Comparative Study Of Time-Frequency Analysis Approaches With Application To Economic Indicators

Presented paper deals with comparison of various methods for time-frequency representation of a signal with time-varying behavior. We choose methods such as wavelet analysis, multiple window method using Slepian sequences, time-frequency varying autoregressive process estimation and time-frequency Fourier transform representation (periodogram). We apply these methods first on the simple simulated artificial signal and we assess their performance. Then we proceed with application on the real data which is monthly data of the industry production index of European Union in the period 1990/M1-2011/M11. During the evaluation we focus on the results with respect to the time of global crisis. The results of the experiments are represented in the the graphical form and briefly discussed. INTRODUCTION The description of time-frequency structure of signal has wide range of usage. Its application can be seen in many scientific areas such as engineering (Xu et al., 2011), medicine (Xu et al., 1999), economy and many others. In last several years these techniques are in the front of economic researchers which analyze comovement of economic indicators. In this sense the papers of (Rua, 2010), (Yogo, 2008) or (Hallett and Richter, 2007) and many others were written. Estimation of spectrogram or scalogram of input signal or time series depends on used methods and their parameters. We investigate in this article four basic methods such as wavelet analysis (Jan, 2002), multiple window method using the slepian sequences (MWM) (Xu et al., 1999), time-frequency varying autoregressive (AR) process (Proakis et al., 2002) spectrum estimation and time-frequency Fourier transform estimation (periodogram) (Jan, 2002). On the basis of simulations on the artificial well-know signal we analyze behavior of each method and search for their advantages, disadvantages and recommendations for their usage. Consequently we compare obtained results with the aim to give recommendation for methods application. In order to practically demonstrate and evaluate the performance of the chosen methods we apply them to the analysis of the real data which is the monthly data of the industry production index of the European Union countries in the period 1990/M1-2011/M11. The paper is organized as follows: In the section Methodical Background we describe chosen methods of time-frequency analysis. Consequently, in the section Data, we briefly describe data used both for the simulation as well as for the practical application. After that, in the section Simulation, we show results of an application of chosen methods on simulated artificial data. The section Application presents results of real data analysis of the industry production index of the European Union and its brief economic interpretation. In both later sections results are graphically represented. The paper ends up with the conclusion and the list of used references. METHODICAL BACKGROUND Let us have a signal (a time series) y(n), n = 1, . . . , N . Under assumption that the time series contain a longterm trend, we can apply additive decomposition in the following form y(n) = g(n) + s(n) + c(n) + ε(n), n = 1, . . . , N, (1) where g(n) denotes a long-term trend, s(n) is the seasonal component, c(n) is the cyclical component and ε(n) is the irregular component (a random noise). Focusing on analysis of cyclical movements around its longterm trend it is necessary to remove the long-term trend applying some filtering methods. When the seasonally adjusted data are not available (in other words the analyzed series contains the seasonal component), the seasonality should be removed by applying some corresponding method. The spectrum of the signal (time series) y(n), n = 1, . . . , N can be written as a Fourier sum (Hamilton, 1994)