MaternSpline

a position-error and low-pass correlated velocity smoothing model

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Declaration

classdef MaternSpline < BSpline

Overview

The BSpline

Topics

• Initializing
  • MaternSpline compute the solution to the position-error velocity model
• Primary attributes
  • noiseDistribution distribution of the noise
  • t observation times (array of values for the independent axis)
  • x observed positions (array of values for the dependent axis)
• Manipulating model parameters
  • M number of low-pass splines
  • lambda velocity penalty
  • maternAlpha matern slope parameter
  • maternT matern time scale
• Accessing model properties
  • L Cholesky decomposition of the velocity covariance
  • Phf high-frequency projection operator
  • Plf low-frequency projection operator
  • U Matern-correlated high-pass-filtered velocity of N splines at points tq
  • VV velocity penalty matrix
  • Vhf high-pass filtered velocity of N splines at points tq
  • Vq velocity (first derivative) of N splines at points tq
  • W weight matrix for the position errors
  • X N splines at the N observations time (t)
  • Xm M low-pass splines at the N observations time (t)
  • hfSpline spline representation of the high-frequency part of the solution
  • lfSpline spline representation of the low-frequency part of the solution
  • maxLambda largest value of lambda different from infinity for double precision
  • midLambda a good starting value for lambda
  • minLambda lowest value of lambda different from zero for double precision
  • tKnotM knot points for the M low-pass splines
  • tq the (t)ime (q)uadrature grid
• Estimating error and likelihood
  • epsilon differences between the observed positions (x) and the model positions
  • epsilonInRange differences between the observed positions (x) and the model positions
  • expectedMeanSquareError The expected mean-square error normalized by the
  • expectedMeanSquareErrorFromCV Cross-validation (CV) estimate for the mean square error
  • expectedMeanSquareErrorFromGCV Generalized cross-validation (GCV) estimate for the mean-square error
  • expectedMeanSquareErrorInPercentileRange The expected mean-square error within an expected percentile range, normalized by the variance.
  • logLikelihood likelihood that the observed data fits the model
  • smoothingMatrix smoothing matrix maps observations to their smoothed values
• Minimizing error
  • minimizeOnLambda minimize a penalty function by performing a search on lambda
  • minimizeOnM minimize a penalty function by performing a grid search on M
  • minimizeOnMaternT minimize a penalty function by performing a search on maternT
  • minimizeOnParameters minimize a penalty function by performing a search a defined set of parameter values
  • Parameter structure
    • appendParametersWithPenalty append parameter struct with current values
    • parameters parameter struct with current values
    • parametersForGrid parameter struct with current values
    • parametersWithPenalty parameter struct with current values
    • setParameters set parameters to new values
• Searching parameter space (Static methods)
  • appendParametersToParameters append a parameter structure to another structure
  • gridSearchOnLambda perform a grid search on lambda
  • gridSearchOnM perform a grid search on M
  • gridSearchOnMaternT perform a grid search on maternT
  • logWrapper wraps a penalty function to take log10(lambda)
  • minimizeOnLambdaBounded minimize a penalty function by performing a bounded search on lambda
  • searchOnLambda perform a search on lambda, returning penalty function values
  • searchOnParameters evaluate penalty functions at each set of parameter values
• Methodology (Static methods)
  • knotPointsForDataPoints create knot points appropriate for observation times t
  • pevModelLambdaRange range of lambda values for position-error velocity model
  • pevModelLogLikelihood log-likehood of the position-error-velocity model
  • pevModelSmoothingMatrix smoothing matrix of the position-error velocity model
  • pevModelSolution compute the solution to the position-error velocity model
• Caching variables
  • LFromCache return a Cholesky autocovariance from the cache for given matern
  • LMaternTMap cache of computation of the Matern Cholesky matrix
  • addLToCache add a Cholesky autocovariance to the cache for given matern
  • sanitizeCacheForK clears the self.variableCache of any variables depedent on spline order K
  • sanitizeCacheForM clears the self.variableCache of any variables depedent on M
  • sanitizeCacheForMaternParameters clears the self.variableCache of any variables depedent on the matern
  • variableCache cache of computational expensive variables
• Internal
  • didSetMultipleParameters called after setting multiple parameters
  • kParameterDidChange notification that K changed and opportunity to clear caches,
  • lambdaParameterDidChange notification that M changed and opportunity to clear caches,
  • mParameterDidChange notification that M changed and opportunity to clear caches,
  • maternParametersDidChange notification that maternT or maternAlpha changed and
  • recomputeSolution recompute the solution the position-error velocity model
  • shouldRecomputeSolution boolean indicating whether or not to compute solutions
  • willSetMultipleParameters called before setting multiple parameters to prevent
• Other
  • sliceParameters

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