τ = Φ 11 ( 0 ) − ∫ −∞ [ Q ( τ ) ] 2 d τ
=
1
2 π
∫
∞
−∞
Φ 11 ( ω ) d ω −
1
2 π
∫
∞
−∞
2
Φ 11 ( ω ) + Φ 21 ( ω )
d ω
k ( ω )


Φ 11 ( ω ) + Φ 21 ( ω ) 2
Φ 11 ( ω ) −

 d ω
−∞
+
+
Φ
ω
+
Φ
ω
Φ
Φ
ω
ω
(
)
(
)
(
)
(
)
11
12
21
22


Φ 11 ( ω ) Φ 12 ( ω )
Φ 21 ( ω ) Φ 22 ( ω )
1 ∞
=
d ω
2 π ∫ −∞ Φ 1 1 ( ω ) + Φ 12 ( ω ) + Φ 21 ( ω ) + Φ 22 ( ω )
(3.914)
=
1
2 π
∫
∞
and a part dependent on the lag:118
a
Chapter III
a
2
∫ −∞ [ Q ( τ ) ] dt = ∫ −∞ dt
Φ 11 ( ω ) + Φ 21 ( ω ) i ω t
e d ω
∫ −∞
k ( ω )
∞
2
(3.915)
It will be seen that the mean square error of filtering is a mono-
tonely decreasing function of lag.
Another question which is interesting in the case of messages
and noises derived from the Brownian motion is the matter of
rate of transmission of information. Let us consider for simplic-
ity the case where the message and the noise are incoherent,
that is, when
Φ 12 ( ω ) ≡ Φ 21 ( ω ) ≡ 0
(3.916)
In this case, let us consider
∞
m ( t ) = ∫ M ( τ ) d ξ ( t − τ , γ )  
−∞

∞
n ( t ) = ∫ N ( τ ) d ξ ( t − τ , δ )  
−∞
(3.917)
where γ and δ are distributed independently. Let us suppose we
know m(t) + n(t) over (−A, A); how much information do we
have concerning m(t)? Note that we should heuristically expect
that it would not be very different from the amount of informa-
tion concerning
∫
A
− A
M ( τ ) d ξ ( t − τ , γ )
(3.918)
which we have when we know all values of
∫
A
− A
M ( τ ) d ξ ( t − τ , γ ) + ∫
A
− A
N ( τ ) d ξ ( t − τ , δ )
(3.919)
where γ and δ have independent distributions. It can, however,
be shown that the nth Fourier coefficient of Expression 3.918
has a Gaussian distribution independent of all the other Fourier
coefficients, and that its mean square value is proportional toTime Series, Information, and Communication
 π n τ 
∫ − A M ( τ ) exp   i A   d τ
A
119
2
(3.920)
Thus, by Eq. 3.09, the total amount of information available
concerning M is
2
2
 i π n τ  d τ + A N ( τ ) exp  i π n τ  d τ
M
τ
exp
(
)
∫ − A
∫ − A
 A 
 A 
1
log 2
∑
2
A
2
π
τ
n
n =−∞


∫ − A N ( τ ) e x p  i A  d τ
(3.921)
A
∞
and the time density of communication of energy is this quan-
tity divided by 2A. If now A → ∞, Expression 3.921 approaches
1
2 π
∫
∞
−∞
du log 2
∫
∞
−∞
2
M ( τ ) exp iu τ d τ +
∫
∞
−∞
∫
∞
−∞
N ( τ ) exp iu τ d τ
N ( τ ) exp iu τ d τ
2
2
(3.922)
This is precisely the result which the author and Shannon have
already obtained for the rate of transmission of information in
this case. As will be seen, it depends not only on the width of the
frequency band available for transmitting the message but also
on the noise level. As a matter of fact, it has a close relation to
the audiograms used to measure the amount of hearing and loss
of hearing in a given individual. Here the abscissa is frequency,
the ordinate of lower boundary is the logarithm of the inten-
sity of the threshold of audible intensity—­what we may call the
logarithm of the intensity of the internal noise of the receiving
system—­and the upper boundary, the logarithm of the intensity
of the greatest message the system is suited to handle. The area
between them, which is a quantity of the dimension of Expres-
sion 3.922, is then taken as a measure of the rate of transmission
of information with which the ear is competent to cope.120
Chapter III
The theory of messages depending linearly on the Brownian
motion has many important variants. The key formulae are Eqs.
3.88 and 3.914 and Expression 3.922, together, of course, with
the definitions necessary to interpret these. There are a number
of variants of this theory. First: the theory gives us the best pos-
sible design of predictors and of wave filters in the case in which
the messages and the noises represent the response of linear res-
onators to Brownian motions; but in much more general cases,
they represent a possible design for predictors and filters. This
will not be an absolute best possible design, but it will minimize
the mean square error of prediction and filtering, in so far as this
can be done with apparatus performing linear operations. How-
ever, there will generally be some non-­linear apparatus which
gives a performance still better than that of any linear apparatus.
Next, the time series here have been simple time series, in
which a single numerical variable depends on the time. There
are also multiple time series, in which a number of such vari-
ables depend simultaneously on the time; and it is these which
are of greatest importance in economics, meteorology, and the
like. The complete weather map of the United States, taken from
day to day, constitutes such a time series. In this case, we have
to develop a number of functions simultaneously in terms of the
frequency, and the quadratic quantities such as Eq. 3.35 and the
|k(ω)| 2 of the arguments following Eq. 3.70 are replaced by arrays
of pairs of quantities—­that is, matrices. The problem of deter-
mining k(ω) in terms of |k(ω)| 2 , in such a way as to satisfy cer-
tain auxiliary conditions in the complex plane, becomes much
more difficult, especially as the multiplication of matrices is not
a permutable operation. However, the problems involved in this
multidimensional theory have been solved, at least in part, by
Krein and the author.Time Series, Information, and Communication
121
The multidimensional theory represents a complication of
the one already given. There is another closely related theory
which is a simplification of it. This is the theory of prediction,
filtering, and amount of information in discrete time series.
Such a series is a sequence of functions f n (α) of a parameter a,
where n runs over all integer values from −∞ to ∞. The quantity
a is as before the parameter of distribution, and may be taken to
run uniformly over (0, 1). The time series is said to be in statisti-
cal equilibrium when the change of n to n + ν (ν an integer) is
equivalent to a measure-­preserving transformation into itself of
the interval (0, 1) over which α runs.
The theory of discrete time series is simpler in many respects
than the theory of the continuous series. It is much easier, for
instance, to make them depend on a sequence of independent
choices. Each term (in the mixing case) will be representable as a
combination of the previous terms with a quantity independent
of all previous terms, distributed uniformly over (0, 1), and the
sequence of these independent factors may be taken to replace
the Brownian motion which is so important in the continuous
case.
If f n (α) is a time series in statistical equilibrium, and it is metri-
cally transitive, its autocorrelation coefficient will be
1
φ m = ∫ f m ( α ) f 0 ( α ) d α
0
(3.923)
and we shall have
1 N
∑ 0 f k + m ( α ) f k ( α )
N →∞ N + 1
φ m = lim
1 N
∑ 0 f − k + m ( α ) f − k ( α )
N →∞ N + 1
= lim
for almost all α. Let us put
(3.924)122
Chapter III
φ n =
1
2 π
π
∫ π Φ ( ω ) e
in ω
−
d ω
(3.925)
or
∞
Φ ( ω ) = ∑ φ n e − in ω
(3.926)
−∞
Let
∞
1
log Φ ( ω ) = ∑ p n cos n ω
2
−∞
(3.927)
and let
G ( ω ) =
∞
p 0
+ ∑ p n e in ω
2
1
(3.928)
Let
e G ( ω ) = k ( ω )
(3.929)
Then under very general conditions, k(ω) will be the boundary
value on the unit circle of a function without zeros or singulari-
ties inside the unit circle if ω is the angle. We shall have
k ( ω ) = Φ ( ω )
2
(3.930)
If now we put for the best linear prediction of f n (α) with a lead
of ν
∞
∑ f
n− ν
( α ) W ν
(3.931)
0
we shall find that
∞
1
∞
π
∑ W μ e μω = 2 π k ( ω ) μ ∑ ν e ω ( μ ν ) ∫ π k ( u ) e
0
i
=
i
−
−
− i μ u
du
(3.932)
This is the analogue of Eq. 3.88. Let us note that if we putTime Series, Information, and Communication
k μ =
1
2 π
π
∫ π k ( u ) e
− i μ u
−
123
(3.933)
du
then
∞
∞
∑ W μ e
i μω
= e
− i νω
∑ ν k μ e μω
i
∞
∑ k μ e μω
0
i
0


= e − i νω  1 −

 
ν − 1

∑ ν k μ e 

∞
∑ 0 k μ e i μω  
(3.934)
i μω
It will clearly be the result of the way we have formed k(ω) that
in a very general set of cases we can put
∞
1
= ∑ q μ e i μω
k ( ω )
0
(3.935)
Then Eq. 3.934 becomes
∞
∑ W μ e μω = e
i
− i νω
0
ν − 1
∞

i μω
i λω 
  1 − ∑ k μ e ∑ q λ e  
0
0
(3.936)
In particular, if ν = 1,
∞
∑ W μ e μω = e
i
0
− i ω
∞

i λω 
  1 − k 0 ∑ q λ e  
0
(3.937)
or
W μ = − q λ +1 k 0
(3.938)
Thus for a prediction one step ahead, the best value for f n+1 (α) is
∞
− k 0 ∑ q λ + 1 f n − λ ( α )
0
(3.939)124
Chapter III
and by a process of step-­by-­step prediction, we can solve the
entire problem of linear prediction for discrete time series. As in
the continuous case, this will be the best prediction possible by
any method if
∞
f n ( α ) = ∫ K ( n − τ ) d ξ ( τ , α )
−∞
(3.940)
The transfer of the filtering problem from the continuous to
the discrete case follows much the same lines of argument. For-
mula 3.913 for the frequency characteristic of the best filter takes
the form
∞
π [ Φ 11 ( u ) + Φ 21 ( u ) ] e iu ν du
1
e − i ω ( ν − a ) ∫
∑
−
π
2 π k ( ω ) ν = a
k ( u )
(3.941)
where all the terms receive the same definitions as in the con-
tinuous case, except that all integrals on ω or u are from −π to
π instead of from −∞ to ∞ and all sums on v are discrete sums
instead of integrals on t. The filters for discrete time series are
usually not so much physically constructible devices to be used
with an electric circuit as mathematical procedures to enable
statisticians to obtain the best results with statistically impure
data.
Finally, the rate of transfer of information by a discrete time
series of the form
∫
∞
−∞
M ( n − τ ) d ξ ( t , γ )
(3.942)
in the presence of a noise
∫
∞
−∞
N ( n − τ ) d ξ ( t , δ )
(3.943)
when γ and δ are independent, will be the precise analogue of
Expression 3.922, namely,Time Series, Information, and Communication
1
2 π
π
∫ π du log
−
2
∫
∞
−∞
2
M ( τ ) e iu τ d τ +
∫
∞
−∞
∫
∞
−∞
N ( τ ) e iu τ d τ
N ( τ ) e d τ
iu τ
2
125
2
(3.944)
where over (−π, π),
∫
∞
−∞
M ( τ ) e iu τ d τ
2
(3.945)
represents the power distribution of the message in frequency,
and
∫
∞
−∞
N ( τ ) e iu τ d τ
2
(3.946)
that of the noise.
The statistical theories we have here developed involve a full
knowledge of the pasts of the time series we observe. In every
case, we have to be content with less, as our observation does
not run indefinitely into the past. The development of our the-
ory beyond this point, as a practical statistical theory, involves
an extension of existing methods of sampling. The author and
others have made a beginning in this direction. It involves all
the complexities of the use either of Bayes’ law, on the one hand,
or of those terminological tricks in the theory of likelihood, 6 on
the other, which seem to avoid the necessity for the use of Bayes’
law but which in reality transfer the responsibility for its use to
the working statistician, or the person who ultimately employs
his results. Meanwhile, the statistical theorist is quite honestly
able to say that he has said nothing which is not perfectly rigor-
ous and unimpeachable.
Finally, this chapter should end with a discussion of mod-
ern quantum mechanics. These represent the highest point of
the invasion of modern physics by the theory of time series. In126
Chapter III
the Newtonian physics, the sequence of physical phenomena is
completely determined by its past and in particular by the deter-
mination of all positions and momenta at any one moment. In
the complete Gibbsian theory, it is still true that with a perfect
determination of the multiple time series of the whole uni-
verse the knowledge of all positions and momenta at any one
moment would determine the entire future. It is only because
these are ignored, non-­observed coordinates and momenta that
the time series with which we actually work take on the sort of
mixing property with which we have become familiar in this
chapter, in the case of time series derived from the Brownian
motion. The great contribution of Heisenberg to physics was the
replacement of this still quasi-­Newtonian world of Gibbs by one
in which the time series can in no way be reduced to an assem-
bly of determinate threads of development in time. In quantum
mechanics, the whole past of an individual system does not
determine the future of that system in any absolute way but
merely the distribution of possible futures of the system. The
quantities which the classical physics demands for a knowledge
of the entire course of a system are not simultaneously observ-
able, except in a loose and approximate way, which nevertheless
is sufficiently precise for the needs of the classical physics over
the range of precision where it has been shown experimentally to be
applicable. The conditions of the observation of a momentum
and its corresponding position are incompatible. To observe the
position of a system as precisely as possible, we must observe it
with light or electron waves or similar means of high resolving
power, or short wavelength. However, the light has a particle
action depending on its frequency only, and to illuminate a body
with high-­frequency light means to subject it to a change in its
momentum which increases with the frequency. On the otherTime Series, Information, and Communication
127
hand, it is low-­frequency light that gives the minimum change
in the momenta of the particles it illuminates, and this has not
a sufficient resolving power to give a sharp indication of posi-
tions. Intermediate frequencies of light give a blurred account
both of positions and of momenta. In general, there is no set of
observations conceivable which can give us enough information
about the past of a system to give us complete information as to
its future.
Nevertheless, as in the case of all ensembles of time series,
the theory of the amount of information which we have here
developed is applicable, and consequently the theory of entropy.
Since, however, we now are dealing with time series with the
mixing property, even when our data are as complete as they
can be, we find that our system has no absolute potential bar-
riers, and that in the course of time any state of the system can
and will transform itself into any other state. However, the prob-
ability of this depends in the long run on the relative probability
or measure of the two states. This turns out to be especially high
for states which can be transformed into themselves by a large
number of transformations, for states which, in the language of
the quantum theorist, have a high internal resonance, or a high
quantum degeneracy. The benzene ring is an example of this,
since the two states are equivalent. This suggests that in a sys-
tem in which various building blocks may combine themselves
intimately in various ways, as when a mixture of amino acids
organizes itself into protein chains, a situation where many of
these chains are alike and go through a stage of close association
with one another may be more stable than one in which they
are different. It was suggested by Haldane, in a tentative manner,
that this may be the way in which genes and viruses reproduce
themselves; and although he has not asserted this suggestion of128
Chapter III
his with anything like finality, I see no cause not to retain it as
a tentative hypothesis. As Haldane himself has pointed out, as
no single particle in quantum theory has a perfectly sharp indi-
viduality, it is not possible in such a case to say, with more than
fragmentary accuracy, which of the two examples of a gene that
has reproduced itself in this manner is the master pattern and
which is the copy.
This same phenomenon of resonance is known to be very
frequently represented in living matter. Szent-­Györgyi has sug-
gested its importance in the construction of muscles. Substances
with high resonance very generally have an abnormal capacity
for storing both energy and information, and such a storage cer-
tainly occurs in muscle contraction.
Again, the same phenomena that are concerned in repro-
duction probably have something to do with the extraordinary
specificity of the chemical substances found in a living organ-
ism, not only from species to species but even within the indi-
viduals of a species. Such considerations may be very important
in immunology.





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1.03_-_Time_Series,_Information,_and_Communication

IN CHAPTERS CLASSNAME

IN CHAPTERS TEXT
1.03_-_Time_Series,_Information,_and_Communication

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chapterThere_is_a_large_class_of_phenomena_in_which_what_is_observed_is
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Wikipedia - High-functioning autism -- People with autism who are deemed to be cognitively "higher functioning" (with an IQ of 70 or greater) than other people with autism
Wikipedia - High-heeled shoe -- Footwear that raises the heel of the wearer's foot significantly higher than the toes
Wikipedia - High-pass filter -- Filter that passes signals with a frequency higher than a certain cutoff frequency, and attenuates signals with lower frequencies
Wikipedia - Humboldtian model of higher education -- Concept of academic education
Wikipedia - Hypergamy -- Practice of a person marrying a spouse of higher caste or social status than theirs
Wikipedia - Hyperrectangle -- Generalization of a rectangle for higher dimensions
Wikipedia - IAPCHE: the International Association for the Promotion of Christian Higher Education -- Christian educational organization
Wikipedia - ICFAI Foundation for Higher Education -- Indian university in Hyderabad, Telangana
Wikipedia - IILM Institute for Higher Education -- Education institute in New Delhi, India
Wikipedia - IIT Hyderabad -- Public higher education institute located in Hyderabad, Telangana
Wikipedia - IIT Ropar -- Engineering and technology higher education institute in Rupnagar, Punjab
Wikipedia - Imperial Academy of Fine Arts (Brazil) -- Institution of higher learning in the arts
Wikipedia - Indian Institute of Science -- Public university for scientific research and higher education in Bangalore
Wikipedia - Inside Higher Education
Wikipedia - Inside HigherEd
Wikipedia - Inside Higher Ed -- News website focused on issues of degree-oriented tertiary education
Wikipedia - Institute of Higher Nervous Activity
Wikipedia - Institutes of Eminence -- A group of higher educational institution in India
Wikipedia - Institutes of National Importance -- Premier public higher education institutions in India
Wikipedia - Intel Management Engine -- Firmware and software that runs on all modern Intel CPUs at a higher level than user-facing operating system
Wikipedia - Interfluve -- Area of higher ground between two rivers in the same drainage system
Wikipedia - International Academic Association for the Enhancement of Learning in Higher Education -- Advance learning centred higher education
Wikipedia - International Association of University Presidents -- Higher education association
Wikipedia - International Buddhist College -- Buddhist institution of higher education
Wikipedia - International Design School -- Higher education institution based in Jakarta, Indonesia
Wikipedia - Irma Becerra Fernandez -- Cuban-American higher education leader
Wikipedia - ISO/IEC 2022 -- Higher-level 7-bit and 8-bit character encoding system
Wikipedia - Iwate University -- Higher education institution in Iwate Prefecture, Japan
Wikipedia - Jagiellonian University -- Polish higher education institution
Wikipedia - Janet Anne Galloway -- Scottish advocate for higher education for women
Wikipedia - Journal of Diversity in Higher Education
Wikipedia - Katabatic wind -- A wind that carries high-density air from a higher elevation down a slope under the force of gravity
Wikipedia - Kerala Vidhyalayam Higher Secondary School -- Secondary school in Chennai
Wikipedia - Khalsa College, Amritsar -- Higher education institution in Amritsar, India
Wikipedia - KL University -- Institution of higher education in Andhra Pradesh, India
Wikipedia - Koshien University -- Higher education institution in HyM-EM-^Mgo Prefecture, Japan
Wikipedia - Land-grant university -- institution of higher education in the US that receive benefits by the Morrill Acts
Wikipedia - Leadership Foundation for Higher Education -- Organization
Wikipedia - Leonardo da Vinci University Center -- Higher education institution in La DM-CM-)fense, France
Wikipedia - Let Us March Forward Dynamically Towards Final Victory, Holding Higher the Banner of Songun -- speech by Kim Jong-un
Wikipedia - Liberal arts colleges in the United States -- Type of undergraduate higher education institution
Wikipedia - Liberal arts education -- Traditional academic program in Western higher education
Wikipedia - Linguistic purism -- The practice of defining or recognizing one variety of a language as being purer or of intrinsically higher quality than others
Wikipedia - List of autonomous higher education institutes in India -- Wikipedia list article
Wikipedia - List of colleges in the United Kingdom offering higher education courses -- Wikipedia list article
Wikipedia - List of higher education and academic institutions in Saint Petersburg -- Wikipedia list article
Wikipedia - List of higher education associations and organizations in Canada -- Wikipedia list article
Wikipedia - List of higher education institutions in Hong Kong -- Wikipedia list article
Wikipedia - List of higher education institutions in Maharashtra -- Wikipedia list article
Wikipedia - List of higher virus taxa -- Wikipedia list article
Wikipedia - List of institutions of higher education in Andhra Pradesh -- Wikipedia list article
Wikipedia - List of institutions of higher education in Arunachal Pradesh -- Wikipedia list article
Wikipedia - List of institutions of higher education in Assam -- Wikipedia list article
Wikipedia - List of institutions of higher education in Bangalore -- Wikipedia list article
Wikipedia - List of institutions of higher education in Bihar -- Wikipedia list article
Wikipedia - List of institutions of higher education in Chandigarh -- Wikipedia list article
Wikipedia - List of institutions of higher education in Chhattisgarh -- Wikipedia list article
Wikipedia - List of institutions of higher education in Delhi -- Wikimedia list article
Wikipedia - List of institutions of higher education in Goa -- Wikipedia list article
Wikipedia - List of institutions of higher education in Gujarat -- Wikipedia list article
Wikipedia - List of institutions of higher education in Haryana -- Wikimedia list article
Wikipedia - List of institutions of higher education in Himachal Pradesh -- Wikipedia list article
Wikipedia - List of institutions of higher education in India -- List of Indian higher education institutions
Wikipedia - List of institutions of higher education in Jammu and Kashmir -- Wikipedia list article
Wikipedia - List of institutions of higher education in Jharkhand -- Wikipedia list article
Wikipedia - List of institutions of higher education in Karnataka -- Wikipedia list article
Wikipedia - List of institutions of higher education in Kerala -- Wikipedia list article
Wikipedia - List of institutions of higher education in Madhya Pradesh -- Wikipedia list article
Wikipedia - List of institutions of higher education in Manipur -- Wikipedia list article
Wikipedia - List of institutions of higher education in Mizoram -- Wikipedia list article
Wikipedia - List of institutions of higher education in Nagaland -- Wikipedia list article
Wikipedia - List of institutions of higher education in Odisha -- Wikipedia list article
Wikipedia - List of institutions of higher education in Punjab, India -- Wikimedia list article
Wikipedia - List of institutions of higher education in Rajasthan -- Wikimedia list article
Wikipedia - List of institutions of higher education in Russia -- Wikipedia list article
Wikipedia - List of institutions of higher education in Sikkim -- Wikipedia list article
Wikipedia - List of institutions of higher education in Tamil Nadu -- Wikimedia list article
Wikipedia - List of institutions of higher education in Telangana -- Wikipedia list article
Wikipedia - List of institutions of higher education in Tripura -- Wikimedia list article
Wikipedia - List of institutions of higher education in Uttarakhand -- Wikipedia list article
Wikipedia - List of institutions of higher education in Uttar Pradesh -- Wikipedia list article
Wikipedia - List of institutions of higher education in West Bengal -- Wikipedia list article
Wikipedia - List of longest serving higher education presidents in the United States -- Wikipedia list article
Wikipedia - List of public higher institutions in Ethiopia -- Wikipedia list article
Wikipedia - List of unaccredited institutions of higher education -- Wikimedia list article
Wikipedia - List of universities and higher education colleges in Jodhpur -- Wikipedia list article
Wikipedia - List of universities and higher education colleges in London -- Wikipedia list article
Wikipedia - Lists of institutions of higher education by endowment size -- Wikipedia list article
Wikipedia - Lists of institutions of higher education by endowment
Wikipedia - Louisiana State Seminary of Learning & Military Academy -- State institution of higher education that became Louisiana State University
Wikipedia - Maharashtra University of Health Sciences -- Higher education institute in Nashik, India
Wikipedia - Malayan Colleges Laguna -- Higher education institution in Cabuyao, Laguna
Wikipedia - Manual of the Higher Plants of Oregon -- Flora of Oregon
Wikipedia - Map (higher-order function)
Wikipedia - Marion Ross Fedrick -- American higher education academic and leader
Wikipedia - Mariya Mahmoud Bunkure -- Commisioner for Higher Education, Kano State
Wikipedia - McGraw-Hill Higher Education
Wikipedia - Medieval university -- Corporation organized during the Middle Ages for the purposes of higher education
Wikipedia - Mejai Bola Avoseh -- Nigerian-American professor of adult and higher education at the University of South Dakota
Wikipedia - Metacognition -- Thinking about thinking, higher-order thinking skills
Wikipedia - MICA (institute) -- Higher education institution for Strategic Marketing and Communication skills in India
Wikipedia - Microdrive -- Type of hard drive intended to provide a higher-capacity alternative to memory cards
Wikipedia - Middlebury Institute of International Studies at Monterey -- Institution of higher education
Wikipedia - Middle States Commission on Higher Education -- University accreditation organization in the U.S.A.
Wikipedia - Middlewood Higher railway station -- Former railway station in Cheshire, England
Wikipedia - Military academy -- Higher education institution operated by or for the military
Wikipedia - Millennium Leadership Initiative -- Higher education leadership development program
Wikipedia - Millicode -- Higher level of microcode
Wikipedia - Mineral oil -- Liquid mixture of higher alkanes from a mineral source, particularly a distillate of petroleum
Wikipedia - Minister of Higher Education, Research and Innovation (France) -- Government ministry of France
Wikipedia - Ministry of Science and Higher Education (Poland)
Wikipedia - Missouri Department of Higher Education
Wikipedia - Miyazaki International College -- higher education institution in Miyazaki Prefecture, Japan
Wikipedia - Monkey -- Animal of the "higher primates" (the simians), but excluding the apes
Wikipedia - Monterrey Institute of Technology and Higher Education, Mexico City
Wikipedia - Monterrey Institute of Technology and Higher Education
Wikipedia - Move On Up a Little Higher -- 1948 song performed by Mahalia Jackson
Wikipedia - Multitech Business School -- Institution of higher education in Uganda
Wikipedia - Multivariate normal distribution -- Generalization of the one-dimensional normal distribution to higher dimensions
Wikipedia - Nagasaki University -- Higher education institution in Nagasaki Prefecture, Japan
Wikipedia - National Institution for Academic Degrees and Quality Enhancement of Higher Education -- Government administrative agency in Japan
Wikipedia - National Universities Commission -- Government commission for higher education in Nigeria
Wikipedia - Nedungayil Sankunni Narayanan Matriculation Higher Secondary School -- Higher secondary school in Chromepet, India
Wikipedia - Neocortex -- Mammalian structure involved in higher-order brain functions
Wikipedia - New England Board of Higher Education -- An interstate compact
Wikipedia - Nigel Healey -- British economist, now working on higher education policy and management
Wikipedia - Northeast Ohio Council on Higher Education -- Nonprofit business and higher education collaborative
Wikipedia - Northwest Association of Secondary and Higher Schools -- Educational organizations based in the United States
Wikipedia - Norwich University of the Arts -- Higher education institution and public university in the UK
Wikipedia - Odessa Military Academy -- Ukrainian higher military institution in Odessa
Wikipedia - Oklahoma State Regents for Higher Education -- Higher education governmental agency in Oklahoma, United States
Wikipedia - Olivet University -- Private Christian institution of biblical higher education in Anza, California, U.S.
Wikipedia - Online learning in higher education
Wikipedia - Oregon Higher Education Coordinating Commission -- Volunteer panel to advise the state government on higher education policy decisions
Wikipedia - Oregon State Board of Higher Education -- State government entity providing leadership for public universities
Wikipedia - Osaka University of Commerce -- Higher education institution in Osaka Prefecture, Japan
Wikipedia - Oversampling -- sampling higher than the Nyquist rate
Wikipedia - Oxygen tent -- A canopy placed over a patient to provide oxygen at a higher level than normal
Wikipedia - Pacific Higher Naval School -- Russian naval academy in Vladivostok
Wikipedia - Pacific Northwest Corridor -- Higher-speed rail corridor in the United States
Wikipedia - Pennsylvania Higher Education Assistance Agency -- Student financial aid agency
Wikipedia - Pennsylvania State System of Higher Education -- Agency that oversees Pennsylvania-owned colleges and universities
Wikipedia - Philippine House Committee on Higher and Technical Education -- Standing committee of the House of Representatives of the Philippines
Wikipedia - Philippine Senate Committee on Higher, Technical and Vocational Education -- Standing committee of the Senate of the Philippines
Wikipedia - Pontifical Biblical Institute -- Higher education institution in Rome and Jerusalem
Wikipedia - Poole Gakuin College -- Higher education institution in Osaka Prefecture, Japan
Wikipedia - Portfolio school -- Higher-level learning institution providing education for art and creative advertising
Wikipedia - Post-secondary educational organizations in the United States -- Organizations for higher education
Wikipedia - Potrero (landform) -- A long mesa that at one end slopes upward to higher terrain.
Wikipedia - Premium-rate telephone number -- Telephone numbers for calls that are charged at a higher than normal rate
Wikipedia - Prince Mohammad bin Salman College of Business and Entrepreneurship -- higher education business administration college inM-BM- King Abdullah Economic City, Saudi Arabia
Wikipedia - Private university -- Higher education institution not operated by a government
Wikipedia - Publication bias -- Higher probability of publishing results showing a significant finding
Wikipedia - Public College, Samana -- Higher education institute lin Punjab, India
Wikipedia - Punjabi University -- Higher education institute in Patiala, Punjab, India
Wikipedia - Rajarshi Memorial Higher Secondary School, Vadavucode -- Senior secondary school in Vadavucode, Kerala, India
Wikipedia - Recommendation Concerning the Status of Higher Education Teaching Personnel -- UNESCO instrument
Wikipedia - Red meat -- Types of meat such as beef, goat, pork, or lamb with higher myoglogin content
Wikipedia - Reduce (higher-order function)
Wikipedia - Requinto -- Spanish and Portuguese term to describe a smaller, higher-pitched version of another instrument
Wikipedia - Riga Higher Military Political School -- Soviet military academy in Riga
Wikipedia - Robbins Report -- Report commissioned by the British government to the Committee on Higher Education, chaired by Lord Lionel Robbins, published in 1963
Wikipedia - Ronald Champagne -- American higher education administrator
Wikipedia - Rule according to higher law -- Belief that universal principles of morality override unjust laws
Wikipedia - Russian Presidential Academy of National Economy and Public Administration -- Higher professional education institution in Moscow, Russia
Wikipedia - Russian University of Transport -- Higher education institution
Wikipedia - Sacrifice -- Offering to a higher purpose, in particular divine beings
Wikipedia - Saint Petersburg State University -- Russian federal state-owned higher education institution
Wikipedia - Saitama Prefectural University -- Higher education institution in Saitama Prefecture, Japan
Wikipedia - Sandwich degree -- Academic degree or higher education course involving practical work experience in addition to academic study
Wikipedia - SBI Graduate School -- Higher education institution in Tokyo
Wikipedia - Sciences Po -- Higher education institution in Paris
Wikipedia - Secondary education in the United States -- Last seven years of statutory formal education before higher level education
Wikipedia - Sendai University -- Higher education institution in Miyagi Prefecture, Japan
Wikipedia - Shandong Normal University -- Institutions of higher learning established in Shandong Province since thI founding of the People's Republic of China.
Wikipedia - Shishu Niketan Higher Secondary School -- Secondary school in Tripura, India
Wikipedia - Shola -- Patches of stunted tropical montane forest found in valleys in the higher montane regions of South India
Wikipedia - Shonphi Dashain Higher Secondary School -- High school in Mahottari District of Nepal
Wikipedia - SIT Graduate Institute -- Institution of higher learning
Wikipedia - Smart Cities EMC Network for Training -- Higher education
Wikipedia - South Dakota Public Universities and Research Center -- Cooperative higher education system in Sioux Falls, South Dakota
Wikipedia - Sprengel's deformity -- Congenital abnormality involving a single higher shoulder blade
Wikipedia - St. Anthony's Higher Secondary School, Shillong
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Wikipedia - St Joseph College of Bulacan -- Private higher education institution of the Philippines
Wikipedia - St. Mary's Anglo-Indian Higher Secondary School
Wikipedia - Stocking -- Hosiery that covers the feet and legs to the knee or higher
Wikipedia - Summit -- Point on a surface with a higher elevation than all immediately adjacent points
Wikipedia - Supinfo -- Private institution of higher education in general Computer Science
Wikipedia - Svarga -- One of the seven higher lokas in Hindu cosmology
Wikipedia - SYCL -- Higher-level programming model for OpenCL
Wikipedia - Symbiosis International University -- Private higher-education institute in Maharashtra, India
Wikipedia - Teknokrat -- Indonesian higher education institute
Wikipedia - The Chronicle of Higher Education -- Newspaper
Wikipedia - The Consortium on Financing Higher Education
Wikipedia - The Higher Command -- 1935 film
Wikipedia - The Higher Law (1914 film) -- 1914 film
Wikipedia - The Higher -- American band
Wikipedia - The Journal of Blacks in Higher Education -- Online magazine for African Americans in academia
Wikipedia - The Journal of Higher Education -- American peer-reviewed academic journal
Wikipedia - Theology of Huldrych Zwingli -- Theological view that considered scripture a higher authority then the church fathers
Wikipedia - The Open University of Japan -- Higher education institution in Chiba Prefecture, Japan
Wikipedia - The Times Higher Education Supplement
Wikipedia - Tidal resonance -- Phenomenon that occurs when the tide excites a resonant mode of a part of an ocean, producing a higher tidal range
Wikipedia - Times Higher Education Supplement
Wikipedia - Times Higher Education -- Weekly magazine based in London
Wikipedia - Times Higher Education World University Rankings -- Annual publication of university rankings
Wikipedia - Topographic prominence -- Characterizes the height of a mountain or hill's summit by the vertical distance between it and the lowest contour line encircling it but containing no higher summit within it; it is a measure of the independence of a summit
Wikipedia - Tuition fees in the United Kingdom -- Cost of higher education in the United Kingdom.
Wikipedia - Ufa State Aviation Technical University -- State higher school in Ufa, Russia
Wikipedia - Unaccredited institutions of higher education
Wikipedia - UNICAF -- Higher education organization
Wikipedia - University of Guadalajara -- Public higher education institution in the Mexican city of Guadalajara
Wikipedia - University of Lyon -- Cluster of several higher education institutions in the region of Lyon, France
Wikipedia - University of Music and Performing Arts Munich -- Institution of higher education in Munich, Germany
Wikipedia - University of Phoenix -- American for-profit institution of higher learning
Wikipedia - University of the People -- American nonprofit, tuition-free online institution of higher education
Wikipedia - Upcycling -- Recycling waste into products of higher quality
Wikipedia - Upper middle class -- Class of people in the higher end of the middle of a societal hierarchy
Wikipedia - Utah System of Higher Education -- Education organization in Utah, United States
Wikipedia - Uzbekistan State Institute of Arts and Culture -- Higher education institution in Tashkent, Uzbekistan
Wikipedia - Varsity College (South Africa) -- South African private higher education provider
Wikipedia - Vascular bundle -- Longitudinal strand of vascular tissue in the roots, stems and leaves of higher plants
Wikipedia - Virtual university -- University that provides higher education programs through electronic media, typically the Internet
Wikipedia - Vishveshwarya Group of Institutions -- Group of higher education institutes in Uttar Pradesh, India
Wikipedia - Vocational school -- Higher-level learning institution providing education needed for specific occupations
Wikipedia - Vocational university -- An institution of higher education and sometimes research that grants professional academic degrees
Wikipedia - Volda University College -- Institution of higher education in Norway
Wikipedia - Warm-blooded -- Animal species that can maintain a body temperature higher than their environment
Wikipedia - West Bengal Council of Higher Secondary Education -- Government education exaination organization
Wikipedia - Western Norway University of Applied Sciences -- Norwegian public institution of higher learning
Wikipedia - Wikipedia:WikiProject Higher education -- Wikimedia subject-area collaboration
Wikipedia - Women's colleges in the United States -- Single-sex institutions of higher education
Wikipedia - Yamanashi Prefectural University -- Higher education institution in Yamanashi Prefecture, Japan
Wikipedia - York College (York) -- Further and higher education college in the City of York, North Yorkshire, England
Wikipedia - Zimmer's conjecture -- Conjecture that symmetries exist in higher dimensions that cannot exist in lower dimensions
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Integral World - Love, Evolution and Higher Values in Darwin, Elliot Benjamin
Integral World - Clarifying Perspectives 4: Higher Order Perspectives and Integral Mathematics, Peter Collins
Integral World - Stages of higher consciousness (Integral Esotericisn - Part Seven), Alan Kazlev
Integral World - Consciousness, meditation, and a higher state, David Lane
Integral World - Higher Worlds: Can Integral Theorists be "Integral" without Correcting Misinterpretations on this?, Giorgio Piacenza
Integral World - FOOTPRINTS IN THE SAND: Can We Identify Brain Correlates of Higher Consciousness?, essay by Andrew Smith
Integral World - The Stage-Skipping Problem: How Did Our Ancestors Realize Higher Consciousness?, essay by Andrew Smith
Integral World - EXCELSIOR: Defining Higher in the Holarchy of Life, essay by Andrew Smith
Integral World - Intimations of Higher Dimensional Realities, Gary Stogsdill
Integral World - Subtle bodies, higher worlds, essay by Frank Visser
The Higher Reaches of Success
Spirituality, Purpose, and Meaning in Higher Education
The Higher Stages of Couple Love
selforum - integrating mind body spirit in higher
https://thoughtsandvisions-searle88.blogspot.com/2012/09/austistic-boy-23-with-higher-iq-than.html
https://thoughtsandvisions-searle88.blogspot.com/2012/10/sant-mat-depictions-of-higher-planes.html
https://thoughtsandvisions-searle88.blogspot.com/2012/10/the-divine-soul-or-higher-self.html
https://thoughtsandvisions-searle88.blogspot.com/2013/11/higher-stages-of-conciousness.html
https://thoughtsandvisions-searle88.blogspot.com/2014/05/the-divine-soul-or-higher-self.html
https://thoughtsandvisions-searle88.blogspot.com/2016/01/higher-conciousness.html
dedroidify.blogspot - becoming-our-higher-self
dedroidify.blogspot - youtube-higher-quality
https://circumsolatious.blogspot.com/2010/01/higher-vision-of-time-is-needed-to.html
https://circumsolatious.blogspot.com/2012/02/higher-knowledge-supermind-prophecy.html
https://circumsolatious.blogspot.com/2019/08/unveiling-mysteries-and-higher-purpose.html
https://esotericotherworlds.blogspot.com/2012/11/about-higher-conciousness-society.html
https://esotericotherworlds.blogspot.com/2012/12/the-higher-self.html
https://esotericotherworlds.blogspot.com/2013/10/the-six-planes-of-higher-consciousness.html
https://esotericotherworlds.blogspot.com/2014/05/the-higher-self-in-sri-aurobindos.html
https://esotericotherworlds.blogspot.com/2014/10/the-terms-vertical-and-higher-and-lower.html
wiki.auroville - Higher_Mind
wiki.auroville - Higher_vital
Psychology Wiki - Integral_psychology_(Sri_Aurobindo)#Higher_levels_of_Mind
Stanford Encyclopedia of Philosophy - consciousness-higher
Stanford Encyclopedia of Philosophy - logic-higher-order
https://tvtropes.org/pmwiki/pmwiki.php/Fanfic/HigherFlier
https://tvtropes.org/pmwiki/pmwiki.php/Fanfic/HigherLearning
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https://tvtropes.org/pmwiki/pmwiki.php/Laconic/AscendToAHigherPlaneOfExistence
https://tvtropes.org/pmwiki/pmwiki.php/Main/AscendedToAHigherPlaneOFExistence
https://tvtropes.org/pmwiki/pmwiki.php/Main/AscendedToAHigherPlaneOfExistence
https://tvtropes.org/pmwiki/pmwiki.php/Main/AscendingToAHigherPlaneOfExistence
https://tvtropes.org/pmwiki/pmwiki.php/Main/AscendsToAHigherPlaneOfExistence
https://tvtropes.org/pmwiki/pmwiki.php/Main/AscendToAHigherPlaneOfExistence
https://tvtropes.org/pmwiki/pmwiki.php/Main/AscendtoaHigherPlaneofExistence
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https://tvtropes.org/pmwiki/pmwiki.php/Main/ThereIsNoHigherCourt
https://tvtropes.org/pmwiki/pmwiki.php/PlayingWith/AscendToAHigherPlaneOfExistence
https://tvtropes.org/pmwiki/pmwiki.php/Quotes/AscendToAHigherPlaneOfExistence
https://tvtropes.org/pmwiki/pmwiki.php/Series/HigherGround
https://tvtropes.org/pmwiki/pmwiki.php/VideoExamples/AscendToAHigherPlaneOfExistence
https://tvtropes.org/pmwiki/pmwiki.php/VisualNovel/AnOctaveHigher
http://tvtropes.org/pmwiki/pmwiki.php/Tropers/Higherbrainpattern
http://tvtropes.org/pmwiki/pmwiki.php/Tropers/RagingHigher
https://en.wikiquote.org/wiki/Higher_consciousness
https://en.wikiquote.org/wiki/Higher_education
https://en.wikiquote.org/wiki/Higher_self
TaleSpin (1990 - 1994) - This casts key characters from The Jungle Book into a 1930's pacific setting, where Baloo is a bush pilot, and running a struggling courier company. His air service and plane named the Sea Duck, are bought by Rebecca Cunningham, who renames the company Higher For Hire. Baloo's navigator Kit Cloudki...
The Amazing Spider-Man (1977 - 1981) - This television adaptation of the massively popular comic book series actually had its share of fans, but not enough to keep it alive for very long. The Spiderman of today, made even more popular by the blockbuster movies, is of a...*ahem*... slightly higher ilk than this low budget production. Gr...
Little Golden Book Land (1989 - 1989) - Adventures from the land of Little Golden Book Land. Characters usually consisted of entities that lacked the ability of higher or basic intelligent thought (Locomotives, Lions, Bears, etc) and were given human facial characteristics. A great children's show that taught morals and always had happy e...
Pearl (1996 - 1997) - Pearl was a one-season TV show that aired on CBS. Created by Don Reo, it starred Rhea Perlman as a middle-aged widow seeking a higher education degree.
Can't Buy Me Love(1987) - Ronald Miller is tired of being a nerd, and makes a deal with one of the most popular girls in school to help him break into the "cool" clic. He offers her a thousand dollars to pretend to be his girlfriend for a month. It succeeds, but he soon learns that the price of popularity may be higher than...
Spanish Judges(1999) - One day Jack (Matthew Lillard) shows up at the antiques-laden warehouse apartment of Max (Vincent D'Onofrio) and Jamie (Valeria Golino), two ruthless low-level criminals with higher aspirations than the two-bit thieving they do now. Jack insists he's Max's brother, but Jamie, whose hobby is collecti...
She-Devil(1989) - Hell hath no fury like a woman scorned...and that's just what happened to fat and frumpy Ruth Patchett. A devoted housewife and mother she tries to please her husband, Bob, an accountant who is trying to climb the social ladder a little bit higher. Her only means of escape is through the romance n...
Man Bites Dog(1992) - Woah! This film is so darkly comical it's set a higher level for films to work at. This movie actually won an oscar in the cannes film festival and was said to be probably the best film o
Higher Learning(1995) - This drama examines the personal, political, and racial dilemmas facing a group of college freshmen as they begin their first semester at Columbus University. Malik (Omar Epps) is an African-American student attending on a track scholarship; academics are not his strong suit, and he goes in thinking...
Shaft in Africa(1973) - the third movie in the shaft trilogy floowes jhon shaft privite detective from harlem on a trip to africa were he must agian bust a crime ring of some sort. the movies budget was significanly higher then the other two but grossed a lot less so metro golwin meyer quickly sold the rights to television...
Thumbelina(1992) - The dam where Thumbelina and her father live is breaking due to the rising waters in the nearby pond. Father worries that when spring comes, the melting snows will rise the water higher than ever causing the dam to crack and water to flow over the meadow, thus drowning the little people who live the...
Busting(1974) - LA cops get in over their heads when they don't heed orders from above and go after a big crime boss. While higher ups in the police department want the cop duo to just focus on nabbing petty criminals, the team does so while still going after LA kingpin Rizzo. Various fist fights, chases, shootouts...
Legal Eagles(1986) - District Attorney Tom Logan is set for higher office, at least until he becomes involved with defence lawyer Laura Kelly and her unpredictable client Chelsea Deardon. It seems the least of Chelsea's crimes is the theft of a very valuable painting, but as the women persuade Logan to investigate furth...
Accepted (2006) ::: 6.4/10 -- PG-13 | 1h 33min | Comedy | 18 August 2006 (USA) -- A high school slacker who's rejected by every school he applies to opts to create his own institution of higher learning, the South Harmon Institute of Technology, on a rundown piece of property near his hometown. Director: Steve Pink Writers:
Don't Be a Menace to South Central While Drinking Your Juice in the ::: 6.6/10 -- Don't Be a Menace to South Central While Drinking Your Juice in the Hood Poster -- A parody of several U.S. films about being in the 'Hood', for instance Boyz n the Hood (1991), South Central (1992), Menace II Society (1993), Higher Learning (1995) and Juice (1992). Director: Paris Barclay
Educating Rita (1983) ::: 7.2/10 -- PG | 1h 50min | Comedy, Drama | 28 October 1983 (USA) -- An alcoholic professor has been hired by a working-class girl for higher education. Director: Lewis Gilbert Writers: Willy Russell (screenplay), Willy Russell (stage play) Stars:
Higher Learning (1995) ::: 6.5/10 -- R | 2h 7min | Crime, Drama, Thriller | 11 January 1995 (USA) -- People from all different walks of life, encounter racial tension, rape, responsibility, and the meaning of an education on a university campus. Director: John Singleton Writer:
Ice Age: The Meltdown (2006) ::: 6.8/10 -- PG | 1h 31min | Animation, Adventure, Comedy | 31 March 2006 (USA) -- Manny, Sid, and Diego discover that the ice age is coming to an end, and join everybody for a journey to higher ground. On the trip, they discover that Manny, in fact, is not the last of the woolly mammoths. Director: Carlos Saldanha Writers:
The Out of Towners (1970) ::: 7.1/10 -- G | 1h 41min | Comedy | 28 May 1970 (USA) -- An Ohio sales executive accepts a higher position within the company and travels to New York City with his wife for his job interview but things go wrong from the start. Director: Arthur Hiller Writer:
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Baka to Test to Shoukanjuu -- -- SILVER LINK. -- 13 eps -- Light novel -- Comedy Romance School Super Power -- Baka to Test to Shoukanjuu Baka to Test to Shoukanjuu -- Fumizuki Academy isn't a typical Japanese high school. This unique institution has implemented a new and innovative system to sort its students. At the end of their freshman year, students take a test that divides up the student body. The highest scorers are placed into A class, all the way down until F class, for the lowest of the low. -- -- Unfortunately for Akihisa Yoshii, his supposedly "great" intellect wasn't quite enough for such a test, and he's now stuck at the bottom of F class. Naturally, F class has the worst facilities: not only rotten tatami mats and broken tables, but also outdated equipment and worn out furniture. On the bright side, his friend Yuuji Sakamoto is in the same class, and to everyone's surprise, the genius girl Mizuki Himeji has also ended up in the same class due to an unforeseen fever on the day of the test. -- -- Unsatisfied with their perquisites, F class rallies behind Yuuji, determined to take on the higher-tiered classes in order to seize their perks by using the school's Examinations Summon Battle system. The participants can summon fantasy characters—whose power levels are equal to their student's test scores—in an all-out battle. Will F class be able to rise to the top, or will they live up to everyone's expectations and fail? -- -- -- Licensor: -- Funimation -- 542,160 7.58
Chainsaw Man -- -- MAPPA -- ? eps -- Manga -- Action Adventure Demons Shounen -- Chainsaw Man Chainsaw Man -- Denji has a simple dream—to live a happy and peaceful life, spending time with a girl he likes. This is a far cry from reality, however, as Denji is forced by the yakuza into killing devils in order to pay off his crushing debts. Using his pet devil Pochita as a weapon, he is ready to do anything for a bit of cash. -- -- Unfortunately, he has outlived his usefulness and is murdered by a devil in contract with the yakuza. However, in an unexpected turn of events, Pochita merges with Denji's dead body and grants him the powers of a chainsaw devil. Now able to transform parts of his body into chainsaws, a revived Denji uses his new abilities to quickly and brutally dispatch his enemies. Catching the eye of the official devil hunters who arrive at the scene, he is offered work at the Public Safety Bureau as one of them. Now with the means to face even the toughest of enemies, Denji will stop at nothing to achieve his simple teenage dreams. -- -- TV - ??? ??, ???? -- 67,759 N/A -- -- Naruto Narutimate Hero 3: Tsuini Gekitotsu! Jounin vs. Genin!! Musabetsu Dairansen Taikai Kaisai!! -- -- Studio Pierrot -- 1 ep -- Game -- Game Adventure Comedy Shounen -- Naruto Narutimate Hero 3: Tsuini Gekitotsu! Jounin vs. Genin!! Musabetsu Dairansen Taikai Kaisai!! Naruto Narutimate Hero 3: Tsuini Gekitotsu! Jounin vs. Genin!! Musabetsu Dairansen Taikai Kaisai!! -- A contest is made by the Fifth Hokage called Jonin vs Genin. The point is to collect crystals for points, with the higher-ranked Chunin and Jonin holding crystals worth more points. The Genin have blue crystals, while the Chunin and Jonin have red crystals. -- -- The video shows various fights between the Genin and Jonin, which each instance ending in the Jonin unknowingly losing their crystal (or discarding it). -- -- (Source: Wikipedia) -- OVA - Dec 22, 2005 -- 67,031 6.77
Fate/stay night Movie: Heaven's Feel - II. Lost Butterfly -- -- ufotable -- 1 ep -- Visual novel -- Action Fantasy Magic Supernatural -- Fate/stay night Movie: Heaven's Feel - II. Lost Butterfly Fate/stay night Movie: Heaven's Feel - II. Lost Butterfly -- The Fifth Holy Grail War continues, and the ensuing chaos results in higher stakes for all participants. Shirou Emiya continues to participate in the war, aspiring to be a hero of justice who saves everyone. He sets out in search of the truth behind a mysterious dark shadow and its murder spree, determined to defeat it. -- -- Meanwhile, Shinji Matou sets his own plans into motion, threatening Shirou through his sister Sakura Matou. Shirou and Rin Toosaka battle Shinji, hoping to relieve Sakura from the abuses of her brother. But the ugly truth of the Matou siblings begins to surface, and many dark secrets are exposed. -- -- Fate/stay night Movie: Heaven's Feel - II. Lost Butterfly continues to focus on the remaining Masters and Servants as they fight each other in the hopes of obtaining the Holy Grail. However, as darkness arises within Fuyuki City, even the state of their sacred war could be in danger. -- -- -- Licensor: -- Aniplex of America -- Movie - Jan 12, 2019 -- 224,860 8.59
Fate/stay night Movie: Heaven's Feel - II. Lost Butterfly -- -- ufotable -- 1 ep -- Visual novel -- Action Fantasy Magic Supernatural -- Fate/stay night Movie: Heaven's Feel - II. Lost Butterfly Fate/stay night Movie: Heaven's Feel - II. Lost Butterfly -- The Fifth Holy Grail War continues, and the ensuing chaos results in higher stakes for all participants. Shirou Emiya continues to participate in the war, aspiring to be a hero of justice who saves everyone. He sets out in search of the truth behind a mysterious dark shadow and its murder spree, determined to defeat it. -- -- Meanwhile, Shinji Matou sets his own plans into motion, threatening Shirou through his sister Sakura Matou. Shirou and Rin Toosaka battle Shinji, hoping to relieve Sakura from the abuses of her brother. But the ugly truth of the Matou siblings begins to surface, and many dark secrets are exposed. -- -- Fate/stay night Movie: Heaven's Feel - II. Lost Butterfly continues to focus on the remaining Masters and Servants as they fight each other in the hopes of obtaining the Holy Grail. However, as darkness arises within Fuyuki City, even the state of their sacred war could be in danger. -- -- Movie - Jan 12, 2019 -- 224,860 8.59
Free!: Dive to the Future -- -- Animation Do, Kyoto Animation -- 12 eps -- Original -- Comedy Drama School Slice of Life Sports -- Free!: Dive to the Future Free!: Dive to the Future -- With the seniors having graduated from high school, the determined swimmers eagerly take on their futures with a dream to fulfill. -- -- Now attending Hidaka University in Tokyo, Haruka Nanase unexpectedly runs into Shiina Asahi, an old teammate and friend from his middle school days. Consequently, the troubling memories regarding his middle school swim team resurface, as it was a time when Haruka's views on swimming became negative and led him to quit the team. Haruka later reconnects with his other middle school classmates; all except for Ikuya Kirishima, who still resents Haruka for quitting the team, resulting in its disbandment. Aware of the issues between them, Haruka resolves to improve his friendship with Ikuya. However, he quickly realizes that making amends with an old friend isn't his only obstacle. -- -- Facing the reality and challenges of encountering higher calibre swimmers, Haruka must work hard to establish himself if he dreams of competing on an international level. -- -- -- Licensor: -- Funimation -- 123,137 7.58
Ga no Iru Tokoro -- -- - -- 1 ep -- Original -- Dementia -- Ga no Iru Tokoro Ga no Iru Tokoro -- A Place Where There Are Moths depicts the conflict between drab concrete block apartment living and the natural environment in Japanese cities. The forces of nature are represented by the motif of a tree whose leaves metamorphose into orange moths and take over a middle-aged woman's apartment, pushing her room higher and higher within the building. -- -- (Source: Midnight Eye) -- Movie - ??? ??, 2001 -- 831 4.44
Godzilla 3: Hoshi wo Kuu Mono -- -- Polygon Pictures -- 1 ep -- Original -- Action Adventure Sci-Fi -- Godzilla 3: Hoshi wo Kuu Mono Godzilla 3: Hoshi wo Kuu Mono -- A door opens, and a golden seal shatters a star. -- -- It is the early 21st century. Mankind has lost the battle for planet Earth to Godzilla, and has taken to the stars in search of a new home. But the search ends in vain, forcing them and their alien allies back to Earth. But 20,000 years have passed in their absence, and the Earth is a wholly different place. -- -- The planet's flora and fauna now embody and serve Godzilla. Earth is a monster's planet, ruled by the largest Godzilla ever at 300 meters in height. Godzilla Earth. -- -- Human protagonist, Captain Haruo, yearns to defeat Godzilla and retake the planet for mankind. There, he meets aboriginal descendants of the human race, the Houtua tribe. The Houtua twin sisters, Maina and Miana, lead him to the skeletal remains of Mecha-Godzilla, an old anti-Godzilla weapon, which to everyone's surprise is still alive in the form of self-generating nanometal. Taken from the Mecha-Godzilla carcass, the nanometals have gradually been rebuilding a "Mecha-Godzilla City," a potential weapon capable of destroying Godzilla Earth. -- -- As the strategy develops, a rift forms between the humans and the Bilusaludo, one of several alien races that had joined the humans on their exodus from Earth. Their leader, Galu-gu, believes that the secret to defeating Godzilla lies in the use of superhuman powers – namely, the nanometal integration – but Haruo resists, fearing that in defeating monsters, they must not become monsters themselves. Haruo ultimately uses his means for defeating Godzilla Earth to destroy the Mecha-Godzilla city so as to prevent nanometal assimilation, killing Galu-gu. However, his childhood friend, Yuuko, has been absorbed by the nanometal integration and has fallen into a brain dead coma. -- -- The human race, once again, is lost. Metphies, commander of the priestly alien race, Exif, marvels at the miraculous survival of Haruo, he begins to attract a following. The Exif has secretly harbored this outcome as their "ultimate goal." Miana and Maina issue warnings against Metphies, while Haruo begins to question mankind's next move. -- -- With no means for defeating Godzilla Earth, mankind watches as King Ghidorah, clad in a golden light, descends on the planet. The earth shakes once again with as war moves to a higher dimension. -- -- What is Godzilla exactly? Does mankind stand a chance? Is there a future vision in Haruo's eyes? Find out in the finale. -- -- (Source: Official site) -- Movie - Nov 9, 2018 -- 23,950 6.26
Kaleido Star -- -- Gonzo, Production I.G -- 51 eps -- Original -- Comedy Sports Drama Fantasy Shoujo -- Kaleido Star Kaleido Star -- The Kaleido Stage is known throughout the world for captivating audiences with its amazing acrobatics, innovative routines, and extravagant costumes and sets. It is a place for guests to believe in magic, and Sora Naegino wants nothing more than to be a part of that magic—by becoming an acrobat for the famed circus herself. -- -- To realize her dream, she travels from Japan to California to audition for a place in the group. However, Sora learns that she needs much more than her natural talent to bring joy to the faces in the crowd. She quickly discovers just how difficult it is to be a professional performer where the stakes—and the stunts—are higher and mistakes spell danger! To put on performances worthy of the Kaleido Stage, she will need to endure rigorous training, unconventional assignments, fierce competition, and the antics of a mischievous spirit named Fool. -- -- Can Sora reach new heights, make new friends, conquer her fears, and surpass her limits to become a Kaleido Star? -- -- 70,745 7.94
Kaleido Star -- -- Gonzo, Production I.G -- 51 eps -- Original -- Comedy Sports Drama Fantasy Shoujo -- Kaleido Star Kaleido Star -- The Kaleido Stage is known throughout the world for captivating audiences with its amazing acrobatics, innovative routines, and extravagant costumes and sets. It is a place for guests to believe in magic, and Sora Naegino wants nothing more than to be a part of that magic—by becoming an acrobat for the famed circus herself. -- -- To realize her dream, she travels from Japan to California to audition for a place in the group. However, Sora learns that she needs much more than her natural talent to bring joy to the faces in the crowd. She quickly discovers just how difficult it is to be a professional performer where the stakes—and the stunts—are higher and mistakes spell danger! To put on performances worthy of the Kaleido Stage, she will need to endure rigorous training, unconventional assignments, fierce competition, and the antics of a mischievous spirit named Fool. -- -- Can Sora reach new heights, make new friends, conquer her fears, and surpass her limits to become a Kaleido Star? -- -- -- Licensor: -- ADV Films, Funimation -- 70,745 7.94
Kingdom 2nd Season -- -- Studio Pierrot -- 39 eps -- Manga -- Action Military Historical Seinen -- Kingdom 2nd Season Kingdom 2nd Season -- A year after the devastating battle against the formidable Zhao, the State of Qin has returned its focus to pursuing King Ying Zheng's ambition of conquering the other six states and unifying China. Their next target is Wei, a smaller state which stands as a geographic stepping stone for the sake of conquest. -- -- Li Xin, now a three hundred man commander of the swiftly rising Fei Xin Unit, continues to seek out lofty achievements in order to garner recognition for himself and his soldiers, motivated by those previously lost in battle. In the preliminary battles ahead of Qin's invasion of Wei, Xin finds competition in other young commanders who are of a higher social status than him. Back in Qin, the royal palace faces turmoil as opposing factions begin to make their move against Ying Zheng's regime. -- -- With their hands full both abroad and at home, Zheng and Xin must lead the way in this era of unending war, resolved to etch their names in history by creating a unified China. -- -- -- Licensor: -- Funimation -- 82,402 8.38
Kono Oto Tomare! 2nd Season -- -- Platinum Vision -- 13 eps -- Manga -- Drama Music Romance School Shounen -- Kono Oto Tomare! 2nd Season Kono Oto Tomare! 2nd Season -- The Tokise High School Koto Club has courageously pushed through their fractured and unsynchronized performance at the Kanto Region Traditional Japanese Music Festival. Clubmembers Chika Kudou, Satowa Houzuki, Takezou Kurata, Hiro Kurusu, Kouta Mizuhara, Saneyasu Adachi, and Michitaka Sakai are devastated to learn the negative results of their performance, leaving them crushed. Nonetheless, the group recognizes their potential and enthusiastically agree to collectively sharpen their skills, improve their flaws, and develop higher caliber playing to succeed in the upcoming national qualifiers in winter. -- -- With the help of their now willing club advisor Suzuka Takinami, the group's goal gradually becomes achievable as they begin to grasp the foundations of good music and refine their koto-playing abilities, with the suggestion of performing more often to gain what they lack most—experience. -- -- However, as their journey to nationals is underway, the koto club members face challenges that obstruct their focus and progress. Not only does the threat of other powerhouse schools and musicians remain, but the high school issues of budding romance and soon-to-be-graduating seniors also begin to push the limits of the determined group of teenagers and the future of the koto club. -- -- 90,539 8.42
Mobile Suit Gundam: Iron-Blooded Orphans 2nd Season -- -- Sunrise -- 25 eps -- Original -- Action Drama Mecha Sci-Fi Space -- Mobile Suit Gundam: Iron-Blooded Orphans 2nd Season Mobile Suit Gundam: Iron-Blooded Orphans 2nd Season -- Tekkadan has now become a direct affiliate of Teiwaz after procuring a new trade agreement with Arbrau. With its newfound funds and prestige, Tekkadan finds both its list of allies and enemies growing. Meanwhile, the flames of the Gjallarhorn power struggle continue to rage in full force. As a part of her efforts to make Mars financially independent from Earth, Kudelia Aina Bernstein founds the Admoss Company and enlists Tekkadan as her business partner. -- -- The stakes are getting higher as the Tekkadan family continues to grow. Will Orga, Mikazuki, and the rest of the Tekkadan faction be able to keep up, or will Kudelia's dream of Martian independence die out? -- -- 102,471 8.23
Mobile Suit Gundam: Iron-Blooded Orphans 2nd Season -- -- Sunrise -- 25 eps -- Original -- Action Drama Mecha Sci-Fi Space -- Mobile Suit Gundam: Iron-Blooded Orphans 2nd Season Mobile Suit Gundam: Iron-Blooded Orphans 2nd Season -- Tekkadan has now become a direct affiliate of Teiwaz after procuring a new trade agreement with Arbrau. With its newfound funds and prestige, Tekkadan finds both its list of allies and enemies growing. Meanwhile, the flames of the Gjallarhorn power struggle continue to rage in full force. As a part of her efforts to make Mars financially independent from Earth, Kudelia Aina Bernstein founds the Admoss Company and enlists Tekkadan as her business partner. -- -- The stakes are getting higher as the Tekkadan family continues to grow. Will Orga, Mikazuki, and the rest of the Tekkadan faction be able to keep up, or will Kudelia's dream of Martian independence die out? -- -- -- Licensor: -- Funimation -- 102,471 8.23
Muv-Luv Alternative: Total Eclipse -- -- ixtl, Satelight -- 24 eps -- Visual novel -- Action Military Sci-Fi Mecha -- Muv-Luv Alternative: Total Eclipse Muv-Luv Alternative: Total Eclipse -- Since 1973, an invasion of aliens upon Earth known as BETA has driven human civilization to destruction. In order to defend themselves from this enormous mass of enemy force, mankind has developed large humanoid arms called Tactical Surface Fighters and deployed them to its defense lines through out the world. However, its efforts could only slow down the impending defeat, and mankind has been forced to abandon the major areas of the Eurasian Continent. For 30 years, they have been caught in an endless war against BETA without any hopes of victory. -- -- Now in 2001, Imperial Japan faces difficulties in the development of the next-generation of Tactical Surface Fighters (TSF) as it defends the front lines of the Far East. The UN has proposed a joint development program between Imperial Japan and the United States as a part of its TSF international mutual development unit, the Prominence Project. -- -- Yui Takamura (a surface pilot of the Imperial Royal Guards of Japan) is given the responsibility of the project and sets off to Alaska. Meanwhile, Yuya Bridges, also a surface pilot of the US Army, heads to the same destination. -- -- They never had any idea just how drastically their encounter would change their fates. -- -- This story of internal dilemma takes place during the development of the new Tactical Surface Fighters, the most crucial and effective weapons against BETA. And this time, the stakes are much higher than a handful of lives and our sanity. -- -- All we can do is fight. -- -- (Source: Muv-Luv Total Eclipse Official English Website, edited) -- 81,759 7.11
Muv-Luv Alternative: Total Eclipse -- -- ixtl, Satelight -- 24 eps -- Visual novel -- Action Military Sci-Fi Mecha -- Muv-Luv Alternative: Total Eclipse Muv-Luv Alternative: Total Eclipse -- Since 1973, an invasion of aliens upon Earth known as BETA has driven human civilization to destruction. In order to defend themselves from this enormous mass of enemy force, mankind has developed large humanoid arms called Tactical Surface Fighters and deployed them to its defense lines through out the world. However, its efforts could only slow down the impending defeat, and mankind has been forced to abandon the major areas of the Eurasian Continent. For 30 years, they have been caught in an endless war against BETA without any hopes of victory. -- -- Now in 2001, Imperial Japan faces difficulties in the development of the next-generation of Tactical Surface Fighters (TSF) as it defends the front lines of the Far East. The UN has proposed a joint development program between Imperial Japan and the United States as a part of its TSF international mutual development unit, the Prominence Project. -- -- Yui Takamura (a surface pilot of the Imperial Royal Guards of Japan) is given the responsibility of the project and sets off to Alaska. Meanwhile, Yuya Bridges, also a surface pilot of the US Army, heads to the same destination. -- -- They never had any idea just how drastically their encounter would change their fates. -- -- This story of internal dilemma takes place during the development of the new Tactical Surface Fighters, the most crucial and effective weapons against BETA. And this time, the stakes are much higher than a handful of lives and our sanity. -- -- All we can do is fight. -- -- (Source: Muv-Luv Total Eclipse Official English Website, edited) -- -- Licensor: -- Sentai Filmworks -- 81,759 7.11
Naruto Narutimate Hero 3: Tsuini Gekitotsu! Jounin vs. Genin!! Musabetsu Dairansen Taikai Kaisai!! -- -- Studio Pierrot -- 1 ep -- Game -- Game Adventure Comedy Shounen -- Naruto Narutimate Hero 3: Tsuini Gekitotsu! Jounin vs. Genin!! Musabetsu Dairansen Taikai Kaisai!! Naruto Narutimate Hero 3: Tsuini Gekitotsu! Jounin vs. Genin!! Musabetsu Dairansen Taikai Kaisai!! -- A contest is made by the Fifth Hokage called Jonin vs Genin. The point is to collect crystals for points, with the higher-ranked Chunin and Jonin holding crystals worth more points. The Genin have blue crystals, while the Chunin and Jonin have red crystals. -- -- The video shows various fights between the Genin and Jonin, which each instance ending in the Jonin unknowingly losing their crystal (or discarding it). -- -- (Source: Wikipedia) -- OVA - Dec 22, 2005 -- 67,031 6.77
Piano no Mori (TV) 2nd Season -- -- Gaina -- 12 eps -- Manga -- Comedy Drama Music School Seinen -- Piano no Mori (TV) 2nd Season Piano no Mori (TV) 2nd Season -- With the start of the Chopin piano competition, Kai Ichinose, Shuuhei Amamiya, and many other hopeful musicians from around the world strive to reach the top. The stakes have never been higher, and the judges are rigorous when it comes to selecting the winner out of the plethora of talented pianists. This competition is so harsh that even famous prodigies can be easily eliminated. -- -- Some play for the money, some play to fulfill their duty to their families, and yet others play for their music to be heard. However, the only one who can reach the top is the one who embodies the spirit of the music Frédéric Chopin crafted for future generations. With the stakes higher than ever before, rivalries, friendships, and family ties will be tested, and each pianist will find their own sound. -- -- 34,359 7.39
Seikaisuru Kado -- -- Toei Animation -- 12 eps -- Original -- Sci-Fi -- Seikaisuru Kado Seikaisuru Kado -- Cool-headed and rational, Koujirou Shindou is a government official and master negotiator with a well-earned reputation. While departing on a business trip, a giant cube materializes and his plane is taken undamaged into the mysterious, indestructible structure. -- -- As Japanese authorities attempt to identify the cube's properties and origins, Shindou encounters an otherworldly entity known as Yaha-kui zaShunina, who materializes in the form of a human man. He assures Shindou that the passengers are not in any danger and requests help in negotiations with the human world. -- -- Hailing from a higher dimensional universe known as Novo, Yaha-kui zaShunina is able to transfer information between Novo and Shindou's universe through a cube called Kado. Despite having these unfathomable abilities, he does not appear hostile. Instead, he announces that he has come to this world with only one intention: to "advance" humanity—starting with Japan. -- -- 95,698 6.80
Sol Bianca -- -- AIC -- 2 eps -- Original -- Action Sci-Fi Adventure Space -- Sol Bianca Sol Bianca -- Five female pirates pilot the Sol Bianca, a starship with a higher level of technology than any other known. With it, they seek out riches, such as the Gnosis, an legendary item of power, and pasha, the most valuable mineral in the galaxy. Along the way, they must consider a stowaway's quest to save the one he loves, and seek revenge against those that have wronged them. -- -- (Source: ANN) -- -- Licensor: -- ADV Films -- OVA - Mar 21, 1990 -- 4,975 6.34
https://wiki.archlinux.org/index.php/Hwdetect#Higher_level_modules
https://commons.wikimedia.org/wiki/File:Herbert_Hoover_on_the_value_of_higher_education,_1927_-_NARA_-_187098.tif
20182020 UK higher education strikes
80th National Guard Higher Command (Greece)
95th National Guard Higher Command (Greece)
96th National Guard Higher Command (Greece)
98th National Guard Higher Command (Greece)
Access to Higher Education
Active Learning in Higher Education
Advanced Higher
Afghanistan Institute of Higher Education
A Higher Form of Killing
A Higher Loyalty
Aimhigher
Alabama Commission on Higher Education
Alexandria Higher Institute of Engineering and Technology
All Saints Church, Higher Walton
Al-Maktoum College of Higher Education
American Higher Education Development
Analytics in higher education
Ancient higher-learning institutions
Arab Higher Committee
Arts and Humanities in Higher Education
Assam Higher Secondary Education Council
Assessment in higher education
Association for Biblical Higher Education
Association for Theatre in Higher Education
Association of Higher Civil and Public Servants
Association of Reformed Institutions of Higher Education
Association of Southeast Asian Institutions of Higher Learning
Azerbaijan Higher Naval Academy
Bachelor's degree or higher
Back to Higher Ground
Bah Institute for Higher Education
Bakhtar Institute of Higher Education
Bedford College of Higher Education
Besat Institute of Higher Education of Kerman
Bharath Institute of Higher Education and Research
Bideford Higher Cemetery
Bloch's higher Chow group
Bromley College of Further & Higher Education
Buckland v Bournemouth University Higher Education Corp
Bullying of students in higher education
Cairo Higher Institute of Cinema
California Master Plan for Higher Education
Cape Higher Education Consortium
Caritas Institute of Higher Education
Carnegie Classification of Institutions of Higher Education
Catholic higher education
Center for Higher National Studies
Center for International Higher Education
Center for the Study of Higher Education
Central Institute of Higher Tibetan Studies
Centre for Higher Education
Centre for Higher Secondary Education
Certificate of Higher Education
Change: The Magazine of Higher Learning
Chief information officer (higher education)
Chu Hai College of Higher Education
Church of the Higher Life
Coalition of Higher Education Students in Scotland
Coity Higher
Colorado Commission on Higher Education
Commission on Higher Education (Philippines)
Commission on the Future of Higher Education
Commonwealth System of Higher Education
Connecticut Board of Regents for Higher Education
Consortium for North American Higher Education Collaboration
Consortium of Higher Education LGBT Resource Professionals
Consortium on Financing Higher Education
Copa Higher Power
Council for Higher Education Accreditation
Council for Higher Education in Israel
Council for Higher Education in Newark
Council for Industry and Higher Education
Council for the Advancement of Standards in Higher Education
Council of Higher Education (Turkey)
Council of Higher Secondary Education
Council of Higher Secondary Education, Odisha
Dadabhoy Institute of Higher Education
Dartmouth Higher Ferry
Degenerate Higher-Order Scalar-Tensor theories
Department of Further and Higher Education, Research, Innovation and Science
Department of Higher Education
Department of Higher Education and Training
Department of Higher Education, Odisha
Diploma of Higher Education
Do Women Have a Higher Sex Drive?
Echthighern mac Cenntig
Ensenada Center for Scientific Research and Higher Education
European Association for Quality Assurance in Higher Education
European Higher Education Area
Even Higher
Filter (higher-order function)
Fold (higher-order function)
Foundation for Independent Higher Education
Friends Association for Higher Education
Further and Higher Education Act 1992
Gateway to Higher Education (program)
Glion Institute of Higher Education
Governance in higher education
Government Higher Secondary Institute Botingoo
Great Lakes Higher Education Corporation
Grimsby Institute of Further & Higher Education
Guangzhou Higher Education Mega Center
Guangzhou Higher Education Mega Center Central Stadium
Gujarat Secondary and Higher Secondary Education Board
Havering College of Further and Higher Education
Health promotion in higher education
Helderberg College of Higher Education
Hessian Ministry of Higher Education, Research and the Arts
Higher
Higher (Ala Boratyn album)
Higher alkanes
Higher and Higher
Higher and Higher (film)
Higher and Higher The Best of Heaven 17
Higher and Higher (The Moody Blues song)
Higher Boscaswell
Higher Brothers
Higher category theory
Higher Certificate
Higher Colleges of Technology
Higher Condurrow
Higher consciousness
Higher-dimensional algebra
Higher-dimensional Einstein gravity
Higher-dimensional gamma matrices
Higher-dimensional supergravity
Higher diploma
Higher Diploma of Technical Studies (France)
Higher education
Higher Education Academy
Higher education accreditation
Higher Education Act
Higher Education Act 2004
Higher Education Act of 1965
Higher Education and Research Act 2017
Higher Education and Research Opportunities in the UK
Higher Education and Training Awards Council
Higher Education Commission
Higher Education Commission (Pakistan)
Higher education controversy in Odisha
Higher Education (disambiguation)
Higher Education European Masters
Higher Education Evaluation and Accreditation Council of Taiwan
Higher Education for the Future
Higher Education Funding Council for England
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Higher Education GAA
Higher education in Alberta
Higher education in British Columbia
Higher education in Canada
Higher education in China
Higher education in Erie, Pennsylvania
Higher education in Hong Kong
Higher education in India
Higher education in Iran
Higher education in Italy
Higher education in Japan
Higher education in Manitoba
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Higher education in Myanmar
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Higher education in Portugal
Higher education in Quebec
Higher education in Saskatchewan
Higher education in Spain
Higher education in Sri Lanka
Higher Education Institute of Brasilia
Higher education in the Northwest Territories
Higher education in the Philippines
Higher education in Ukraine
Higher education in Yukon
Higher Education (journal)
Higher education leadership
Higher Education Press
Higher Education Price Index
Higher Education Recruitment Consortium
Higher Education Research Data Collection
Higher Education Research Institute
Higher Education Review
Higher Education Snooker and Pool Council
Higher Education Support Act 2003
Higher End
Higher (Erik Grnwall song)
Higher Etiquette
Higher Folds
Higherford
Higher formation insignia of the British Army
Higher Futures
Higher Ground
Higher Ground (Jennifer Rush song)
Higher Ground Productions
Higher Ground (Stevie Wonder song)
Higher Ground (TV series)
Higher Institute for Applied Sciences and Technology
Higher Institute of Ballet
Higher Institute of Dramatic Arts (Damascus)
Higher Institute of Engineering
Higher Institute of Iranian Studies
Higher Institute of Technology of Ivory Coast
Higher Intelligence Agency
Higher Learning
Higher Learning Commission
Higher Learning (disambiguation)
Higher Learning Vol. 2
Higherlife Foundation
Higher Life movement
Higher local field
Higher Love
Higher National Certificate
Higher National Diploma
Higher National Technical Education Certificate
Higher Octave Music
Higher-order abstract syntax
Higher-Order and Symbolic Computation
Higher-order compact finite difference scheme
Higher-order differential cryptanalysis
Higher-order function
Higher order grammar
Higher-order logic
Higher order message
Higher-order operad
Higher-Order Perl
Higher-order programming
Higher-order singular value decomposition
Higher-order sinusoidal input describing function
Higher-order statistics
Higher-order thinking
Higher-order volition
Higher (Peter Jback song)
Higher Plane (Kool & the Gang song)
Higher Population Council (Jordan)
Higher Porthpean
Higher Power
Higher Power (disambiguation)
Higher Power (film)
Higher Power (seaQuest DSV)
Higher Preparatory Examination (HF)
Higher Principle
Higher (Regina Belle album)
Higher Scientific Institute for Diocesan Priests at St. Augustine's
Higher (Scottish)
Higher self
Higher-speed rail
Higher-spin theory
Higher sulfur oxides
Higher Superstition
Higher (Taio Cruz song)
Higher Technical Examination Programme
Higher Technical Institute of Cyprus
Higher Technological Institute
Higher Than High
Higher Than the Eiffel
Higher than the Sky
Higher Than the Stars
Higher than the Sun
Higher than the Sun (song)
Higher (The Game song)
Higher (The Horrors album)
Higher (The Naked and Famous song)
Higher (The Overtones album)
Higher (The Ready Set song)
Higher Topos Theory
Higher Truth
Higher University of San Andrs
Higher Walton
Hong Kong Higher Level Examination
Honiara Solomon Islands College of Higher Education
Humboldtian model of higher education
IAPCHE: the International Association for the Promotion of Christian Higher Education
IILM Academy of Higher Learning
IILM Institute for Higher Education
Inside Higher Ed
Institute for Learning and Teaching in Higher Education
Institute of Higher International Studies
Institute of Higher National Diploma in Engineering
Institute of Higher Nervous Activity and Neurophysiology
International Centre for Higher Education Research Kassel
International Institute for Higher Education in Morocco
Internationalization of higher education
Inversions higher than third
I Want to Take You Higher
Journal of Diversity in Higher Education
Journal of Higher Criticism
Journal of Higher Education Outreach and Engagement
Journal of Hispanic Higher Education
JSS Academy of Higher Education & Research
Karpagam Academy of Higher Education
La Sainte Union College of Higher Education
Leadership Foundation for Higher Education
Lean higher education
Let's Go Higher
List of African-American pioneers in desegregation of higher education
List of Dominican Ministers of Higher Learning, Science and Technology
List of higher education associations and alliances
List of higher education associations and organizations in Canada
List of higher education institutions in Denver
List of higher education institutions in Hong Kong
List of higher education institutions in Maharashtra
List of higher virus taxa
List of Independent Baptist higher education institutions
List of institutions of higher education in Andhra Pradesh
List of institutions of higher education in Arunachal Pradesh
List of institutions of higher education in Assam
List of institutions of higher education in Bangalore
List of institutions of higher education in Bihar
List of institutions of higher education in Chandigarh
List of institutions of higher education in Chhattisgarh
List of institutions of higher education in Delhi
List of institutions of higher education in Goa
List of institutions of higher education in Gujarat
List of institutions of higher education in Haryana
List of institutions of higher education in Himachal Pradesh
List of institutions of higher education in India
List of institutions of higher education in Jammu and Kashmir
List of institutions of higher education in Jharkhand
List of institutions of higher education in Karnataka
List of institutions of higher education in Kerala
List of institutions of higher education in Madhya Pradesh
List of institutions of higher education in Manipur
List of institutions of higher education in Mizoram
List of institutions of higher education in Nagaland
List of institutions of higher education in Odisha
List of institutions of higher education in Punjab, India
List of institutions of higher education in Rajasthan
List of institutions of higher education in Russia
List of institutions of higher education in Sikkim
List of institutions of higher education in Tamil Nadu
List of institutions of higher education in Telangana
List of institutions of higher education in Tripura
List of institutions of higher education in Uttarakhand
List of institutions of higher education in Uttar Pradesh
List of institutions of higher education in West Bengal
List of Monterrey Institute of Technology and Higher Education faculty
List of recognized higher education accreditation organizations
List of unaccredited institutions of higher education
List of universities and higher education colleges in London
List of unrecognized higher education accreditation organizations
Lists of institutions of higher education by endowment size
Macarthur Institute of Higher Education
Mahapola Higher Education Scholarship Trust Fund
Manipal Academy of Higher Education
Manipal Academy of Higher Education, Dubai
Manual of the Higher Plants of Oregon
Map (higher-order function)
Master of Arts in Higher Education and Student Development (MAHE)
Midwestern Higher Education Compact
Minister for Further Education, Higher Education and Science
Minister of Higher Education, Research and Innovation (France)
Minister of Higher Education, Science and Technology
Ministry of Basic, Higher and Technical Education (Bangsamoro)
Ministry of Education and Higher Education (Lebanon)
Ministry of Education and Higher Education (Quebec)
Ministry of Higher and Tertiary Education (Zimbabwe)
Ministry of higher education
Ministry of Higher Education (Afghanistan)
Ministry of Higher Education and Science (Denmark)
Ministry of Higher Education and Scientific Research (Iraq)
Ministry of Higher Education and Scientific Research (Ivory Coast)
Ministry of Higher Education and Scientific Research (UAE)
Ministry of Higher Education (Egypt)
Ministry of Higher Education (Malaysia)
Ministry of Higher Education (Oman)
Ministry of Higher Education, Science and Technology
Ministry of Higher Education, Science and Technology (Dominican Republic)
Ministry of Higher Education, Science, Research and Innovation
Ministry of Higher Education (Soviet Union)
Ministry of Higher Education, Technology and Innovation
Ministry of Science and Higher Education (Poland)
Ministry of Science, Technology and Higher Education
Monterrey Center for Higher Learning of Design
Monterrey Institute of Technology and Higher Education
Move On Up a Little Higher
National Academy of Higher Education
National Agency for Higher Education
National Agency for Quality Assurance in Higher Education
National Association of Teachers in Further and Higher Education
National Board for Higher Mathematics
National Computer Center for Higher Education (France)
National Council of Academic Evaluation and Accreditation of Higher Education Institutions
National Institute for Higher Education
National Institution for Academic Degrees and Quality Enhancement of Higher Education
National Security Higher Education Advisory Board
Nevada System of Higher Education
Newbold College of Higher Education
Newcastle Higher
New England Board of Higher Education
New Jersey Commission on Higher Education
Noorul Islam Centre for Higher Education
Northeast Ohio Council on Higher Education
North of England Council for Promoting the Higher Education of Women
Office of the Higher Education Commission
One Step Higher
Online learning in higher education
Oregon Higher Education Coordinating Commission
Palestinian Ministry of Education and Higher Education
Pennsylvania Higher Education Assistance Agency
Piled Higher and Deeper
Postgraduate Certificate in Higher Education
Potchefstroom University for Christian Higher Education
Quality Assurance Agency for Higher Education
Quality Assurance Agency of Higher Education
Raise Your Spirit Higher
Raise Your Spirit Higher (2003 album)
Raise Your Spirit Higher (2004 album)
Rana Institute of Higher Studies
Rule according to higher law
Scottish Further and Higher Education Association
Self-financing Higher Education in Hong Kong
Shell higher olefin process
Shiraz Pasargad Higher Education Institute
Should Be Higher
Sleep deprivation in higher education
Society for Research into Higher Education
Sora no Uta ~Higher and Higher~/Hisbi
Sri Devaraj Urs Academy of Higher Education and Research
Sri Ramachandra Institute of Higher Education and Research
Sri Sathya Sai Institute of Higher Learning
Sri Siddhartha Academy of Higher Education
Statal Institute of Higher Education Isaac Newton
St. Peter's Institute of Higher Education and Research
Strengthening Transparency in Higher Education Act
Student rights in higher education
Studies in Higher Education
Suzhou Dushu Lake Higher Education Town
Take Me Higher
Take You Higher
Take You Higher (Wilkinson song)
Taking Me Higher
Tamaulipas Institute of Higher Education
Teaching and Higher Education Act 1998
Teaching in Higher Education
Technological and Higher Education Institute of Hong Kong
Texas Higher Education Coordinating Board
The Chronicle of Higher Education
The Higher Command
The Higher Institute of Computer Technology
The Higher Law
The Higher Power of Lucky
The Higher They Climb
The Hispanic Outlook in Higher Education
The Journal of Blacks in Higher Education
The Journal of Higher Education
The Review of Higher Education
Times Higher Education
Times Higher Education World University Rankings
Tresham College of Further and Higher Education
Unaccredited institutions of higher education
Universities and higher education in Brazil
Vyas Institutes of Higher Education
Weaver v National Association of Teachers in Further and Higher Education
West Bengal Council of Higher Secondary Education
West London College (further and higher education)
West London Institute of Higher Education
Wikipedia talk:WikiProject Higher education/Archive 11
World Declaration on Higher Education
(Your Love Keeps Lifting Me) Higher and Higher
Zand Institute of Higher Education


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