Old and New Methods in the Age of Big Data

Transcription

1 Old and New Methods in the Age of Big Data Zoltán Somogyvári1,2 1Department of Theory Wigner Research Center for Physics of the Hungarian Academy of Sciences 2National Institute for Clinical Neuroscience

2 How do we find structures behind the data?

3 Transformation of time series into connections

4 How to find connection between data series? The traditional method: Correlation (more precisely, the linear correlation coefficient) USD vs GBP Exchange rate 2*EUR vs GBP Days

5 How to find connection between data series? USD vs GBP The traditional method: Correlation (more precisely, the linear correlation coefficient) R=0.6 2*EUR vs GBP

6 What does the correlation tells us? Problem 1: it is possible, that there is a clear connection between the two time series, but the correlation is 0 because of the non-linear form of connection.

7 Causality or common cause? Is the temporal delay shows us the direction of the causality? EUR vs GBP leads USD vs GBP leads Cross correlation function: Correlation between delayed signals Unfortunately not, because it assumes, that we observe the two Signals with the same delays. Delay

8 Is there any way to infer causality?

9 Granger-causality The original idea came from Norbert Winer x y, if the inclusion of past x values improves the prediction quality on y? Clive Granger Publication 1969 Nobel price in Economic Sciences 2003

10 ? Granger-causality Linear autoregression: Presumtions: Stationary processes Zero-mean Uncorrelated Gaussian noise We have data of every important valiable

11 Application rat hippocampus Data: Local Field Potential Microelectrode-array 256 channels 20 khz freq Information stream in the hippocamus options: State-dependent differences ie. sleep-awake Event-related Spike-related information transfer

12 Problems with the Granger-causality Uses linear models (there are nonlinear extensions) Assumes weak interactions (separability) Unreliable results in case of circular causality Has problems in deterministic (non-stochastic) cases

13 Cross Convergence Map: A new framework for causality analysis Science 338, 496 (2012) A new approach, promising Detection of circular causality Causality in nonlinear system Deterministic (chaotical) system

14 The model system: The logistic map xn+1=rxxn(1-xn) A one dimensional, discreet-time dynamical system implementing stretching an folding transformations.

15 The model system: The logistic map It can exhibit different dynamical behavior, from stable fixpoint, through periodic oscillations to chaos, depending on the parameter r.

16 Taken's time delay embeding theorem The trajectory reconstructed in the state space is topologically equivalent With the trajectory of the system's original trajectory in its real space. First coordinate: Second coordinate: Third coordinate:.... the data itself the data delayed by tau the data delayed by 2 tau

17 Our model system: Two coupled logistic maps xn+1=xn(rx(1-xn)+byxyn) yn+1=yn(ry(1-yn)+bxyxn) rx=ry=3.8 so both maps are in the chaotic regime

18 Phase-space reconstruction based on delayed maps xn+1=xn(rx(1-xn)+byxyn) first data series in 3 embedding dimension yn+1=yn(ry(1-yn)+bxyxn) second data series in 3 embedding dimension Both dataset formed a 2D manifold in the 3D embedding space

19 In case of causal connections, the the reconstructed manifold sholud be topologically equivalent according to the Takens' theorem. But, how to test it? first data series in 3 embedding dimension second data series in 3 embedding dimension Both dataset formed a 2D manifold in the 3D embedding space

20 Sugihara's method: Convergent Cross mapping Choose a point first data series in 3 embedding dimension second data series in 3 embedding dimension Both dataset formed a 2D manifold in the 3D embedding space

21 Sugihara's method: Convergent Cross mapping Find its neighborhood first data series in 3 embedding dimension second data series in 3 embedding dimension Both dataset formed a 2D manifold in the 3D embedding space

22 Sugihara's method: Convergent Cross mapping Find the same time points in the other state space first data series in 3 embedding dimension second data series in 3 embedding dimension Lets do it for many points! If the neighbors in the first space are neighbors in the the second space as well, then the second variable is causal to the first one.

23 Sugihara's method: Convergent Cross mapping In case of circular causality the mapping should work in both directions first data series in 3 embedding dimension second data series in 3 embedding dimension Let us do it into the other direction!

24 Sugihara's method: Convergent Cross mapping The chosen point first data series in 3 embedding dimension second data series in 3 embedding dimension Let us do it into the other direction!

25 Sugihara's method: Convergent Cross mapping The neighborhood first data series in 3 embedding dimension second data series in 3 embedding dimension Let us do it into the other direction!

26 Sugihara's method: Convergent Cross mapping Mapping first data series in 3 embedding dimension second data series in 3 embedding dimension The mapping worked well into both directions! This is the sign of circular causality.

27 Cross mapping in case of unidirectional interactions How can be the topological equivalence is an asymmetric relation? xn+1=xn(rx(1-xn)+byxyn) first data series in 3 embedding dimension yn+1=ryyn(1-yn) second data series in 3 embedding dimension While the first dataset formed a 2D manifold, the second dataset resulted an only 1D manifold in the 3D embedding space!

28 Cross mapping in case of unidirectional interactions How can be the topological equivalence is an asymmetric relation? Mapping works well in this direction first data series in 3 embedding dimension second data series in 3 embedding dimension While the first dataset formed a 2D manifold, the second dataset resulted an only 1D manifold in the 3D embedding space!

29 Cross mapping in case of unidirectional interactions How can be the topological equivalence is an asymmetric relation? But spread out in the other direction! first data series in 3 embedding dimension second data series in 3 embedding dimension The mapping worked well from x to y but failed from y to x, showing, that y is causal to x but x is not causal to y.

30 MRI with implanted electrodes 4*8 channels in the grid plus 2*8 channels In two strip electrodes, 1024 Hz sampling

31 Electric Potential [mv] EEG signal of an epileptic seizure recorded on 48 channels Time [ms]

32 Electric Potential [mv] The initiation of the seizure Time [ms]

33 It is clear, that the right hippocampus has large LFP effect to the Left hippocampus, while there is only mild effect in the backward direction. Left FO This seizure appeared only on in the right hippocampus. Right FO Connection dynamics during seizure Causality Right Left Left Right Time [s]

34 Connection dynamics in seizure Right FO The seizure was more pronounced in the left hippocampus, Although, The right hippocampus drove the left during the first period of the seizure, then a circular connection structure emerged. Left FO LFP Causality May help in surgical preparation Right Left Left Right Time [s]