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If you flipped a coin 100 times,
it is possible that you would get exactly 50 ``heads'' and 50 ``tails''.
More likely it would be something between 6040 and 4060.
Typically, how much deviation from 50 would you expect to see?
The average (mean) value should be 50,
but some other value is almost
always obtained from a random sample.
The other value is called the sample mean.
We would like to know how much difference to expect between the
sample mean and the true mean.
The average squared difference is called the
variance of the sample mean.
For a very large sample,
the sample mean should be proportionately much closer to the true mean
than for a smaller sample.
This idea will lead to an uncertainty relation between the
probable error in the estimated mean and the size of the sample.
Let us be more precise.
The ``true value'' of the mean could be defined by flipping the coin n times
and conceiving of n going to infinity.
A more convenient definition of ``true value'' is that the experiment
could be conceived of as having been done
separately under identical conditions
by an infinite number of people (an ensemble).
Such an artifice will enable us to define
a timevariable mean for coins which change with time.
The utility of the concept of an ensemble is often subjected to serious attack
both from the point of view of the theoretical foundations of statistics and
from the point of view of experimentalists applying the techniques of
statistics.
Nonetheless a great body of geophysical literature uses the
artifice of assuming the existence of an unobservable ensemble.
The advocates
of using ensembles (the Gibbsians) have the advantage over their adversaries
(the Bayesians)
in that their mathematics is more tractable (and more explainable).
So, let us begin!
A conceptual average over the ensemble,
called an expectation, is denoted by the symbol E.
The index for summation over the ensemble is never shown explicitly;
every random variable is presumed to have one.
Thus, the true mean at time t may be defined as
 
(13) 
If the mean does not vary with time, we may write
 
(14) 
Likewise, we may be interested in a property of x_{t}
called its variance which is a measure of variability
about the mean defined by
 
(15) 
The x_{t} random numbers could be defined in such a way that or
m or both is either timevariable or constant.
If both are constant, we have
 
(16) 
When manipulating algebraic expressions
the symbol E behaves like a summation sign, namely
 
(17) 
Notice that the summation index is not given,
since the sum is over the ensemble, not time.
Now let x_{t} be a time series made up from (identically distributed,
independently chosen)
random numbers in such a way that m and do not depend on time.
Suppose we have a sample of n points of x_{t} and are
trying to determine the value of m.
We could make an estimate of the mean m with the formula
 
(18) 
A somewhat more elaborate method of estimating the mean
would be to take a weighted average.
Let w_{t} define a set of weights normalized so that
 
(19) 
With these weights the more elaborate estimate of the mean is
 
(20) 
Actually (18) is just a special case of (20) where the
weights are w_{t} = 1/n; .
Our objective in this section is to determine how far the estimated mean
is likely to be from the true mean m for a sample of length n.
One possible definition of this excursion is
 
(21) 
 (22) 
Now utilize the fact that
 
(23) 
 (24) 
 (25) 
Now the expectation symbol E may be regarded as a summation sign and brought
inside the sums on t and s.
 
(26) 
By the randomness of x_{t} and x_{s} the expectation on the right,
that is, the sum over the ensemble,
gives zero unless s = t.
If s = t, then the expectation is the variance defined by (16).
Thus we have
 
(27) 
 (28) 
or
 
(29) 
Now let us examine this final result for n weights each of size 1/n.
For this case, we get
 
(30) 
This is the most important property of random numbers
which is not intuitively obvious.
For a zero mean situation it may be expressed in words:
``n random numbers of unit magnitude add up to a magnitude of about the
square root of n.''
When one is trying to estimate the mean of a random series which has a
timevariable mean, one faces a basic dilemma.
If one includes a lot of numbers in the sum to get small,
then m may be changing while one is trying to measure it.
In contrast, measured from a short sample of the series
might deviate greatly from the true m (defined by an
infinite sum over the ensemble at any point in time).
This is the basic dilemma faced by a stockbroker when a client tells him,
``Since the market fluctuates a lot I'd like you to sell my stock
sometime when the price is above the mean selling price.''
If we imagine that a time series is sampled every seconds
and we let
denote the length of the sample then (30) may be
written as
 
(31) 
It is clearly desirable
to have both and as small as possible.
If the original random numbers x_{t} were correlated with one another,
for example, if x_{t} were an approximation to a continuous function,
then the sum of the n numbers would not cancel to root n.
This is expressed by the inequality
 
(32) 
The inequality (32) may be called an uncertainty relation between
accuracy and time resolution.
In considering other sets of weights one may take a definition of which is more physically sensible than times the number of weights.
For example, if the weights w_{t} are given by a sampled gaussian function as
shown in Figure 2 then could be taken as the separation of
halfamplitude points,
1/e points, the time span which includes 95 percent of the area,
or it could be given many other ``sensible'' interpretations.
Given a little slop in the definition of and ,it is clear that the inequality of (32) is not to be strictly applied.
41
Figure 2 Binomial coefficients tend to the gaussian
function. Plotted are the coefficients of Z^{t} in (.5 + .5Z)^{20}.

 
Given a sample of a zero mean random time series x_{t},
we may define another series y_{t} by y_{t} = x^{2}_{t}.
The problem of estimating the variance
of x_{t} is identical to the problem of estimating the mean
m of y_{t}.
If the sample is short,
we may expect an error in our estimate of the variance.
Thus, in a scientific paper one would like to write for the mean
 
(33) 
 (34) 
but since the variance often is not known either, it is
necessary to use the estimated , that is
 
(35) 
Of course (35) really is not right
because we really should add something to indicate additional uncertainty
due to error in . This estimated error would again have an error, ad infinitum.
To really express the result properly,
it is necessary to have a probability
density function to calculate all the E(x^{n}) which are required.
The probability function can be either estimated from the data
or chosen theoretically.
In practice, for a reason given in a later section,
the gaussian function often occurs.
In the exercise it is shown that
 
(36) 
Since , by squaring we have
 
(37) 
The inequality applies if the random numbers x_{t} are not totally
unpredictable random numbers.
If x_{t} is an approximation to a continuous function,
then it is highly predictable and there will be a lot of slack
in the inequality.
Correlation is a concept similar to cosine.
A cosine measures the angle between two vectors.
It is given by the dot product of the two vectors divided by their magnitudes
Correlation is the same sort of thing,
except x and y are scalar random variables,
so instead of having a vector subscript their subscript is
the implicit ensemble subscript.
Correlation is defined
In practice one never has an ensemble.
There is a practical problem when the
ensemble average is simulated by averaging over a sample.
The problem arises with small samples
and is most dramatically illustrated for a sample with only one element.
Then the sample correlation is
regardless of what value the random number x
or the random number y should take.
In fact, it turns out that the sample correlation will
always scatter away from zero.
No doubt this accounts for many false ``discoveries''.
The topic of bias and variance of coherency estimates is a complicated one,
but a rule of thumb seems to be to expect bias and variance
of on the order of for samples of size n.
EXERCISES:
 Suppose the mean of a sample of random numbers is estimated by a
triangle weighting function, i.e.,
Find the scale factor s so that . Calculate .Define a reasonable . Examine the uncertainty relation.
 A random series x_{t} with a possibly timevariable mean may have the
mean estimated by the feedback equation
 (a)
 Express as a function of and not .
 (b)
 What is , the effective averaging time?
 (c)
 Find the scale factor b so that if m_{t} = m,
then
 (d)
 Compute the random error [answer goes to as goes to zero].
 (e)
 What is in this case?
 Show that
 Define the behavior of an independent zeromeantie series x_{t} by
defining the probabilities that various amplitudes will be attained. Calculate
E(x_{i}), E(x^{2}_{i}), .If you have taken a course in probability theory,
use a gaussian probability density function for x_{i}. HINT:
and
Next: FREQUENCYSTATISTICAL RESOLUTION
Up: Resolution
Previous: TIMEFREQUENCY RESOLUTION
Stanford Exploration Project
10/30/1997