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ELEC 475/575: Homework 3
发布时间:2023-04-18
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ELEC 475/575: Homework 3
Problem 1
Assume we have the following binary sequences. What is the probability of the next symbol being 0 for each sequence?
a) 011011011011011011?
b) 011011011111011011?
c) 0100110100110111001?
Hint: It is most convenient to assume a tree depth of 3.
Problem 2
Generate a Gaussian random sequence with zero mean and variance 1, referred to as X . Generate another random sequence Y with zero mean and variance 1 independent of X . Use k-nn plugin estimator of the joint density of X and Y to estimate I(X; Y).
Problem 3
Imagine that a neuron is tuned by a stimulus s which has a Gaussian distribution with mean µ and variance σ 2 , and it produces the response r = as + n, where n is a Gaussian noise with zero mean and variance η 2
and a being constant. What is the likelihood function p(r|s) and the marginal response p(r)? What are the entropies of these two distributions? How does noise entropy depend on s?
Problem 4
If X , W and Y are wide sense stationary Gaussian random processes and Y is a linear time invariant function
of X and W while X and W are random and independent, show that:
MIx,Y (f, f) = - log[1 - CxY (f)]?
Hint: You could use y(t) = h1 (t) * x(t) + h2 (t) * w(t)
Problem 5
Use one recording from the online neuro data set to estimate the entropy of the recording over a time period, long enough to acquire a good estimate but not too long to lose stationarity.
Problem 6
For a one-dimensional dataset x = {x1 , . . . , xn } with x1 < . . . < xn , the k-nearest-neighbour density model fˆx (x) = can be applied for estimating the distribution.
a) Given k < n, calculate fˆx (x) for x > xn and x < x1 .
b) Show that the k-nearest-neighbour density model fˆx (x) is an improper distribution, i.e. the integral of fˆx (x) over (-o, +o) is divergent (o).