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Lecture 9

Asymptotic Distribution of Parameter Estimators

Topics to be covered include:

- Central Limit Theorem
- The Prediction-Error approach: Basic theorem
- Expression for the Asymptotic Variance
- Frequency-Domain Expressions for the Asymptotic Variance
- Distribution of Estimation for the correlation Approach
- Distribution of Estimation for the Instrumental Variable Methods

Overview

If convergence is guaranteed, then

But, how fast does the estimate approach the limit?

What is the probability distribution of ?

- The variance analysis of this chapter will reveal:
- a) The estimate converges to at a rate proportional to
- b) Distribution converges to a Gaussian distribution: N(0,Q)
- c) Covariance matrix Q, depends on
- - The number of samples/data set size: N,
- - The parameter sensitivity of the predictor:
- - The noise variance

Central Limit Theorem

Topics to be covered include:

- Central Limit Theorem
- The Prediction-Error approach: Basic Theorem
- Expression for the Asymptotic Variance
- Frequency-Domain Expressions for the Asymptotic Variance
- Distribution of Estimation for the correlation Approach
- Distribution of Estimation for the Instrumental Variable Methods

The mathematical tool needed for asymptotic variance analysis is “Central Limit” theorems.

Example: Consider two independent random variable, X and Y, with the same uniform distribution, shown in Figure below.

Define another random variable Z as the sum of X and Y: Z=X+Y. we can obtain the distribution of Z, as :

Further, consider W=X+Y+Z. The resultant PDF is getting close to a

Gaussian distribution

The resultant PDF is getting close to a Gaussian distribution.

In general, the PDF of a random variable approaches a Gaussian

distribution, regardless of the PDF of each , as N gets larger.

Let be a d-dimensional random variable with :

Mean

Cov

Consider the sum of given by:

Then, as N tends to infinity, the distribution of converges to the

Gaussian distribution given by PDF:

The Prediction-Error approach: Basic Theorem

Topics to be covered include:

- Central Limit Theorem
- The Prediction-Error approach: Basic Theorem
- Expression for the Asymptotic Variance
- Frequency-Domain Expressions for the Asymptotic Variance
- Distribution of Estimation for the correlation Approach
- Distribution of Estimation for the Instrumental Variable Methods

The Prediction-Error Approach

Applying the Central Limit Theorem, we can obtain the distribution of estimate

as N tends to infinity.

Let be an estimate based on the prediction error method

Then, with prime denoting differentiation with respect to ,

Expanding around gives:

is a vector “between”

The Prediction-Error Approach

Assume that is nonsingular, then:

To obtain the distribution of , and must be

computed as N tends to infinity.

Where as usual:

The Prediction-Error Approach

For simplicity, we first assume that the predictor is given by a linear regression:

The actual data is generated by (is the parameter vector of the true system)

So:

Therefore:

Let us treat as a random variable. Its mean is zero, since:

The covariance is

Consider:

Appling the central limit Theorem:

The Prediction-Error Approach

The extended result of estimate distribution is summarized in the following

theorem, i.e. Ljun’g Textbook Theorem 9-1.

Theorem 1 Consider the estimate determined by:

Assume that the model structure is linear and uniformly stable and that the

data setsatisfies the quasi stationary requirements. Assume also that

converges with probability 1 to a unique parameter vector involved in :

Also we have:

And that:

Converge to with probability 1

The Prediction-Error Approach

Where is the ensemble mean given by:

Then, the distribution of converges to the Gaussian

distribution given by

The Prediction-Error Approach

As stated formally in Theorem 1, the distribution of converges

to a Gaussian distribution for the broad class of system identification problems.

This implies that the covariance of asymptotically converges to:

- This is called the asymptotic covariance matrix and it depends not only on
- (a) the number of samples/data set size: N, but also on
- (b) the parameter sensitivity of the predictor:

(c) Noise variance

Expression for the Asymptotic Variance

Topics to be covered include:

- Central Limit Theorem
- The Prediction-Error approach: Basic Theorem
- Expression for the Asymptotic Variance
- Frequency-Domain Expressions for the Asymptotic Variance
- Distribution of Estimation for the correlation Approach
- Distribution of Estimation for the Instrumental Variable Methods

Let us compute the covariance once again for the general case:

Unlike the linear regression, the sensitivity is a function of θ,

Assume that is a white noise with zero mean

and variances . We have:

Similarly

Hence:

The asymptotic variance is therefore

a) inversely proportional to the number of samples, b) proportional to the noise variance, and c) inversely related to the parameter sensitivity.

The more a parameter affects the prediction, the smaller the variance becomes.

Since is not known, the asymptotic variance cannot be determined.

A very important and useful aspect of expressions for the asymptotic covariance matrix is that it can be estimated from data. Having N data points and determined we mat use:

sufficient data samples needed for assuming the model accuracy may be obtained.

Example : Covariance of LS Estimates

Consider the system

and are two independent white noise with variances and

respectively

Suppose that the coefficient for is known and the system is identified

in the model structure

Or

We have:

Example : Covariance of LS Estimates

Hence

To compute the covariance, square the first equation and take the expectation:

Multiplying the first equation by and taking expectation gives:

The last equality follows, since does not affect (due to the time delay )

Hence:

Example : Covariance of LS Estimates

Assume a=0.1,

Estimated values for parameter a, for 100 independent experiment using LSE, is shown in the bellow Figure.

Cov (a) = 0.0471

Example : Covariance of an MA(1) Parameter

Consider the system

is white noise with variance . The MA(1) model structure is used:

Given the predictor 4.18:

Differentiation w.r.t c gives

At we have :

If is the PEM estimate of c:

For general Norm

We have:

Similarly:

We can use the asymptotic normality result in this more general form whenever required. The expression for the asymptotic covariance matrix is rather complicated in general.

Assume that . Then under assumption after straightforward

calculations:

Clearly for for quadratic

The choice of in the criterion only acts as scaling of the covariance matrix

Frequency-Domain Expressions for the Asymptotic Variance

Topics to be covered include:

- Central Limit Theorem
- The Prediction-Error approach: Basic Theorem
- Expression for the Asymptotic Variance
- Frequency-Domain Expressions for the Asymptotic Variance
- Distribution of Estimation for the correlation Approach
- Distribution of Estimation for the Instrumental Variable Methods

The asymptotic variance has different expression in the frequency domain,

which we will find useful for variance analysis and experiment design.

Let transfer function and noise model be consolidated into a matrix

The gradient of T, that is, the sensitivity of T to θ, is

For a predictor, we have already defined W(q,θ,) and z(t), s.t.

Therefore the predictor sensitivity is given by

Where

Substituting in the first equation:

At (the true system), note and

where

Let be the spectrum matrix of :

Using the familiar formula:

For the noise spectrum,

Using this in equation below:

We have:

The asymptotic variance in the frequency domain.

Distribution of Estimation for the correlation Approach

Topics to be covered include:

- Central Limit Theorem
- The Prediction-Error approach: Basic Theorem
- Expression for the Asymptotic Variance
- Frequency-Domain Expressions for the Asymptotic Variance
- Distribution of Estimation for the correlation Approach
- Distribution of Estimation for the Instrumental Variable Methods

We shall confine ourselves to the case study in Theorem 8.6, that is,

and linearly generated instruments. We thus have:

By Taylor expansion we have:

This is entirely analogous with the previous one obtained for PE approach, with he difference that in is replaced with in .

Theorem : consider by

Assume that is computed for a linear, uniformly stable model structure

And that:

is a uniformly stable family of filters.

Assume also that that is nonsingular and that

Then

Example : Covariance of an MA(1) Parameter

Consider the system

is white noise with variance . Using PLR method:

At we have :

Distribution of Estimation for the Instrumental Variable Methods

Topics to be covered include:

- Central Limit Theorem
- The Prediction-Error approach: Basic Theorem
- Expression for the Asymptotic Variance
- Frequency-Domain Expressions for the Asymptotic Variance
- Distribution of Estimation for the correlation Approach
- Distribution of Estimation for the Instrumental Variable Methods

- Instrumental Variable Methods Methods

We have:

Suppose the true system is given as

Where e(t) is white noise with variance independent of {u(t)}. then

is independent of {u(t)} and hence of if the system operates in

open loop. Thus is a solution to:

- Instrumental Variable Methods Methods

To get an asymptotic distribution, we shall assume it is the only solution to .

Introduce also the monic filter

Intersecting into these Eqs.,

Example : Covariance of LS Estimates Methods

Consider the system

and are two independent white noise with variances and

respectively

Suppose that the coefficient for is known and the system is identified

in the model structure.

Let a be estimated by the IV method using:

By comparing the above system with

We have

- Instrumental Variable Methods Methods

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