Circular statistics Maximum likelihood Local likelihood Kenneth D. Harris
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Circular statistics Maximum likelihood Local likelihood Kenneth D. Harris 4/3/2015 Relationship of cells to oscillations Klausberger & Somogyi, Science 2008 Relationship of cells to oscillations O’Keefe & Recce, Hippocampus 1993 How do we quantify: • A cell’s average phase of firing? • How phase depends on other variables? Computing instantaneous phase • Hilbert transform • Peak fitting Phase histogram • Different cells prefer different phases • How do we compute each cell’s mean phase? Linear mean doesn’t work Circular mean • Treat angles as points on a circle π§ = ππ π • The mean of these gives you • Circular mean π • Vector strength π • If all angles are the same: • π is this angle • π is 1 • If angles are completely uniform • π is 0 • π is meaningless. π§ = π π ππ R π von Mises distribution π π cos π−π0 π π = 2ππΌ0 π • π0 is central angle • π is concentration parameter • Larger -> more peaked • Zero -> uniform distribution • πΌ0 π is a Modified Bessel function • Needed to make probabilities integrate to 1. Maximum likelihood estimation π π cos π−π0 π π; π0 , π = 2ππΌ0 π • Given data set π1 … ππ , how do we estimate parameters π and π ? • Choose them to maximize π ππ ; π0 , π π Maximum likelihood estimation • Given data set π1 … ππ , how do we estimate parameters π and π ? • Log likelihood πΏ= log π ππ ; π0 , π π =π cos ππ − π0 − π log πΌ0 π π • Solve ππΏ ππ0 = 0 and ππΏ ππ =0 − π log 2π Maximum likelihood von Mises • π0 = π, (π = circular mean). • π is the solution of πΌ1 π πΌ0 π = π , (π = vector strength). • πΌ1 π is another modified Bessel function. How does phase depend on another variable? • Don’t use linear regression! Locally-weighted likelihood • Log likelihood πΏ ππ ; π, π = log π ππ π, π π • Locally-weighted likelihood • Spike phases ππ occur at positions π± π : πΏπ± ππ ; ππ± , π π± = E.g. π€ Δπ± = π π€ π± − π±π log π ππ ππ± , π π± π Δπ± 2 2π2 Kernel smoother ∑π¦π π€ π± − π± π’ π¦(π₯) = ∑π€ π± − π± π’ Is local likelihood for a Gaussian Place field estimation ππππππΆππ’ππ‘πππ ∗ πΎ ππππππ ∗ πΎ Local likelihood estimation for a Poisson distribution Phase field • Local likelihood estimation for von Mises Abstract space Physical space Harris et al Nature 2002 Confirmatory statistics: • How to test that phase is independent of place? • Shuffling method? • Test statistic?
