10/8/2014. The Multivariate Gaussian Distribution. Time-Series Plot of Tetrode Recordings. Recordings. Tetrode

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1 10/8/ INTRODUCTION TO STATISTICS FOR BRAIN AND COGNITIVE SCIENCES Lecture 4 Emery N. Brown The Multivariate Gaussian Distribution Case : Probability Model for Spike Sorting The data are tetrode recordings (four electrodes) of the peak voltages (mv) corresponding to putative spike events from a rat hippocampal neuron recorded during a texture-sensitivity behavioral task. Each of the 15,600 spike events recorded during the 50 minutes is a four vector. The objective is to develop a probability bilit model to describe the cluster of spikes events coming from a single neuron. Such a model provides the basis for a spike sorting algorithm. Acknowledgments: Data provided by Sujith Vijayan and Matt Wilson Technical Assistance Julie Scott Tetrode Recordings Time-Series Plot of Tetrode Recordings Neuron 1

2 10/8/014 Six Bivariate Plots of Tetrode Channel Recordings Histograms of Spike Events By Channel Channel 1 Channel Channel 3 Channel 4 Box Plots of Spike Events By Channel DATA: The Tetrode Recordings Voltage (V) x k,1 x k, x k x k,3 x k,4 Four peak voltages recorded on the k-th spike event for k = 1,, K, where K is the total number of spike events. Channel Channel 4 Channel 1 Channel 3

3 10/8/014 GAUSSIAN PROBABILITY MODEL Four-Variate Gaussian Model Mean ( k,w )= exp - ( x k - f x ) W (x k - ) ( W Covariance Matrix (symmetric) k 1,..., K =( 1,, 3, 4 ) W = N-Multivariate Gaussian Model Factoids 1. A Gaussian probability density is completely defined by its mean vector and covariance matrix.. All marginal probability densities are univariate Gaussian. 3. Frequently used because it is i) analytically and computationally tractable ii) suggested by the Central Limit Theorem 4. Any linear of the components is Gaussian (a characterization). Central Limit Theorem N-Multivariate Gaussian Model Factoids The distribution of the sum of random quantities such that the contribution of any individual quantity goes to zero as the number of quantities being summed becomes large (goes to infinity) will be Gaussian. Cross-section is an ellipse Bivariate i Gaussian Distribution ib ti Marginal distribution is univariate Gaussian Cumulative Distribution Function 3

4 10/8/014 Univariate Gaussian Model Factoids Gaussian Probability Density Function 1 1(x) f( x ) ( ) exp{ }. Standard Gaussian Probability Density Function f ( x ) ( ) exp{ x }. Standard Cumulative Gaussian Distribution Function x 1 1 ( x ) ( ) exp{ u } du. Univariate Gaussian Model Factoids Mu is the mean (location) Standard deviation (scale) Any Gaussian distribution can be converted into a standard Gaussian distribution (mu = 0, sd =1) 68% of the area within ~ 1 sd of mean 95% of the area within ~ sd of mean 99% of the area within ~.58 sd of mean ESTIMATION Joint Distribution of the Four-Variate Gaussian Model K K 1 1 K f( x,w )= f( x k,w )= exp - (x k - ) W -1 (x k - ) k=1 ( W k=1 where x = ( x 1,..., x K ) Log Likelihood ESTIMATION For Gaussian observations the maximum likelihood and method-of-moments estimates are the same. Sample Mean Sample Variance -1 K -1 K ˆi = K k1 x k,i ˆi = K k= 1 (x k,i ˆi ) = Sample Covariance Sample Correlation K 1 K log f( x,w ) = -Klog( ) - log W - k=1 (x k - )W( x k - ) where K is the number of spike events in the data set. ˆi, j = K -1 k K = 1 (x k,i ˆi )(x k, j ˆ j ) for i = 1,, 4 and j = 1,,4. ˆ i, j ˆ i, j = 1 ˆ i ˆ j 4

5 10/8/014 CONFIDENCE INTERVALS FOR THE PARAMETER ESTIMATES OF THE MARGINAL GAUSSIAN DISTRIBUTIONS The Fisher Information Matrix is where, ) i The confidence interval is L I ( ) = -E K / I( )= 4 K / ii ± zα I ( θ ) īi 1 Four-Variate Gaussian Model Parameter Estimates Sample Mean Vector Sample Covariance Matrix e - 07 x Sample Correlation Matrix Marginal Gaussian Parameter Estimates and Confidence Intervals (An Exercise: Compute the Confidence Intervals) Histograms of Spike Events By Channel Sample Mean Vector Channel 1 Channel Sample Variances 1.0 e - 07 x Channel 3 Channel 4 5

6 10/8/014 Empirical and Model Estimates of Marginal Cumulative Probability Densities Six Bivariate Plots of Tetrode Channel Recordings With 95% Probability Contour GOODNESS-OF-FIT Q-Q Plots Q-Q Plots Kolmogorov-Smirnov Tests A Chi-Squared Test Separate the bivariate data into deciles and compute 10 (O i - E i ) 9 ~ d = 1 O i where O i is the observed number of observation in decile i and E i is expected number of observations in decile i. Channel 1 Channel Channel 3 Channel 4 6

7 10/8/014 K-S Plots Six Bivariate Plots of Tetrode Channel Recordings With 95% Probability Contour Channel 1 Channel Channel 3 Channel 4 Bivariate Plots of Tetrode Channel Recordings Bivariate Plots of Tetrode Channel Recordings with Gaussian Equiprobability Contours (An Exercise: Carry Out the Chi-Squared Test) Channel 4 vs 1 Channel 3 vs Channel 4 vs 1 Channel 3 vs 7

8 10/8/014 Linear Combinations of Gaussian Random Variables are Gaussian Linear Combination Analysis If where and where then X ~ N (, W) X ( x1, x, x3, x 4 ) 4 Y c x i i i1 c ( c1, c, c3, c 4 ) 4 Y ~ N (ci i,c 'Wc) i1 Channel Linear Combination CONCLUSION The data seem well approximated with a four-variate Gaussian model. Epilogue Another real example of real Gaussian data in neuroscience data? The marginal probability density of Channel 4 is the best Gaussian fit. The Central Limit Theorem most likely explains why the Gaussian model works here. 8

9 10/8/014 Analysis of Background Magnetoencephalogram Noise Magnetoencephalogram Background Noise Boxplot Q-Q Plot Why are these data Gaussian? Answer: Central Limit Theorem Courtesy of Simona Temereanca MGH Martinos Center for Biomedical Imaging 9

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