ON THE LINEAR LEAST-SQUARE PREDICTION PROBLEM

R. M. Fern

´

andez-Alcal

´

a, J. Navarro-Moreno, J.C. Ruiz-Molina and M. D. Estudillo

Department of Statistic and Operations Research. University of Ja

´

en

Campus Las Lagunillas, s/n. 23071, Ja

´

en (Spain)

Keywords:

Correlated signal and noise, linear least-square prediction problems.

Abstract:

An efﬁcient algorithm is derived for the recursive computation of the ﬁltering and all types of linear least-

square prediction estimates (ﬁxed-point, ﬁxed-interval, and ﬁxed-lead predictors) of a nonstationary signal

vector. It is assumed that the signal is observed in the presence of an additive white noise which can be

correlated with the signal. The methodology employed only requires that the covariance functions involved

are factorizable kernels and then it is applicable without the assumption that the signal veriﬁes a state-space

model.

1 INTRODUCTION

The estimation of a signal in the presence of additive

white noise has been found to be among the central

problems of statistical communication theory.

Application of the linear least mean-square error

criterion leads to a linear integral equation, called

Wiener-Hopf equation, whose solution is the impulse

response of the optimal estimate. Although the lin-

ear least mean-square estimation problem is com-

pletely characterized by the solution to the Wiener-

Hopf equation, a great effort has been made in the

searching of efﬁcient procedures for the computation

of the desired estimator. Roughly speaking two ap-

proaches have been applied.

A ﬁrst via of solution consists in using integral-

equations approaches which provide the solution to

the Wiener-Hopf integral equation for the impulse re-

sponse function of the optimal estimator, from the

knowledge of the covariance functions of the sig-

nal and noise [see, e.g. (Van Trees, 1968), (Kailath

et al., 2000), (Fortmann and Anderson, 1973), (Gard-

ner, 1974), (Gardner, 1975), (Navarro-Moreno et al.,

2003)]. This technique is closely connected to series

representation for stochastic processes and, in gen-

eral, a series representation for the optimal estimate

is provided instead of a recursive computational algo-

rithm. The use of series representation for stochastic

processes only allow to derive recursive procedures

for the computation of suboptimum estimates.

On the other hand, a conventional approach to esti-

mate a signal observed through a linear mechanism

lies in imposing structural assumptions on the co-

variance functions involved. In this framework, the

most representative algorithm is the Kalman-Bucy ﬁl-

ter [see. e.g., (Kalman and Bucy, 1961), (Gelb, 1989)]

which requires that the signal veriﬁes a state-space

model. However, although the Kalman-Bucy ﬁlter

has been widely applied, there are a great number

of physical phenomena that cannot be modelled by a

state-space system. For problems with covariance in-

formation, linear least mean-square estimation algo-

rithms have been designed under less restrictive struc-

tural conditions on the processes involved [(Sugisaka,

1983), (Fern

´

andez-Alcal

´

a et al., 2005)]. Speciﬁcally,

the only hypothesis imposed is that the covariance

functions of the signal and noise are expressed in the

factorized functional form.

Therefore, under the assumption that the covari-

ance functions of the signal and noise are factoriz-

able kernels, we aim to derived a recursive solution

to the linear least-square estimation problem involv-

ing correlation between the signal and the observation

noise. Speciﬁcally, using covariance information, an

imbedding method is employed in order to design re-

cursive algorithms for the ﬁlter and all kinds of pre-

dictors (ﬁxed-point, ﬁxed-interval, and ﬁxed-lead pre-

dictors). Moreover, recursive formulas are designed

for the error covariances associated with the above es-

timates.

332

M. Fernández-Alcalá R., Navarro-Moreno J., C. Ruiz-Molina J. and D. Estudillo M. (2005).

ON THE LINEAR LEAST-SQUARE PREDICTION PROBLEM.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and

Control, pages 332-335

DOI: 10.5220/0001160303320335

Copyright

c

SciTePress

Then, the paper is structured as follows. In the

next section, a general formulation of the linear least-

squares ﬁltering and prediction problem is consid-

ered. Finally, in Section 3, the recursive algorithms

for the ﬁlter and all types of predictors as well as their

error covariances are derived.

2 PROBLEM STATEMENT

Let {x(t), 0 ≤ t < ∞} be a zero-mean signal vector

of dimension n which is observed through the follow-

ing equation:

y(t) = x(t) + v(t), 0 ≤ t < ∞

where y(t) represents the n-dimensional observation

vector and v(t) is a centered white observation noise

with covariance function E[v(t)v

′

(s)] = r δ(t − s),

with r a positive deﬁnite covariance matrix of dimen-

sion n × n, and correlated with the signal.

We assume that the autocovariance function of the

signal and the cross-covariance function between the

signal and the observation noise are factorizable ker-

nels which can be expressed in the following form:

R

x

(t, s) =

A(t)B

′

(s), 0 ≤ s ≤ t

B(t)A

′

(s), 0 ≤ t ≤ s

R

xv

(t, s) =

α(t)β

′

(s), 0 ≤ s ≤ t

γ(t)λ

′

(s), 0 ≤ t ≤ s

(1)

where A(t), B(t), α(t), β(t), γ(t), and λ(t) are

bounded matrices of dimensions n × k, n × k, n × l,

n × l, n × l

′

, and n × l

′

, respectively.

We consider the problem of ﬁnding the linear least

mean-square error estimator, ˆx(t/T ), with t ≥ T , of

the signal x(t) based on the observations {y(s), s ∈

[0, T ]}. It is known that such an estimate is the or-

thogonal projection of x(t) onto H(y, t) (the Hilbert

space spanned by the process {y(s), s ∈ [0, T ]}).

Hence, ˆx(t/T ) can be expressed as a linear function

of all the observed data of the form

ˆx(t/T ) =

Z

T

0

h(t, s, T )y(s)ds, 0 ≤ s ≤ T ≤ t

(2)

As a consequence of the orthogonal projection the-

orem, we obtain that the impulse response function

h(t, s, T ) must satisfy the Wiener-Hopf equation

R

xy

(t, s) =

Z

T

0

h(t, σ, T )R(σ, s)dσ + h(t, s, T )r

(3)

for 0 ≤ s ≤ T ≤ t, where R

xy

(t, s) = R

x

(t, s) +

R

xv

(t, s), and R(t, s) = R

x

(t, s) + R

xv

(t, s) +

R

vx

(t, s).

From (1), it is easy to check that R

xy

(t, s) and

R(t, s) can be written as follows:

R

xy

(t, s) =

F (t)Γ

′

(s), 0 ≤ s ≤ t

G(t)Λ

′

(s), 0 ≤ t ≤ s

R(t, s) =

Λ(t)Γ

′

(s), 0 ≤ s ≤ t

Γ(t)Λ

′

(s), 0 ≤ t ≤ s

(4)

where F (t) = [A(t), α(t), 0

n×l

′

], G(t) =

[B(t), 0

n×l

, γ(t)], Λ(t) = [A(t), α(t), λ(t)], and

Γ(t) = [B(t), β(t), γ(t)] are matrices of dimensions

n × m with m = k + l + l

′

, and 0

p×q

denotes the

(p × q)-dimensional matrix whose elements are zero.

Note that, we can expressed the optimal linear ﬁl-

ter and all kinds of predictors through the equations

(2) and (3). Speciﬁcally, by considering T = t

we have the ﬁltering estimate ˆx(t/t), the ﬁxed-point

predictor ˆx(t

d

/T ) is derived by taking a ﬁxed in-

stant t = t

d

> T , for the ﬁxed-interval predictor,

we consider a ﬁxed observation interval [0, T

d

], with

T

d

< t, and ﬁnally the ﬁxed-lead prediction estimate

ˆx(T + d/T ), is given by (2) and (3) with t = T + d,

for any d > 0.

Likewise, the error covariances associated with the

above estimates can be deﬁned as

P (t/T ) = E[(x(t) − ˆx(t/T ))(x(t) − ˆx(t/T ))

′

] (5)

with a suitable estimation instant, t, and a speciﬁc ob-

servation interval [0, T ].

Therefore, in the next section, the Wiener-Hopf

equation (3) will be used, with the aid of invariant

imbedding, in order to design recursive procedures for

the ﬁlter and all kinds of predictors of the signal vec-

tor x(t) as well as their associated error covariances.

We must note that the only hypothesis assumed is that

the covariance functions involved are factorizable ker-

nels of the form (1).

3 RECURSIVE LINEAR

ESTIMATION ALGORITHMS

Under the hypotheses established in Section 2, an ef-

ﬁcient recursive algorithm for the linear least-square

ﬁlter, and the ﬁxed-point, ﬁxed-interval and ﬁxed-lead

prediction estimates of the signal and their associated

error covariance functions is presented in the follow-

ing theorem.

Theorem 1 The ﬁlter and the ﬁxed-point, ﬁxed-

interval and ﬁxed-lead prediction estimates of the sig-

nal x(t) are recursively computed as follows:

ˆx(t/t) =F (t)L(t)

ˆx(t

d

/T ) =F (t

d

)L(T )

ˆx(t/T

d

) =F (t)L(T

d

)

ˆx(T + d/T ) =F (T + d)L(T )

(6)

ON THE LINEAR LEAST-SQUARE PREDICTION PROBLEM

333

where the m-dimensional vector L(T ) obeys the dif-

ferential equation

∂

∂T

L(T ) =J(T ) [y(T ) − Λ(T )L(T )]

L(0) =0

m

(7)

with 0

m

the m-dimensional vector with zero elements,

and where J(T ) is given by the expression

J(T ) = [Γ

′

(T ) − Q(T )Λ

′

(T )] r

−1

(8)

with Q(T ) satisfying the differential equation

∂

∂T

Q(T ) =J(T ) [Γ(T ) − Λ(T )Q(T )]

Q(0) =0

m×m

(9)

Moreover, the optimal linear estimation error co-

variance functions associated with the ﬁltering esti-

mate, P (t/t), the ﬁxed-point predictor, P (t

d

/T ), the

ﬁxed-interval predictor, P (t/T

d

), and the ﬁxed-lead

predictor, P (T + d/T ), are formulated as follows:

P (t/t) =R

x

(t, t) − F (t)Q(t)F

′

(t)

P (t

d

/T ) =R

x

(t

d

, t

d

) − F (t

d

)Q(T )F

′

(t

d

)

P (t/T

d

) =R

x

(t, t) − F (t)Q(T

d

)F

′

(t)

P (T + d/T ) =R

x

(T + d, T + d)

− F (T + d)Q(T )F

′

(T + d)

(10)

proof 1 From (4), the Wiener-Hopf equation (3) can

be rewritten as

h(t, s, T )r = F (t)Γ

′

(s) −

Z

T

0

h(t, σ, T )R(σ, s)dσ

Now, we introduce an auxiliary function J(s, T )

satisfying the equation

J(s, T )r = Γ

′

(s) −

Z

T

0

J(σ, T )R(σ, s)dσ (11)

Then, it is obvious that the impulse response func-

tion is given by the expression

h(t, s, T ) = F (t)J(s, T ) (12)

Next, differentiating (11) with respect to T , we ob-

tain that J(s, T ) obeys the following partial differen-

tial equation:

∂

∂T

J(s, T ) = −J(T )Λ(T )J(s, T ) (13)

where J(T ) = J(T, T ).

On the other hand, from (4) and (11), it is easy to

check that

J(T )r = Γ

′

(T ) −

Z

T

0

J(σ, T )Γ(σ)dσΛ

′

(T )

Then, the deﬁnition of a function Q(T ) as

Q(T ) =

Z

T

0

J(σ, T )Γ(σ)dσ (14)

leads to the equation (8).

The equation (9) is obtained by differentiating (14)

with respect to T and using (13) in the resultant equa-

tion.

Next, introducing a new auxiliary function

L(T ) =

Z

T

0

J(σ, T )y(σ)dσ (15)

and substituting (12) in (2), we have that

ˆx(t/T ) = F (t)L(T ), ∀t ≥ T (16)

Then, by considering a suitable estimation instant,

t, and a speciﬁc observation interval [0, T ] in (16),

the ﬁlter and all kinds of predictors are given by the

expressions (6).

Moreover, differentiating (15) with respect to T and

considering (13) in the resultant equation, it is easy to

check that the above function L(T ) satisﬁes the differ-

ential equation (7).

Finally, in order to derived the expressions (10) for

the error covariances associated with the above es-

timates, we remark that, from the orthogonal projec-

tion lemma, the error covariance function (5), can be

rewritten as

P (t/T ) = R

x

(t, t) − E[ˆx(t/T )ˆx

′

(t/T )]

Then, substituting (16) in the above equation and

using (11), it is easy to check that

P (t/T ) = R

x

(t, t) − F (t)Q(T )F

′

(t)

As consequence, the expressions given in (10) can

be obtained.

ACKNOWLEDGMENT

This work was supported in part by Project

MTM2004-04230 of the Plan Nacional de I+D+I,

Ministerio de Educaci

´

on y Ciencia, Spain. This

project is ﬁnanced jointly by the FEDER.

REFERENCES

Fern

´

andez-Alcal

´

a, R. M., Navarro-Moreno, J., and Ruiz-

Molina, J. C. (2005). Linear Least-Square Estimation

Algorithms Involving Correlated Signal and Noise.

IEEE Trans. Signal Processing. Accepted for publi-

cation.

Fortmann, T. E. and Anderson, B. D. O. (1973). On the

Approximation of Optimal Realizable Linear Filters

Using a Karhunen-Lo

`

eve Expansion. IEEE Trans. In-

form. Theory, IT-19:561–564.

ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

334

Gardner, W. A. (1974). A Simple Solution to Smoothing,

Filtering, and Prediction Problems Using Series Rep-

resentations. IEEE, Trans. Inform. Theory, IT-20:271–

274.

Gardner, W. A. (1975). A Series Solution to Smoothing,

Filtering, and Prediction Problems Involving Corre-

lated Signal and Noise. IEEE, Trans. Inform. Theory,

IT-21:698–699.

Gelb, A. (1989). Applied Optimal Estimation. The Analytic

Sciences Corporation.

Kailath, T., Sayed, A., and Hassibi, B. (2000). Linear Esti-

mation. Prentice Hall.

Kalman, R. E. and Bucy, R. S. (1961). New Results

in Linear Filtering and Prediction Theory. Trans.

ASME, J. Basic Engineering, Ser. D, 83:95–108. In:

Epheremides, A. and Thomas, J.B. (Ed.) 1973. Ran-

dom Processes. Multiplicity Theory and Canonical

Decompositions.

Navarro-Moreno, J., Ruiz-Molina, J., and Fern

´

andez, R. M.

(2003). Approximate Series Representations of

Second-Order Stochastic Processes. Applications to

Signal Detection and Estimation. IEEE, Trans. In-

form. Theory, 49(6):1574–1579.

Sugisaka, M. (1983). The Design of On-line Least-Squares

Estimators Given Covariance Speciﬁcations Via an

Imbedding Method. Applied Mathematics and Com-

putation, (13):55–85.

Van Trees, H. L. (1968). Detection, Estimation, and Modu-

lation Theory-Part I. Wiley, New York.

ON THE LINEAR LEAST-SQUARE PREDICTION PROBLEM

335