Recursive Supervised Estimation of a Markov-Switching GARCH Process in the Short-Time Fourier Transform Domain

In this paper, we introduce a Markov-switching generalized autoregressive conditional heteroscedasticity (GARCH) model for nonstationary processes with time-varying volatility structure in the short-time Fourier transform (STFT) domain. The expansion coefficients in the STFT domain are modeled as a multivariate complex GARCH process with Markov-switching regimes. The GARCH formulation parameterizes the correlation between sequential conditional variances while the Markov chain allows the process to switch between regimes of different GARCH formulations. We obtain a necessary and sufficient condition for the asymptotic wide-sense stationarity of the model, and develop a recursive algorithm for signal restoration in a noisy environment. The conditional variance is estimated by iterating propagation and update steps with regime conditional probabilities, while the model parameters are evaluated a priori from a training data set. Experimental results demonstrate the performance of the proposed algorithm.

[1]  Hamidreza Amindavar,et al.  GARCH coefficients as feature for speech recognition in Persian isolated digit , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..

[2]  Yariv Ephraim,et al.  A Bayesian estimation approach for speech enhancement using hidden Markov models , 1992, IEEE Trans. Signal Process..

[3]  Shih-Ping Han.  A globally convergent method for nonlinear programming , 1975 .

[4]  Neri Merhav,et al.  Lower and upper bounds on the minimum mean-square error in composite source signal estimation , 1991, IEEE Trans. Inf. Theory.

[5]  Jonathan D. Cryer,et al.  Time Series Analysis , 1986 .

[6]  Jun Cai.  A Markov Model of Switching-Regime ARCH , 1994 .

[7]  Israel Cohen,et al.  Asymptotic Stationarity of Markov-Switching Time-Frequency Garch Processes , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

[8]  Juri Marcucci.  Forecasting Stock Market Volatility with Regime-Switching GARCH Models , 2005 .

[9]  Klaassen.  Improving GARCH Volatility Forecasts with Regime-Switching GARCH Klaassen, F.J.G.M , 2001 .

[10]  Israel Cohen,et al.  Speech enhancement for non-stationary noise environments , 2001, Signal Process..

[11]  Israel Cohen,et al.  Noise spectrum estimation in adverse environments: improved minima controlled recursive averaging , 2003, IEEE Trans. Speech Audio Process..

[12]  Naftali Z. Tisby.  On the application of mixture AR hidden Markov models to text independent speaker recognition , 1991, IEEE Trans. Signal Process..

[13]  Philip E. Gill,et al.  Practical optimization , 1981 .

[14]  Ephraim.  Speech enhancement using a minimum mean square error short-time spectral amplitude estimator , 1984 .

[15]  Marc S. Paolella,et al.  A New Approach to Markov-Switching GARCH Models , 2004 .

[16]  Israel Cohen.  From Volatility Modeling of Financial Time-Series to Stochastic Modeling and Enhancement of Speech Signals , 2005 .

[17]  William J. J. Roberts,et al.  Revisiting autoregressive hidden Markov modeling of speech signals , 2005, IEEE Signal Processing Letters.

[18]  Harris Drucker.  Speech processing in a high ambient noise environment , 1967 .

[19]  Franc J. G. M. Klaassen,et al.  Improving GARCH volatility forecasts with regime-switching GARCH , 2002 .

[20]  Stephen Gray.  Modeling the Conditional Distribution of Interest Rates as a Regime-Switching Process , 1996 .

[21]  Rainer Martin,et al.  Noise power spectral density estimation based on optimal smoothing and minimum statistics , 2001, IEEE Trans. Speech Audio Process..

[22]  Israel Cohen,et al.  ON THE STATIONARITY OF MARKOV-SWITCHING GARCH PROCESSES , 2007, Econometric Theory.

[23]  Olivier Cappé,et al.  Elimination of the musical noise phenomenon with the Ephraim and Malah noise suppressor , 1994, IEEE Trans. Speech Audio Process..

[24]  David Malah,et al.  Speech enhancement using a minimum mean-square error log-spectral amplitude estimator , 1984, IEEE Trans. Acoust. Speech Signal Process..

[25]  Biing-Hwang Juang,et al.  On the application of hidden Markov models for enhancing noisy speech , 1989, IEEE Trans. Acoust. Speech Signal Process..

[26]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[27]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[28]  Neri Merhav,et al.  Hidden Markov processes , 2002, IEEE Trans. Inf. Theory.

[29]  Israel Cohen,et al.  Speech spectral modeling and enhancement based on autoregressive conditional heteroscedasticity models , 2006, Signal Process..

[30]  James D. Hamilton,et al.  Autoregressive conditional heteroskedasticity and changes in regime , 1994 .

[31]  L. R. Rabiner,et al.  An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition , 1983, The Bell System Technical Journal.

[32]  Israel Cohen,et al.  State smoothing in Markov-switching time-frequency GARCH models , 2006, IEEE Signal Processing Letters.

[33]  Israel Cohen,et al.  Relaxed statistical model for speech enhancement and a priori SNR estimation , 2005, IEEE Transactions on Speech and Audio Processing.

[34]  Israel Cohen.  Modeling speech signals in the time-frequency domain using GARCH , 2004, Signal Process..