1. Frequency Domain Causality Analysis Method for Multivariate Systems in Hypothesis Testing Framework Hao Ye Department of Automation, Tsinghua University

2. Outline A Brief Introduction to Causal Analysis A Brief Introduction to Causal Analysis Two Problems of PDC Two Problems of PDC Frequency Domain Causal Analysis Methods Based on Two Statistics Frequency Domain Causal Analysis Methods Based on Two Statistics Simulation Examples Simulation Examples Concluding Remarks Concluding Remarks

3. A Brief Introduction to Causal Analysis Causal relationship among the time series of an industrial process is of great use for fault detection, alarm management, synthesis design, and modeling, etc. Causal relationship among the time series of an industrial process is of great use for fault detection, alarm management, synthesis design, and modeling, etc. But it is often complicated and unknown in case of lack of a perfect knowledge of system structure But it is often complicated and unknown in case of lack of a perfect knowledge of system structure Causality Analysis methods Causality Analysis methods Time domain Time domain Frequency domain Frequency domain Wiener(1956): Concept based on data Granger(1963): Granger causality Geweke(1982): Conditional Granger causality Kaminski(1991): Frequency domain Granger causality Baccala(2001) ： Partial directed coherence (PDC) Wide application Some issues still need further discussions

4. , A Brief Introduction to Causal Analysis

5. Two Problems of PDC, Baccala and Sameshima (2001) : PDC has the ability to rank the relative interaction strength with respect to a given signal source because of the normalization. Reference: (Barrett and Seth, 2009): Granger causality can measure the strength Problem 1: PDC cannot correctly rank the causal strength

6. Two Problems of PDC Few conclusions about what information the distribution of PDC in frequency domain can further offer. It is natural to guess that it represent how the strength of causal influence changes with ω. monotonically decreases as ω grows monotonically increases as ω grows Can be extended to a general first order multivariate system Problem 2: PDC cannot describe how the causal strength changes with frequency Only affects the dynamics of x 5 Does not affect the qualitative causal relationship

7. Frequency domain Causal Analysis Methods based on Two Statistics Schelter et al. (2009): Renormalized PDC (RPDC) Schelter et al. (2009): Renormalized PDC (RPDC) Schelter et al. (2009): Statistical property of RPDC Schelter et al. (2009): Statistical property of RPDC Schelter et al. (2009): Detection rule Schelter et al. (2009): Detection rule How to measure the strength of causality from x j to x i at ω was not discussed Baccala and Sameshima (2001) : Due to the normalization, PDC(x j →x i ) may change if more (or less) signals are influenced by x j, reflects the relative rather than the absolute strength of influence

8. Smaller probability under null hypothesis Greater probability under null hypothesis Stronger causal strength Weaker causal strength RPDC can be directly used to measure the strength of causality from x j to x i at ω measure the strength of the causality between each pair of time series Frequency domain Causal Analysis Methods based on Two Statistics

9. Schelter et al. (2005): Schelter et al. (2005): Statistical property Statistical property Measure the strength of causality from x j to x i at ω Measure the strength (and the existence) of the causality between each pair of time series Frequency domain Causal Analysis Methods based on Two Statistics (Schelter et al.,2005) : To solve the over fitting problems in model estimation Lower computation load compared with RPDC

10. Simulation Examples

11. , 0.2002 0.3932 0.20990.3056 0.10880.11750.0998 0.1951 Granger causality 24.5012 39.2280 27.701032.0942 16.268419.676515.7450 22.9304 S PRDC 33.0849 43.8977 38.869839.5011 22.102326.995519.9758 32.8947 SϒSϒ The calculated strengths based on S PRDC and S ϒ are consistent with those given by Granger causality

12. Simulation Examples monotonically decreases as ω grows monotonically increases as ω grows The distributions of RPDC or of these two processes are roughly similar as expected

13. Frequency ω/0.01 π (0 to π ) Granger causality The distribution of Granger causality in the frequency domain is consistent with those given by PRDC and ϒ -statistics Simulation Examples

14. Concluding Remarks Discussion of Problem 1: PDC cannot correctly rank the causal strength (contribution) Discussion of Problem 2: PDC cannot describe how the causal strength changes with frequency (contribution ) Baccala and Sameshima (2001) : PDC reflects the relative rather than the absolute strength of influence Solve the two problems respectively (clear physical meaning in the hypothesis testing framework, simulation examples ) Complex computation (Schelter et al.,2005) : To solve the over fitting problems in model estimation Similar advantages to PRDC Simpler computation

15. Concluding Remarks 1.Zhang J, Yang F, Ye H. Frequency domain causality analysis method for multivariate systems in hypothesis testing framework. The 19 th IFAC World Progress, 2014