Time- varying signals: cross- and auto- correla5on, correlograms. NEU 466M Instructor: Professor Ila R. Fiete Spring 2016

Transcription

1 Time- varying signals: cross- and auto- correla5on, correlograms NEU 466M Instructor: Professor Ila R. Fiete Spring 2016

2 Sta5s5cal measures We first considered simple sta5s5cal measures for single variables (mean, variance). We next considered measures for the rela5onship between two (sta5onary) random variables (covariance, Pearson s correla5on coefficient; regression). Extension to K variables: pairwise rela5onships (covariance matrix). Now, extension to 5me- series: rela5onships between different 5me- varying signals.

3 Time- series data { g t 1,g t,g t+1 } { h t 1,h t,h t+1 } g: a temporally varying signal sampled at discrete intervals h: another 5me- varying signal, sampled at the same 5mes How many variables? In g alone?

4 Time- series data { g t 1,g t,g t+1 } g: a temporally varying signal sampled at discrete intervals Time- series not sta5onary. Could think of response at each 5me point as separate (though typically not independent) variable. If length(g) = T, then T variables in g. Same 5me- point in repe55ons of the series from same ini5al condi5on: mul5ple samples of that variable.

5 Finding structure between 5me- series { g t 1,g t,g t+1 } { h t 1,h t,h t+1 } g: a temporally varying signal sampled at discrete intervals h: another 5me- varying signal, sampled at the same 5mes How about trying previously seen sta5s5cal measures? compute cov(g, h)

6 1 0.8 Example of two 5me- series dt = t =[0 : dt : 1 dt]; g = sin(20 pi t); h = cos(20 pi t); figure; hold on; plot(g); plot(h, 0 r 0 ); ylabel( 0 response 0 ) xlabel( 0 time 0 ) response time C(g, h) =0, hghi 0 Correct but unsa5sfying: g, h similarly 5me- varying func5ons: the same func5on with a π/2 shi[.

7 Defini5on: cross- correla5on func5on C g,h (n) = 1X m= 1 g (m)h(m + n) n =0: [ g 3 g 2 g 1 g0 g1 g2 g3 ] [ h 3 h 2 h 1 h 0 h 1 h 2 h 3 ] take time-by-time product, add all terms

8 Defini5on: cross- correla5on func5on 1X C g,h (n) = g (m)h(m + n) m= 1 n =1: [ g 3 g 2 g 1 g 0 g 1 g 2 g 3 ] [ h 2 h 1 h 0 h 1 h 2 h 3 h 4 ] h tape shifted leftwards by 1

9 Measure of relatedness at different 5me shi[s C g,h (n) = 1X m= 1 g (m)h(m + n) The cross- correla5on func5on is a measure of covariance between g, h at different rela5ve 5me- shi[s (for zero- mean or mean- subtracted signals). How is g at any 5me (linearly) related to h n 5me- steps away?

10 Cross- correla5on func5on for finite- length signals {g 1,,g N } {h 1,,h N } g, h: 5me- series of length N C g,h (n) = NX n m=1 g (m)h(m + n) average over (N- n ) terms Total length of cross- correla5on: 2N- 1 Zero- shi[ed entry: N

11 Proper5es of the cross- correla5on C gh (n) 6= C hg (n): does not commute (contrast with covariance). In fact, C gh (n) =C hg ( n) : shi[ing h to right rela5ve to g: equivalent to shi[ing g to le5 rela5ve to h. (Same plot, flipped 5me axis.) Ordering mabers: tells which signal leads the other.

12 (Previous) example response time 5me index

13 Cross- correla5on of example series 500 figure; plot(xcorr(h, g), 0 k 0 ) xlabel( 0 time index (n) 0 ) ylabel( 0 C gh (n) 0 ) C gh (n) time index (n) 5me index Signals of length N = Cross- correla5on of length 2N- 1.

14 500 Cross- correla5on of example series Matlab: signals of length N have 0- shi[ at N in xcorr figure; plot(xcorr(h, g), 0 k 0 ) xlabel( 0 time index (n) 0 ) ylabel( 0 C gh (n) 0 ) C gh (n) time index (n) In C gh (n): peak at le[. Interpreta5on: h leads g, or h must be shi[ed right (nega5ve n) to line up with g. In present example, cosine (h) leads sine (g) in phase. Mul5ple peaks: periodic re- alignment of cos with sin at mul5ples of period Cau5on! C gh (t) is xcorr(h,g) in Matlab: note reversed order of g, h!

15 Cross- correla5on of example series 500 figure; plot(xcorr(h, g), 0 k 0 ) xlabel( 0 time index (n) 0 ) ylabel( 0 C gh (n) 0 ) C gh (n) time index (n) 5me index Decay in amplitude due to finite length of g, h: shi[ n is a sum over N- n terms, so amplitude will go to 0 as shi[ goes to N.

16 Autocorrela5on func5on Special case of cross- correla5on: signal correla5on with itself at all 5me shi[s. Commonly used to detect temporal paberns (periodic or otherwise) within noisy 5me- series data. Symmetric; central peak always at 0 5me- lag.

17 Autocorrela5on example C gg (n) time index (n) 5me index

18 Overview BACK TO A MODELING PERSPECTIVE

19 Time- series data {g 1,,g N } g: a temporally varying signal sampled at discrete intervals If g is 5me- series of length N, then N variables within g. But then should construct NxN covariance matrix with (α,β) entry given by cov(g α g β ), and N(N+1)/2 dis5nct entries. Autocorrela5on: only (2N- 1)/2 dis5nct entries (1/2 because of symmetry about 0-5me lag).

20 Autocorrela5on and 5me- series data What are we throwing out when studying only the autocorrelaion of a Ime- series g (N disinct entries), compared to the NxN covariance matrix of its components (N(N+1)/2 disinct entries)?

21 Autocorrela5on and 5me- series data One entry in cross- correla5on measures rela5onship between terms in g at a fixed 5me- lag, summed over all =mes. Assump5on of =me transla=on- invariance: rela5onship between terms in g at lag n is similar, regardless of star5ng 5me. (E.g. whenever a cell spikes, it will tend not to spike for 2 ms: refractory period. Pabern independent of actual spike 5me.) Same assump5on in cross- correla5on.

22 Summary Time- series inherently more complex than random draws from a sta5onary system. Look for: transla5on- invariant temporal paberns within and across 5me- series. Auto- and cross- correla5on func5ons. Each term an average across the en5re 5me- series: reduced noise.

23 Applica5on SPIKE TRAINS: CROSS- CORRELATION AND SIMPLE STATISTICS

24 Mo5on detec5on in the blowfly Image by Muhammad Mahdi Karim, published under GNU Free Documenta5on License, Version 1.2

25 H1 neuron: horizontal mo5on sensing Single H1 neuron in lobula plate of each hemisphere Lobula plate: highest visual area, before motor output.

26 H1 response during horizontal visual mo5on Data: Rob de Ruyter van Steveninck 500 Hz, spikes and whole- field horizontal mo5on s5mulus stim, H1 spks sample number (500 Hz)

27 Spike autocorrela5on x C(rho, rho) sample number x 10 5

28 Spike autocorrela5on x 10 4 Large narrow peak at zero. Why? What does its height mean? C(rho, rho) sample number x 10 5

29 Spike autocorrela5on x C(rho, rho) sample number x 10 5 Sharp narrow dip. What does this mean? Where could it come from?

30 Spike autocorrela5on x 10 4 C(rho, rho) Posi5ve slowly decaying transient. What does this mean? Where could it come from? sample number x 10 5

31 S5mulus autocorrela5on x C(stim, stim) sample number x 10 5

32 S5mulus autocorrela5on Medium- width peak at zero. What does its height mean? x C(stim, stim) sample number x 10 5

33 S5mulus, spike autocorrela5ons C(rho, rho), C(stim, stim) Can s5mulus correla5ons explain spike correla5ons? sample number x 10 5

34 S5mulus, spike cross- correla5on 16 x When s5mulus goes up (more posi5ve velocity), response increases: posi5ve peak. 10 C(stim, rho) Response lags s5mulus (peak to right of zero- shi[) sample number (500 Hz) x 10 5

35 S5mulus, spike cross- correla5on 16 x C(stim, rho) sample number (500 Hz) x 10 5 What does s5mulus, response cross- correla5on really mean, from modeling perspec5ve?

36 Back to original goal: Modeling WHAT DOES IT MEAN TO BUILD A MODEL OF OBSERVATIONS IN THIS EXPERIMENT?

37 Modeling Rela5vely simple/compact descrip5on of data, good predic5on performance. Extrac5ng features of data as a way to model it. To determine predictability, important to cross- validate models/fits.

38 Modeling spike train data Model: Simple, predic5ve descrip5on. But of what? Given s5mulus, predict spikes? Given spikes, predict s5mulus?

39 Modeling spike train data Model: Simple, predic5ve descrip5on. But of what? Given s5mulus, predict spikes? Encoding model Given spikes, predict s5mulus? Decoding model Yes, both!

40 Summary Autocorrela5ons of s5mulus, response tell us about structure within s5mulus, response. Comparison of the autocorrela5ons helps understand differences in the structure and mo5vates us to search for causes for these differences. Cross- correla5on tells us about some rela5onships between s5mulus and response: 5me- lags, sign of rela5onship, etc. Beber understanding of what s5mulus, response cross- correla5on is telling us?