Asymptotic properties of parameter estimates

3.2.2 Separation of Filtering and Parameter Estimation

The estimation method requires access to the residuals, the dierence be-tween the predictions and the measurements. For the general class of mod-els, the computation of the optimal predictions amounts to solving the general ltering problem. Exact solutions to this problem is only possible in special cases, e.g. for linear models, otherwise it is necessary to employ certain approximations to obtain implementable algorithms. However, it turns out, that while the properties of the ML parameter estimates depend upon the residuals and their properties, they do not depend on how they were found. It is therefore natural to separate the ltering problem from the parameter estimation problem. In that way we may develop the parameter estimator independently of the predictor, which according to the previous discussion, Ljung (1987), isthe model itself. In the present derivation of the ML estimator there is only certain requirements on the residuals. As stated earlier the problem of treating a general class of models, is focused on the development of implementable predictors. This problem is treated in Chapter 4.

that the model set contains the true system, ^{S 2 M}, for a given set of experimental conditions, ^X, will often not be fullled for real systems.

Usually the system is far more complex than we would allow the model to be.

For practical applications the objective of system identication is often to nd an approximate description, catching the relevant features of the system. The discussion of consistency is therefore mainly of theoretical interest. On one hand if the identication method is not able to estimate the true system within a model set, it is probably not a good method.

On the other hand, knowing that a method is consistent implies nothing a priori about its performance inapproximatinga more complex system, and the latter property is the most important when identifying a real process.

Even though we mainly consider the ML method, most of the results in this section apply to the more general prediction error method applied to the same model structure. This implies that if the conditions on the distribution of innovations required by the ML method cannot be obtained, we are still using a prediction error method. The asymptotic model thus obtained gives under very general conditions, the best average prediction performance. This result also implies a strong robustness property of the method (Ljung, 1978).

3.3.1 Convergence

Lets introduce some regularity assumptions from Ljung (1978) and Ljung &

Caines (1979) which are relevant for analyzing the convergence properties of the parameter estimates.

The rst condition is on the system. It requires that the expectationE(^yk) exists and the system^Sis described by

yk=E(^yk^jZk^,1) +k (3.19)

where the sequence of innovations,^fk^gis a stochastic process withE(k^jZk^,1) = 0. In 3.19 the conditional expectation isE(^yk^jZk^,1) =^gS(tk;^yk,1;^uk,1), i.e. a deterministic function of old data.

A second condition is on the data, which concerns the system ^S and the experimental condition^X. The closed-loop system (3.3), (3.19) should be exponentially stable. That is, the past is forgotten at an exponential rate.

If the system is linear, this condition simply requires that the poles of the system is inside the unit circle.

The next condition concerns the model set^M. It is required that^g(tk;;^zk,1) in (3.6) is three times continuously dierentiable with respect to. There is also a restriction on how fast^gmay increase with^uk and^yk for nonlin-ear models. Eectively, it may not increase faster than linnonlin-early. Another restriction is that the model and its derivatives with respect to are ex-ponentially stable.

Consider a stochastic, linear state-space model given by the equations

xk+1 =^A(tk;)^xk+^B(tk;)^uk+^E(tk;)^vk (3.20)

yk=^C(tk;)^xk+^D(tk;)^uk+^F(tk;)^ek (3.21) where ^v N(⁰;^I) and ^e N(⁰;^I) are independent. Assume that the matrix elements are continuously dierentiable with respect to ² D^M, where D^M is a compact set. If the system is completely observableand controllable, uniformly in t and in ² D^M, then the model fullls the conditions on the model set previously described, and the predictor is the

associated Kalman lter, see (Ljung, 1978).

The nal conditions to set up, concerns the loss functionl(tk;;) as given in the general prediction error formulation (3.9). It is required that the loss function is three times continuous dierentiable with relation to and and that these dierentials are bounded, (Ljung & Caines, 1979). In our case, where the maximum likelihood method is used, the loss function is given by the negative logarithm of the prediction error density, usually assumed to be Gaussian (3.17) and (3.18). Then the requirements on the derivatives in are fullled and the condition becomes mainly a question of parameterization (Graebe, 1990b).

When studying convergence of parameter estimates, we consider the general prediction error criterion

V(;^zN) = 1 N

N

X

k=1l(tk;;(tk;)) (3.22) which also contains the maximumlikelihood method based on the Gaussian distribution. The limit

V() = lim_N

!

1

V(;^zN) (3.23)

exist under weak conditions on^Sand^X, e.g. if the processes are asymptoti-cally stationary or periodic (Ljung & Caines, 1979). However this may not be valid for general time-varying and adaptive feedback, but even when the limit in (3.23) does not exist it is still possible to make statements about the convergence of the parameter estimates. If the limit exist the parameter estimate, ^N that minimizes the criterion converges to the set

DV =^fj2D^M; V min

2D^MV( )^g (3.24)

If however the limit does not exist, we dene the set DV=^fj2D^M;liminf_N

!1

E(V()) min

2D^Mlimsup_N

!1

E(V( ))^g (3.25) This is clearly more general than (3.24) in that it does not require the existence of the limit V, if however the limit exists thenDV=DV. We are now able to formulate the general theorem about parameter convergence.

Theorem 3.1 (Convergence) Let the conditions previously described in this section be satised. Then

sup²D^MjV(;^zN)^,E(V(;^zN))^j!0 w.p.1 asN^!1 (3.26) uniformly in ² D^M. Moreover, since the estimate ^{^}N minimizes V(;^zN), it follows that

^N^!DV w.p.1 asN^!1 (3.27)

if the limit (3.24) exists thenDV=DV. For a proof see (Ljung, 1978).

The theorem states that the value of the loss function calculated from a realization of data will become arbitrarily close to its expected value as the data length approaches innity, and that parameter estimates computed by minimizing the loss function will converge into appropriate sets. Note that in the conditions of the theorem it is not required that the model set contains a system equivalent member.

3.3.2 Distribution of Parameters

Having established the conditions for convergence of the parameter esti-mates it will now be established that the parameter estiesti-mates are asymp-totically normally distributed under similar conditions. Also in this case the approximate modelling approach of Ljung & Caines (1979) is adopted.

Theorem 3.2 (Distribution) Assume that the conditions of theorem 3.1 are satised. Consider the general loss function (3.9)

V(;^zN) = 1 N

N

X

k=1l(tk;;(tk;)) (3.28) Let ^{^}N be the global minimum of (3.28) over the compact set D^M. Introduce the functionWN() = E(V(;^zN)), where the expectation is with respect to the data. Let _N be the global minimum of WN(). Now introduce the term

PN= (W_N⁰⁰(_N))^,1UN(W_N⁰⁰(_N))^,1 (3.29) where a quote denotes dierentiation w.r.t. the parameters, and

UN =E(NV⁰(_N;^zN)(V⁰(_N;^zN))^T) (3.30) and assume that W_N⁰⁰(_N) and UN are invertible. Then the quantity

pN^P,_N¹⁼²(^N^,_N)is asymptotically normal with zero mean and unit covariance matrix. Assume furthermore that

WN()^!W() (3.31)

uniformly in D^M as N^!1. LetW() have a global minimum at , and assume that W⁰⁰() is invertible. Let UN be dened as (3.30),

and assume the limitU= limN^!1UN exists and is invertible. If now

pNW_N⁰()^!0 asN^!1; (3.32)

then ^pN(^N ^,) is asymptotically normal with zero mean and co-variance matrix

P= ( W⁰⁰())^,1U( W⁰⁰())^,1 (3.33)

The theorem is an adaption of theorem 1 and the corollary in Ljung &

Caines (1979), and a proof can be found there. The basic conditions are the same as required for the convergence theorem, but the requirement of a unique global minimum of the criterion function is fairly restrictive. From theorem 3.1 we know that the minimization of V will lead us close to a local minimum of W. We may then think of this theorem as applying to the neighborhood of this local minimum point. The assumption, in the second part of the theorem, of convergence ofWN() asNgoes to innity is satised, for example, if the processes are asymptotically stationary or periodic.

If the true parameter0is contained in the model set such that^f(tk;0)^g is a sequence of independent random vectors, results from the asymptotic distribution of the deviation of the estimate from 0 can be used to ob-tain condence intervals for the parameters etc. If the limiting model does not give a true description of the system (i.e., if ^f(t;)^g are not independent) but only the best approximation available in ^M, then the distribution may still be used for model validation, concerning e.g. the relevance of certain parameters in the model, see (Ljung & van Overbeek, 1978).

Cram´er-Rao Bound

As a quality measure of an estimator we use its mean square error matrix

P(^N) =E(^N^,0)(^N^,0)^T (3.34) where 0 denote the true parameter. We are interested in selecting an estimator that makesPsmall. There is a lower limit to the values ofPthat can be obtained by any unbiased estimator. This is shown by the following.

Theorem 3.3 (Cram´er-Rao inequality) Let^{^}Nbe an estimator of²D^M, where D^M is a compact subset in ^Rd such that E(^N) = 0. The expectation is with relation to the data, assuming that the probability density function of the data isp(^yNj0), for all values of0. Suppose that ^yN may take values in a subset of ^RNs, whose boundary does not depend on . Then under mild regularity conditions

P(^N)^M,1 (3.35)

where

M,E(_@@logp(^yNj)(_@@logp(^yNj))^T)^j=0 (3.36) is known as the Fisher information matrix.

Proof: see e.g. Goodwin & Payne (1977).

Note that the evaluation ofM requires knowledge of0, so the exact value ofM is usually not available to the user. If the covariance of an unbiased estimator achieves the lower bound of the Cramer-Rao inequality, it is said to be ecient.

We now return to the ML estimator (3.17), assuming independent innova-tions with a known probability density function. Assuming that^S2Mthis estimator converges to a normal random variable according to theorem 3.2, with a covariance matrix^P(^) =^M,1, i.e. it attains, asymptotically, the lower limit of the Cramer-Rao inequality, see Goodwin & Payne (1977).

Another interesting property about ecient estimators is that whenever there exists an unbiased ecient estimator, then it is also the maximum likelihood estimator. This is shown e.g. by Goodwin & Payne (1977).

3.3.3 Consistency

Suppose, that the model setcontainsa system equivalent member, corre-sponding to a parameter value0, does then the estimate ^N tend to this truevalue of the parameters as the number of data tends to innity? This matter of consistency is discussed in this section.

A suitable way to express that the true system is representable within the model set^Mis to require that the set

DT(^S;^M;X) =^fj2D^M; Nlim^!1 1

N ^N

X

k=1E^jy^^S(tk)^,y^^M(tk^j)^j2 =0;^8X2X^g (3.37) is nonempty, where X is the set of experimental conditions, under which we would like our model to be valid. Here ^^yS(tk) =E(^y_k^jyk,1) assuming condition 3.19 is satised, and ^^yM(tk^j) is the predictor as given in (3.6).

Hence if DT is nonempty and ² D^M, then the corresponding model

M() is, in the mean square sense of predicted output, indistinguishable from the true system, (Ljung, 1978).

Theorem 3.4 (Consistency) Assume that the conditions of Theorem 3.1 apply. Assume also the following condition on the criterion functions 0²DT(^S;^M;X)⁾⁸k;²D^M:El(tk;0;)El(tk;;) (3.38) The condition is satised e.g. for the ML criterion (3.18). Assume that the parameter set of mean square equivalence,DT(^S;^M;X), dened in (3.37) is nonempty. For a particular experiment,^X2X, we dene the set

DI(^S;^M;^X) =

f^j2D^M;liminf_N

!1

N1

N

X

k=1E^{jy^S}(tk)^,y^^M(tk^j)^j2=0^g (3.39) Then

^N^!DI(^S;^M;^X) w.p.1 asN^!1 (3.40)

The theorem is an adaption of lemma 4.1 in Ljung (1978). The basic conditions of the theorem are the same as required for the asymptotic con-vergence and distribution theorems, but additionally it is required thatDT from (3.37) is nonempty. This is a strong condition, but the discussion of consistency is only meaningful if the model set contains a system equiva-lent member in some sense. The setDTrepresents thecorrectdescriptions of the system in the sense of mean square equivalence. It is obvious that

8X2X:DT(^S;^M;X)DI(^S;^M;^X) (3.41) sinceDI is dependent of a certain experimental condition^X. If, under this experimental condition, we are not able to get a suciently representative picture of the system, then we are not guaranteed convergence to the right

model, according the theorem. The desired relation is that

DT(^S;^M;X) =DI(^S;^M;^X) (3.42)

for the chosen experimental condition^X. This is a matter of choosing the right ^X, which includes properties like persistently excitinginputs, and not too special feedback mechanisms. This discussion of experimental de-sign is postponed to Chapter 6. Even when (3.42) is satised, and theorem 3.4 guarantees the convergence to a system equivalent model, there might still be several parameter values yielding this property. In order to obtain uniqueness of the solution further restrictions on the model structure and parameterization must be applied. This is a matter of structural identia-bility which is considered in Section 6.1.

In document IMM LYNGBY
Ph.D.THESISNO. ByHenrikMelgaard Identi (Sider 58-68)