Time series causality inference using the Phase Slope Index. Florin - PowerPoint PPT Presentation

Time series causality inference using the Phase Slope Index. Florin Popescu Guido Nolte Fraunhofer Institute FIRST, Berlin Mini-Symposium on Time Series Causality 1 Popescu NIPS 2009

Introduction  Linear time series analysis techniques can be useful in analyzing data that is actually generated by nonlin inear ear stochastic processes (i.e. in the real world).  Linear time series analysis can be conducted in the time domain (e.g. autoregressive models) or in the frequency domain (e.g. discrete Fourier transform, coherency among spectra) – theoretically both approaches are equivalent but numerically they are not. Causal estimation in time domain (AR): Granger 1973, Kaminski Blinowska 1991, Schreiber 2000, Rosenblum & Pikovsky 2001. Frequency domain method: Phase Slope Index (Nolte et al. 2008, Nolte et al. 2009) . Connection: partially directed coherence (Baccala & Sameshima 1998, 2001).  Separating correlation relation from causation sation is hard, even if the data is time-labeled. There can be correlations among non-interacting time-series variables. Mini-Symposium on Time Series Causality 2 Popescu NIPS 2009

Outline  Overview of different types of data generating processes (DGPs), which are stochastic generative models of time series  Highlight causality assessment challenges in neuroscience and economics.  AR estimation challenges for covariate innovations processes (needed for GC).  PSI - Phase Slope Index  PSI and AR results for bi-variate simulations available on Causality Workbench.  Structural causality estimation in multivariate time series. Mini-Symposium on Time Series Causality 3 Popescu NIPS 2009

DGP: Data Generating Process Data Generating Process DGP Symbolic representation u(t) o DGPs are abstractions of real-world dynamic processes which generate data: not necessarily are they regressive, recursive or stochastic, but are more powerful when they are. y(t-2) y(t-1) y(t) y(t) o They can be inferred ed from data directly or by bottom om-up up modeling of the underlying physical /social processes (in neuroscience, economics very hard) Mini-Symposium on Time Series Causality 4 Popescu NIPS 2009

Stochastic DGP Data Generating Process DGP Symbolic representation o If the DGP is stochastic and noise in an input it is generally called innovati tion ons s process cess and it is independently distributed if it is y(t-2) y(t-1) y(t) independently distributed. o If, also then the system u(t) is station onary. 5 Popescu NIPS 2009 Mini-Symposium on Time Series Causality

DGP equivalence Equivalence: DGP Symbolic representation 2 DGPs are output ut equivalent if, for all t : y(t-2) y(t-1) y(t) u(t) DGPs are stochastic hasticall ally equivalent if, for all t : z y 1 (t) Cano nonical nical representation (non-unique) u 1 (t) Mini-Symposium on Time Series Causality 6 Popescu NIPS 2009

DGP variations Potential DGP ‘upgrades’ DGP Symbolic representation o covariat ariate e or mixed innovati tion ons u 1 (t)  * u 2 (t) o endogen enou ous/ s/exogenous inputs o cointe tegration ration Mini-Symposium on Time Series Causality 7 Popescu NIPS 2009

DGP variations Data Generating Process DGP Symbolic representation o covariat ariate e or mixed innovati tion ons z y 1 (t) o endogen enou ous/ s/exogenous inputs some inputs are stochastic but observable le, or z (t) u 1 (t) non-stochastic, or excluded from potential effects o co co-inte ntegratio ration d(t) Mini-Symposium on Time Series Causality 8 Popescu NIPS 2009

DGP variations Potential DGP ‘upgrades’ DGP Symbolic representation o covariat ariate e or mixed innovati tion ons z z y 1 (t) o endogen enou ous/ s/exogenous inputs some inputs are stochastic but observable le, or z (t) u 1 (t) simply non-stochastic o co co-inte ntegratio ration d(t) Some states are simple dynamic transformations of i.i.d processes -this can be taken into account Mini-Symposium on Time Series Causality 9 Popescu NIPS 2009

Structural / G - Causality G - Causality DGP Symbolic representation Granger causality inference z requires derivation of a predictive model (can of worms…) y 1 (t) u 1 (t) u 2 (t) u 2,0 (t) y 2 (t) y 2 (t) z z Mini-Symposium on Time Series Causality 10 Popescu NIPS 2009

Structural / G - Causality G-causality DGP Symbolic representation o G-causality is inferred by comparing conditional entropy in competing structural models z z y 1 (t) u 1 (t) y 2 (t) 1  2 1  2 u 2 (t) 1  2 1  2 z z Mini-Symposium on Time Series Causality 11 Popescu NIPS 2009

Structural / G - Causality Covariate innovations? DGP Symbolic representation o In many instances it is reasonable to z assume that the innovations process is y 1 (t) covariate. For example: yearly weather variability and historical shocks on aggregate indicators. u 1 (t) o Also possible is that other u 2 (t) unobservable factors actually provide root causes for correlations among y 2 (t) innovations processes. z Mini-Symposium on Time Series Causality 12 Popescu NIPS 2009

Structural / G - Causality Mixed outputs: EEG DGP Symbolic representation y 1 (t) o In some instances it is the physical process of observation that separates us from the time-series z of interest. For example cortical sources and scalp based sensors (the mixing problem). x 1 (t) x 2 (t) z y 2 (t) Mini-Symposium on Time Series Causality 13 Popescu NIPS 2009

Structural / G - Causality Stochastic equivalence DGP Symbolic representation y 1 (t) o It is also possible that there is both a non- diagonal observation matrix and covariate noise z but these situations correspond to stochastically equivalent DGPs and cannot be disambiguated without further assumptions Covariate innovations  * Mixed output z R is a rotation matrix y 2 (t) S is a diagonal (scaling) matrix Mini-Symposium on Time Series Causality 14 Popescu NIPS 2009

Structural / G - Causality Stochastic equivalence DGP Symbolic representation y 1 (t) o It is also possible that there is both a non- diagonal observation matrix and covariate noise z but these situations correspond to stochastically equivalent DGPs and cannot be disambiguated without further assumptions Covariate innovations  * Mixed output z R is a rotation matrix y 2 (t) S is a diagonal (scaling) matrix Mini-Symposium on Time Series Causality 15 Popescu NIPS 2009

Structural / G - Causality Noise covariance estimation DGP Symbolic representation o Instantaneous mixing / innovations covariance can be used to establish „source‟ causality z y 1 (t) (Moneta 2008), (to follow!) o If a triangular structure is imposed on the instantaneous „mixing‟ matrix of a linear SVAR the estimate of the equivalent noise covariance is unbiased (Popescu, 2008) y 2 (t) z Mini-Symposium on Time Series Causality 16 Popescu NIPS 2009

Structural / G - Causality Data Generating Process DGP Symbolic representation K      x A x A x b S e  n U ,0 n U i , n i U U U n , z  i 1 y 1 (t)   Zero-lag AR system A 0 if q p strictly upper diagonal : U ,0, , p q Can be solved by standard 2-norm linear regression Strictly upper diagonal means resulting residuals are not correlated. K         1 1 1 S ( I A ) x S A x S b e  U U n U U i n i U U U n ,0 , ,  i 1 K        T 1 1 U V x S A x S b e  U U U n U U i , n i U U U n ,  i 1   K             T T T T 1 1 1 1 1 V x U S A x U S b U e  U U U U U U i , n i U U U U U U U n ,  i 1  T y V x n n K             1 T 1 1 T 1 1 Mixed output y U S A V y U S b e  n U U U U i , U n i U U U U U n y 2 (t)  z i 1 Mini-Symposium on Time Series Causality 17 Popescu NIPS 2009

Phase slope index Basic principle: mixing does not affect the imaginary part of the complex coherency of a multivariate time series (Nolte 2004) z z z z Mini-Symposium on Time Series Causality 18 Popescu NIPS 2009

Phase slope index Let us consider the case of a dynamically interacting system with correlated noise observations z z Mini-Symposium on Time Series Causality 19 Popescu NIPS 2009

Phase slope index Let us consider the case of a dynamically interacting system with correlated noise observations. Relative influence of covariate noise? z z Mini-Symposium on Time Series Causality 20 Popescu NIPS 2009

Phase Slope Index 1 0.8 f nyqu 0.6 f nyquist 0.4  0.2 im(coherency) Im(coherency) f nyquist /2 F nyquist 0.5f nyquist 0 0 -0.2 Complex coherency C ij ij is -0.4 calculated from the complex -0.6 spectral density S ij 0 0 ij -0.8 -1 -1 -0.5 0 0.5 1 re(coherency) Re(coherency) Mini-Symposium on Time Series Causality 21 Popescu NIPS 2009

Phase slope index 1 1 0.8 0.8 0.6 0.6 0.4 0.4 Im(coherency) Im(coherency) 0.2 0.2 im(coherency) im(coherency) 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 re(coherency) re(coherency) Re(coherency) Re(coherency) Mini-Symposium on Time Series Causality 22 Popescu NIPS 2009