1. of the theoretical mean. The sample means

1.                              
Random signals and stochastic processes

 

1.1           
Statistical Estimation

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rand(m, n) is a Matlab
function that generates a matrix of size m x n, containing random numbers
uniformly distributed between 0,1.

Figure 1:
rand(1,N) plot for N=1000

The data in the plot displayed is completely random and
distributed between values of 0 and 1. To gather more information about the
random process in question, it’s statistical properties must be investigated.
Those properties are its mean, standard deviation and probability density
function (pdf).

Below are comparisons between the equations of the
theoretical mean and the sample mean.

Theoretical Mean

Sample Mean

Using the Matlab function mean(x) multiple times, various values
for the mean were returned – 0.5130, 0.5148, 0.492 etc. Since the values of the
mean returned all fluctuate in an even manor around the theoretical mean, it
can be said that the estimator of the sample mean described above is unbiased.
If the number of samples, N, is large enough then the sample mean will converge
to the theoretical mean. This is seen as N->, the sample mean equation can be written
as  which translates to the integral of the of the
theoretical mean. The sample means accuracy is therefore proportional to the
sample size, N.

Theoretical Standard Deviation

 =  =  ? 0.289

Sample Standard Deviation

 

Using the Matlab function std(x),
it was found that the standard deviation varied between each trial and
fluctuated around the theoretical standard deviation in an unbiased manner.
Just like the sample mean, the accuracy of the sample standard deviation was
proportional to the sample size, N.

Estimator Bias

To further probe into the statistical properties, the bias of the
two estimators was created and investigated by generating 10 realisations of X.

                                Figure
2: Sampled Mean and Standard Deviation Compared to the Theoretical Values

From looking at these graphs, it is shown that the sample mean
and sample standard deviation fluctuate above and below their theoretical
value. This facilitates and supports the notion that the estimators are
unbiased.

PDF Approximations

The first order statistics of our set of
random variables were inspected using the histogram()
function in Matlab.

a)     
N = 100000, nbins =
1000

b)     
N = 100000, nbins = 100

 

Figure 3: Comparison
Between the Approximxated PDFs and the Theoretical PDF

From the two histograms displayed it’s evident that the accuracy
is inversely proportional to the number of bins. This is so because for fewer
bins, each bin contains more samples and the accuracy is proportional the
number of samples present.

On the contrary, fewer bins reduces the plot’s resolution
significantly because each bin groups a greater number of x values onto the same
bar on the graph. Thus, the number of bins present is proportional to the
resolution. From this, a proposed optimal solution is to use a large sample
size and limit the number of bins used.

Normal Distribution

A normal distribution is a Gaussian distribution with zero mean
and a standard deviation equal to 1. It is described by the following pdf:

Estimator Bias of
Normal Distribution

 

                                Figure 4:  Comparison of 2nd Order Statistics
Compared with their Theoretical Values

Just like the sample mean and sample
standard variation above, the normal distribution’s sample mean and sample
standard deviation also fluctuate in and around the theoretical value and are
unbiased estimators.                                                                                                                               a) N = 1000, bins =
100                                                                                   b)
N = 1000, bins = 1000      

Figure 5: Comparison between Approximated
PDF and Theoretical PDF

Approximated PDF                 

This figure shows clearly that the left-hand graph has too few
bins which affects its accuracy and resolution – although this is contrary to
what was said before, the accuracy of the right graph is less due to too few
bins resulting in quantisation error. The right figure has the same number of
bins as it does samples so they fit perfectly into the normal distribution.

1.2          
Stochastic Processes

The Matlab code for each RP function was
run and the corresponding graphs were plotted together.

 

Figure 6: 2nd
Order Statistics of RP1, RP2 and RP3

 

In order to determine the Stationarity
of a random process it must be shown that its statistical properties are time
invariant. If the random processes second order statistics are time invariant
then this implies the random process is Wide Sense Stationary (WSS).

Referring to the above random processes; RP1’s ensemble mean is
linearly increasing as time increases. Time is represented with the sample
number.  The ensemble standard deviation
shows time varying properties as well. Therefore, it is non-stationary.

Whereas RP2 and RP3 display time invariant statistics in their
ensemble statistics. They can therefore be labelled as stationary. RP3 shows
much greater fluctuation than RP2 from its steady state mean which is showcased
by its standard deviation being more than double that of RP3.

A WSS process can be described as ergodic if a single realisation
of adequate length is enough to deduce its stochastic properties. Thus, the
random process is said to be ergodic if the time average approaches the
ensemble average when evaluated as the following: .

 

 

 Figure 7: 2nd Order Statistics of
RP1, RP2 and RP3

RPI is not ergodic because it is non-stationary. This can be
seen because the sample mean is not the same as the ensemble mean just as the
standard deviation doesn’t either. However, RP3 has sample statistics that
match its ensemble statistics significantly. It may be ergodic but this needs
to be confirmed by further mathematical analysis. RP2 has a significant amount
of fluctuation in its sample mean and standard deviation; significantly larger
than the other two processes. It can’t be concluded whether RP2 is non-ergodic
though due to too small of a sample size. Further mathematical analysis will
clarify this uncertainty.

Random Process 1

RP1 can be expressed as the following when the random variable
is defined as X ~ U(0,1);

The theoretical mean of V:

 =
0.02n

And the standard deviation:

Both statistics depend on n, the sample number. Thus, RP1 is neither
stationary or ergodic.

Random Process 2

RP2 can be expressed as the following when the random variable
is defined as X ~ U(0,1);

M and A are both random variables constant within any given
realisation.

Theoretical Mean

Standard Deviation

Neither the mean nor standard deviation depend on the sample number
and therefore RP2 is stationary. However, for this random process the sample
mean settles at whatever random value is generated by A for all the
realisations and the value for A varies between each realisation and therefore
RP2 is not ergodic.

Random Process 3

RP3 can be expressed as the following when the random variable
is defined as X ~ U(0,1);

Both
a and m are constants.

Theoretical Mean

Standard Deviation

Once
again, the mean and standard deviation do not depend on the sample number and
thus RP3 is stationary just as RP2 is. However,