Elastic Regression

Warping Invariant Regression using SRSF

moduleauthor:: J. Derek Tucker <jdtuck@sandia.gov>

class regression.elastic_logistic(f, y, time)

This class provides elastic logistic regression for functional data using the SRVF framework accounting for warping

Usage: obj = elastic_logistic(f,y,time)

Parameters:

• f (np.ndarray) – numpy ndarray of shape (M,N) of N functions with M samples

• y – numpy array of N responses

• time (np.ndarray) – vector of size M describing the sample points

• B – optional matrix describing Basis elements

• alpha – alpha parameter of model

• beta – beta(t) of model

• fn – aligned functions - numpy ndarray of shape (M,N) of M functions with N samples

• qn – aligned srvfs - similar structure to fn

• gamma – calculated warping functions

• q – original training SRSFs

• b – basis coefficients

• Loss – logistic loss

Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 29-Oct-2021

calc_model(B=None, lam=0, df=20, max_itr=20, cores=-1, smooth=False)

This function identifies a regression model with phase-variability using elastic pca

Parameters:

• B – optional matrix describing Basis elements

• lam – regularization parameter (default 0)

• df – number of degrees of freedom B-spline (default 20)

• max_itr – maximum number of iterations (default 20)

• cores – number of cores for parallel processing (default all)

predict(newdata=None)

This function performs prediction on regression model on new data if available or current stored data in object Usage: obj.predict(newdata)

Parameters:

• newdata (dict) – dict containing new data for prediction (needs the keys below, if None predicts on training data)

• f – (M,N) matrix of functions

• time – vector of time points

• y – truth if available

• smooth – smooth data if needed

• sparam – number of times to run filter

class regression.elastic_mlogistic(f, y, time)

This class provides elastic multinomial logistic regression for functional data using the SRVF framework accounting for warping

Usage: obj = elastic_mlogistic(f,y,time)

Parameters:

• f (np.ndarray) – numpy ndarray of shape (M,N) of N functions with M samples

• y – numpy array of N responses

• time (np.ndarray) – vector of size M describing the sample points

• B – optional matrix describing Basis elements

• alpha – alpha parameter of model

• beta – beta(t) of model

• fn – aligned functions - numpy ndarray of shape (M,N) of N functions with M samples

• qn – aligned srvfs - similar structure to fn

• gamma – calculated warping functions

• q – original training SRSFs

• b – basis coefficients

• Loss – logistic loss

Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 29-Oct-2021

calc_model(B=None, lam=0, df=20, max_itr=20, delta=0.01, cores=-1, smooth=False)

This function identifies a regression model with phase-variability using elastic pca

Parameters:

• B – optional matrix describing Basis elements

• lam – regularization parameter (default 0)

• df – number of degrees of freedom B-spline (default 20)

• max_itr – maximum number of iterations (default 20)

• cores – number of cores for parallel processing (default all)

predict(newdata=None)

This function performs prediction on regression model on new data if available or current stored data in object Usage: obj.predict(newdata)

Parameters:

• newdata (dict) – dict containing new data for prediction (needs the keys below, if None predicts on training data)

• f – (M,N) matrix of functions

• time – vector of time points

• y – truth if available

• smooth – smooth data if needed

• sparam – number of times to run filter

class regression.elastic_regression(f, y, time)

This class provides elastic regression for functional data using the SRVF framework accounting for warping

Usage: obj = elastic_regression(f,y,time)

Parameters:

• f – numpy ndarray of shape (M,N) of N functions with M samples

• y – numpy array of N responses

• time – vector of size M describing the sample points

• B – optional matrix describing Basis elements

• alpha – alpha parameter of model

• beta – beta(t) of model

• fn – aligned functions - numpy ndarray of shape (M,N) of M functions with N samples

• qn – aligned srvfs - similar structure to fn

• gamma – calculated warping functions

• q – original training SRSFs

• b – basis coefficients

• SSE – sum of squared error

Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 29-Oct-2021

calc_model(B=None, lam=0, df=20, max_itr=20, cores=-1, smooth=False)

This function identifies a regression model with phase-variability using elastic pca

Parameters:

• B – optional matrix describing Basis elements

• lam – regularization parameter (default 0)

• df – number of degrees of freedom B-spline (default 20)

• max_itr – maximum number of iterations (default 20)

• cores – number of cores for parallel processing (default all)

predict(newdata=None)

This function performs prediction on regression model on new data if available or current stored data in object Usage: obj.predict(newdata)

Parameters:

• newdata (dict) – dict containing new data for prediction (needs the keys below, if None predicts on training data)

• f – (M,N) matrix of functions

• time – vector of time points

• y – truth if available

• smooth – smooth data if needed

• sparam – number of times to run filter

regression.logistic_warp(beta, time, q, y)

calculates optimal warping for function logistic regression

Parameters:

• beta – numpy ndarray of shape (M,N) of N functions with M samples

• time – vector of size N describing the sample points

• q – numpy ndarray of shape (M,N) of N functions with M samples

• y – numpy ndarray of shape (1,N) responses

Return type:

numpy array

Return gamma:

warping function

regression.logit_gradient(b, X, y)

calculates gradient of the logistic loss

Parameters:

• b – numpy ndarray of shape (M,N) of N functions with M samples

• X – numpy ndarray of shape (M,N) of N functions with M samples

• y – numpy ndarray of shape (1,N) responses

Return type:

numpy array

Return grad:

gradient of logistic loss

regression.logit_hessian(s, b, X, y)

calculates hessian of the logistic loss

Parameters:

• s – numpy ndarray of shape (M,N) of N functions with M samples

• b – numpy ndarray of shape (M,N) of N functions with M samples

• X – numpy ndarray of shape (M,N) of N functions with M samples

• y – numpy ndarray of shape (1,N) responses

Return type:

numpy array

Return out:

hessian of logistic loss

regression.logit_loss(b, X, y)

logistic loss function, returns Sum{-log(phi(t))}

Parameters:

• b – numpy ndarray of shape (M,N) of N functions with M samples

• X – numpy ndarray of shape (M,N) of N functions with M samples

• y – numpy ndarray of shape (1,N) of N responses

Return type:

numpy array

Return out:

loss value

regression.mlogit_gradient(b, X, Y)

calculates gradient of the multinomial logistic loss

Parameters:

• b – numpy ndarray of shape (M,N) of N functions with M samples

• X – numpy ndarray of shape (M,N) of N functions with M samples

• y – numpy ndarray of shape (1,N) responses

Return type:

numpy array

Return grad:

gradient

regression.mlogit_loss(b, X, Y)

calculates multinomial logistic loss (negative log-likelihood)

Parameters:

• b – numpy ndarray of shape (M,N) of N functions with M samples

• X – numpy ndarray of shape (M,N) of N functions with M samples

• y – numpy ndarray of shape (1,N) responses

Return type:

numpy array

Return nll:

negative log-likelihood

regression.mlogit_warp_grad(alpha, beta, time, q, y, max_itr=8000, tol=1e-10, delta=0.008, display=0)

calculates optimal warping for functional multinomial logistic regression

Parameters:

• alpha – scalar

• beta – numpy ndarray of shape (M,N) of N functions with M samples

• time – vector of size M describing the sample points

• q – numpy ndarray of shape (M,N) of N functions with M samples

• y – numpy ndarray of shape (1,N) responses

• max_itr – maximum number of iterations (Default=8000)

• tol – stopping tolerance (Default=1e-10)

• delta – gradient step size (Default=0.008)

• display – display iterations (Default=0)

Return type:

tuple of numpy array

Return gam_old:

warping function

regression.phi(t)

calculates logistic function, returns 1 / (1 + exp(-t))

Parameters:

t – scalar

Return type:

numpy array

Return out:

return value

regression.regression_warp(beta, time, q, y, alpha)

calculates optimal warping for function linear regression

Parameters:

• beta – numpy ndarray of shape (M,N) of M functions with N samples

• time – vector of size N describing the sample points

• q – numpy ndarray of shape (M,N) of M functions with N samples

• y – numpy ndarray of shape (1,N) of M functions with N samples responses

• alpha – numpy scalar

Return type:

numpy array

Return gamma_new:

warping function