Neyman-Pearson detection and stationary time series data
Throughout this project you will require data and routines which can be obtained via HTTP from http://www.dip.ee.uct.ac.za/ nicolls/eee515z/comass1. You can get them from me if you can't find a browser.
I expect a report from you detailing the results and conclusions you have reached. Don't hand back reams of code - a brief outline of the algorithm used in the implementations is more than sufficient. Results must be presented for sensible and interesting cases; analytical derivations must be precise and complete. Insightful comments and shattering observations will be more than appreciated.
1. Random vector generator
Write a function which generates random sample vectors from a normal distribution . At the very least, the function should take the vector and the matrix as parameters. The dimensionality of the random vectors returned should be determined by the dimensions of these two parameters.
Test you function for the 2-D case. Consider a covariance matrix of the form
for some and some , and generate a number of samples from this distribution. Plot these on a scatter diagram and confirm that you get what you expect.
2. Detection in white noise
Design a UMP Neyman-Pearson test for detecting the signal in white Gaussian noise with power . Use the principle of sufficiency to reduce the dimensionality of the detection statistic. Find the power of the test as a function of the false alarm probability and the parameter .
Use the signal from the web page to experimentally verify your results for say 10 sensible values of and . Do this by generating random samples from the relevant distributions and finding the average detection rate. Check that the false alarm rate is also as expected.
3. Detection with known covariance matrix
MATLAB users: fetch the files required for the random covariance matrix generator (markovcm.m and randmcm.m). Generate a random Markov covariance matrix for your use with the command
R = randmcm.
Suppose the signal to be detected is still , but this time the noise is distributed . The observed signal then has the distribution
If this signal was a sample from a stationary random process, what would the autocovariance function be? What would the autocorrelation function be under each hypothesis? The hypothesis testing problem is to test versus in the model above, with the covariance matrix you have been presented with.
Analytical work: Suppose you didn't know this covariance matrix, and assumed the independent uncorrelated case for the design of your detector. What would the actual false alarm rate that you achieve be as a function of the desired false alarm rate that you used in the detector design? How would the detection probability be affected by the mismatch between the model and the actual data distribution? Make some plots demonstrating the effects for some particular configuration of parameters.
Experimental work: Approach the problem now from the standpoint that you do know , and design the Neyman-Pearson detector accordingly. Find expressions for the false alarm probability and detection probability. Repeat the experiment you performed in the previous question for one of the values of , and verify that you get what you expect. How has the correlation between the noise samples decreased the efficiency of your detector?
4. Invariant detectors
We will assume that the signal you are detecting has been distorted in two ways:
Test your result in the following way: again for each generate say 500 samples of noise and signal plus noise. To each of these samples add a linear trend with some offset (let the slope of the trend come from the distribution U[-1,1] and the offset from U[0,10] - it doesn't really matter; for the detector you assume you don't know this). Confirm that for some useful value of and some interesting values of you get what you expect.
Having done this, note the following:
5. Detection in time-series For time-series problems, unless you have some synchronous mechanism for signalling the observation interval, you generally have a situation of detecting an instance of a signal at an unknown position in a continuous stream of sample points. The way to deal with this is usually to consider a window of the required length sliding across the data, and to perform a hypothesis test for every case. For window positions where there is some signal present which doesn't coincide with the template the hypothesis test breaks down partially, but nevertheless we can usually expect a peak in the detector statistic for the case when they do coincide.
If the (discrete) time-series is stationary, the autocovariance function is defined and the covariance matrix for any finite sample of the data has a Toeplitz structure. A natural and simple formulation follows for this case.
Generate a 512 element sample from a zero-mean stationary time-series which has the same autocovariance function as for the correlated noise you used in the previous questions (assume that the correlations are zero for all correlation lags greater than you have data for). Add a slowly varying sinusoid to this time-series (period = 512 samples), as well as a DC offset.
For some interesting values of , perform the following tasks: