Spectral Analysis - Introduction

Transcription

1 Spectral Analysis - Introduction Initial Code Introduction Summer Semester 011/01 Lukas Vacha In previous lectures we studied stationary time series in terms of quantities that are functions of time. The value of a variable X t at time t was described as a sequence of innovations. The main focus was on the covariance structure between X t and X t-d at a distinct dates t and t - d. This approach is known as analyzing the properties of a process X t in the time domain. Another approach is to describe the value of a variable (signal) X t as a weighted sum of periodic functions of the form coshwl and sinhwl, where w denotes an angular frequency: X t = m + Ÿ 0 p ahwl coshw tl w + Ÿ 0 p dhwl sinhw tl w The goal is to determine how important are cycles of different frequencies in the variable X t. This is known as frequency domain analysis. The spectrum of a stationary time series hhwl in frequency domain, is the counterpart of a covariance function in the time domain. Theoretically it provides a different representation of the same information, this approach can yield both powerful numerical methods of analysis and new insights. Frequency, Period f is frequency of cycles per second, c.p.s., Hz w = p f - angular frequency (radinas per second) T = 1êf - period The complex exponential A sine wave e i x = coshxl + i sinhxl Nice example of sin function and the angular frequency:

2 QF_II_Lecture 3 Spectra1.cdf Plot@Sin@xD,8x,0,Pi<D Plot@Sin@4xD,8x,0,Pi<D Plot@Sin@1xD,8x,0,Pi<D

3 QF_II_Lecture 3 Spectra1.cdf 3 Examples Plot@Sin@xD+Sin@8xD,8x,0,Pi<D Plot@Sin@xD+Sin@4xD+Sin@1xD+Sin@0xD,8x,0,4Pi<D Sampling The Nyquist frequency is equal to one-half of the sampling frequency. w N Y Q U I S T = T-1, T denotes the sample length. The Nyquist frequency is the highest frequency that can be measured in a signal. An undersampled signal Expel of undersampled signal:

4 4 QF_II_Lecture 3 Spectra1.cdf t/nyquist_limit_java_plugin.html The Fourier transform A transform takes one function (or signal) and turns it into another function (or signal). Any infinite sequence xhtl may be viewed as a combination of periodic functions. The Fourier coefficients of a time series (signal) x(t) are given by XHfL = Ÿ - xhtl e - p i f t d t XHfL determines the relative weight of each complex sinusoids. inverse Fourier transform: xhtl = 1 p Ÿ p -p XHfL e p i f t d f The original sequence is a linear combination of complex sinusoids. Discrete Fourier Transform The Discrete Fourier Transform (DFT) transforms one function into another, which is called the frequency domain representation of the original function (which is often a function in the time domain). The input to the DFT is a finite sequence of real or complex numbers.the only requirements of these conventions are that the DFT and Inverse Discrete Fourier Transform (IDFT) have opposite-sign exponents and that the product of their normalization factors be 1êN. A normalization of 1í N for both the DFT and IDFT makes the transforms unitary, which has some theoretical advantages, but it is often more practical in numerical computation to perform the scaling all at once (and a unit scaling can be convenient in other ways). DFT: X k = N-1 t=0 x t e - p i f k t, k = 0, 1,... N - 1 IDFT:

5 QF_II_Lecture 3 Spectra1.cdf 5 IDFT: x t = 1 N-1 N k=0 X k e p i f k t, t = 0, 1,... N - 1 How Does FT Work FT uses complex exponentials (sinusoid) as building blocks. e j w t = coshw tl + j sinhw tl For each frequency of complex exponential, the sinusoid at that frequency is compared to the signal. If the signal consists of that frequency, the correlation is high ö large FT coefficients. XHfL = Ÿ - xhtl e - p i f t dt If the signal does not have any spectral component at a frequency, the correlation at that frequency is low / zero, ö small / zero FT coefficient. Example FT At Work Remove comments (*...*) for modification of the example signal=table@sin@4xdh*+sin@8xd*l,8x,0,pi,.04<d; ListPlot@signal,PlotLabelØ"signal",FillingØAxis,PlotStyleØ8PointSize@MediumD<D ListPlot@Abs@Fourier@signalDD@@1;;Floor@HLength@signalD-1LêDDD^,PlotLabelØ,Pl signal

6 6 QF_II_Lecture 3 Spectra1.cdf Example with noise signal=table@sin@4xd+1.5*randomreal@normaldistribution@0,1ddh*+sin@8xd*l,8x,0,pi,.04<d ListPlot@signal,PlotLabelØ"signal",FillingØAxis,PlotStyleØ8PointSize@MediumD<D ListPlot@Abs@Fourier@signalDD@@1;;Floor@HLength@signalD-1LêDDD^,PlotLabelØ,Pl signal

7 QF_II_Lecture 3 Spectra1.cdf 7 Example: Complete time series reconstruction I Example shows that the original sequence is a linear combination of complex sinusoids. signal=table@sin@4xd+sin@8xd+sin@4xdh*+randomreal@normaldistribution@0,1dd*l,8x,0,pi ListPlot@signal,PlotLabelØ"Original signal",fillingøaxis,plotstyleø8pointsize@mediumd<d ListPlot@Abs@Fourier@signalDD@@1;;Floor@HLength@signalD-1LêDDD^,PlotLabelØ,Pl InverseFourier@Fourier@signalDD; ListPlot@InverseFourier@Fourier@signalDD,PlotLabelØ"Reconstructed signal",fillingøaxis, 4 Original signal Reconstructed signal Nice wiki example:

8 8 QF_II_Lecture 3 Spectra1.cdf Example: Complete time series reconstruction II (Stationarity) FT identifies all spectral components present in the signal, however it does not provide any information regarding the temporal (time) localization of these components. Why? signal1=table@sin@xd,8x,0,5pi,.1<d; signal=table@sin@0xd,8x,0,5pi,.1<d; signal=join@signal1,signald; ListPlot@signal,PlotLabelØ"Original signal",fillingøaxis,plotstyleø8pointsize@mediumd<d ListPlot@Abs@Fourier@signalDD@@1;;Floor@HLength@signalD-1LêDDD^,PlotLabelØ,Pl H*InverseFourier@Fourier@signalDD;*L ListPlot@InverseFourier@Fourier@signalDD,PlotLabelØ"Reconstructed signal",fillingøaxis, Original signal Reconstructed signal What happens when we use for the reconstruction only the most significant frequencies?

9 QF_II_Lecture 3 Spectra1.cdf 9 Example: time series reconstruction - a real time series There is a problem with the trend!. Why? signal=log@financialdata@"^dji","nov. 1, 009"D@@All,DDD; ListLinePlot@signal,PlotLabelØ"Original signal"d ListPlot@Abs@Fourier@signalDD@@1;;Floor@HLength@signalD-1LêDDD^,PlotLabelØ,Pl InverseFourier@Fourier@signalDD; ListLinePlot@InverseFourier@Fourier@signalDDH*@@1;;10DD*L,PlotLabelØ"Reconstructed sign Original signal Reconstructed signal Parseval's theorem The energy spectral density describes how the energy (variance) of a signal (time series) is distributed with frequency. Total energy of the signal may be obtained by integrating the energy per unit frequency 1ê p XHfL over the p interval. t=- x t = 1 p Ÿ p -p XHfL df The squared magnitude of the Fourier transform X HfL is called the power spectrum, denoted hhwl, (or

10 10 QF_II_Lecture 3 Spectra1.cdf the p interval. t=- x t = 1 p Ÿ p -p XHfL df The squared magnitude of the Fourier transform X HfL is called the power spectrum, denoted hhwl, (or energy-density spectrum, or spectrum) of the signal x t. (Periodogram is another name for essentially the same quantity) Parseval's theorem for the finite sequence: N-1 t=0 x t = 1 N-1 N k=0 X k Estimation of the The spectrum of a stationary time series hhwl is the counterpart of a covariance function in frequency domain. That is, it is the Fourier transform of the covariance function ghkl and vice versa. hhwl = 1 p k=- ghkl e -i k w. p ghkl =Ÿ -p hhwl e i w k w In practice, h`hwl is computed only at the Fourier frequencies: w j = p jên for j œ I1,,..., n è M where n è = dn - 1êt h`iw j M = 1 p n n t=1 xhtl e iw j t = 1 p XIw j M Examples: Random Numbers, ARFIMA ListLinePlot@Abs@Fourier@RandomReal@NormalDistribution@0,1D,300DDD@@1;;Floor@H300-1LêDD

11 QF_II_Lecture 3 Spectra1.cdf 11 ListLinePlot@Abs@Fourier@ARFIMA@0.6,0.4,-0.0,100DDD@@1;;Floor@H100-1LêDDD^,PlotLabelØ Appendix Sample periodogram The periodogram S(w) is a basic estimator for the spectrum (spectral density) of stationary stochastic processes. The periodogram S(w) is often calculated only for N equidistant points in the frequency domain from zero until (N-1)/N Hz. w j = p jêt for j = 1,,..., HT - 1Lê [ a j = T T t=1 xhtl cos Aw j Ht - 1LE [ b j = T T t=1 xhtl sin Aw j Ht - 1LE [ Sx Hw j L = :a j + [ b j >

12 1 QF_II_Lecture 3 Spectra1.cdf Examples: spectral densities of AR and MA processes ü 1) AR signal=arfima@0.9,0,0.0,300d; ListLinePlot@signal,PlotLabelØSignalD ListLinePlot@Abs@Fourier@signalDD@@1;;Floor@H300-1LêDDD^,PlotLabelØ,PlotRange 6 Signal

13 QF_II_Lecture 3 Spectra1.cdf 13 signal=arfima@-0.8,0,0.0,300d; ListLinePlot@signal,PlotLabelØSignalD ListLinePlot@Abs@Fourier@signalDD@@1;;Floor@H300-1LêDDD^,PlotLabelØ,PlotRange 4 Signal

14 14 QF_II_Lecture 3 Spectra1.cdf ü ) MA signal=arfima@0.0,0,0.8,300d; ListLinePlot@signal,PlotLabelØSignalD ListLinePlot@Abs@Fourier@signalDD@@1;;Floor@H300-1LêDDD^,PlotLabelØ,PlotRange 3 Signal

15 QF_II_Lecture 3 Spectra1.cdf 15 signal=arfima@0.0,0,-0.8,300d; ListLinePlot@signal,PlotLabelØSignalD ListLinePlot@Abs@Fourier@signalDD@@1;;Floor@H300-1LêDDD^,PlotLabelØ,PlotRange 3 Signal

16 16 QF_II_Lecture 3 Spectra1.cdf ü MA(q) Needs@"TimeSeries`TimeSeries`"D tsma=timeseries@mamodel@8-0.5, 0.9<, 1D,300D; ListLinePlot@Abs@Fourier@tsMADD@@1;;Floor@HLength@tsMAD-1LêDDD^,PlotLabelØ, tsma=timeseries@mamodel@80.3, 0.4, 0.5, 0.4, 0.3<, 1D,300D; ListLinePlot@Abs@Fourier@tsMADD@@1;;Floor@HLength@tsMAD-1LêDDD^,PlotLabelØ, References: Priestley (1981) Hamilton (1994) Homework #4 Deadline: Monday.4.01, 16:00 1 ) Estimate power spectrum (Periodogram) of simulated AR, MA and ARMA process. ) Estimate power spectrum (Periodogram) of a real financial market time series. 3) to be announced... Homework may be returned in class, or sent via to vachal@utia.cas.cz