Classification of stellar spectral data based on Kalman filter and RBF neural networks

K e y w o r d s : Kalman Filter; Principal Component Analysis; Stellar Spectral Data; RBF Neural Networks. 1 I n t r o d u c t i o n Current and future large astronomical surveys will yield multiparameter databases on millions or even billions of objects. The scientific exploitation of these will require powerful, robust, and automated classification tools tailored to the specific survey. Partly motivated by this, the past five to ten years has seen a significant increase in the amount of work focused on automated classification and its application to astronomical data. At its most general level, the objective of classification is to identify similarities and differences between objects, with the goal of efficiently reducing the number of types of objects one has to deal with. In the case of stellar astrophysics, physical stellar parameters show continuous distributions, so some researchers often think it is appropriate to parameterize spectra on continuous parameter scales rather than classify them into discrete classes. Recognition some specific class stellar spectra becomes more difficult task in the continuous distribution parameter space. Recognition usually adopts classification techniques, traditional method is * 0 7 8 0 3 7 9 5 2 7 / 0 3 / $ 1 7 . 0 0 c 2 0 0 3 I E E E . matching spectra to be recognized to the known model spectra, accordingly, divides the spectrum into respective classes. Almost all of the recent work on automated stellar classification has used one of four techniques: principal component analysis (PCA); neural networks; minimum distance methods and Gaussian probabilistic models[2]. Stellar spectra are extremely noisy and voluminous. Consequently, the acceptable method of classification must be both computationally efficient and robust to noise. As we know, back propagation (BP) neural network has already had many applications on astronomical data reduction[6]. However, there are two main drawbacks in BP neural network, one is its slow convergence rate, and the other is that it sometimes falls into local minimum instead of global minimum. The slow convergence rate will pose a computational problem, and local minima will degrade the classification accuracy. To overcome these two drawbacks, one of methods is to use radial basis function (RBF) neural network[10]. It is in virtue of RBF model's non-linearity and localization, recognition is implemented through building nonlinear relationship between feature spectra and classes. To acquire better performance in both the training time and the recognition accuracy, we adopt the RBF network in this work. For stellar spectral data, the number of variables (wavelengths) is much higher than the number of training samples, it is said that stellar data are severely ill-posed. Due do such high dimensionality, the common multivariate classification methods of linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA) cannot be directly applied because of the matrix singularity problem[I]. Principle component analysis technique[7] is usually used directly for such illconditioned data by extracting the latent variables. The number of latent variables is lower than the number of training samples[9]. Another possible approach to solve the singularity problem is to use Kalman filter[12, 13]. The Kalman filter is a recursive, point-by-point, leastsquares fitting technique[8]. The parameters of the model are calculated iteratively, and no matrix inversion operator is necessary in this technique. When the number of variables is higher than the number of objects, the Kalman filter still yields a solution, but with a higher risk of overfitting. It builds a linear model between an independent matrix X and a dependent matrix Y, Y = X • B + Error, (1) and the B matrix is estimated by least-squares fitting techniques. In order to get higher classification, we propose to cascade Kalman filter and RBF network to form a new classifier. As a data pre-processor, Kalman filter can reduce both the noise and the dimensionality of stellar data. Then RBF neural network is employed as a classifier to recognize stellar data. The rest part of this paper is organized as follows: Section 2 briefly review the Kalman filter and RBF netral network structure. Section 3 describes experimental investigating the stellar spectral data classification problem by proposed composite classifier. Finally, conclusions of the paper are given in Section 4. 2 B a c k g r o u n d 2.1 Kalman filter The Kalman filter was first created by R.E. Kalman to process data for problems in satellite orbit mechanics[8]. In essence, it is a recursive linear parameter estimation filter, or a least-squares fitting technique. It has been used in calibration, multi-component analysis, kinetic analysis, outlier detection and so on[9]. Assuming that x(k) is the measuring value at the time of k, then ~)(k) is the estimate value at the time of k. The Kalman recursion filter for one dimension is shown below[14]. ~)(k) = afl(k 1) + b(k)[x(k) ac~)(k 1)] (2) in which the filter gaining equation is: c [ a 2 P ( k 1 ) + a 2] (3) b(k) = c[a2p( k _ 1) + a 2] + or2 and the filter deviation equation is: