# 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.

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