Spectral Inference under Complex Temporal Dynamics Jun Yang joint - - PowerPoint PPT Presentation

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

• ver 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?

time 500 1000 1500 2000

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 Earthquake Data 10 Time-frequency Estimates time 6 2 5 freq 10 15 0.02 0.04 0.01 0.03 time 2 4 6 8 10 12 freq 2 4 6 8 10 12 14 16 Time-frequency Estimates time 500 1000 1500 2000

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 Explosion Data 10 Time-frequency Estimates time 6 2 5 freq 10 15 0.03 0.02 0.01 time 2 4 6 8 10 12 freq 2 4 6 8 10 12 14 16 Time-frequency Estimates

◮ Is Explosion time-frequency separable?

H0 : f (u, θ) ∝ g(u)h(θ).

Spectral Inference (Jun Yang) 3

SP500 Daily Returns

time 1000 2000 3000 4000 5000 6000

• 0.1
• 0.05

0.05 0.1 0.15 Daily Returns of SP500 20 15 Time-frequency Estimates time 10 5 5 freq 10 ×10-4 15 0.5 1 1.5 time 5 10 15 20 freq 2 4 6 8 10 12 14 16 Time-frequency Estimates

◮ Time-varying white noise?

H0 : f (u, θ) ∝ g(u).

◮ Volatility forecasting (goodness of fit test)

time 1000 2000 3000 4000 5000 6000

• 0.02

0.02 0.04 0.06 0.08 0.1 Absolute SP500 20 15 Time-frequency Estimates time 10 5 5 freq 10 ×10-4 15 1 2 time 5 10 15 20 freq 2 4 6 8 10 12 14 16 Time-frequency Estimates

Spectral Inference (Jun Yang) 4

Absolute SP500 Daily Returns

20 15 time 10 tv-AR(1) 5 5 frequency 10 15 ×10-4 2 1 20 15 time 10 tv-AR(4) 5 5 frequency 10 15 ×10-4 2 4 20 15 time 10 tv-AR(5) 5 5 frequency 10 15 ×10-4 4 2 20 15 time 10 tv-ARMA(1,1) 5 5 frequency 10 15 ×10-4 0.5 1 20 15 time 10 tv-ARMA(2,1) 5 5 frequency 10 15 ×10-4 4 2 20 15 time 10 tv-ARMA(3,1) 5 5 frequency 10 15 ×10-4 4 2

◮ 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

20

Time-frequency Estimates time

15 10 5 5

freq

10 ×10-4 15 1 20

95% Confidence Region time

15 10 5 5

freq

10 ×10-4 15 2 4

95% Confidence Region

20 15

time

10 5 5 10 15 ×10-4 2 4

freq 95% Confidence Region

20 15

time

10 5 ×10-4 2 4

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).

H0 Stationarity TV White Noise Separability Earthquake 0.0011∗∗ 0.012∗ 0.064+ Explosion 0.0005∗∗∗ 0.033∗ 0.61 SP500 0.0001∗∗∗ 0.99 0.99 SP500 (Abs) 0.0004∗∗∗ 0.037∗ 0.048∗ (∗ ∗ ∗) < 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 tv-AR(1) 0.0066∗∗ tv-ARMA(1, 1) 0.019∗ tv-AR(2) 0.0015∗∗ tv-ARMA(2, 1) 0.79 tv-AR(3) 0.0015∗∗ tv-ARMA(3, 1) 0.77 tv-AR(4) 0.0012∗∗ tv-ARMA(4, 1) 0.78 tv-AR(5) 0.0012∗∗ 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 2π

• k∈Z

r(u, k) exp( √ −1kθ).

STFT-based spectral density estimator

Let a(·) be an even, Lipschitz continuous kernel function with support [−1, 1] and a(0) = 1; let Bn be a sequence of positive integers with Bn → ∞ and Bn/n → 0, then the STFT-based spectral density estimator is defined by ˆ fn(u, θ) := 1 2π

Bn

• k=−Bn

ˆ r(u, k)a(k/Bn) exp( √ −1kθ). (1)

Spectral Inference (Jun Yang) 10

Theoretical Results (cont.)

Theorem (Y. and Zhou’18)

Under certain conditions, one can have Pr

• max

(u,θ)∈GN

n Bn |ˆ fn(u, θ) − E(ˆ fn(u, θ))|2 f 2(u, θ) 1

−1 a2(t)dt

−2 log Bn − 2 log Cn + log(π log Bn + π log Cn) ≤ x] → e−e−x/2.

◮ Nearly optimal dense grids (u, θ) ∈ GN ◮ 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