International Journal of Control

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International Journal of Control

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Refined instrumental variable methods of recursive time-series analysis Part I. Single input, single output systems a

a

Peter Young ; Anthony Jakeman a Centre for Resource and Environmental Studies, Australian National University. Canberra, ACT. Australia To cite this Article: Young, Peter and Jakeman, Anthony , 'Refined instrumental variable methods of recursive time-series analysis Part I. Single input, single output systems', International Journal of Control, 29:1, 1 - 30 To link to this article: DOI: 10.1080/00207177908922676 URL: http://dx.doi.org/10.1080/00207177908922676

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INT. J. CONTROL,

1979,

VOL.

29,

NO. I,

1-30

Refined instrumental variable methods of recursive time-series analysis Part I. Single input, single output systems PETER YOUNGt and ANTHONY JAKEMANt This paper is the first in a series concerned with a comprehensive evaluation of the refined instrumental variable-approximate maximum likelihood (IVAML) method of time-series analysis. The implementation of a recursive/iterative version of the refined IV Al\1L algorithm for single input, single output systems is discussed in detail and the performance of the algorithm is evaluated by Monte-Carlo simulation analysis applied to five simulated stochastic systems. As conjectured, the algorithm appears to yield asymptotically efficient estimates of the time-series model parameters and, indeed, it seems to approach minimum variance estimation of the basic system model parameters for even low sample size and low signal/noise ratios. The noise model parameters are not estimated so well at the smaller sample sizes but the estimation performance appears similar to that of other competing methods of analysis, such as recursive maximum likelihood (Rl\IL). Subsequent papers on this same general topic will deal with extensions of the refined IVAML procedure to handle multdvaeiable systems, time-variable parameters and the estimation of continuous-time systems described by ordinary differential equations.

1.

Introduction In a previous paper (Young 1976) the instrumental variable (IV) and approximate maximum likeiihood (AML) methods of time-series analysis for single input, single output systems were discussed within the context of maximum likelihood estimation. It was shown that a ' refined' IVAM:L procedure could be evolved which held the promise of improved statistical efficiency. In the present paper this refined IVAML method is examined further and its statistical properties are evaluated empirically using Monte-Carlo simulation analysis. As expected, the new approach provides for improved statistical efficiency (i.e. lower estimation error variance) when used in either its recursive or iterative modes of operation (see Young 1976). Indeed, in the iterative mode it appears not only asymptotically efficient, but in the case of the basic (deterministic) system it also seems to yield close to minimum error-variance estimates for moderate sample size and low signal/noise conditions. One disadvantage of the refined IV AM:L approach exposed by the present analysis is that convergence of the noise model parameter estimates cannot be guaranteed in all cases. This confirms the theoretical analyses of Ljung (1974) and Holst (1977) which indicate that the AML noise model estimator will fail, in certain apparently rare circumstances, to completely converge on the true values. This disadvantage is discussed in the paper and minor modifications to the refined IVAM:L procedure are suggested which obviate the difficulty whilst maintaining the computational simplicity of the solution.

Received 19 December 1977, t Centre for Resource and Environmental Studies, Australian National University, Canberra ACT, Australia. CON.

I

A


P. Young and A. Jakeman

2

The refined IVAML approach to recursive time-series analysis is particularly flexible and can be extended in various ways. Part II of this paper (Jakeman and Young 1978) will show, for instance, how the same approach can be used for multivariable (multi-input, multi-output) systems, where it also seems to be asymptotically efficient. Other extensions to handle the possibility of parametric variations and the case of continuous-time systems described by ordinary differential equations are also straightforward and will be described in a subsequent paper (Young and Jakeman 1979). The refined IVAML algorithm The refined IVAML algorithm is designed to estimate the parameters in the time-series model shown within the dotted lines in Fig. I, where the measured output variable Yk is related to a deterministic input variable Uk and a stochastic , white noise' variable ek by the equation 2.

B(Z-l) D(z-l). Yk= A(Z-l) u k+ G(Z-l) ek

(I)

I

Here, the nomenclature is the same as in Young (1976): in particular A(Z-l), B(Z-l), G(Z-l) and D(z-l) are the following polynomials in the backward shift operator Z-l : A(Z-I)=I+a1z-1+ +anz- n l+ B(Z-l)=bo+blz+bnz- n G(z-1)=I+CIZ-1+ +cnz- n (2) I+ D(Z-I)=I+d1z+dnz- n where ek is assumed to have a normal Gaussian amplitude distribution and possess the following statistical properties: E{ek}=O;

E{e jed=u20jk;

E{ej1td=O,

for all j, k

(3)

where Ojk is the Kronecker delta function. The derivation of the refined IVMiL algorithm has been described in detail by Young (1976) and we will concentrate here on its implementation in computational terms. The algorithm is based on an approximate decomposition of the maximum likelihood solution to the problem of estimating the parameters characterizing the polynomials in (2). This decomposition is shown diagrammatically in Fig. 2, where it is seen that the algorithm has two principal components: the refined IV and AML sub-algorithms, respectively. These sub-algorithms continuously communicate with each other and it is this communication or ' coordination' which marks the principal difference between refined IV AML and basic IVAML and which leads to the improvement in statistical efficiency. The coordination involves additional adaptive prefiltering operations in both sub-algorithms, the physical functions of which have been discussed by Young (1976). Both the recursive and iterative versions of the refined algorithm utilize recursive updating of the parameter estimates. The major difference between the two formulations lies in the nature of the coordination between the IV and AML sub-algorithms. In the recursive case the parameters of the various prefilters, including the auxiliary model (see Young 1976) used to generate the instrumental variables, are themselves updated recursively on the basis of the


Recursive time-series analysis

3

,

_ _ _ _ _ .J

PRE·FILTERS

PRE-FILTERS

-t

..A.DA

REFINED AML

REFINED IV equation

equation 20

19

REFINED IVAML ESTIMATES

Figure l.

W\XIMi.I'I LI KELlHOOD

C and D assumed known

BASIC IV ALGal I1»1

I

REFINED

~

IV

PROBLEM

\

~

REFINED

A"tL

4

ALGORI1H'1

I

A and B assumed known

~

BASIC AML ALGOR ITH'I

ALGORITI-I'I COORD INATJON BEniEEN ALGORITHMS

i

REFINED IV AML ESTIMATES

Figure 2. Decomposition of the maximum likelihood problem. 1A 2


P. Young and A. Jokeman

4

current recursive IV and AML estimates. In the iterative case, on the other . hand, these prefilter parameters are kept constant during each iteration through the entire data set and are updated only subsequent to the completion of each iteration. The convergence of this iterative. procedure is normally . complete in less than six iterations and the number of iterations can be prespecified by the user. Alternatively, iteration can be assumed complete when consecutive iterations yield no appreciable change in some norm associated . With the estimated values.

2.1. Iterative version For simplicity of exposition let us consider in detail only the iterative version of the refined algorithm. Here, at the kth sampling instant on some arbitrary jth iteration, the refined IV sub-algorithm generates an estimate a k = [all ... , an, bo' ... , bn]k T of the system model parameter vector a = [all"" an' bo, •.. , bn]T from the following recursive algorithm: . (i) 1

~ ~ ak=a k_1-

R " *['2 R "*]-1( Zk*T a~ k-1-Yk *) rk-1"'k a +Zk *T rk_1"'k

where

. Here 6 2 is an estimate of a 2 in (3) while the vectors Zk * -- [- Yk-l' *

"'J

- Y k-n' *

U ~ *,

Zk

* and i k * are

defined as

, U k-n *]T}

(5)

i k* = [ - Xk:"'1*, ... , -xk_n*,uk*,

,uk_n*)T

where xk is the instrumental variable generated by the following auxiliary model: (6)

while the star superscript indicates that the variables are prefiltered as shown in Fig. 1 by a filter F j of the form Fj

=

(7)

(Jj-l

D j _ 1A j _ 1

In (6) and (7) Aj _lI 13 j_1, (Jj_1 and /)j-1 are, respectively, estimates of the polynomials A(Z-1), B(Z-1), O(Z-1) and D(Z-1) obtained at the end of the previous (j - 1)th iteration and are denoted by

. Aj _ 1 = 1 + "'IZ-I + 13j_ 1 = #0 + #IZ-I + (Jj_1 = 1 + Y1Z-1 + /)j_1 = 1 + 31z- 1 +

+ "'nz-n

+ #nz-n

+ Ynz-n

) (8)

+ 0nz-n

In other words, "'i' #;, Y; and 3; are set equal to the estimates of the equivalent a;, b;, c; and di parameters in (2) obtained at the previous (j -l)th iteration of the complete algorithm.



5

The estimates of the noise model parameters required for the adjustment of the prefilters F j are obtained from the refined AML sub-algorithm. Here, at the kth sampling instant on the jth iteration, the estimate ek = Eel' ..., en, d1> ... , dnF of the noise model parameter vector c = [C1> ... , cn' d1> ... , dnF is given byt (i) ek = ek_ l - Pk_IN ~k[&2 + fikT Pk_IN ~d-l(fikT ek_l - tk)

I

where (ii) P/'=Pk_IN_Pk_IN ~k[&2+fikT Pk_1N ~d-lfikT Pk_1N and the vectors fik and ~k are defined as follows: fik = [- tk-1> ~k=[

, - tk-n, ek_l ,

... ,

-L:». , -L:». ek_

1*,

ek-nF

(9)

} (10)

... , ek_n*]T

Here, tk is an estimate of the noise variable gk generated by subtracting the auxiliary model output k in (6) from the observed output Yk' i.e,

x

tk=Yk-Xk

(11)

and ek is a recursive estimate of the' white noise' input ek obtained from the equation (12) which can be considered as an ' inverse noise model' as shown in Fig. 1. Once again, the star superscript indicates that the variables are prefiltered, but here the filter is simply Fjn, where 1 (13) Ft=J5":": i-I

Note also that, while the auxiliary model and prefilter parameters are updated iteratively and kept constant throughout each iteration, the estimate ek in (12) is generated with the help of the current recursive estimate of c. In the iterative solution suggested here, the sub-algorithms (4) and (9) are utilized alternately starting with the refined IV estimator (4). During the first iteration through the complete data set (say T samples), it is clear that this estimator must be initialized in some manner. As indicated in Figs. 1 and 2, we believe that this is best achieved by reference to the basic IVAML results obtained in prior analysis. Here, the noise variance estimate &2, the elements of So and the parameters of both the auxiliary model (6) and the prefilters (7) are all set to the appropriate prior IVAML estimates, while the matrix Po is assumed to be diagonal with all elements set to a high level, say 10 6 • It is well known (see, e.g. Young 1974) that this latter assumption is equivalent to informing the algorithm that the analyst has little confidence in the accuracy of the initial estimates So and &2. While this is not the case if the estimates have been obtained by prior IVAML analysis, it does prevent overconfidence in the estimates, which could in turn lead to poor convergence. It. t Note that we use a slightly different nomenclature here than in Young (1976) with PkN and ~k replacing iV k and m k , respectively. This avoids confusion in Part 11 of the present paper.


P. Young and A. J akeman

6

is, in other words, a conservative safety measure which seems worthwhile if the algorithm is to be used in routine day to day application. As we shall see later, however, this setting for j> 0 is not used on iterations subsequent to the first and it is also not advisable in the purely recursive form of the refined algorithm. In addition to the initialization of those aspects of the algorithm (4) linked with the system and noise model parameters, it is also necessary to initialize the elements of the vectors Zk * and i k * before estimation can proceed. This is achieved by solving the filtering equations for Yk*, Uk *, and x k* recursively until all required elements of Zk * and i k * are fully specified. In discrete-time terms, these filtering operations can be written:

T Yk*=Yk P+Yk Uk*=U kT p+u k

I

(14)

xk*=ik*T a where

Here the definition of the 3n dimensional vector P arises because the denominator of the prefilter F i in (7) is defined as the product of the two polynomials f)i-l and Ai_I' The recursive algorithm (4) is not called until the initialization of the vectors Zk * and i k * is complete. As a result, al to an are all set to ao and only at sample n + 1 is this initial estimate updated. Application of the algorithm (4) then continues until all the available data, say T samples, have been processed: at this point aT provides the estimate of a at the end of the first refined IV iteration and allows for the updating of the polynomials Ao and 130 to Al and 13" respectively. The updated auxiliary model based on Al and ,81' i.e. (16)

x

can then be used to generate k , k= 1,2, ... , T, which, in turn, can be incorporated in eqn. (11) to provide the estimated noise sequence tk' k= 1,2, ... , T. It is this estimated noise sequence which provides the coordination between the IV and AML sub-algorithms. tk' so obtained, provides the input to the filtering operations for tk * and k*. As in the IV case, these filtering operations can be written as follows in discrete-time terms:

e

(i)

(ii)

}

( 17)


Recursive time-series analysis where

y = [Yv Y2' ... , Yn'

"1' .,., "n]T

;k

7

I

(18)

Here, the zero elements in the vectors and e arise because the filter Ft in (13) has a numerator of unity, so that no moving average operations are specified. The filtering operation (17) (i) is carried out over the first n + 1 samples in a similar manner to the equivalent operation in the IV sub-algorithm and this yields the tk * elements required for the initialization of the mk vector in (10). This is not possible in the case of (17) (ii ), however, since 13 k , k = 1, 2, ... , n, are not known. As in the basic Al\'IL algorithm, therefore, these initial elements in the 13k sequence 'ere set to their expectation of zero (see assumptions (3)) and this in turn means that 13 k *, k= 1,2, ... , n, needs also to be set to zero]. , Following the initialization of "k 'end mk , Co is defined by reference to the prior IV AJ\lL results and recursive estimation proceeds from sample n + 1 using eqns. (9). The 13 k and 13 k * variables required at each recursive step in these equations are obtained by the use of eqns. (12) and (17) (ii ), respectively, and recursion continues until cT is obtained. This marks the end of the first complete refined IVAML iteration. At this point, the cT vector provides the required coord ination between the A~lL and IV sub-algorithms: the noise model parameters defined by cT are communicated to the IV sub-algorithm where they define the filter parameters Yi and "i' i = I, 2, ... , n, needed for the second IV iteration. The second and subsequent iterations through the complete refined IVAML procedure are similar to the first, with the initial estimates and prefilter parameters being provided by the estimates obtained at the previous iteration. It has been found by experience, however, that advantage is gained after the first iteration by initializing the Po and PoN matrices to the values Pk and P"N saved at a specified recursive step in the previous iteration. This is a heuristic method of indicating to the algorithm the increased confidence that accrues from the repeated iteration. As a rule of thumb, it has been found that an appropriate recursive step for saving the matrices is given by TjJ, where J is the number of iterations specified by the programme user which, as mentioned earlier, will normally be less than six. Indeed, we have found in practical applications that a neeel for greater than six iterations is normally associated with poor prior model structure identification.

k

2.2. Recursive version The iterative version of refined IV AML described above includes recursive parameter estimation at each iteration. But this recursive solution does not involve recursive updating of the prefilter and the auxiliary model parameters. A fully recursive solution which does include such operations is clearly feasible, however, and it has maximum potential utility in on-line applications, such as forecasting and adaptive control. In such a recursive algorithm, prefilter and

t This is in contrast to an 'exact maximum likelihood' approach where these initial values are also cstimated (see, e.g. Pagan and Nicholls 1976).


8


auxiliary model parameters are updated in the same manner as in the iterative version but updating takes place after each recursive step. We will not go into the details of this kind of fully recursive algorithm here except to point out that, as in the basic IVAML case, the recursive updating must be implemented carefully to avoid convergence problems. In particular, the recursive adaptation of the prefilters and the auxiliary model parameters should (a) not be initiated until after the initial convergence is complete and reasonably' stable' parameter estimates are being obtained; and (b) include prior treatment of the recursive estimates in the form of a pure time delay, low pass filtration and a Schur-Cohn stability check, as discussed in detail by Young et al, (1971) and Young (1979). Unfortunately, the flexibility of an on-line recursive algorithm of this type is obtained at some cost. The rapid adaptation of the auxiliary model and the prefilters coupled with the complex coordination between the IV and AML sub-algorithms tends to make the recursive algorithm less robust than the iterative algorithm when in general use. It must, for example, be tuned carefully in any particular situation in order to obtain good results and may need modifications, such as the introduction of a fading memory factor, as advised, for example, by Soderstrom et al, (1974). Also, since the fully recursive algorithm involves only a single passage through the data, it will not be as statistically efficient over a finite data set as the iterative algorithm. It seems a reasonable conjecture, however, that the statistical efficiency will approach that of the iterative algorithm for large data sets. This is confirmed in the Monte-Carlo simulation results given later in .§ 4. Finally, it is worth noting that the recursive estimates generated throughout each iteration in the iterative solution can be useful in their own right. Indeed, in both the basic and refined IVAML cases we have found that, after convergence is achieved, it is advisable to make an additional iteration through the data in which the auxiliary model and prefilter parameters are maintained constant at their converged values and the recursive estimation is initiated. from initial Po and PoN matrices set to the values used in the first iteration (i.e. diagonal with elements set to 106 ) . The recursive estimates obtained in this manner can then be examined to see if there is any evidence of parametric nonstationarity or other statistical problems such as collinearity. Also, as we shall see in § 3, the final PT and PTN matrices obtained in this additional iteration are useful in assessing the error-covariance matrices associated with the parameter estimates.

2.3. Some modifications to the basic refined 1 V AillL procedure On the basis of the Monte-Carlo simulation results discussed later in § 4 and theoretical analyses of the refined IVAML algorithm based on the approach of Ljung (see Ljung 1974 and Holst 1977), it would appear that certain improvements to the basic estimation procedure described above are possible. These improvements result in superior performance ingeneral applications and avoid possible convergence problems that can affect the refined AML algorithm in certain rare circumstances. The modifications, which also are suggested by the theoretical results of Pierce (1972) in connection with the non-recursive Maximum Likelihood method of Box and Jenkins (see § 3), involve converting the Pk and '/" matrices into symmetric form.



9

The use of symmetric matrix gains in recursive IV algorithms has been discussed previously by Young (1969; see also 1966) who suggested a symmetric matrix version of the basic IV algorithm. An alternative symmetric matrix version suggested by the results of Pierce (1972) involves replacing the refined IV algorithm (4) by the following equations: (i)

ak = ak_1 -

Pk-1i k*[a2 + i k*T Pk-1i k*J-l(Zk *T ak_1 - Yk*)

where

I

(19)

(ii) Pk=Pk_l-Pk_lik*[a2+ik*T Pk_lik*J-lik*TPk_l

In other words, all Zk * vectors, other than that appearing in the error term (zk *T ak_1 -Yk *) in (19) (i), are replaced by i k*. The advantages of this algorithm over the non-symmetric matrix equivalent will become clear when the results of Pierce are discussed in § 3 in relation to the statistical properties of the estimates ak' As we shall see, the algorithm seems to perform well in general applications where there is a reasonable sample size. Theoretical analysis by Holst (1977) based on the approach of Ljung (1974) indicates that a symmetric matrix version of the refined AML algorithm also has some advantages. While the non-symmetric matrix refined IVAML procedure should normally have good convergence characteristics when applied to ARMAX time-series models (i.e. Model (1) with O(Z-l) assumed equal to A(Z-l)), the refined AML part of the algorithm may experience some minor convergence problems when applied to the errors-in-variables transfer-function model (1) preferred here. For this reason, it is better in general applications to convert the refined AML algorithm (9) to the following symmetric matrix form, which is simply the Recursive Maximum Likelihood algorithm originally suggested by Soderstrom (1973) :

where (ii)

I

(20J

PkN=Pk_.N_Pk_1N

ril k [a2 + ril k T

Pk_.N

ril k J- 1rilk T

Pk_1N

3.

The statistical properties of the refined IVAML estimates Pierce (1972) has shown that the large sample properties of the maximum likelihood (ML) estimates of the parameters in the time-series model (1) can be evaluated in a manner which is pertinent to our present discussion. In the nomenclature of the present paper, and using systems terminology, his results are as follows.

Theorem lfin(l): (i) the 13k are independent and identically distributed with zero mean, variance a 2 and skewness and kurtosis K 1 and K 2 ; (ii) the parameter values are admissible (i.e. the model is stable and observable) and



10

(iii) the Uk are persistently exciting (see Astrom and Bohlin 1965), then the ML estimates aT' cT and 6'2 obtained from a data set of T samples, possess a limiting normal distribution, such that the following results hold: (a) the asymptotic covariance matrix of the estimation errors associated with the estimate aT is of the form

P=~[Plim;L~k*~k*Trl

(21)

(b) the estimate cT is uBymptotically independent of aT and has an error covariance matrix of the form (22)

and (e) the estimate 6'2 has asymptotic variance (2a4JT)(1 +!K2)

and, if

K1

= 0,

is independent of the above estimates.

In (21) and (22) ~k * and ""k are defined as the values of these variables which would result if the auxiliary model, inverse noise model and prefilter parameters were all set at values based on the true model parameters a and c. The relevance of these results is obvious: 'if it is assumed that in all identifiable situaticns'] the refined IVAML algorithms given by the interactive use of eqns. (19) and (20) converge in the sense that ak->a and ck->c for increasing k, then the matrices Pk and PkN will provide good empirical estimates of P and PN. And, noting thc Monte-Carlo results of Pierce (1972), we might also assume that PT and PTN would provide good indication ofthe error oovaria.nce properties of aT and cT for even small sample size T. The Monte-Carlo simulation experiments discussed in the next section confirm that this is indeed the case for aT but the results are not so satisfactory as regards CT , although they appear acceptable for large sample sizes. 4.

Monte-Carlo simulation experiments The iterative version of the refined IV AML algorithm described in § 2.1 (i.e. the non-symmetric gain version given by eqns. (4) and (9)) has been thoroughly evaluated by Monte-Carlo simulation applied to the following five models: 1·0z-I+0·5z- 2 1-0·189z-l . 'u+ e I _ 1'5z-1 + 0'7z-2 k 1 _ 1'027z-1 + 0.264z-2 k

Model I

1J=

Model 2:

-0·102z- 1+0·173z- 2 Yk = 1 _ 1.425z-1 + 0'496z-2

• k

Uk

1-0'775z- 1 + 1 _ 1'027z-1 + 0'264z- 2 ek·

t l.n simple terms, an identifiable situation is one in which the assumed model structure and time-series data are such that a unique solution to the maximum likelihood problem is possible. We will not discuss the conditions for identifiability in this paper but refer the reader to, e.g. Staley and Vue (1970).


Recursive time-series analysis Model 3: y

Model 4:

Model 5:

k

=

0'065z- 1 + 0'048z- 2 - 0'008z- 3 u 1 _ 1.5z 1 + 0.705z 2 k

l-Oz>! Yk

Yk=

----,-----,------,--------C0 Uk 1 + 0'9z- 1 + 0'95z- 2

1'0z- 1 + 0'5z- 2

l-I·5z- 1+O·7z 2

1t k

+

II

1 - 0'826z- 1

+ 1- 0'527z 1 + 0.0695z

1 + 1'5z- 1 + 0'75z- 2

1 + 0'9z

1

+ 0'95z

1 - 1'0z- 1 + 0'2z- 2

+ l-l'5z

1+0'7z- 2

2

e

k

ek

2

ek

The first two models are those used by Isermann et al. (1973) in a comparative study of six time-series estimation procedures, while the last two have been used by Soderstrom et al. (1974) and Ljung et al. (1975). Model 3 is a modification of a model used by Isermann et al. (1973), for reasons discussed in § 5. It will be noted that both Models 4 and 5 have C(Z-l) equal to A(Z-1). This is a consequence of the models used by Soderstrom et al. and Ljung et al. being in the ARMAX rather than the transfer-function form of (1). In the Monte-Carlo analysis, each model was evaluated for three different sample sizes and two signal-noise (S) ratios. In the case of the first three models, the three sample sizes selected were 75, 300 and 1000; while in the latter two, this was changed to 100, 500 and 2000 so that the results could be compared directly with those of Soderstrom et al. (1974), who also used these sample sizes in their Monte-Carlo analysis. In all cases, the input Uk was selected as a random binary signal with levels plus or minus 1'0, and ek was generated as a sequence of normal random variables with variance cr 2 which was adjusted to provide S ratios of either 10 (low noise) or 1 (high noise). Here, S is defined as N

S fo

LN X k 2 L ~k2

(23)

The results of the Monte-Carlo analysis are shown in Tables 1 to 5. In Tables 1 to 4, the refined IVAML results (A and C) for the two chosen S ratios are compared with the basic IVAML results (B and D); while in Table 5 they are compared with the results obtained by Soderstrom et al. (1974). In each of these tables, the mean and standard deviation of the estimates obtained from the Monte-Carlo simulations are given for each sample size in the first and second unlabelled columns respectively. Also shown in the third column of the refined IVAML tables are the standard errors of the estimates obtained by reference to the Pierce theoretical error-covariance matrices defined in eqns. (21) and (22). Here the expected value in (22) is replaced by the sample average to allow for numerical computation and the results should be viewed with this in mind. The complete Pierce matrices P and pN for Models 3 and .5 are compared with the estimates of these same error-covariance matrices as obtained in the refined IVAML algorithm in Tables 3 and 5, parts E, F and G. Similar errorcovariance results were obtained for all models but only those for Models 3 and 5 are given here in order to conserve space. The interested reader can obtain the complete results, however, by reference to Young and Jakeman (1977).

-1'5 0·7 1·0 0·5 -1-027 0·264 -0,189

a, a2

- 1·5008 ± 0·0204 ± 0·0207 0·7000 ± 0·0183 ± 0·0186 1·0080 ± 0·0592 ± 0·0576 0·5059 ± 0·0680 ± 0·0752 - 0·8695 ± 0·2960 ± 0'4322 0·1502 ± 0·2285 ± 0·3211 - 0·0406 ±0·2859 ± 0·4437

-1'4947 ±0·0449 ±0·0415 0·6960 ± 0·0430 ± 0·0377 1·0011 ±0·1204 ±0·1l54 0·5195 ± 0·1757 ± 0·1512 - 0·7243 ± 0·4464 ± 0·8922 0·0576 ± 0·3522 ± 0·6643 0·0722 ±0·4090 ±0·9157

-1-5004 ± 0'01l3 ± 0'01l0 0·7004 ± 0·0101 ± 0·0101 0·9984 ± 0·0282 ± 0·0315 0·5018 ± 0·0399 ± 0·0407 - 0·9556 ± 0·1742 ± 0·2348 0·2116 ± 0·1305 ± 0·1742 -0'1l41 ± 0·1770 ± 0·2411

1000

Tables 4, 5 and 6 B show averages for 10 experiments.

300

75

Number of samples

Refined IVAML, 8=10.

Tables 1-3 and 6 A show averages for 100 experiments.

d,

C, C2

b2

b,

True value

Parameter

Table 1 A.

i

~

~

~ ~

~

~

."

......



-1-425 0·496 -0,102 0'173 -1·027 0·264 -0'775

a, a2

c, c. d,

b, b2

-1-4104 ± 0·0844 0·4808 ± 0·0889 -0,1036 ± 0·0193 0·1732 ± 0·0147 - 0·3327 ± 0·7936 -0,0327 ± 0·3664 0·0541 ±0·9175

75

True value

Parameter

c, c2 d,

Table 2 B.

-1,4240 ± 0·0488 ± 0·0448 0·4952 ± 0·0459 ± 0·0420 -0,1019 ± 0·0128 ±0·0127 0·1731 ±0·0154±0·0149 -0·3174 ± 0·8496 ± 0·4311 0·1356 ± 0·2379 ± 0·1259 -0·0884 ± 0·8082 ± 0·4375

-1,425 0·496 -0·102 0·173 -1·027 0·264 -0'775

a, a2

b, b2

75

True value

-1-4214 ± 0·0381 0·4943 ± 0·0413 -0'1009 ± 0·0095 0·1725 ± 0·0071 -0,2934 ± 0·7764 0·0330 ± 0·2528 0·0108 ± 0·8272

300

Number of samples

IVAML, 8 = 10.

-1'4195 ± 0·0217 ± 0·0224 0·4913 ± 0·0202 ± 0·0210 - 0·1011 ± 0·0064 ± 0·0062 - 0·1729 ± 0·0071 ± 0·0073 - 0·5289 ± 0·5609 ± 0·1979 0·1566 ± 0·1398 ± 0·0599 -0·2771 ± 0·5614 ± 0·1995

300

Number of samples

--

Refined IVAML, 8=10.

Parameter

Table 2 A.

- 1·4271 ± 0·0238 0·4989 ± 0·0262 -0·1022±0·0053 0·1731 ± 0·0036 -0·4725±0·6420 0·1122 ± 0·1860 -0'1923 ± 0·6678

1000

-1·4239 ± 0·0124 ± 0·0126 0·4949 ± 0·0117 ± 0·0118 -0'1022 ± 0·0031 ±0·0034 0·1733 ± 0·0036 ± 0·0039 - 0·6242 ± 0·4975 ± 0·1057 0·1760 ± 0·1217 ± 0·0324 -0·3641 ±0'5037 ± 0·1064

1000

;:l

'"

?:"

'"';:§"

.....

;:l

'" "':>-"

Ii,

;3

...'" '" ...1::'"...

ee,

0:: W.,. f-

.

W'"

::;:

IT 0:: IT 0...",

.,;

'",

I

I

8.0

TIME

12.0

*10 2

I

16.0

I

20.0

Figure 3. It is clear from the results in Tables 4 Band 4 D that the basic AML does have problems in the case of Model 4, as predicted by Ljung et al. and yields poor estimation of the noise model parameters even for large sample size. The refined IVAML algorithm (Tables 4 A and 4 0) corrects this deficiency, however, and yields much reduced error variances on the noise parameter estimates. These results accord with theoretical analysis of the problem using the approach of Ljung (1974), which indicates that the non-symmetric matrix refined AML, when combined with refined IV estimation of the basic system parameters, will be able to handle ARMAX models without convergence problems. This arises because the refined IV algorithm then provides an estimate of the common A(Z-l)=C(Z-l) polynomial in Model 4, so that refined AML is only applied to an effective pure moving average noise process, where there appears to be no convergence problems (Soderstrom et al. 1974).


P. You.ng and A . Jokeman.

26

It is worth noting that the convergence problems with this particular model are only associated with the basic AML algorithm: the basic IV algorithm is unaffected; indeed, it appears to provide error variances for til and ti2 that are very similar to those of refined IV (and therefore M.L), even for low sample size. Thus, if the objective of the analysis is to estimate the parameters of the system model and not those of the noise model, then the basic IV results might well be considered good enough for most purposes. Finally, the results obtained from the symmetric matrix refined IV AML (19) and (20) when applied to Models 2 and 4 are given in Tables 6 A and 6 B. In 6 B we see that even if C(Z-l) is assumed unknown (i.e. C(Z-l) not necessarily equal to A(z-I)) the algorithm achieves convergence. On the other hand, the non-symmetric matrix version of the refined AML algorithm (4) and (9) fails to converge in these circumstances.

Tuble 6 B.

(Symmetric) refined [VAML, 8=10, process 4 with C not, constrained to be A. Number of samples

Parameter al

a,

h, f,t

f"

d[ d,

True

value

100

500

2000

0·9 0·95 1·0 0·9 0,9.5 1·5 0·75

0·8977 ± 0·0092 0·9508 ± 0·0150 1·0066 ± 0·0166 0·9079 ± 0·0300 0·9188 ± 0·0627 1·2657 ± 0·2631 0,.5825 ± 0·3841

0·9017 ± 0·0049 0·9520 ± 0·0059 1·0015± 0·0108 0·8932 ± 0·0150 0·9407± 0·0195 1·3700±0·1628 0·1)542 ± 0·1504

0·8995 ± 0·0022 0·9491 ± 0·0025 0·9997 ± 0·0031 0·8997 ± 0·0094 0·9486 ± 0·0062 1·4703± 0·0539 0·7241 ± 0·0204

Other Monte-Carlo results were obtained (see Young and including those for smaller sample sizes, different inputs specified noise model structures. In all cases: however, the algorithm performed in a predictable and satisfactory fashion therefore, report the full results here.

J akeman 1977), and incorrectly refined IV AML and we will not,


Recursive time-series analysis 5.

27

Discussion of results The most notable general observation on the results given in Tables 1 to 5 is that, except for Model 3, the system model parameters (i.e. ai and bi ) are estimated extremely well at all sample sizes and signal/noise ratios, with zero estimation bias and close to minimum error variance all of the time. Moreover, the empirically estimated error-covariance matrix PT (and, therefore, the resultant estimated standard errors on the parameters) agree both with the theoretical' minimum variance' results of Pierce for the ML estimates and with the experimentally calculated statistical properties obtained from the MonteCarlo analysis. In other words, the refined IVAML algorithm appears to converge on the maximum likelihood estimates of the ai and b; parameters as conjectured by Young (1976). The experimental results would suggest that the refined IVAML algorithm does not perform so well in regard to the estimation of the noise model parameters Ci and di . Here, for small and moderate sample size, the estimates quite often appear to be biased and minimum error variance is not achieved; indeed, it is only for large sample size that the algorithm seems to achieve anything like optimum performance in this regard. Although this is in line with the results obtained by Soderstrom et al. (1974), who also only achieve reasonable results for large sample size, it seems somewhat surprising at first sight, since the noise model parameters are, of course, utilized to achieve the statistical efficiency of the ai and bi parameter estimates via their use in the prefilters (7). It can only be assumed that, while the parametric estimates associated with the noise model are not particularly good for low and moderate sample size, the filtering capacity of the prefilters (7) obtained with the help of these estimates is sufficient to ensure statistical efficiency for the system parameter estimates. In a sense, therefore, one might conjecture that the estimated noise model, although parametrically ill-defined, is defined sufficiently well in frequency-domain terms for most filtering (and, therefore, control) applications. It will also be noted that, in general, the empirically estimated errorcovariance matrices for the noise model parameters do not show good agreement with the theoretical minimum variance results, although they sometimes broadly agree with the experimentally determined error variances. In most cases, however, the estimated error variances are larger than the achieved error variances and they tend, therefore, to err on the conservative 'safe' side. -_./ The refined IV AML results are almost always an improvement on the equivalent IVAML results, although the improvement is sometimes quite small and rarely large. Only in Model 4 does the refined algorithm achieve a distinct improvement and then only in terms of the AML noise model parameter estimates. Here, for 2000' samples, the C1 and C2 parameters are estimated much better by refined IVAML, with zero bias and standard errors (about the mean) reduced by a factor of about 6. This is only to be expected, however, since Model 4 was evolved by Ljung et al. (1975) as a counter example to the general convergence of the basic AML procedure (termed by them RMLI). We see that the prefilters of refined IV AML have corrected the convergence problems in this ARMAX model case and reasonable results are obtained even for low sample size (although zero bias and minimum variance is achieved only for 2000 samples at S = 10). These results are in general agreement with those


28

P. Young and A. Jakeman.

of Ljung et 0.1. (1975) in connection with their modified AML procedure (which is termed by them RML2 and is equivalent to eqn. (20)). Another comparison of refined IVAI\:lL and RML2 is shown in Table 5, where it will be seen that the refined IV AML in the presently tested iterative mode of operation out-performs the fully recursive, non-iterative, Rl\1L2 for low sample size although, as might be expected, the results arc similar for large sample size. The improvement obtained from off-line iterative operations for small sample size is quite clear from these results and it demonstrates the importance of iteration whenever there is a paucity of data and the iterative mode of operation is acceptable in practical terms. As regards the results for Model 3, the system model parameters at and b, are again estimated well for 300 and 1000 samples at S = 10, and for 1000 samples at S = I, but high standard errors result for 0.'1 and 0. 2 at lower sample sizes (indeed, the estimates of these parameters even appear biased at 75 samples for S = I). These relatively poor results are probably due to the nature of the model in this case, which is the Model III used by Isermann et. 0.1. (1973) with the 0. 3 parameter of their model omitted completely. This modification was incorporated because, as lsermann ei 0.1. also found, extremely poor estimation results were obtained with data generated from the original model. Indeed, Isermann et 0.1. noted that the recursive estimates' converge or diverge simultaneously during short time periods so as to compensate each other, resulting in approximately the same input-output behaviour'. And again, , for very small model impulse response errors, very different errors of parameters resulted'. It would appear from these observations that the original Isermann ei 0.1. Model lIJ is poorly identifiable, in the sense that there is no clearly defined unique parametrization. AmI these identifiability problems, which seem similar in some regards to that noted for certain continuous-time models by Young (1968), are not entirely overcome by the removal of the 0. 3 parameter: the results in 'I'able 3 for low sample size are not nearly as good as the excellent results obtained for all the other models. It may be, therefore, that some mild identifiability problem is the cause of the slightly impaired estimation performance for this model. It would be interesting, to investigate further the nature of this problem both for the Model 3 here and, more particularly, for the original Model III used by Isermann ei 0.1. (1973). Table 6 indicates that the symmetric matrix refined IV AML algorithm (19) and (20) performs in a very similar manner to the non-symmetric matrix equivalent but has superior convergence characteristics as regards noise model estimation when applied to the counter example of Ljung ei 0.1. (1975). For this reason, we recommend its use rather than algorithm (4) and (9) in general applications. The symmetric gain version also has the advantage of directly generating good estimates of the Pierce error-covariance matrices P and P'" associated with the estimated parameters. It is necessary, however, to give one caveat, with regard to the use of the symmetric matrix form of the refined IV algorithm. In the investigations of the multivariable version of the algorithm (Jakeman and Young 1978), poor performn.nce of the algorithm has been encountered under' small' sample size



29

conditions where the ratio of the sample size to the number of unknown parameters is very low « 4). We discuss possible reasons for this poor performance in the subsequent paper but the results suggest that caution should be exercised in the application of the symmetric matrix IV algorithm when there is a paucity of data. In such situations, it is probably safer to utilize the non-symmetric algorithm until convergence is achieved and then use the symmetric matrix form during an additional iteration in order to generate the j> and j>N estimates of the error covariance matrices. Finally, reference to the recursive estimates shown in Fig. 3 shows that the convergence of the ai and b, parameters is very rapid. This seems to be the case even in the fully recursive situation not investigated thoroughly here, and seems to be an advantageous general property of the IVAML type algorithms, particularly at low sample size (see, e.g. Soderstrom et al. 1974).

6. Conclusions The exhaustive Monte-Carlo simulation experiments discussed in this paper, together with theoretical analyses of the constituent algorithms (Holst 1977, Solo 1978), demonstrate the efficacy of the refined IVAML method of time-series analysis for single input, single output systems. In the examples investigated, the refinement introduced by the use of adaptive prefilters is seen to yield asymptotic efficiency and also to produce near minimum variance estimates of the basic system model parameters for even low sample size and low signal/noise ratios. While the noise model parameters are clearly estimated well enough to ensure such performance, the parametric estimates associated with the noise model are themselves not particularly satisfactory for small sample sizes, although they seem comparable with those obtained by other competing methods of time-series analysis such as the Recursive Maximum Likelihood (RML2) method of Soderstrom. The paper has also discussed a symmetric matrix version of the refined I VAML algorithm which corrects a minor deficiency in the original AML part of the algorithm and, at the same time, offers certain advantages in relation to the recursive estimation of the error-covariance matrices associated with the refined estimates. This new version appears to work well in general, identifiable situations where there is reasonable sample size. There are various other comments that can be made regarding the refined IVAML approach to time-series analysis, in particular its performance in very low sample size situations and its relationship with other methods of time-series estimation. But these are best delayed until we have discussed the multivariable version of the algorithm in Part II of the paper. REFERENCES ASTROM, K. J., and BOHLIN, T., 1965, Theory of Self Adaptive Control Systems, edited by P. H. Hammond (New York: Plenum Press) (Proc. of IFAC Symposium, Teddington, 1965). HOLST, J., 1977, Report LUTF D2((TRFT-1013)/1-206/(1977), Lund Institute of Technology, Division of Automatic Control. ISERMANN, R., BAUB, D., BAMBERGER, W., KNEPPO, P., and SIEBERT, H., 1973, Identification and System Parameter Estimation, edited by P. Eykhoff (New York: American Elsevier; Amsterdam: North-Holland), p. 1086.


30


.JAKEMAN, A. J., and YOUNG, P. C., 1979, Int. J. Control (to be published) (see also Report NU'I~ber AS/RI3(1977), Centre for Resource and Environmental Studies, Australian National University). LJUNG, L., 1974, Report 7403, Lund Institute of Technology, Division of Automatic Control. LJUNG, L., SODERSTROM,. T., and GUSTAVSSON, I., 1975, I.E.E.E. Trans. autom. Control, 20, 643. PAGAN, A. R., and NICHOLLS, D. F., 1976, Rev. Econ. Studies, 43, 383. PIERCE, D. A., 1972, Biometrika, 59, 73. SODERSTROM, T., 1973, Report 7308, Lund Institute of Technology,. Division of Automatic Control. SODERSTROM, T., WUNG, L., and GUSTAVSSON, 1., 1974, Report 7427, Lund Institute of Technology, Division of Automatic Control. SOLO, V., 1978, Report No. AS/R20 (1978), Centre for Resource and Environmental Studies, Australian National University. STALEY, R. 1\1., and YUE, P. C., 1970, Inform. Sci., 2,127. YOUNG, P. C., 1966, Proc. l.E.E.E., 54, 1965; 1968, Proc. Second Asilomar Conference in Circuits and Systems, New York, p. 416; 1969, Ph.D. Thesis, Department of Engineering, University of Cambridge, 1-6-14; 1974, Bull. Inst. Math. Applic., 10, 209; 1976, Int. J. Control, 23, 593; 1979, Recursive Estimation (Springer Verlag), (in preparation). YOUNG, P. C., SHELLSWELL, S. H., and NEETHLING, C. G., 1971, Report Number CUED/B-Control TRl6, Control and Systems Division, Department of Engineering, University of Cam bridge. YOUNG, P. C., and JAKEMAN, A. J., 1977, Report No. AS/RI2 (1977), Centre for Resource and Environmental Studies, Australian National University. YOUNG, P. C., and JAKEMAN, A. J., 1978, Report Number AS/RI8 (1978), Centre for Resource and Environmental Studies, Australian National University.