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A Note on the Prediction Error for Small - IEEE Xplore

**IEEE** TRANSACTIONSON INFORMATIONTHEORY,VOL.IT-31,N0. 5,SEPTEMBER1985 611 Correm**on**dence A **on** **the** Predicti**on** **Error** **for** **Small** Time Lags into **the** Future JAMES A. BUCKLEW, MEMBER, **IEEE** Abstract -Explicit expressi**on**s are derived **for** derivatives at zero lag of **the** mean-square predicti**on** error **for** a class of random processes that includes those with rati**on**al power spectra. A simple random process not in **the** class is dem**on**strated. A sufficient c**on**diti**on** **for** membership in **the** class is given. I. INTRODUCTION Finding **the** minimum variance linear predictor of a c**on**tinu- ous-time, wide-sense-stati**on**ary random process is a classical problem in **the** electrical engineering and ma**the**matics literature. Wiener [l] solved **the** problem of predicti**on** given that **the** infinite past of **the** process is observed. Krein [2] solved **the** problem in principle **for** **the** much harder case of observing **on**ly a finite porti**on** of **the** entire past. A difficulty is that **the** predicti**on** **for**mula is difficult to evaluate in practice. Rozanov [3] gives **for**mulas **for** **the** rati**on**al power-spectrum case, but **the**se still require solving complex differential equati**on**s to obtain certain c**on**stants. Alternatively, **on**e can use **the** complex analysis ap- proach of Yaglom [4], which is similar in computati**on**al diffi- culty. Given that **the** predictors may be difficult to calculate, **on**e may choose to resort to suboptimal ad hoc schemes. If so it is of crucial importance to be able to calculate **the** mean-square error per**for**mance of such a suboptimal predictor and compare it to **the** per**for**mance of **the** optimal **on**e, Cuzick [5] gives a nice overview of **the** problem and derives an upper bound to **the** predicti**on** error that approaches zero at **the** correct rate as **the** “lag” or time interval into **the** future ap- proaches zero. This bound depends **on** an unknown c**on**stant that hampers (as **the** author states) **the** practical utility of **the** results. An idea to be investigated here, **for** **the** small time-lag predic- ti**on** of a c**on**tinuous random process, is **the** use of Taylor series to extrapolate into **the** future from **the** last point observed. If **on**ly m mean-square derivatives of **the** random process are available, **the**n **the** series is truncated after those terms. One would imagine that as **the** time lag goes to zero, this should be a very good estimator of **the** future. This reas**on**ing turns out to be essentially correct **for** a large class of random processes. More interesting (perhaps un**for**tunately), this is an incorrect way of thinking about some random processes. In o**the**r words, Taylor series predictors can be very bad in comparis**on** to optimal predicti**on** even as **the** time lag approaches zero. We will give approximate expressi**on**s **for** **the** mean-squared predicti**on** error **for** small extrapolati**on**s **for** a class of random processes that includes those with rati**on**al power spectra. Much of our intuiti**on** is based up**on** how filtering and predicti**on** works **for** **the** rati**on**al case. As a fur**the**r interesting sidelight, it is dem**on**strated that **the** predictors (and **the**ir error) can depart very widely from those that would be expected based up**on** a rati**on**al approximati**on**. Manuscript received October 25, 1984; revised February 5, 1985. This work, d**on**e while **the** author was visiting **the** Center **for** Stochastic Processes, Depart- ment of Statistics, University of North Carolina, Chapel Hill, NC, was sup- ported by **the** Air Force Office of Scientific Research under C**on**tract AFOSR F49620-82-C-309. The author is with **the** Electrical and Computer Engineering Department, University of Wisc**on**sin, Madis**on**, WI 53706, USA. Notati**on** {x(t) - cc < - T I t < 0) An observati**on** of a wide-sense- stati**on**ary n**on**deterministic zero- mean random process. ,(A’( t) The kth mean-square derivative of x(t) (if it exists). h(r) The minimum mean-squared error predictor of X(T), T 2 0. ;yw - %“(~V > p m.!!er(7). , The power spectrum of **the** ran- dom process x(t); 9-l{F’(A)} = R(t), **the** autocorrelati**on** func- ti**on** **the** random process where 9{.} and g-l(.) denote **the** Fourier and inverse Fourier Trans- **for**m operators, respectively. R(t) =f(t) .f(- t), where f(t) is a causal square integrable functi**on** that is guaranteed to exist by **the** Paley- Wiener **the**orem [6]. II. DEVELOPMENT Suppose F’(X) is rati**on**al and F’(X) f 0; **the**n .@{ f( t)} is also rati**on**al. Then we have F’(X) = i A:( iX)j j=O P c Bpx) j=O where A; and B; are real, and /? 2 (Y + 1, and f A:(iX)j ‘F{ f( t)} = j;O . c B’qiX)’ j=O Let k be such that /3 2 (Y + k + 1. Then **the** Laplace trans**for**m of **the** k th derivative of f(t) is given by (Sk) Iit A;( &)’ j=O SF{ f’k’( t)} = s .i i B’ .(-) j=O ’ 28 By **the** Abelian **the**orem **for** Laplace trans**for**ms we have pk’(0) = sl-n/{ f’k’( t)} s = 0, P>a+k+l However, from [7, p. 5441 we have R’k’(O+) = 0, 2/3>2a+k+2 00189448/85,‘0900-0677$01.00 01985 **IEEE** = 32.)“.‘, P=cr+k+l. ’ 2/3 = 201t k + 1.

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