Noise Reduction in Time Series using F-transform Michal Holˇ capek and Vil´ em Nov´ ak and Irina Perfilieva Centre of Excellence IT4Innovations Division of University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, 701 03 Ostrava 1, Czech Republic Email: Michal.Holcapek@osu.cz, Vilem.Novak@osu.cz, Irina.Perfilieva@osu.cz Abstract—In this paper, we will focus on the application of fuzzy transform (F-transform) in the analysis of time series. We assume that the time series is decomposed into two constituents: the trend-cycle and random noise. We will demonstrate that using the F-transform we can reduce the variability of random noise which consequence is an extraction of the trend-cycle. Keywords—Fuzzy transform, random process, noise reduction I. I NTRODUCTION This paper is devoted to analysis of time series using fuzzy transform (in short, F-transform — see [1]) and it is a continuation of recent research in this field (see, e.g., [2], [3], [4], [5]). Here, we will assume the simplest scheme where a time series X is a stochastic process that is decomposed into a trend-cycle TC and a random noise R, i.e., X(t)= TC(t)+ R(t). (1) In this case, the trend-cycle constituent TC(t) is a real function and R(t) is a random process which is not necessary stationary. In what follows, for the sake of simplicity, we suppose that the time t belongs to the set R of reals. 1 Note that usually a presence of seasonal constituent is also assumed in (1), however, the goal of this paper is not a filtering out this constituent but the reduction of noise which does not need any assumption on the seasonal constituent. Filtering out the seasonal constituent of time series has been investigated in [5]. The noise reduction is the goal of nearly all filters. In the case of F-transform, the first attempt to remove a deterministic noise 2 using this technique was published in [6]. A reduction of white noise using the discrete direct F-transform and the continuous inverse F-transform was investigated in [7] (see also [8]). In that paper, it was shown that under some nat- ural assumptions the variance of discrete stochastic process (function) after the application of F-transform is less than the variance of white noise presented in that process at the beginning. In this paper, we will show that using the higher-degree F-transform (F m -transform), we can satisfactorily reduce the noise described by the stochastic process R(t). This result significantly supports the argument that the F-transform is a powerful tool which can be effectively applied in the analysis of time series. 1 If one wants to deal within a closed real interval, it is sufficient to apply the restriction to this interval. Moreover, the complex case can be investigated similarly, only we must separately consider the real and imaginary part. 2 This noise is described by a deterministic function. II. RANDOM PROCESS R(t) In this section, we give the assumptions on R(t) used in this paper and provide a necessary background for the analysis of noise reduction by the F-transform. A. Assumptions on R(t) In what follows, we assume that a probability space (Ω, F ,P ) is fixed and we consider a real random process R(t) such that for any finite sequence t 1 ,...,t n (n =1, 2,... ) of times there is a joint distribution function given by F t1,...,tn (x 1 ,...,x n )= P ({R(t 1 ) ≤ x 1 ,...,R(t n ) ≤ x n }). (2) The distribution functions (2) must satisfy the following two conditions: (D1) The symmetry condition, according to which F ti 1 ,...,t in (x i1 ,...,x in )= F t1,...,tn (x 1 ,...,x n ), where i 1 ,...,i n is a permutation of the indices 1,...,n; (D2) The compatibility condition, according to which F t1,...,tm,tm+1,...,tn (x 1 ,...,x m , ∞,..., ∞)= F t1,...,tm (x 1 ,...,x m ) for any t m+1 ,...,t n if m<n; We use E, Var, Cov to denote the expected value, variance and covariance of random variables. Further, let us define a real function B(t, s) called correlation function of R(t) (see [9]) by B(t, s)= E[R(t)R(s)]. (3) We assume that, for any t, it holds (i) E[R(t)] = 0; (ii) B(t, t)= σ 2 (i.e., the variance of R(t) at each time t is equal to σ 2 ); (iii) B(t, s) is Lebesgue integrable. Obviously, the covariance of random variables R(t) and R(s) is equal to B(t, s), i.e., Cov(R(t),R(s)) = B(t, s). (4) Specifically, we have Var(R(t)) = B(t, t)= σ 2 . Then, |B(t, s)|≤ B(t, t)= σ 2 (5)