Fractal functional filtering ad regularization

Transcription

1 Fractal functional filtering ad regularization R. Fernández-Pascual 1 and M.D. Ruiz-Medina 2 1 Department of Statistics and Operation Research, University of Jaén Campus Las Lagunillas Jaén, Spain ( rpascual@ujaen.es) 2 Department of Statistics and Operation Research, University of Granada Facultad de Ciencias Granada, Spain ( mruiz@ugr.es) Abstract. Pseudodifferential models have been widely applied in the description of fractal biological systems. This paper studies the filtering and prediction problems associated with functional data defined from such models. Specifically, functional data are given here in terms of the functional process of interest (e.g. biological cell structural changes and movements), solution to a pseudodifferential evolution model, plus a functional additive fractal noise. The functional filtering and extrapolation problems associated must be regularized to remove the ill-posed nature of such problems. In this paper, the optimal functional geometry for regularization is selected according to the local singularity properties of the fractal observation noise and the process of interest. A simulation study is developed to illustrate the regularization methodology proposed. Keywords: Fractality, functional spatio-temporal estimation, pseudodifferential evolution models. 1 Introduction Fractal pseudodifferential models are usually considered for representation of anomalous diffusion processes (see Ruiz-Medina et al. [4] and Ruiz-Medina and Fernández-Pascual [6]). In this paper we consider the pseudodifferential evolution model Y t (t,x) + L xy (t,x) = ε(t,x), (t,x) R + D, (1) Y (0,x) 0, (2) where ε represents spatio-temporal white noise, and L x represents a fractal pseudodifferential operator of order γ > 0. For example, L x can reflect the fractal nature of neurology structures. The functional data are given by Z(t,x) = Y (t,x) + N(t,x), x D OBS Z, (3)

2 2 Fernández-Pascual and Ruiz-Medina where N represents spatio-temporal fractal noise uncorrelated with Y and DZ OBS denotes the set of observable locations. The following notation will be considered: The random initial condition Y 0 takes its values in the space H Y0 = H α (D), for a certain α > 0, which is considered to be an element of the fractional Sobolev space scale on D. The realizations of Y belongs to a functional space H Y which is considered to be an element of the fractional Sobolev space scale on D. Thus, H Y = H s (D), for certain α R +. The realizations of functional noise N lie on a fractional Sobolev space H N = H β (D), for a certain β > 0. Thus, processes Y and N can display fractality, for certain ranges of parameters α and β. The functional filtering and prediction problems are solved by inversion of the associated Wiener-Hopf equation, which in this case, involves an unbounded inverse observation covariance operator, associated with the dependence structure of the functional fractal data. We propose to formulate the functional estimation problem in a functional space (fractional Sobolev space) with suitable geometry to get a bounded inversion. In this paper, we consider the weaker topology, associated with parameter s β, to remove the ill-posed nature of the problem by regularization in terms of an interpolating geometry between the ones defined on H Y and H N (see Fernández-Pascual et al. [2] and Ruiz-Medina and Fernández-Pascual [6]. 2 Least-squares functional estimator of fractal signals The functional normal equations defining the functional least-squares predictor, and filtering estimator of the process of interest Y are derived from the identity ] E [[Y (t,x) Ŷ (t,x)]z(s,y) = 0, y D OBS, s [0, T], (4) where Ŷ (t,x) = L t,xz(t,x), for x D,t R +. The filtering L t,x defining the functional estimator Ŷ then satisfies the following equation R ZZ = L t,x R Y Z, (5) where R ZZ is the integral covariance operator of Z, with kernel the covariance function of Z, and R Y Z is the integral cross-covariance operator, with kernel the cross covariance function between Y and Z. By inversion of the above equation, the formal expression of Ŷ is obtained from L t,x = R Y Z R 1 ZZ. (6)

3 Fractal functional filtering ad regularization 3 Thus, the functional least-squares linear estimator of Y is defined as Ŷ (t,x) = L t,x Z(t,x) = [R Y Z R 1 ZZY ](t,x). (7) Since L t,x is defined between the spaces H Z and H Y associated with the observed process Z and the interest process Y, its stable computation depends on the parameters s and β involved in the definition of the process of interest and the noise process. A suitable topology for its stable computation can be found when s β > 0. This condition means that the fractional topologies associated with parameter s and β must be different to define a regularizer topology. Indeed, the norm of R 1 ZZ is upper bounded by s β 1. Thus, the stability of the functional estimation problem increases when s β increases. 3 Simulations: Influence of smoothness of functional data One of the main drawback of functional statistical computational techniques is the dependence of such techniques on the discretization level of the functional data. In this section, we illustrate this fact in the case of sequences of fractal spatial functional data. The density of the spatial locations defining our spatial functional data sequence is fundamental in the empirical approximation of the fractality parameter β characterizing our functional observation model. Therefore, in the regularization of the functional filtering and prediction problems, the parameter ranges considered are affected by the smoothness of our functional data, which depends on the density of our spatial grid. In the simulation study developed in this section, we illustrate this fact, regarding quality of the functional estimates depending on the fractality of the noise, and on the self-similarity and long-range dependence of the random initial condition (see Ruiz-Medina and Fernández-Pascual [6]). Let us consider the evolution model Y t (t,x) = Y (t,x), (t,x) R + D R + R 2, (8) with Gaussian random initial condition Y 0 having covariance operator r Y0 r Y0, with denoting convolution, and r Y0 being defined as r Y0 (C, θ,x y) = 1 (C + x y ) 2 θ. Here, as usual, denotes the Laplacian operator. The additive noise N is a Gaussian temporal white noise with spatial covariance operator R NN = ( ) β. The parameter ranges studied are 1/C (0,0.5], θ (0,0.5), and β (1,2). The influences of the fractality of the functional observation noise, parameter β, of the self-similarity, parameter C of the random initial condition,

4 4 Fernández-Pascual and Ruiz-Medina and of the spatial dependence range, parameter θ of the random initial condition, on the stability of the functional estimation method are investigated in the simulation study developed in this section. Specifically, for fixed values θ = 0.1 and C = 1/2 of the structural dependence parameters of the random initial condition, the following decreasing values of parameter β, β = 1.8,1.5,1,2 are considered to illustrate that the quality of the estimates increases when β decreases for a fixed s, which depends on the fixed parameters θ and C involved in the definition of the random initial condition. This fact can be appreciated in Figures 1 and 2. On the other hand, for θ = 0.1 and β = 1.8, fixed, decreasing parameter values of C, the quality of the estimates decreases. This fact is illustrated with the parameter values C = 0.5,0.4,0.3,0.2 (see Figures 3 and 4). Finally, for C = 1/2 and β = 1.2, the effect of increasing the parameter values of θ, that is, the effect of increasing the spatial dependence range of the random initial condition is studied, considering the parameter values θ = 0.1,0.3,0.45. In Figures 5 and 6, it can be appreciated that the quality of functional estimates decreases when θ increases. In the scenarios described above, a clearer effect of the fractality of the noise, and self-similarity and strong dependence of the random initial condition on the quality of the functional estimates can be appreciated increasing the density of the spatial grid defining the elements of the functional data sequence. 4 Final Comments The functional estimation of a fractal biological signal can be performed, in a stable way, when the structural and noise local singularities are controlled, and the discretization level defining the functional data sequence available allows a clear discrimination between them. Additionally, the spatial dependence range and the self-similarity characterizing the initial functional state of the signal of interest can destroy the stability of the functional filter defining its approximation. This fact will be be more pronounced when the mesh density increases. 5 Acknowledgments This work has been supported in part by projects MTM of the DGI, MEC and P06-FQM of CICE. 6 Bibliography 1 Bosq, D. (2000). Linear Processes in Function Spaces. Springer-Verlag, New York.

5 Fractal functional filtering ad regularization 5 Fig. 1: S1. CASE C (θ = 0.1, C = 1/2, and β = 1.2). Spatial realizations Y 3 (top) and functional estimations Ŷ3, (bottom) of the process of interest. Fig. 2: S1. Averaged Functional Quadratic Errors in cases β = 1.8 (left), β = 1.5, (center) and β = 1.2 (right) for the three grids considered. 2 Fernandez-Pascual, R., Ruiz-Medina, M.D. and Angulo, J.M. (2006). Estimation of intrinsic processes affected by additive fractal noise. Journal of Multivariate Analysis 97, Kato, T. (1995). Perturbation Theory of Linear Operators. Springer- Verlag, Berlin. 4 Ruiz-Medina, M.D., Angulo, J.M. and Fernández-Pascual (2007). Waveletvaguelette decomposition of spatiotemporal random fields. Stochastic Environmental Research and Risk Assessment 21, Ruiz-Medina, M.D. and Angulo, J.M. and Anh, V.V. (2008). Multifractality in Space-Time Statistical Models. Stoch. Environm. Res. Risk Assess. 22,

6 6 Fernández-Pascual and Ruiz-Medina Fig. 3: S2. CASE A (θ = 0.1, C = 0.5, and β = 1.8). Spatial realizations Y 3 (top) and functional estimations Ŷ3, (bottom) of the process of interest. Fig. 4: S2. Averaged Functional Quadratic Errors in cases C = 0.5 (left), C = 0.3 (center) and C = 0.2 (right) for the three grids considered.

7 Fractal functional filtering ad regularization 7 Fig. 5: S3. CASE C (θ = 0.45, C = 1/2, and β = 1.2). Spatial realizations Y 3 (top) and functional estimations Ŷ3, (bottom) of the process of interest. Fig. 6: S2. Averaged Functional Quadratic Errors in cases θ = 0.1 (left), θ = 0.3 (center) and θ = 0.45 (right) for the three grids considered. 6 M.D. Ruiz-Medina and R. Fernández-Pascual (2009). Spatiotemporal filtering from fractal spatial functional data sequence (sumitted to special issue of SERRA associated with METMA4 conference, in process).