D- Ai5S 762 SIEVES ND FILTER FOR AUSSI N PROCESSES (U WISCONSIN 1/1 UNIY-NILUAUKEE J SEDER 25 APR 85 AFOSR-TR-85-062i I IEEE". RFOSR-84-9329 UNCLASSIFIED F/G 12/1 NL
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AFOSii- 7 85-0 6 2 1 Retearch Progress and Forecast Report: April 25, 1985 Sieves and a Filter for Gaussian Processes: AFOSP-84-0329 Results on Conjecture 1. We use the definitions and notation of [1]. Let H be the subspace of 2, L.,,p spanned b 'X, tej, and let H = H(R,T', be the corresponding reproducing kernel Hilbert space. Assume that H has countable dimension, with CON basis :Ui. op. Fix a = {ak}, and consider a sample of size n. Proposition 1. Under P a' the Uk are independent, and Uk has the n(a k, 1/n) distribution. L" We consider the sieve estimator defined in [I] p. 7. Then for each m and for a sample of size n, the sieve estimator of the mean g(t) is (1) - I,' g(t) = Z kg k ( t ) l where,gki is the CON basis of H corresponding to {Ukl. We wish to pick m as a function of n so that Ili-gll -* 0 (norm in H). But ig= -a 2 (norm in 2 ) = a 2 + 2 a H) Bt ksm fk-ak k>m ak = X n m k' ' say. The non-stochastic tail vanishes as-n if we require m By Proposition 1, nx is x 2 (m), and the following weak result describes C) nm the limitations we will need to impose on m = mn. Proposition 2. If m/n - 6, then X - 0 in P -probability iff B=O. * I.. ni a C.3 The Borel-Cantelli Lemma gives a sufficient condition for 9 a (X-0) Xnm - = 1: Proposition 3. Let {m n } be chosen so that m - nn sequence {cn} defined by and such that the D T IC.-1 u ELECTER% c,, = n 1 u-m) e - u du AU G 2 919 5 Lq is summable for every c > 0. Then P a (Xnm4O) :1I. G as If m is chosen so that it is always even, then we may rewrite c n n cn e e,_,(an), f o.,. o".......,........... -..".. -....,..............., - ', '-. -.. "* =-.', * 5. "- ". -. "-""- ""S.
* - - - Unclassified SECURITY C.LASSIFICAT ION OF-THIS PAGE I&la REPORT SECURITY CLASSIFICATION REPORT DOCUMENTATION PAGE lb. RESTRICTIVE MARKINGS Unclassified 2a. SECURITY CLASSIFICATION AUTHORITY 3. DISTRIBUTION/AVAILASILITY OF REPORT Approved for public release; distribution * 2b. DECLASSI F ICATI ON/OOWNGRA DING SCHEDULE unlimited N/A * 4. PERFORMING ORGANIZATION REPORT NUMBER(S) S. MONITORING ORGANIZATION REPORT NUMBER(S) A1'SR-TR 85-0 62 1 Ga. NAME OF PERFORMING ORGANIZATION b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION University of Wisconsin-Milwa~kee AF05R 6c. ADDRESS (City. State and ZIP Code) 7b. ADDRESS (City. State an~d ZIP Code) P. 0. Box 413 Milwaukee, WI 53201 Bldg. 410 Bolling AFB, D.C. 20332-6448 go8. NAME OF FUNDING/SPONSORING 8 b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER *ORGANIZATION itplcbe *AFQSR NM AFOSR-84-0329 & c. ADDRESS (City. State and ZIP Code) 10. SOURCE OF FUNDING NOS. * Bldg. 410 PROGRAM PROJECT TASK WORK UNIT Bolling AFB, D.C. 20332-6448 ELEMENT NO. NO. NO. NO. 11. TITLE (Include Security Classificain) Sieves and a Filter for Gaussian ProcessesI _ 12. PERSONAL AUTHOR(S) 13a. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT lyr.. M~o.. Day) 15. PAGE COUNT Tn~prim IFROM TO _ 25_pri_195_ 1S. SUPPLEMENTARY NOTATION 17. COSATI CODES IS. SUBJECT TERMS (Con tinue on reverse if necessary and identify by block number) IED GROUP sub. GR. Gaussian Processes 19. ABST RACT (Contintue on reverse if necessary and identify by block number) /Results are reported on the first two conjectures that were to be investigated as described in the research proposal. Conjecture 1 has been established and follow-on results are obtained. What remains to be investigated is the use of these results to make confidence statements and to test hypotheses. Results which establish the truth of conjecture 2 are also reported. 17' -6J 20. DISTRl BUTIONIA VAILASILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATION UNCLASSIPIGD/UNLIMITIED 3SAME AS RPT. MOTIC USERS 03 Unclassified * 22&. NAME OFP- 9'-ONSIBLE INDIVIDUAL 22b. TELEPHONE NUMBER 22c. OFF ICE SYMBOL (Include Are Code) BinW. Woodruff, Maj, USAF (202)767-5027 NM 00 FORM 1473,83 APR E1DITION OF I JAN 73 IS OBSOLETE. Uncassified SECURITY CLASSIFICATION OF THIS PAGEa... %'%...
-2- n k where a - c/ 2, n - 29, and en (x)- E X Invert -t(n), say n -(l). n kno I-01; Now we must choose A(Z) so that at least X/1 - and so that ee (ca) is. summable for every m > 0. Write el e (cax) + A Using results of Buckholtz (2], we may prove the following: Theorem 1. Let X = X(9.) be such that lk/ -k as 2. -. Then for each aci(o,-) we have A.(a) + +- +0( 9) t From this and Stirling's formula we have Corollary 1. If X/ - as. -, then for each cic(o,) the sequence e a e 2 (cx) is summable. Translating back, we get the following consistency result. Theorem 2. If m = 2.(n) = o(n) and m - -, then a - a a.s. At first this holds a.s. P 8. but since the measures in P are equivalent, we may assert a.s. convergence without qualification. In any case, this settles Conjecture 1. In particular, the condition m = o(n), which was sufficient for weak convergence, also gives strong convergence, and since m/n - 0 > 0 does not even give weak convergence, our condition is in some sense best possible. Moreover, a result similar to Theorem 1 allows us to see just what does hold almost surely if we only require m/n 0 > 0:
". -3- Theorem 3. If m/n - > 0, then for all c sufficiently large (depending only on 0) we have Pa (Xnm C i.o.) = 0 The distribution of the estimator g given by (1) is easy to describe. Consider g(t) as a stochastic process. From Proposition 1 we have the following. Theorem 4. Under P, the process {i(t), tct} is Gaussian with mean m m function I ag(t) and covariance function - E g.(s)g.(t). i=laigi g=11 i t For comparison we note that the reproducing kernel R(s,t) of H may be written R(s,t) - gi(s)gt(t) (as long as {gi } is CON in H), and that under Pa the true mean function is E aigt(t). What needs to a i=l be investigated now is the use of Theorem 4 to make confidence statements and to test hypotheses. Results on Conjecture 2. The set-up here is given in [l], pp. 7-8. Let us * fix a countably infinite orthonormal set {gkl (not necessarily complete) in H, and consider the subset PO c P consisting of measures corresponding to covariances of from S(s,t) = R(st) + Zlkgk(s)gk(t), where i = {l is in the subset e 2 of 93. Now the likelihood function depends only on Let UkcH correspond to 9k as usual, and let A = {wcfl: Uk(cO)to for every k. Certainly A is an event in A. Theorem 5. Q(A) = 1 for every QcP, and for each outcome wn the corresponding likelihood is unbounded over P 0. Thus Conjecture 2 is established.
References [1] Jay H. Seder. Sieves and a filter for Gaussian processes, AFOSR grant proposal. [2) J. D. Buckholtz. Concerning an approximation of Copson, Proc. Amer. Math. Soc. 14 (1961), 564-568. Accessi~on For "SGRA&I DTIC TAB Lz "'n Ounee 0 d Justiticati 0 Dy JAvai~j and/or 'c
FILMED 10-85 DTIC