2011 COURSE IN NEUROINFORMATICS MARINE BIOLOGICAL LABORATORY WOODS HOLE, MA Introduction to Spline Models or Advanced Connect-the-Dots Uri Eden BU Department of Mathematics and Statistics August 16, 2011 Slide acknowledgements: D. Keren, Dept. of Computer Sci, Haifa U.
Motivating Problem Compute a smooth firing rate function from repeated spike train data Time (ms) Trial
Motivating Problem Compute a smooth firing rate function from repeated spike train data Firing Rate (Hz) Time (sec)
Motivating Problem Compute a smooth firing rate function from repeated spike train data Time (sec) Firing Rate (Hz)
Motivating Problem Compute a smooth firing rate function from repeated spike train data Time (sec) Firing Rate (Hz)
Problem Statement Given a set of control points, P i, known to be on a curve, find a parametric function that interpolates or approximates the curve.
For interpolating 2 points we need a polynomial of degree 1. Polynomial interpolation (x 1,y 1 ) (x 2,y 2 )
For interpolating 2 points we need a polynomial of degree 1. Polynomial interpolation (x 1,y 1 ) (x 2,y 2 )
For interpolating 2 points we need a polynomial of degree 1. For interpolating 3 points we need a polynomial of degree 2. In general, for interpolating n points, we need a polynomial of degree n-1. Polynomial interpolation (x 1,y 1 ) (x 2,y 2 )
Polynomial interpolation Let’s try it on our example control points: Time (sec) Firing Rate (Hz)
Polynomial interpolation Let’s try it on our example control points: Time (sec) Firing Rate (Hz)
Polynomial interpolation Disadvantages: –Coefficients are difficult to interpret geometrically –No local operations – adjusting parameters to change one region will change all others –If polynomial degree is high: Resulting function will have unwanted fluctuations Numerical precision problems may arise. –If polynomial degree is low, the resulting curve will not interpolate the points
Spline interpolation Another approach: Polynomial splines –Advantages: Rich, flexible representation Geometrically meaningful coefficients –Local, piecewise, low-degree, polynomial curves with continuous, smooth joints
Spline interpolation Simplest Example: Zeroth order ‘spline’ –Constant function between control points –Simple mathematical representation Local (x i+1,y i+1 ) (x i,y i )
Motivating Problem Compute a smooth firing rate function from repeated spike train data Time (sec) Firing Rate (Hz)
Motivating Problem Compute a smooth firing rate function from repeated spike train data Time (sec) Firing Rate (Hz) Local but not continuous
Spline interpolation Simple Example: Linear ‘spline’ –Linear combination of the two nearest control points –Simple mathematical representation where
Spline interpolation Simple Example: Linear ‘spline’
Spline interpolation Simple Example: Linear ‘spline’
Spline interpolation Simple Example: Linear ‘spline’
Spline interpolation Simple Example: Linear ‘spline’
Spline interpolation Simple Example: Linear ‘spline’
Motivating Problem Compute a smooth firing rate function from repeated spike train data Time (sec) Firing Rate (Hz)
Motivating Problem Compute a smooth firing rate function from repeated spike train data Time (sec) Firing Rate (Hz) Local and continuous, but not smooth
Spline interpolation Cubic Splines –Let be cubic functions. –Produces continuous, differentiable curves –Need to decide how to model derivatives at control points Natural Splines: Derivative is zero at endpoints Cardinal Splines: Derivative determined by nearby control points Hermitian Splines: Derivatives determined by separate parameters
Spline interpolation Cardinal splines –Linear combination of 4 nearest control points –Features: Derivatives determined by adjacent points Shape controlled by tension parameter,
Spline interpolation Cardinal splines
Spline interpolation Cardinal splines –Basis/blending functions for cardinal spline with s = 0.5.
Spline interpolation Cardinal splines
Spline interpolation Cardinal splines
Spline interpolation Cardinal splines
Spline interpolation Cardinal splines
Motivating Problem Compute a smooth firing rate function from repeated spike train data Time (sec) Firing Rate (Hz)
Motivating Problem Compute a smooth firing rate function from repeated spike train data Time (sec) Firing Rate (Hz)
Fitting Spline Models To define a spline model, we must specify –Control point locations (x-coordinates) –Tension parameter Often these are selected ad-hoc, but adaptive Bayesian procedures (e.g. BARS) to determine optimal values typically produce better results.
Fitting Spline Models Under a specified noise model, the spline parameters (heights of control points) can be estimated by maximum likelihood. –E.g. If, where is Gaussian white noise, and is a spline function, the ML estimator,, can be found by simple linear regression. –For spike data, ML spline parameters can be estimated using point process models.
Conclusions Splines offer one approach to modeling smooth functional relationships in neuroscience data. Cubic splines are local 3 rd degree polynomials that are piecewise continuous and differentiable. Under common noise distributions, spline models can be fit easily using ML methods. Goodness of spline model fits depends on selection of control point locations and tension parameters.