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Classification of cyclical time series using complex demodulation

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Abstract

A new and innovative procedure based on time varying amplitudes for the classification of cyclical time series is proposed. In many practical situations, the amplitude of a cyclical component of a time series is not constant. Estimated time varying amplitudes obtained through complex demodulation of the time series are used as the discriminating variables in classical discriminant analysis. The aim of this paper is to demonstrate through simulation studies and applications to well-known data sets, that time varying amplitudes have very good discriminating power and hence their use in classical discriminant analysis is a simple alternative to more complex methods of time series discrimination.

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Acknowledgements

The author acknowledges the support of a grant from the Faculty of Business and Economics, Monash University, and the helpful comments by Professor Mervyn Silvapulle of the Department of Econometrics and Business Statistics, Monash University.

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Authors and Affiliations

  1. Department of Econometrics and Business Statistics, Monash University, 900 Dandenong Road, Caulfield East, Victoria, 3145, Australia

    Elizabeth Ann Maharaj

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  1. Elizabeth Ann Maharaj
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Correspondence to Elizabeth Ann Maharaj.

Appendix: Distribution of the estimated time varying parameter, \(\widehat{R}_{t}\)

Appendix: Distribution of the estimated time varying parameter, \(\widehat{R}_{t}\)

The estimated time varying amplitude from complex demodulation \(\widehat {R}_{t} = |y_{t}|\) is equivalent to \(\widehat{R}_{t} = \sqrt{\hat{A}_{t}^{2} + \hat{B}_{t}^{2}}\) from local harmonic regression analysis. Consider the harmonic regression model in Eq. (1). As the least squares estimates, \(\hat{A}\) and \(\hat{B}\) of A and B respectively, are unbiased, it has been shown (Bloomfield 2000) that for Fourier frequencies, i.e., frequencies that are harmonic with the span of the data, their variances and covariances simplify to \(\mathit{var}(\hat{A}) = \mathit{var}(\hat{B}) = 2\sigma^{2}/n\), and \(\mathit{cov}(\hat {A},\hat{B})=0\), respectively. If it is assumed that the errors ε t are white noise, then as a result of the central limit theorem (Feller 1968, pp. 244–254), \(\hat{A}\) and \(\hat{B}\) as linear functions of the errors would be expected to be normally distributed with respective means and variances. However, while the variable \(\widehat{R} = \sqrt{\hat{A}^{2} + \hat{B}^{2}}\) is a function of other normally distributed variables, it distribution is not of a closed form as shown below.

Let \(X = \hat{A}^{2}\), \(Y = \hat{B}^{2}\) and Z=X+Y, then

$$\begin{aligned} f_{X}(x) =& \varPhi \biggl(\frac{\sqrt{x}-A}{\sigma\sqrt{2/n}} \biggr) \frac{1}{2\sigma\sqrt{2x/n}} \\ &{} - \varPhi \biggl(\frac{-\sqrt{x}-A}{\sigma\sqrt {2/n}} \biggr)\frac{1}{-2\sigma\sqrt{2x/n}} \end{aligned}$$

f Y (y) is similarly obtained. Since X and Y are independent, by the convolution technique

$$f_Z(z) = \int_{-\infty}^{\infty}f_X(x)f_Y(z-x) \,dx $$

Let \(W = \widehat{R} = \sqrt{\hat{A}^{2} + \hat{B}^{2}}\). Then \(W = \sqrt{Z} = g(Z)\) and w=g(z) defines a one-to-one transformation of the sample space of Z onto that of W and the derivative of z=g −1(w) with respect to w, is continuous and nonzero for w in the sample space of W. Then by the transformation technique, the density function of W=g(Z) is

$$f_{W}(w) = \biggl|\frac{d}{dw}g^{-1}(w) \biggr| f_{Z} \bigl(g^{-1}(w) \bigr) $$

Hence while \(W = \widehat{R} = \sqrt{\hat{A}^{2} + \hat{B}^{2}}\) is a function of normally distributed variables, it’s distribution is not of a closed form. Similarly the time varying amplitude \(W_{t} = \widehat {R}_{t} = \sqrt{\hat{A}_{t}^{2} + \hat{B}_{t}^{2}}\) and filtered versions of it do not have closed form distributions.

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Maharaj, E.A. Classification of cyclical time series using complex demodulation. Stat Comput 24, 1031–1046 (2014). https://doi.org/10.1007/s11222-013-9418-0

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