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Statistical analysis and modeling of neural data Lecture 4 Bijan Pesaran 17 Sept, 2007
Goals Develop probabilistic description of point process. Characterize properties of observed sequences of events. Illustrate more applications of non- parametric estimates
Recap Non-parametric histogram estimates
Recap Linear, Gaussian model for neuronal response Input CovarianceSpike-triggered sum Receptive field
Recap Polynomial model Problem: Can’t fit higher than 2 nd order model because dimensionality of parameter space too high.
Parametric formulation Non-parametric formulation Basis function for every data point
Bias-variance trade-off = Bias^2 + Variance Cross-validation Score
Density estimation Estimate with as few assumptions as possible
Density estimation Estimate with as few assumptions as possible Cross-validation Score
Risk decreases to zero Histogram estimate converges like Kernel estimate converges like 0.21, 0.05, 0.01 0.16, 0.03, 0.004
Recap Linear, non-linear model Non-linearity 1D scalar function
Recap Linear, non-linear, Poisson model Poisson spike generator
Poisson process – Interval function Waiting time Probability density
Poisson process – Intensity function
Integrate and fire neuron with Poisson inputs so
Wait for k events with rate
Renewal process Independent intervals Completely specified by interspike interval density
Characterization of renewal process Parametric: Model ISI density. –Choose density function, Gamma distribution: –Maximize likelihood of data No closed form. Use numerical procedure.
Characterization of renewal process Non-parametric: Estimate ISI density –Select density estimator –Select smoothing parameter
Non-stationary Poisson process – Intensity function
Conditional intensity function
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Model assessment and cross-validation - overview
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Resampling techniques Why resampling? Jacknife Cross-validation Bootstrap Examples of application of bootstrap.
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