The Spectral Density Estimation of Stationary Time Series with Missing Data

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1 The Spectral Density Estimation of Stationary Time Series with Missing Data Jian Huang and Finbarr O Sullivan Department of Statistics University College Cork Ireland Abstract The spectral estimation of unevenly sampled data has been widely investigated in astronomical and medical areas. However the investigations are usually carried out in the context of periodicity detection and deterministic signal. Here we consider estimating the spectral density of stationary time series with missing data. An asymptotically unbiased estimation approach is proposed. The simulations are used to compare it to the classical periodogram, the Lomb periodogram (widely used for irregularly sampled data) and the SVD based periodogram. The results show that the new method substantially reduced the bias and slightly increased the variance. Overall the new approach significantly reduced the mean squared percentage error. As an example, the approach is applied to rainfall data in Irelan. Keywords: stationary time series, singular value decomposition, spectral density, the Lomb periodogram. 1 Introduction The Lomb periodogram(lomb, 1976 and Scargle, 1982) has been proposed for processing of irregularly sampled data. Its successful application to detecting periodicities in data and to estimating the spectral density of deterministic signal has been reported in literature (see, e.g. Green et al., 2002, Kaneoke and Vitek, 1996, Fortin and Mackey, 1999, and Laguna et al., 1998). Here we consider estimating the spectral density of stationary time series with missing data with focus on the estimation accuracy. This is motivated by the following facts: In climatological studies, the anomaly data can usually be modeled as stationary time series (after removing some trend and seasonal components). common in climatological data. Missing observations are 1

2 Now consider a missing data set {x t1,..., x tn } sampled from the stationary time series {X t }. The discrete Fourier transform (DFT) can be defined as NX FT(ω) = x tj exp iωt j, j=1 where t j s are non-missing sampling times and N is the number of non-missing values. The periodogram is then conventionally defined as P (ω) = = 1 2Nπ FT((ω) 2 (1) 1 N 2Nπ [( X NX x j cos(ωt j )) 2 +[( x j sin(ωt j )) 2 ]. j=1 j=1 Although the periodogram can be evaluated for any frequency, it is traditionally evaluated only at a special set of evenly spaced frequencies (the Fourier frequencies) ω k =2kπ/N k =0, 1,...,[N/2], where [N/2] is the largest integer less than or equate to N/2. It is well known that at the Fourier frequencies sinusoid components in (2) are orthogonal. However this property does not hold with missing data. To recovery the orthogonality, Lomb (1976) introduced a time delay τ, defined for each frequency ω by tan(2ωτ) = P Nj=1 sin(2ωt j ) P Nj=1 cos(2ωt j ), his redefinition of the periodogram called the Lomb periodogram is P (ω) = 1 P Nj=1 2Nπ [( x j cosω(t j τ)) 2 P Nj=1 cos 2 ω(t j τ) + (P N j=1 x j sinω(t j τ)) 2 P Nj=1 ]. sin 2 ω(t j τ) The Lomb periodogram is preferable to the classical periodogram for two reasons: it has a simple statistical behavior, and is equivalent to the reduction of the sum of squares fitting of sine waves to the data (Scargle, 1982). However the actual values are typically not changed much (Scargle, 1982). Hence we cannot expect the Lomb periodogram will lead a significant improvement in the estimation accuracy. In the non-missing case, alternatively, the classical periodogram can be obtained by solving the linear equation system resulted from fitting sine waves with the Fourier frequencies to the data. (Brockwell and Davis, 1993). Now if some values are missing, an under-determined linear equation system results (that is the number of equations less than 2

3 the number of unknown coefficients). A nature remedy is using the signgular values decomposition to solve this equation system. The SVD based approach (may called the SVD periodogram) is investigated using simulation studies. The results show that the actual values are quite close to the classical periodogram (see Section 3). In this paper we investigate a two-step spectral estimation approach. The basic idea of the method is to refine the classical estimator based on the effect on the spectral density of misssing data. This approach is described in Section 2. In Section 3 we compare the refined periodogram to the classical periodogram, the lomb periodogram and the SVD periodogram using simulated data. Conclusions and further research are given in Section 4. 2 Refined periodogram The missing data will be modeled as Y t = W t X t (2) where W t are stationary random weights with 0-1 values (0 for missing and 1 for nonmissing). {X t } and {Y t } are zero mean stationary time series with the spectral densities f X (λ) andf Y (λ), respectively. The problem is to estimate the spectral density of {X t } using observations on {Y t } and {W t }. We assume that {W t } is independent of {X t }.Thenwehave R Y (h) =EY t+h Y t = E(W t+h W t )E(X t+h X t )=(R W (h)+e 2 W )R X (h), where E 2 W and R W (h) arethemeanandtheautocovarincefunctionof{w t }, respectively. So It follows that R X (h) = R Y (h) EW 2 (1 R W (h) R W (h)+ew 2 ) f X (ω) = 1 EW 2 f Y (ω) 1 EW 2 f Y (ω) l W (ω) (3) R Where l W (ω) is the Fourier transformation of W (h). Hence we proposed to estimate R W (h)+ew 2 the spectral density of {X t } by the following two stage procedure: Step 1. Estimating the spectral density of {Y t } using the observations on {Y t }.-ˆf Y (ω) 3

4 Step 2. Refining the estimate obtained in Step 1 using (2) with R W (h) ande 2 W estimated from the observations on {W t }. ˆf X (ω) = 1 Ê 2 W ˆf Y (ω) 1 Ê 2 W ˆf Y (ω) ˆl W (ω), where ˆl W (ω) the Fourier transformation of ˆR W (h) ˆR W (h)+ê 2 W Remark. (a) The spectral density estimation of {Y t } isanonmissingdataproblem, hence the classical spectral estimation methods can be used in Step 1. (b) Due to the sampling variability, refining the spectral estimate using (2) may produce negative values. If this happens, the value zero should be used. Note if we assume that {W t } is IID, then we have VW 2 f W (λ) = (VW 2 + E2 W ). Hence (2) becomes F X (λ) = 1 V EW 2 W 2 F Y (λ) V Y 2 MW 2 (V W 2 + (4) E2 W ), where VW 2 and V Y 2 are the variances of {W t}, respectively. Refining spectral estimates using (4) will be very simple. 3 Evaluation of the Approach In this section we use simulated data to compare the two-step approach described in Section 2 to the classical periodogram, the Lomb periodogram and the svd periodogram. Although the two-step approach can (should) be used with smoothed periodogram estimators, for fair comparison (since the classical periodogram, the Lomb periodogram and the svd periodogram are not involed smoothing), we use the periodogram in Step 1. In such case the approachcanbecalledtherefined periodogram. The simulated data are generated by the followin two steps: Step 1. The S-PLUS function arima.sim is used to generate data {x 1,..., x T } with T =128 from the AR(2) model X t = a 1 X t 1 + a 2 X t 2 + Z t, where a 1 =0.4, a 2 = 0.8 and{z t } are IID N(0,1). The spectral density of the model is f(λ) = 1 2π{(1 a 2 ) 2 + a a 1(1 + a 2 )cos(λ)+4a 2 cos 2 (λ)} 4

5 This density function has a peak at the medium frequency (see Figure 1). Step 2. The missing data are generated by deleting x t from the data set if u t > 0.8, for t=1,...,t. Where {u 1,..,u T } are a random sample from the uniform distribution U[0,1]. The refined periodogram is calculated using the simulated data. To use (4), E W is estimated as Ê W = the number of missings divided by T,andVW 2 is estimated as ˆV W 2 = Ê W (1 Ê W ). As comparison the classical periodogram, the SVD periodogram and the Lomb periodogram are computed using the simulated data. This process is repeated 100 times. The sample mean and variance of each estimate are calculated and recorded. Figure 1 gives the sample mean of each spectral estimate. In this figure we can see that the sample mean of the refined periodogram is much closer to the true spectrum than the ones of the other methods. Hence the refined periodogram substantially reduced the estimation bias. Figure 1 also show that the sample means of the SVD periodogram and the Lomb periodogram and the classical periodogram are very close to each other. It seems that the estimation bias cannot bt improved by the Lomb periodogram and the SVD periodogram. In fact Scargle (1982) pointed out that for the Lomb periodogram and the classical periodogram the actual values are typically not changed much, even though the form is significantly changed. Figure 2 shows the sample variance of each spectral estimate. In this figure we can see that the refined periodogram has larger variance than the SVD periodogram and the Lomb periodogram and the classical periodogram. Figure 2 also show there no big difference among the SVD periodogram and the Lomb periodogram and the classical periodogram. Comparing to the other periodograms, the refined periodogram has the smaller bias but larger variance. Since the standard deviations of the spectral estimate are (approximately) proportional to the power spectrum, Parzen (1957) suggested to use mean squared percentage error to measure the estimation accuracy, which is defined by K(ω) ={v(ω) 2 + b(ω) 2 }/f(ω) 2 In this simulation study, b(w) iscomputedasthesamplemean-f(ω), and v(w) 2 is taken as the sample variance. Figure 3 shows the mean squared percentage error of each spectral estimate. From Figure 3, we can see that the refined periodogram has smaller MSPE than the other periodograms 5

6 Power spectrum Classical Modified Lomb SVD Frequency Figure 1: The sample mean of of the spectral estimates. The results based on 100 simulations with T=128 and 20% missing Modified Classical Lomb SVD Frequency Figure 2: The variance of the spectral estimates. The results based on 100 simulation with T=128 and 20% missing. 6

7 percentage error Modified Classical Lomb SVD Frequency Figure 3: The mean squared percentage error of the spectral estimates. The results based on 100 simulations with T=128 and 20% missing. at most frequencies. The average value of MSPE over the frequencies is reported in Table 1 Theaveragevaluesofthepercentagebiassquired(PBS)andthepercentagevariance(PV) are also reported in Table 1. Table 1. The average values of mean squared percentage error and percentage bias squared and the percentage variance over the frequencies. The results are based on 100 simulations with T=128 and 20% missing. Periodograms PBS PV MSPE(SE) Classical (0.11) Lomb (0.11) SVD (0.11) Refined (0.09) 4 Application to a real data set The rainfall data in Ireland consist of monthly average daily rainfall records at 1294 stations over the period of 37 years. Data missing patterns are quite different among the stations. As an example to illustrate the application of our technique we select rainfall data in 7

8 Figure 4: The Monthly average daily rainfall anomaly at DURROW G.S. DURROW G.S. (7 o , 52 o ). Annual rainfall pattern are defind as the monthly (daily) rainfall amount averaged over 37 years. The anomaly are defined by subtracting annual rainfall pattern from the data and shown in Figure 4. Figure 4 seems show that the data have randomly (26Formula (4) is used to compute the refined periodogram with Ê W =0.74 and ˆV W 2 = the refined periodogram of the anormaly is shown in Figure 5. Figure 5 seems indicate that the anormaly are white noise. 5 Conclusion and discussion We have investigated a new approach for estimating the power spectrum of stationary time series with missing data. The new approach is the refined spectral estimate of non missing data, its asymptotical unbiasness is easy to be established. This property does not hold for the classical periodogram and the Lomb periodogram and the SVD periodogram for missing data. The simulation results show that the new approach substantially reduced the bias and slightly increased the variance (relative to the other periodogram methods). Overall the new approach significantly reduced the mean squared percentage error. The simulation results also show that the actual values of the classical periodogram and the Lomb periodogram and the SVD periodogram are quite similar. 8

9 Frequency Figure 5: The refined periodogram of the anomaly in Figure 4 In climatological studies, data missing is usually time dependent. For example we have more missing in winter than in summer in rainfall data in Ireland. Hence further work can be focused on smoothed periodogram estimators and non stationary weights {W t }. References Brockwell PJ and Davis RA (1993). Time series: theory and methods. Springer-Verlag, New York. Fortin P and Mackey MC (1999). Periodic chronic myelogenous leukaemia: spectral analysis of blood cell counts and aetiological implications. BritishJ.ofHaematology, 102(2), Green D, Maclaren DA, Allison W, and Dastoor PC. (2002). Developments in the analysis of helium atom scattering data: the sequential filtering Lomb periodogram technique. J. of Physics D-Applied Physics 24, Kaneoke Y and Vitek JL (1996) Burst and oscillations as disparate neuronal properties. J. of Neuroscience Methods, 68(2),

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