Observed Random Process
Let {x_n} be an observed random process which is generated by the following nonlinear recursion
x_n = (theta_1)(x_n-1) + (theta_2g)(x_n-2, x_n-3) + w_n
where {w_n} is an i.i.d. zero-mean sequence and g(.,.) is a known deterministic function. We are interested in estimating the two parameters theta_1, theta_2. (a) Propose a fixed sample size and an adaptive estimator for the parameter vector [theta_1, theta_2] that does not require knowledge of the statistics of {w_n}. (b) Analyze the convergence properties of your adaptive estimator.
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SOLUTION This solution is FREE courtesy of BrainMass!
The explanations are in the attached file.
Notations:
RLSE = Recursive least Squares Estimator
Patras Notes = http://www.ssp.ece.upatras.gr/courses/detest/noexternalweb/chapter4.pdf
IAState Notes = http://www.cs.iastate.edu/~cs577/handouts/recursive-least-squares.pdf
Explanations:
We propose to use RLSE, as this method does not require knowledge of the statistics of {w_n}.
Helpful notes on RLSE can be seen in Patras Notes for arbitrary number of estimated parameters, and in IAState Notes which deal with a single parameter and so their presentation of the method is relatively transparent.
Both texts suffer from typos, but convey the ideas to some extent.
To write the RLSE setup in a standard form we define the following.
Parameter vector
theta = (theta_1, theta_2)
Observations vector after nth observation Xn = x_4, x_5, x_n-1, x_n
It starts from x_4 rather than x_1, because of the recursion that uses three previous values of x to produce the next value.
Noise vector after nth observation
Wn = W_4, W_5 ... W_n-1, W_n
See attached for complete solution.
© BrainMass Inc. brainmass.com December 24, 2021, 10:54 pm ad1c9bdddf>https://brainmass.com/statistics/ordinary-least-squares/observed-random-process-518518