Spectral Inference under Complex Temporal Dynamics Jun Yang joint work with Zhou Zhou Department of Statistical Sciences University of Toronto SSC 2019, University of Calgary Spectral Inference (Jun Yang) 1
Motivation Frequency content of many real-world stochastic processes evolves over time. Time-frequency analysis ◮ One of the major research areas in applied mathematics and signal processing. ◮ has been developed independently with non-stationary spectral domain theory and methodology in time series. Statistical inference ◮ has been paid little attention to in time-frequency analysis, such as confidence region construction and hypothesis testing. ◮ To date, there is no results on the joint and simultaneous inference of the evolutionary spectrum. Spectral Inference (Jun Yang) 2
Earthquake versus Explosion ◮ How to distinguish Earthquake and Explosion? Earthquake Data Time-frequency Estimates Time-frequency Estimates 0.6 16 14 0.4 0.04 12 0.2 0.03 10 freq 0 0.02 8 0.01 -0.2 6 0 15 4 -0.4 10 10 2 5 6 -0.6 2 500 1000 1500 2000 freq 2 4 6 8 10 12 time time time Explosion Data Time-frequency Estimates Time-frequency Estimates 0.6 16 14 0.4 12 0.03 0.2 10 0.02 freq 0 8 0.01 -0.2 6 0 4 15 -0.4 10 10 2 5 6 -0.6 2 500 1000 1500 2000 freq 2 4 6 8 10 12 time time time ◮ Is Explosion time-frequency separable? H 0 : f ( u , θ ) ∝ g ( u ) h ( θ ) . Spectral Inference (Jun Yang) 3
SP500 Daily Returns Daily Returns of SP500 Time-frequency Estimates Time-frequency Estimates 0.15 16 × 10 -4 14 0.1 1.5 12 1 0.05 10 freq 8 0.5 0 6 0 -0.05 4 15 10 20 15 2 5 10 -0.1 5 1000 2000 3000 4000 5000 6000 5 10 15 20 freq time time time ◮ Time-varying white noise? H 0 : f ( u , θ ) ∝ g ( u ) . ◮ Volatility forecasting (goodness of fit test) Absolute SP500 Time-frequency Estimates Time-frequency Estimates 0.1 16 × 10 -4 14 0.08 2 12 0.06 10 1 freq 0.04 8 0.02 6 0 15 4 0 10 20 2 15 5 10 -0.02 5 1000 2000 3000 4000 5000 6000 freq 5 10 15 20 time time time Spectral Inference (Jun Yang) 4
Absolute SP500 Daily Returns tv-AR(1) tv-AR(4) × 10 -4 × 10 -4 2 4 1 2 0 0 15 15 20 20 10 10 15 15 5 10 5 10 5 5 frequency time frequency time tv-AR(5) tv-ARMA(1,1) × 10 -4 × 10 -4 4 1 2 0.5 0 0 15 15 10 20 10 20 15 15 5 10 5 10 5 5 frequency time frequency time tv-ARMA(2,1) tv-ARMA(3,1) × 10 -4 × 10 -4 4 4 2 2 0 0 15 15 20 20 10 10 15 15 10 10 5 5 5 5 frequency frequency time time ◮ Validating tv-ARMA models (for volatility forecasting)? Spectral Inference (Jun Yang) 5
Our Contribution ◮ A unified theory and methodology for the inference of evolutionary Fourier power spectra for a general class of locally stationary and possibly nonlinear processes. ◮ Simultaneous confidence regions (SCR) with asymptotically correct coverage rates are constructed for the evolutionary spectral densities on a nearly optimally dense grid of the joint time-frequency domain. ◮ A simulation based bootstrap method is proposed to implement the SCR. The SCR serves as a unified tool for a wide range of statistical inference problems in time-frequency analysis. Spectral Inference (Jun Yang) 6
Time-Frequency Inference ◮ Simultaneous Confidence Regions (SCR) Figure: SCR for SP500 daily returns Time-frequency Estimates 95% Confidence Region × 10 -4 × 10 -4 4 1 2 0 0 15 15 10 20 10 20 15 15 5 10 5 10 5 5 freq time freq time × 10 -4 × 10 -4 95% Confidence Region 95% Confidence Region 4 4 2 2 0 0 15 10 5 5 10 15 20 5 10 15 20 freq time time Spectral Inference (Jun Yang) 7
Time-Frequency Inference ◮ Hypothesis Testing (p-values) Table: Real Data: p-values for testing (a) stationarity, (b) time-varying white noise (TV White), (c) time-frequency separability (correlation stationarity). Stationarity TV White Noise Separability H 0 0 . 064 + 0 . 0011 ∗∗ 0 . 012 ∗ Earthquake 0 . 0005 ∗∗∗ 0 . 033 ∗ Explosion 0 . 61 0 . 0001 ∗∗∗ SP500 0 . 99 0 . 99 0 . 0004 ∗∗∗ 0 . 037 ∗ 0 . 048 ∗ SP500 (Abs) ( ∗ ∗ ∗ ) < 0 . 001 ≤ ( ∗∗ ) < 0 . 01 ≤ ( ∗ ) < 0 . 05 ≤ (+) < 0 . 1. Spectral Inference (Jun Yang) 8
Time-Frequency Inference ◮ Validating time-varying linear models Table: p-values for fitting time-varying parametric models to absolute SP500 Model p-value Model p-value 0 . 0066 ∗∗ 0 . 019 ∗ tv-AR(1) tv-ARMA(1 , 1) 0 . 0015 ∗∗ tv-AR(2) tv-ARMA(2 , 1) 0 . 79 0 . 0015 ∗∗ tv-AR(3) tv-ARMA(3 , 1) 0 . 77 0 . 0012 ∗∗ tv-AR(4) tv-ARMA(4 , 1) 0 . 78 0 . 0012 ∗∗ tv-AR(5) tv-ARMA(5 , 1) 0 . 84 ( ∗ ∗ ∗ ) < 0 . 001 ≤ ( ∗∗ ) < 0 . 01 ≤ ( ∗ ) < 0 . 05 ≤ (+) < 0 . 1. Spectral Inference (Jun Yang) 9
Theoretical Results Instantaneous spectral density Let u ∈ [0 , 1], the spectral density at u is defined by √ f ( u , θ ) := 1 � r ( u , k ) exp( − 1 k θ ) . 2 π k ∈ Z STFT-based spectral density estimator Let a ( · ) be an even, Lipschitz continuous kernel function with support [ − 1 , 1] and a (0) = 1; let B n be a sequence of positive integers with B n → ∞ and B n / n → 0, then the STFT-based spectral density estimator is defined by B n √ f n ( u , θ ) := 1 ˆ � r ( u , k ) a ( k / B n ) exp( ˆ − 1 k θ ) . (1) 2 π k = − B n Spectral Inference (Jun Yang) 10
Theoretical Results (cont.) Theorem (Y. and Zhou’18) Under certain conditions, one can have � | ˆ f n ( u , θ ) − E (ˆ f n ( u , θ )) | 2 n Pr max � 1 B n f 2 ( u , θ ) − 1 a 2 ( t ) d t ( u ,θ ) ∈G N − 2 log B n − 2 log C n + log( π log B n + π log C n ) ≤ x ] → e − e − x / 2 . ◮ Nearly optimal dense grids ( u , θ ) ∈ G N ◮ Bootstrap procedure based on the maximum deviation result. Spectral Inference (Jun Yang) 11
Summary Our Contribution ◮ A unified theory and methodology for the inference of evolutionary Fourier power spectra. ◮ Simultaneous confidence regions (SCR) with asymptotically correct coverage rates on a nearly optimally dense grid of the joint time-frequency domain. ◮ A simulation based bootstrap method is proposed to implement the SCR. ◮ The SCR serves as a unified tool for a wide range of statistical inference problems in time-frequency analysis. Reference ◮ Jun Yang and Zhou Zhou, Spectral Inference under Complex Temporal Dynamics , arXiv:1812.07706 Spectral Inference (Jun Yang) 12
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