Adaptive Neural Filters with Fixed Weights

Transcription

1 Adaptive Neural Filters wit Fixed Weigts James T. Lo and Justin Nave Department of Matematics and Statistics University of Maryland Baltimore County Baltimore, MD 150, U.S.A. Abstract By te fundamental neural filtering teorem, a properly trained recursive neural filter wit fixed weigts tat processes only te measurement process generates recursively te conditional expectation of te signal process wit respect to te joint probability distributions of te signal and measurement processes and any uncertain environmental process involved. Tis means tat said recursive neural filter wit fixed weigts as te ability to adapt to te uncertain environmental parameter. Tis ability is called accommodative ability. Tis paper sows tat if te uncertain environmental process is observable (not necessarily constant) from te measurement process, ten te estimate of te signal process generated by said recursive neural filter wit fixed weigts approaces te estimate of te signal process tat would be generated as if te precise value of te uncertain environmental process were given and processed togeter wit te measurement process by a minimal-variance filter. 1 Introduction If te signal and measurement processes involve an uncertain environmental process, an adaptive filter is needed to adapt to te environmental process in processing te measurements to estimate te signals. An adaptive filter usually requires online adjustment of its parameters, wic is difficult or impossible in many applications especially if te precised values of te signals are unavailable. In 199, a very general fundamental teorem on recursive neural filtering was proven [, 6, 7] in a most general formulation of filtering. Te teorem states tat a recursive neural network exists tat inputs te measurement process and outputs an estimate of te signal process, were te estimate can be made as close as desired to te conditional expectation of te signal process given te past istory of te measurement process tat as been processed. It was observed [3, 10] tat as an immediate corollary of tis fundamental teorem, a properly trained recurrent neural network wit fixed weigts can adapt to an environmental process tat is observable from te measurement process. In fact, suc adaptation can be viewed as a manifestation of an estimation of te observable environmental process performed internally inside te recurrent neural network. Tis adaptive capability of recurrent neural networks wit fixed weigts was used for active engine exaust noise control [4] and engine idle speed control and time series prediction [1]. To distinguis adaptive capability to adapt witout online processor adjustment from te ordinary adaptive capability to adapt wit online processor adjustment, te former is called accommodative capability. A neural network wit accommodative capability is called an accommodative neural network. In [5], te accuracy of accommodative neural networks for adaptive identification of dynamical systems was analyzed. It was found tat under mild conditions, if an uncertain environmental parameter is observable from te measurement process or if it is nonobservable but constant, ten a recursive neural network wit fixed weigts exists tat inputs te current dynamical state of a dynamical system and outputs a predicted value of te dynamical state for te next time point, were te predicted value approaces to te predicted value tat could be generated as if te precise value of te uncertain environmental parameter were given and processed togeter wit te current dynamical state. In tis paper, some similar results for accommodative filtering, namely adaptive filtering by recursive neural networks wit fixed weigts, are obtained. Altoug it was discussed in [3, 10, 8] tat an accommodative neural network does not generalize as well as an adaptive neural network (wit long- and sort-term memories), not requiring online adjustment for adaptation is a igly desirable advantage especially wen online training data is not available. In recent years, anoter approac to nonlinear filtering, called particle filtering, as been developed. However, it performs Monte Carlo online and tus involves excessive amount of online computation. Moreover, for adaptive filtering, te particle filter as to augment te signal process to include te environmental process and

2 estimate it online, wic increases te dimensionality of te signal process and tus requires even more particles and online computation. In contrast, an accommodative neural filter is syntesized prior to its deployment and functions recursively muc like te Kalman filter does for linear signal and measurement processes [9]. Te syntesis of te adaptive neural filter can be viewed as Monte Carlo, but no Monte Carlo is needed to run te filter online. How well an environmental process can be adapted to by an adaptive processor depends on ow muc information te input to te adaptive processor contains about te environmental process. Obviously, te best performance tat te adaptive processor can acieve is te performance acievable given te precise value of te environmental process. If an adaptive processor exists wose performance converges to tis best performance acievable given te value of te environmental process, te environmental process is said to be adaptation-fit forte processing involved. A precise statement of tis definition of adaptation-fitness for filtering is given in Section 3aftertefiltering problem is stated in Section. It is proven in Section 3 tat tere exists an accommodative neural filter wit a filtering accuracy allowed by te degree of adaptation-fitness of te environmental process. Te proof is based on te fundamental neural filtering teorem[,6,7]. Observability of a stocastic environmental process from a measuremet process is defined in Section 4. As expected, te degree of adaptation-fitness of te environmental process depends on te degree of its observability as well as te smootness of te conditional expectation of te signal process wit respect to te environmental process. Te main teorem in Section 4 gives an intuitively appealing quantification of te dependence. A numerical example is given in Section 5. In te example, te signal process is a Henon system wit an observable environmental parameter, and te measurement process is te cubic sensor. Te Monte Carlo test results are consistent wit te teoretical results discussed above. Problem of Adaptive Filtering A signal process to be estimated is described by te vector equation: For t =0, 1,..., x(t, θ t ) = f(x t 1 t p (θ),θ t,w t ) (1) x t 1 t p(θ) : = (x (t 1,θ t 1 ),...,x(t p, θ t p )) () wit te initial condition (or state), (x(0,θ),...,x(1 p, θ)) = (x 0,...,x 1 p ), (3) were te vector-valued function f and te integer p are given; x(t) is a known input vector at time t; w is a random vector sequence wit given joint probability distributions; θ (t) denotes te vector-valued uncertain environmental parameter at time t; and te initial state, (x 0,...,x 1 p ), is a random vector wit given probability distribution (reflecting te relative frequencies of te actual initial states of te system, (1), in operations). A measurement y (t) of te vector output x (t) is made available at time t, tat satisfies y(t, θ t )= x t t q+1(θ),θ t,v(t), (4) were v is a random vector sequence wit given joint probability distributions. Te problem of adaptive filtering is to design and implement an adaptive recursive processor tat adapts to te uncertain operating environment represented by te equations, (1), (3) and (4), and produce an estimate of x (t, θ t ) tat makes a best use of te information about θ contained in te measurements y. During te operation of suc an adaptive processor, no part of te signal process is directly measurable (i.e., signal plus noise unavailable) and at time t, only te measurement, y (t, θ t ), is available for processing by te adaptive processor. Te adaptive processor inputs y(t, θ t ) and outputs an estimate ˆx (t, θ t ) of x (t, θ t ) at eac time t =1,,,T,wereT is a positive integer or infinity. Te most widely used estimation error criterion is te conditional mean square error criterion, kx (t, θ t ) ˆx (t)k y t (θ) i, were y t (θ) := {y (s, θ s ),s=1,...,t}, ˆx (t) denotes an estimate of x (t, θ t ) given y t (θ), [ y t (θ)] is te conditional expectation given y t. An optimal estimate wit tis criterion is [x (t, θ t ) y t (θ)], wic is called te minimum-variance estimate. For notational simplicity, x (t, θ t ), y(t, θ t ), x t s(θ) and y t (θ) will be denoted by x(t), y(t), x t s and y t respectively in te sequel. 3 Accommodating an Adaptation- Fit nvironmental Process Successful adaptation to an uncertain environmental processrequiressufficient information about it. If sufficient information about te environmental process is contained in te measurement process suc tat an adaptive filter exists tat generates an estimate of te signal process tat approaces tat acievable wit a filter given te precise value of te unvironmental process, te environmental process is called adaptation-fit forfiltering. A rigorous definition of an adaptation-fit environmental process for filtering is stated as Definition 1 below. In te main teorem of tis section, it is proven tat if te environmental process is adaptation-fit forfiltering to a certain accuracy, adaptive filtering to te same accuracy can be realized

3 wit a recurrent neural network wit fixed weigts. Suc a recurrent neural network does not require online weigt adjustment and is called an accommodative neural filter. Definition 1. Letx, y and θ be signal, measurement and environmental processes respectively, and let ˆx (t) and ˆx (t θ) denote te conditional expectations [x (t) y t ] and x (t) y t,θ t,wereθ t := {θ τ,τ =1,...,t} and y t := {y τ,τ =1,...,t}. Assume tat θ is a stocastic process, wic is not necessarily time-varying. Te environmental process θ is said to be adaptation-fit towitinanerrorof ε for filtering, if tere is a positive integer N (ε) suc tat for all t>n(ε), kˆx (t θ) ˆx (t)k i <ε We need te fundamental neural filtering teorem [, 6, 7] to prove te main teorem of tis section. Teorem 1. Consider an n-dimensional stocastic process x(t) and an m-dimensional stocastic process y(t), t =1,,T defined on a probability space (Ω, A,P). Assume tat te range {y(t, ω) t =1,,T,ω Ω} R m is compact and ψ is an arbitrary k-dimensional Borel function of x(t) wit finite second moments [kψ(x(t))k ], t =1,,T.Let α(t) denote te k-dimensional output at time t of a recurrent neural network wic as taken te inputs, y(1),,y(t), integivenorder. 1. Given >0, tere exists a recurrent neural network wit one idden layer of fully interconnected neurons suc tat 1 T TX [kα(t) [ψ(x(t)) y t ]k ] <. t=1. If te recurrent neural network as one idden layer of N neurons, wic are fully interconnected, and te output α(t) is written as α(t; N) ere to indicate its dependency on N, ten r(n) :=min w 1 T TX [kα(t, N) [ψ(x(t)) y t ]k ] (5) t=1 is monotone decreasing and converges to 0 as N approaces infinity. Te following corollary follows immediately te above fundamental neural filtering teorem. Corollary 1. Consider an n-dimensional stocastic process x(t) and an m-dimensional stocastic process y(t), t =1,,T defined on a probability space (Ω, A,P). Assume tat te range {y(t, ω) t =1,,T,ω Ω} R m is compact and ψ is an arbitrary k-dimensional Borel function of x(t) wit finite second moments [kψ(x(t))k ], t =1,,T.Letα(t) denote te k-dimensional output at time t of a recurrent neural network wic as taken te inputs, y(1),,y(t), integivenorder. Given >0, tere exists a recurrent neural network wit one idden layer of fully interconnected neurons suc tat for all t =1,...,T, [kα(t) [ψ(x(t)) y t ]k ] <. We are now ready to state te first main teorem of tis paper: Teorem. Let x, y and θ be signal, measurement and environmental processes respectively, were θ is adaptation-fit towitinanerrorofε for filtering. For filtering over a time interval, 1 t T,afixed-weigt MLPWIN, tat as only one idden layer of neurons; inputs y(t) and outputs an estimate α (t) of te signal x(t, θ t ) at time t, existsasanadaptivefilter suc tat kˆx(t θ) α (t)k i <ε for all t less tan T and greater tan some positive integer N (ε), wereˆx(t θ) denotes te conditional expectation x (t) y t,θ t. Proof. By Definition 1, for te given ε, tere is a positive integer N (ε) suc tat for all t>n(ε), kˆx (t θ) ˆx (t)k i <ε (6) Let ε 1 be a positive number less tan ε kˆx (t θ) ˆx (t)k i > 0. Notice tat bot ˆx (t) and α (t) aremeasurablewitrespecttoy t. It follows tat (ˆx (t θ) ˆx (t)) T (ˆx (t) α (t)) y ti = (ˆx (t θ) ˆx (t)) T y ti (ˆx (t) α (t)) = ˆx T (t θ) y t ˆx T (t) ³ˆx ³ (t θ) ˆx t ˆθ = ˆx T (t) ˆx T (t) ³ˆx ³ (t θ) ˆx t ˆθ =0 (7) By te smooting property of te conditional expectation, we ave, for t>n(ε), kˆx (t θ) α (t)k i = k(ˆx (t θ) ˆx (t)) + (ˆx (t) α (t))k i = kˆx (t θ) ˆx (t)k i + kˆx (t) α (t)k i + (ˆx (t θ) ˆx (t)) T (ˆx (t) α (t)) y tii = kˆx (t θ) ˆx (t)k i + kˆx (t) α (t)k i <ε+ kˆx (t) α (t)k i

4 were te last equality and last inequality follow from (7) and (6) respectively. By Corollary 1, for any ε 1 > 0, tere is a recurrent neural network wose output α (t) satisfies kˆx (t) α (t)k i <ε 1 for all t =1,...,T. It follows tat kˆx (t θ) α (t)k i <ε, wic completes te proof. 4 Observability Implies Adaptation-Fitness Observability is a well-developed concept in te statespace system teory. However, it is defined only for deterministic systems. For our purpose, we adopt te following definition of an observable environmental process for a stocastic system: Definition. Let an environmental process be a stocastic process and let te conditional expectation [θ t y t ] of θ t given y t := {y τ,τ =1,...,t} be denoted by ˆθ t. Ten te environmental process θ is said to be observable from te measurement process y to witin an error of ε>0, if tere is a positive integer N (ε) suc tat for all t>n(ε), θt ˆθ t <ε Teorem 3. If te environmental process θ is observable from te measurement process y to witin an error of ε, and if te uclidean norm kd [x (t) y t,θ t ] /dθ t k <M uniformly for all y t and θ t,wered [x (t) y t,θ t ] /dθ t is te derivative of te conditional expectation [x (t) y t,θ t ] wit respect to θ t, ten te environmental process is adaptation-fit to witin an error of M ε for filtering. Proof. It follows from (1) tat ˆx (t θ) := x (t) y t,θ t = [x (t) y t,θ t ]. Hence ˆx (t θ) is a Borel measurable function of y t and θ t. Denoting tis function by g (y t,θ t ) and substituting ˆθ t for θ t in it yield g ³y t, t ˆθ,wicismeasurable wit respect to y t.notetat ³ˆx (ˆx (t θ) ˆx (t)) T (t) g ³y t, t ˆθ ³ˆx = (ˆx (t θ) ˆx (t)) T (t) g ³y t, t ˆθ y tii = (ˆx (t θ) ˆx (t)) T y ti³ˆx (t) g ³y t, t i ˆθ = ˆx T (t θ) y t ˆx T (t) ³ˆx (t) g ³y t, t i ˆθ = ˆx T (t) ˆx T (t) ³ˆx (t) g ³y t, t i ˆθ =0 Tis sows tat te cross term resulting from expanding te second expression below is zero and establises te second equality below. ˆx (t θ) g ³y t, t ˆθ ³ = (ˆx (t θ) ˆx (t)) + ˆx (t) g ³y t, t ˆθ ³y t, t ˆθ = kˆx (t θ) ˆx (t)k + ˆx (t) g Hence, kˆx (t θ) ˆx (t)k ˆx t θ t g ³y t, t ˆθ = g y t ³,θ t g y t, t ˆθ By te mean value teorem, g y t,θ t g ³y t, t ˆθ ³ dg y t, θ θt /dθ t ˆθ t M θ t ˆθ t. Terefore, kˆx (t θ) ˆx (t)k i θt M ˆθ t M ε,forsomen (ε), completing te proof. Remark. ε in Teorem 3 reflects te predictability of θ, and M reflects te smootness of te conditional expectation [x (t) y t,θ t ] wit respect to θ t. Teorem 3 only provides a quantification of our intuition tat adaptationfitness increases as te predictability of θ and te smootness of te conditional expectation [x (t) y t,θ t ] wit respect to θ t increase. 5 An Numerical xample In te numerical example, te signal process is te wellknown Henon system: x (t +1) = bx (t 1) + 1 θx (t)+0.1w(t) y (t) = x 3 (t)+0.1v(t) were w and v are wite sequences wit w (t) and v (t) aving normal distributions wit mean 0 and variance 1 and wit any samples greater tan or equal to 3 in absolute value discarded; b =0.3;andθ [0, 0.5] is te uncertain environmental parameter. Te training dataset consisted of 1000 realizations of x and y tat were 30 time points long including 30 for priming. Te teta values were cosen randomly from te set {0.1, 0., 0.3, 0.4, 0.5}. Starting values for x (0) and x ( 1) were selected from [0.5, 0.9]. Anoter dataset was generated similarly and used for cross-validation during training. Te weigts tat produced te minimum RMS for te cross-validation data set were used. Training was <

5 preformed on various arcitectures, but a recurrent neural network wit a single idden layer of 15 neurons was selected as te accommodative neural filter. To see weter and ow te output of te accommodative neural filter approaces te conditional expectation of x (t) given bot y t and θ, a separate MLPWIN was similarly trained for eac given teta value from te set { , 0.5, 0.35, 0.45}. A testing data set was generated wit 500 streams also 30 time points long. It contained 100 streams for eac of te teta values in te set { }. Te RMS of te accommodative neural filter on tis test data set is Te RMS of te MLPWINs trained for te exact teta values is Te standard deviation of te signal is Tese numbers sow tat te accommodative neural filter performed satisfactorily. Attaced are two plots. Figure 1 sows te RMS vs. time averaged over all te streams in te testing data set. Figure sows one realization of te signal process at θ = 0.5 and its estimate generated by te accommodative neural filter over 50 time points. 6 Conclusion Te adaptive capability of recurrent neural networks wit fixed weigts was observed as a consequence of te fundamental neural filtering teorem in 1994 [3, 10]. Tis capability is called accommodative capability. Tis paper defines adaptation-fitness and observability of an environmental process, and proves tat if te environmental process is adaptation-fit, a recurrent neural network wit fixed weigts exists wose filtering performance approaces te performance tat would be acievable as if te environmental process were given, and sows tat observability of an environmental process implies it being adaptation-fit. Altoug recurrent neural networks do not generalize as well as adaptive neural networks (wit long- and sortterm memories), te former ave te unique advantage of not requiring online adjustment. Tis advantage is important in many applications, especially wen no data is available online for adjustment of te parameters or weigts of a processor (e.g., filter or controller) to adapt to an uncertain environmental process. [] J. T. Lo. Neural network approac to optimal filtering. Invited paper presented at te 199 World Congress of Nonlinear Analysts, Tampa, Florida, Aug [3] J. T. Lo. Neural network approac to optimal filtering. Tecnical Report RL-TR , Rome Laboratory, Air Force Material Command, [4] J. T. Lo. Neural network approac to active engine noise cancellation. Tecnical report, Maryland Tecnology Corporation, [5] J. T. Lo. Matematical underpinning of adaptive capability of recurrent neural networks wit fixed weigts. In Proceedings of te 003 International Joint Conference on Neural Networks, pages , Portland, Oregon, July 003. [6] J. T. Lo. Syntetic approac to optimal filtering. In Proceedings of te 199 International Simulation Tecnology Conference and 199 Worksop on Neural Networks, pages , Clear Lake, Texas, November 199. [7] J. T. Lo. Syntetic approac to optimal filtering. I Transactions on Neural Networks, 5: , September [8] J. T. Lo and D. Bassu. Adaptive versus accommodative neural networks for adaptive identification of dynamic systems. In Proceedings of te 001 International Joint Conference on Neural Networks, Wasington, D.C., July, 001. [9] J. T. Lo and L. Yu. Recursive neural filters and dynamical range transformers. Proceedings of Te I, 9, No. 3: , 004. [10] J. T. Lo and L. Yu. Adaptive neural filtering by using te innovations process. Proceedings of te 1995 World Congress on Neural Networks, II:9 35, July References [1] L. A. Feldkamp and G. V. Puskorious. Training of robust neural controllers. In Proceedings of te 33rd Conference on Decision and Control, pages , Lake Buena Vista, Florida, December 1994.

6 Figure 1: Te RMSs of te estimates generated by te accommodative neural filter compared wit tose generated using given teta values. Figure : A single realization of te signal process compared wit its estimate generated by te accommodative neural filter.