Spectral Density Estimation of Continuous Time Series

This paper studies spectral density estimation of a strictly stationary r-vector valued continuous time series including missing observations. The finite Fourier transform is constructed in L-joint segments of observations. The modified periodogram is defined and smoothed to estimate the spectral density matrix. We explore the properties of the proposed estimator. Asymptotic distribution is discussed.

1. Introduction

Although spectral analysis is one of the oldest tools for time series analysis, it is still one of the most widely used analysis techniques in many branches of sciences, [1] - [6] . For zero mean r-vector valued strictly stationary time series, the spectral estimation has been studied, [7] - [17] . Time series with missing observations frequantly appear in paractice. If a block of observations is periodically unobtainable, Jones [18] provides a development for spectral estimation of a stationary time series. The theory of amplitude-modulated stationary processes has been developed by Parzen [19] and applied to periodic missing observations problems [20] . The case where an observation is made or not according to the out come of a Bernoulli trial has been discussed by Scheinok [21] . Bloomfield [22] considered the case where a more general random mechanism is involved. Broersen et al. [23] and [24] developed models for time series with missing observation and discussed their use for spectral estimation. Unbiased spectral estimators have been formulated assuming wavelet models of stationary time series by [25] . Their asymptotic properties have been also investigated.

In this paper, we will discuss the spectral analysis of a strictly stationary r-vector valued continuous time series with randomly missing observations in joint segments of observations. The paper is organized as follows. Section 2 introduces the basic definitions and assumptions. The modified series is defined in Section 3. Section 4 considers the expanded finite Fourier transform and its properties. The modified periodogram, the spectral density estimator and its properties are given in Section 5.

2. Observed Series

Let be a zero mean r-vector valued strictly stationary time series with

(2.1)

and

(2.2)

where denotes the matrix of absolute values, the bar denotes the complex conjugate and '' denotes the matrix transpose. We may then define the matrix of second order spectral densities by

(2.3)

Using the assumed stationary, we then set down

Assumption I. is a strictly stationary continuous series all of whose moments exist. For each and any k-tuple we have

(2.4)

where

(2.5)

(;).

Because cumulants are measures of the joint dependence of random variables, (2.4) is seen to be a form of mixing or asymptotic independence requirement for values of well separated in time. If satisfies Assumption I we may define its cumulant spectral densities by

(2.6)

(). If the cross-spectra are collected together in the matrix of (2.3).

Assumption II. Let is bounded, is of bounded variation

and vanishes for all t outside the interval, that is called data window.

3. Modified Series

Let be a process independent of such that, for every t

note that

The success of recording an observation not depend on the fail of another and so it is independent. We may then define the modified series

with components,

where

4. Expanded Finite Fourier Transform in L-Joint Segments of Observations

In the case when there are some randomly missing observations, Elhassanein [17] constructed the expanded finite Fourier transform on disjoint segments of observations. In this section the expanded finite Fourier transform is constructed in L-joint segments of observations for a strictly stationary r-vector valued time series. Expression for its mean, variance and cumulant will be derived. The results introduced here may be regarded as a generalization to [13] and [17] . Let be an observed stretch of data with some randomly missing observations. Let, where L is the number of joint segments and N is the length of each segment and M is the length of joint parts, , where we get the results in [17] . The expanded finite Fourier transform of a given stretch of data, is defined by

(4.1)

where and is the data window satisfies Assump- tion II.

Theorem 4.1. Let be a strictly stationary r-vector valued time series with mean zero, and satisfy Assumption I. Let be defined as (3.1), and satisfy Assumption II, for then

(4.2)

(4.3)

where

and

where

for then

(4.4)

(4.5)

where is uniform in as, and

Proof. We will prove (4.5), by (4.1) we get

let and since

for some constants and, we get

where

since satisfy Assumption II for then

which implies to, using (2.6) the proof is completed. ,

5. Estimation

Using expanded finite Fourier transform (4.1), we construct the modified periodogram as

(5.1)

such that

where the bar denotes the complex conjugate. The smoothed spectral density estimate is constructed as

(5.2)

Theorem 5.1. Let be a strictly stationary r-vector valued continuous time series with mean zero, and satisfy Assumption I. Let be given by (3.6), and satisfy Assumption II for then

(5.3)

(5.4)

(5.5)

where the summation extends over all partitions

into pairs of the quantities

excluding the case with for some, where is uniform in.

Proof. By (5.1), we have

then by (4.3) the proof of (5.3) is completed. From (5.1), and by Theorem (2.3.2) in [10] p. 21, we have

By Theorem (4.1) the proof of (5.4) is completed. From (5.1), we have

By Theorem (2.3.2) in [10] p. 21, we get

where the summation extends over all indecomposable partitions of the transformed table

Then, by Theorem (4.1), we get the proof of (5.5). ,

Theorem 5.2. Let be a strictly stationary r-vector valued time series

with mean zero, and satisfy Assumption I. Let be

given by (3.6), for and satisfy Assumption II for Then are asymptotically independent variates. Also if. then is asymptotically independent of the previous variates. Where, denotes an symmetric matrix-valued Wishart variate with covariance matrix and degree of freedom and denotes an Hermitian matrix-valued complex Wishart variate with covariance matrix and degree of freedom.

Proof. The proof comes directly from Theorem (4.2), for more details about Wishart distribution see [26] . ,

Theorem 5.3. Let be a strictly stationary r-vector valued time series with mean zero, and satisfy Assumption I. Let be given by (3.7), then

(5.6)

(5.7)

Proof. By (5.2), we have

then by (5.3) the proof of (5.6) is completed. From (5.2), we get

which completes the proof of (5.7). ,

Theorem 5.4. Let be a strictly stationary r-vector valued time series with mean zero, and satisfy Assumption I. Let be given by (5.2),

, for, Then

are asymptotically independent variates. Also if. then is asymptotically indepen- dent of the previous variates.

Proof. The proof comes directly by Theorem (5.3) and Theorem (7.3.2) in [26] p. 162. ,

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Elhassanein, A. (2016) Spectral Density Estimation of Continuous Time Series. Applied Mathematics, 7, 2140-2148. doi: 10.4236/am.2016.717170.

 [1] Olafsdttira, K.B., Schulz, M. and Mudelsee, M. (2016) REDFIT-X: Cross-Spectral Analysis of Unevenly Spaced Paleoclimate Time Series. Computers & Geosciences, 91, 11-18. http://dx.doi.org/10.1016/j.cageo.2016.03.001 [2] Fong, S., Cho, K., Mohammed, O., Fiaidhi, J. and Mohammed, S. (2016) A Time Series Pre-Processing Methodology with Statistical And Spectral Analysis for Classifying Non-Stationary Stochastic Biosignals. The Journal of Supercomputing, 72, 3887-3908. http://dx.doi.org/10.1007/s11227-016-1635-9 [3] Huang, N.E., et al. (2016) On Holo-Hilbert Spectral Analysis: A Full Informational Spectral Representation for Nonlinear and Non-Stationary Data. Philosophical Transactions of the Royal Society A, 374, 20150206. http://dx.doi.org/10.1098/rsta.2015.0206 [4] Kotoku, J., Kumagai, S., Uemura, R., Nakabayashi, S. and Kobayashi, T. (2016) Automatic Anomaly Detection of Respiratory Motion Based on Singular Spectrum Analysis. International Journal of Medical Physics, Clinical Engineering and Radiation Oncology, 5, 88-95. [5] Miller, K.J., Schalk, G., Hermes, D., Ojemann, J.G. and Rao, R.P.N. (2016) Spontaneous Decoding of the Timing and Content of Human Object Perception from Cortical Surface Recordings Reveals Complementary Information in the Event-Related Potential and Broad-band Spectral Change. PLOS Computational Biology, 12, e1004660. http://dx.doi.org/10.1371/journal.pcbi.1004660 [6] Kim, J., Park, S., Jeung, G. and Lee, J. (2016) Estimation of a Menstrual Cycle by Covariance Stationary-Time Series Analysis on the Basal Body Temperatures. Journal of Medical and Bioengineering, 5, 63-66. http://dx.doi.org/10.12720/jomb.5.1.63-66 [7] Brillinger, D.R. (1969) Asymptotic Properties of Spectral Estimate of Second Order. Biometrika, 56, 375-390. http://dx.doi.org/10.1093/biomet/56.2.375 [8] Dahlhaus, R. (1985) On Spectral Density Estimate Obtained by Averaging Periodograms. Journal of Applied Probability, 22, 598-610. http://dx.doi.org/10.1017/S0021900200029351 [9] Bloomfield, P. (2000) Fourier Analysis of Time Series an Introduction. 2ond Edition, John Wiley & Sons, Inc., Hoboken. http://dx.doi.org/10.1002/0471722235 [10] Brillinger, D.R. (2001) Time Series Data Analysis and Theory. Society for Industrial and Applied Mathematics. http://dx.doi.org/10.1137/1.9780898719246 [11] Broersen, P.M.T. (2006) Automatic Autocorrelation and Spectral Analysis. Springer-Verlag London Limited, London. [12] Ghazal, M.A. and Elhassanein, A. (2006) Periodogram Analysis with Missing Observations. Journal of Applied Mathematics and Computing, 22, 209-222. http://dx.doi.org/10.1007/BF02896472 [13] Ghazal, M.A. and Elhassanein, A. (2007) Nonparametric Spectral Analysis of Continuous Time Series. Bulletin of Statistics and Economics, 1, 41-52 [14] Ghazal, M.A. and Elhassanein, A. (2008) Spectral Analysis of Time Series in Joint Segments of Observations. Journal of Applied Mathematics & Informatics, 26, 933-943. [15] Ghazal, M.A. and Elhassanein, A. (2009) Dynamics of EXPAR Models for High Frequency Data. IJAMAS, 14, 88-96. [16] Elhassanein, A. (2011) Nonparametric Spectral Analysis on Disjoint Segments of Observations. JAMSI, 7, 81-96. [17] Elhassanein, A. (2014) On the Theory of Continuous Time Series. Indian Journal of Pure and Applied Mathematics, 45, 297-310. [18] Jones, R.H. (1962) Spectral Analysis with Regularly Missed Observations. The Annals of Mathematical Statistics, 33, 455-461. http://dx.doi.org/10.1214/aoms/1177704572 [19] Parzen, E. (1962) Spectral Analysis of Asymptotically Stationary Time Series. Bulletin de International de Statistique, 33rd Session, Paris. [20] Parzen, E. (1963) On Spectral Analysis with Missing Observations and Amplitude Modulation. Sankhya A, 25, 383-392 [21] Scheinok, P.A. (1965) Spectral Analysis with Randomly Missed Observations: The Binomial Case. The Annals of Mathematical Statistics, 36, 971-977. http://dx.doi.org/10.1214/aoms/1177700069 [22] Bloomfield, P. (1970) Spectral Analysis with Randomly Missing Observations. Journal of the Royal Statistical Society, 32, 369-380 [23] Broersen, P.M.T., de Waele, S. and Bos, R. (2004) Autoregressive Spectral Analysis When Observations Are Missing. Automatica, 40, 1495-1504. http://dx.doi.org/10.1016/j.automatica.2004.04.011 [24] Broersen, P.M.T. (2006) Automatic Spectral Analysis with Missing Data. Digital Signal Processing, 16, 754-766. http://dx.doi.org/10.1016/j.dsp.2006.01.001 [25] Mondal, D. and Percival, D.B. (2008) Wavelet Variance Analysis for Gappy Time Series. Annals of the Institute of Statistical Mathematics, 62, 943-966. [26] Anderson, T.W. (1972) An Introduction to Multivariate Statistical Analysis. Wiley Eastern Limited, New Delhi.