FDASRSF(1)

NAME
fdasrsf - fdasrsf Documentation

A python package for functional data analysis using the square root slope framework and curves using the square root velocity framework which performs pair-wise and group-wise alignment as well as modeling using functional component analysis and regression.

FUNCTIONAL ALIGNMENT

Group-wise function alignment using SRSF framework and Dynamic Programming

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

time_warping.align_fPCA(f, time, num_comp=3, showplot=True, smoothdata=False, cores=- 1)
    aligns a collection of functions while extracting principal components. The functions are aligned to the principal components

Parameters

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

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

•   num_comp -- number of fPCA components

•   showplot -- Shows plots of results using matplotlib (default = T)

•   smooth_data (bool) -- Smooth the data using a box filter (default = F)

•   cores -- number of cores for parallel (default = -1 (all))

Return type
    tuple of numpy array

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

Return qn
    aligned srvfs - similar structure to fn

Return q0
    original srvf - similar structure to fn

Return mqn
    srvf mean or median - vector of length M

Return gam
    warping functions - similar structure to fn

Return q_pca
    srsf principal directions

Return f_pca
    functional principal directions

Return latent
    latent values

Return coef
    coefficients

Return U
    eigenvectors

Return orig_var
    Original Variance of Functions

Return amp_var
    Amplitude Variance

Return phase_var
    Phase Variance

time_warping.align_fPLS(f, g, time, comps=3, showplot=True, smoothdata=False, delta=0.01, max_itr=100)
    This function aligns a collection of functions while performing principal least squares

Parameters

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

•   g (np.ndarray) -- numpy ndarray of shape (M,N) of N functions with M samples

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

•   comps -- number of fPLS components

•   showplot -- Shows plots of results using matplotlib (default = T)

•   smooth_data (bool) -- Smooth the data using a box filter (default = F)

•   delta -- gradient step size

•   max_itr -- maximum number of iterations

Return type
    tuple of numpy array

Return fn
    aligned functions - numpy ndarray of shape (M,N) of N

    functions with M samples :return gn: aligned functions - numpy ndarray of shape (M,N) of N functions with M samples :return qfn: aligned srvfs - similar structure to fn :return qgn: aligned srvfs - similar structure to fn :return qf0: original srvf - similar structure to fn :return qg0: original srvf - similar structure to fn :return gam: warping functions - similar structure to fn :return wqf: srsf principal weight functions :return wqg: srsf principal weight functions :return wf: srsf principal weight functions :return wg: srsf principal weight functions :return cost: cost function value

class time_warping.fdawarp(f, time)
    This class provides alignment methods for functional data using the SRVF framework

    Usage: obj = fdawarp(f,t)

Parameters

•   f -- (M,N): matrix defining N functions of M samples

•   time -- time vector of length M

•   fn -- aligned functions

•   qn -- aligned srvfs

•   q0 -- initial srvfs

•   fmean -- function mean

•   mqn -- mean srvf

•   gam -- warping functions

•   psi -- srvf of warping functions

•   stats -- alignment statistics

•   qun -- cost function

•   lambda -- lambda

•   method -- optimization method

•   gamI -- inverse warping function

•   rsamps -- random samples

•   fs -- random aligned functions

•   gams -- random warping functions

•   ft -- random warped functions

•   qs -- random aligned srvfs

•   type -- alignment type

•   mcmc -- mcmc output if bayesian

    Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 15-Mar-2018

gauss_model(n=1, sort_samples=False)
    This function models the functional data using a Gaussian model extracted from the principal components of the srvfs

Parameters

•   n (integer) -- number of random samples

•   sort_samples (bool) -- sort samples (default = T)

joint_gauss_model(n=1, no=3)
    This function models the functional data using a joint Gaussian model extracted from the principal components of the srsfs

Parameters

•   n (integer) -- number of random samples

•   no (integer) -- number of principal components (default = 3)

multiple_align_functions(mu, omethod='DP2', smoothdata=False, parallel=False, lam=0.0, cores=- 1, grid_dim=7)
    This function aligns a collection of functions using the elastic square-root slope (srsf) framework.

Usage: obj.multiple_align_functions(mu)
    obj.multiple_align_functions(lambda)

    obj.multiple_align_functions(lambda, ...)

Parameters

•   mu -- vector of function to align to

•   omethod -- optimization method (DP, DP2, RBFGS) (default = DP)

•   smoothdata (bool) -- Smooth the data using a box filter (default = F)

•   parallel -- run in parallel (default = F)

•   lam (double) -- controls the elasticity (default = 0)

•   cores -- number of cores for parallel (default = -1 (all))

•   grid_dim -- size of the grid, for the DP2 method only (default = 7)

plot()
    plot plot functional alignment results

    Usage: obj.plot()

srsf_align(method='mean', omethod='DP2', smoothdata=False, parallel=False, lam=0.0, cores=- 1, grid_dim=7)
    This function aligns a collection of functions using the elastic square-root slope (srsf) framework.

Parameters

•   method -- (string) warp calculate Karcher Mean or Median (options = "mean" or "median") (default="mean")

•   omethod -- optimization method (DP, DP2, RBFGS) (default = DP2)

•   smoothdata (bool) -- Smooth the data using a box filter (default = F)

•   parallel -- run in parallel (default = F)

•   lam (double) -- controls the elasticity (default = 0)

•   cores -- number of cores for parallel (default = -1 (all))

•   grid_dim -- size of the grid, for the DP2 method only (default = 7)

    Examples >>> import tables >>> fun=tables.open_file("../Data/simu_data.h5") >>> f = fun.root.f[:] >>> f = f.transpose() >>> time = fun.root.time[:] >>> obj = fs.fdawarp(f,time) >>> obj.srsf_align()

time_warping.normal(loc=0.0, scale=1.0, size=None)
    Draw random samples from a normal (Gaussian) distribution.

    The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2], is often called the bell curve because of its characteristic shape (see the example below).

    The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2].

    NOTE: New code should use the normal method of a default_rng() instance instead; please see the random-quick-start.

loc float or array_like of floats Mean ("centre") of the distribution.

scale
    float or array_like of floats Standard deviation (spread or "width") of the distribution. Must be non-negative.

size
    int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.

out ndarray or scalar Drawn samples from the parameterized normal distribution.

scipy.stats.norm
    probability density function, distribution or cumulative density function, etc.

    Generator.normal: which should be used for new code.

    The probability density for the Gaussian distribution is

    p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },

    where \mu is the mean and \sigma the standard deviation. The square of the standard deviation, \sigma^2, is called the variance.

    The function has its peak at the mean, and its "spread" increases with the standard deviation (the function reaches 0.607 times its maximum at x + \sigma and x - \sigma [2]). This implies that normal is more likely to return samples lying close to the mean, rather than those far away.

[1] Wikipedia, "Normal distribution", https://en.wikipedia.org/wiki/Normal_distribution

[2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability, Random Variables and Random Signal Principles", 4th ed., 2001, pp. 51, 51, 125.

    Draw samples from the distribution:

    >>> mu, sigma = 0, 0.1 # mean and standard deviation
    >>> s = np.random.normal(mu, sigma, 1000)

    Verify the mean and the variance:

    >>> abs(mu - np.mean(s))
    0.0  # may vary

    >>> abs(sigma - np.std(s, ddof=1))
    0.1  # may vary

    Display the histogram of the samples, along with the probability density function:

    >>> import matplotlib.pyplot as plt
    >>> count, bins, ignored = plt.hist(s, 30, density=True)
    >>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
    ...                np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
    ...          linewidth=2, color='r')
    >>> plt.show()

    Two-by-four array of samples from N(3, 6.25):

    >>> np.random.normal(3, 2.5, size=(2, 4))
    array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
           [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random

time_warping.pairwise_align_bayes(f1i, f2i, time, mcmcopts=None)
    This function aligns two functions using Bayesian framework. It will align f2 to f1. It is based on mapping warping functions to a hypersphere, and a subsequent exponential mapping to a tangent space. In the tangent space, the Z-mixture pCN algorithm is used to explore both local and global structure in the posterior distribution.

    The Z-mixture pCN algorithm uses a mixture distribution for the proposal distribution, controlled by input parameter zpcn. The zpcn$betas must be between 0 and 1, and are the coefficients of the mixture components, with larger coefficients corresponding to larger shifts in parameter space. The zpcn["probs"] give the probability of each shift size.

Usage: out = pairwise_align_bayes(f1i, f2i, time)
    out = pairwise_align_bayes(f1i, f2i, time, mcmcopts)

Parameters

•   f1i -- vector defining M samples of function 1

•   f2i -- vector defining M samples of function 2

•   time -- time vector of length M

•   mcmopts -- dict of mcmc parameters

    default mcmc options: tmp = {"betas":np.array([0.5,0.5,0.005,0.0001]),"probs":np.array([0.1,0.1,0.7,0.1])} mcmcopts = {"iter":2*(10**4) ,"burnin":np.minimum(5*(10**3),2*(10**4)//2), "alpha0":0.1, "beta0":0.1,"zpcn":tmp,"propvar":1, "initcoef":np.repeat(0,20), "npoints":200, "extrainfo":True}

    :rtype collection containing :return f2_warped: aligned f2 :return gamma: warping function :return g_coef: final g_coef :return psi: final psi :return sigma1: final sigma

    if extrainfo :return accept: accept of psi samples :return betas_ind :return logl: log likelihood :return gamma_mat: posterior gammas :return gamma_stats: posterior gamma stats :return xdist: phase distance posterior :return ydist: amplitude distance posterior)

time_warping.rand(d0, d1, ..., dn)
    Random values in a given shape.

    NOTE: This is a convenience function for users porting code from Matlab, and wraps random_sample. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

    Create an array of the given shape and populate it with random samples from a uniform distribution over [0, 1).

d0, d1, ..., dn
    int, optional The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

out ndarray, shape (d0, d1, ..., dn) Random values.

    random

    >>> np.random.rand(3,2)
    array([[ 0.14022471,  0.96360618],  #random
           [ 0.37601032,  0.25528411],  #random
           [ 0.49313049,  0.94909878]]) #random

FUNCTIONAL PRINCIPAL COMPONENT ANALYSIS

Vertical and Horizontal Functional Principal Component Analysis using SRSF

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

class fPCA.fdahpca(fdawarp)
    This class provides horizontal fPCA using the SRVF framework

    Usage: obj = fdahpca(warp_data)

Parameters

•   warp_data -- fdawarp class with alignment data

•   gam_pca -- warping functions principal directions

•   psi_pca -- srvf principal directions

•   latent -- latent values

•   U -- eigenvectors

•   coef -- coefficients

•   vec -- shooting vectors

•   mu -- Karcher Mean

•   tau -- principal directions

    Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 15-Mar-2018

calc_fpca(no=3)
    This function calculates horizontal functional principal component analysis on aligned data

Parameters
    no (int) -- number of components to extract (default = 3)

Return type
    fdahpca object of numpy ndarray

Return q_pca
    srsf principal directions

Return f_pca
    functional principal directions

Return latent
    latent values

Return coef
    coefficients

Return U
    eigenvectors

plot()
    plot plot elastic horizontal fPCA results

    Usage: obj.plot()

class fPCA.fdajpca(fdawarp)
    This class provides joint fPCA using the SRVF framework

    Usage: obj = fdajpca(warp_data)

Parameters

•   warp_data -- fdawarp class with alignment data

•   q_pca -- srvf principal directions

•   f_pca -- f principal directions

•   latent -- latent values

•   coef -- principal coefficients

•   id -- point used for f(0)

•   mqn -- mean srvf

•   U -- eigenvectors

•   mu_psi -- mean psi

•   mu_g -- mean g

•   C -- scaling value

•   stds -- geodesic directions

    Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 18-Mar-2018

calc_fpca(no=3, id=None)
    This function calculates joint functional principal component analysis on aligned data

Parameters

•   no (int) -- number of components to extract (default = 3)

•   id (int) -- point to use for f(0) (default = midpoint)

Return type
    fdajpca object of numpy ndarray

Return q_pca
    srsf principal directions

Return f_pca
    functional principal directions

Return latent
    latent values

Return coef
    coefficients

Return U
    eigenvectors

plot()
    plot plot elastic vertical fPCA result

    Usage: obj.plot()

class fPCA.fdavpca(fdawarp)
    This class provides vertical fPCA using the SRVF framework

    Usage: obj = fdavpca(warp_data)

Parameters

•   warp_data -- fdawarp class with alignment data

•   q_pca -- srvf principal directions

•   f_pca -- f principal directions

•   latent -- latent values

•   coef -- principal coefficients

•   id -- point used for f(0)

•   mqn -- mean srvf

•   U -- eigenvectors

•   stds -- geodesic directions

    Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 15-Mar-2018

calc_fpca(no=3, id=None)
    This function calculates vertical functional principal component analysis on aligned data

Parameters

•   no (int) -- number of components to extract (default = 3)

•   id (int) -- point to use for f(0) (default = midpoint)

Return type
    fdavpca object containing

Return q_pca
    srsf principal directions

Return f_pca
    functional principal directions

Return latent
    latent values

Return coef
    coefficients

Return U
    eigenvectors

plot()
    plot plot elastic vertical fPCA result Usage: obj.plot()

ELASTIC FUNCTIONAL BOXPLOTS

Elastic Functional Boxplots

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

class boxplots.ampbox(fdawarp)
    This class provides amplitude boxplot for functional data using the SRVF framework

    Usage: obj = ampbox(warp_data)

Parameters

•   warp_data (fdawarp) -- fdawarp class with alignment data

•   Q1 -- First quartile

•   Q3 -- Second quartile

•   Q1a -- First quantile based on alpha

•   Q3a -- Second quantile based on alpha

•   minn -- minimum extreme function

•   maxx -- maximum extreme function

•   outlier_index -- indexes of outlier functions

•   f_median -- median function

•   q_median -- median srvf

•   plt -- surface plot mesh

    Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 15-Mar-2018

construct_boxplot(alpha=0.05, k_a=1)
    This function constructs the amplitude boxplot using the elastic square-root slope (srsf) framework.

Parameters

•   alpha -- quantile value (e.g.,=.05, i.e., 95%)

•   k_a -- scalar for outlier cutoff (e.g.,=1)

plot()
    plot box plot and surface plot

    Usage: obj.plot()

class boxplots.phbox(fdawarp)
    This class provides phase boxplot for functional data using the SRVF framework

    Usage: obj = phbox(warp_data)

Parameters

•   warp_data (fdawarp) -- fdawarp class with alignment data

•   Q1 -- First quartile

•   Q3 -- Second quartile

•   Q1a -- First quantile based on alpha

•   Q3a -- Second quantile based on alpha

•   minn -- minimum extreme function

•   maxx -- maximum extreme function

•   outlier_index -- indexes of outlier functions

•   median_x -- median warping function

•   psi_median -- median srvf of warping function

•   plt -- surface plot mesh

    Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 15-Mar-2018

construct_boxplot(alpha=0.05, k_a=1)
    This function constructs phase boxplot for functional data using the elastic square-root slope (srsf) framework.

Parameters

•   alpha -- quantile value (e.g.,=.05, i.e., 95%)

•   k_a -- scalar for outlier cutoff (e.g.,=1)

plot()
    plot box plot and surface plot

    Usage: obj.plot()

FUNCTIONAL PRINCIPAL LEAST SQUARES

Partial Least Squares using SVD

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

fPLS.pls_svd(time, qf, qg, no, alpha=0.0)
    This function computes the partial least squares using SVD

Parameters

•   time -- vector describing time samples

•   qf -- numpy ndarray of shape (M,N) of N functions with M samples

•   qg -- numpy ndarray of shape (M,N) of N functions with M samples

•   no -- number of components

•   alpha -- amount of smoothing (Default = 0.0 i.e., none)

Return type
    numpy ndarray

Return wqf
    f weight function

Return wqg
    g weight function

Return alpha
    smoothing value

Return values
    singular values

ELASTIC REGRESSION

Warping Invariant Regression using SRSF

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

regression.elastic_logistic(f, y, time, B=None, df=20, max_itr=20, cores=- 1, smooth=False)
    This function identifies a logistic regression model with phase-variability using elastic methods

Parameters

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

•   y -- numpy array of labels (1/-1)

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

•   B -- optional matrix describing Basis elements

•   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)

Return type
    tuple of numpy array

Return alpha
    alpha parameter of model

Return beta
    beta(t) of model

Return fn
    aligned functions - numpy ndarray of shape (M,N) of M

    functions with N samples :return qn: aligned srvfs - similar structure to fn :return gamma: calculated warping functions :return q: original training SRSFs :return B: basis matrix :return b: basis coefficients :return Loss: logistic loss

regression.elastic_mlogistic(f, y, time, B=None, df=20, max_itr=20, cores=- 1, delta=0.01, parallel=True, smooth=False)
    This function identifies a multinomial logistic regression model with phase-variability using elastic methods

Parameters

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

•   y -- numpy array of labels {1,2,...,m} for m classes

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

•   B -- optional matrix describing Basis elements

•   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)

Return type
    tuple of numpy array

Return alpha
    alpha parameter of model

Return beta
    beta(t) of model

Return fn
    aligned functions - numpy ndarray of shape (M,N) of N

    functions with M samples :return qn: aligned srvfs - similar structure to fn :return gamma: calculated warping functions :return q: original training SRSFs :return B: basis matrix :return b: basis coefficients :return Loss: logistic loss

regression.elastic_prediction(f, time, model, y=None, smooth=False)
    This function performs prediction from an elastic regression model with phase-variability

Parameters

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

•   time -- vector of size M describing the sample points

•   model -- identified model from elastic_regression

•   y -- truth, optional used to calculate SSE

Return type
    tuple of numpy array

Return alpha
    alpha parameter of model

Return beta
    beta(t) of model

Return fn
    aligned functions - numpy ndarray of shape (M,N) of N

    functions with M samples :return qn: aligned srvfs - similar structure to fn :return gamma: calculated warping functions :return q: original training SRSFs :return B: basis matrix :return b: basis coefficients :return SSE: sum of squared error

regression.elastic_regression(f, y, time, B=None, lam=0, df=20, max_itr=20, cores=- 1, smooth=False)
    This function identifies a regression model with phase-variability using elastic methods

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

•   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)

Return type
    tuple of numpy array

Return alpha
    alpha parameter of model

Return beta
    beta(t) of model

Return fn
    aligned functions - numpy ndarray of shape (M,N) of M

    functions with N samples :return qn: aligned srvfs - similar structure to fn :return gamma: calculated warping functions :return q: original training SRSFs :return B: basis matrix :return b: basis coefficients :return SSE: sum of squared error

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 :param alpha: numpy scalar

Return type
    numpy array

Return gamma_new
    warping function

ELASTIC PRINCIPAL COMPONENT REGRESSION

Warping Invariant PCR Regression using SRSF

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

class pcr_regression.elastic_lpcr_regression(f, y, time)
    This class provides elastic logistic pcr regression for functional data using the SRVF framework accounting for warping

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

Parameters

•   f -- (M,N) % matrix defining N functions of M samples

•   y -- response vector of length N (-1/1)

•   warp_data -- fdawarp object of alignment

•   pca -- class dependent on fPCA method used object of fPCA

    :param information :param alpha: intercept :param b: coefficient vector :param Loss: logistic loss :param PC: probability of classification :param ylabels: predicted labels

    Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 18-Mar-2018

calc_model(pca_method='combined', no=5, smooth_data=False, sparam=25, parallel=False)
    This function identifies a logistic regression model with phase-variability using elastic pca

Parameters

•   pca_method -- string specifing pca method (options = "combined", "vert", or "horiz", default = "combined")

•   no -- scalar specify number of principal components (default=5)

•   smooth_data -- smooth data using box filter (default = F)

•   sparam -- number of times to apply box filter (default = 25)

•   parallel -- calculate in parallel (default = F)

predict(newdata=None)
    This function performs prediction on regression model on new data if available or current stored data in object Usage: obj.predict() 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 pcr_regression.elastic_mlpcr_regression(f, y, time)
    This class provides elastic multinomial logistic pcr regression for functional data using the SRVF framework accounting for warping

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

Parameters

•   f -- (M,N) % matrix defining N functions of M samples

•   y -- response vector of length N

•   Y -- coded label matrix

•   warp_data -- fdawarp object of alignment

•   pca -- class dependent on fPCA method used object of fPCA

    :param information :param alpha: intercept :param b: coefficient vector :param Loss: logistic loss :param PC: probability of classification :param ylabels: predicted labels :param

    Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 18-Mar-2018

calc_model(pca_method='combined', no=5, smooth_data=False, sparam=25, parallel=False)
    This function identifies a logistic regression model with phase-variability using elastic pca

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

•   pca_method -- string specifing pca method (options = "combined", "vert", or "horiz", default = "combined")

•   no -- scalar specify number of principal components (default=5)

•   smooth_data -- smooth data using box filter (default = F)

•   sparam -- number of times to apply box filter (default = 25)

•   parallel -- run model in parallel (default = F)

predict(newdata=None)
    This function performs prediction on regression model on new data if available or current stored data in object Usage: obj.predict() 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 pcr_regression.elastic_pcr_regression(f, y, time)
    This class provides elastic pcr regression for functional data using the SRVF framework accounting for warping

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

Parameters

•   f -- (M,N) % matrix defining N functions of M samples

•   y -- response vector of length N

•   warp_data -- fdawarp object of alignment

•   pca -- class dependent on fPCA method used object of fPCA

•   alpha -- intercept

•   b -- coefficient vector

•   SSE -- sum of squared errors

    Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 18-Mar-2018

calc_model(pca_method='combined', no=5, smooth_data=False, sparam=25, parallel=False, C=None)
    This function identifies a regression model with phase-variability using elastic pca

Parameters

•   pca_method -- string specifing pca method (options = "combined", "vert", or "horiz", default = "combined")

•   no -- scalar specify number of principal components (default=5)

•   smooth_data -- smooth data using box filter (default = F)

•   sparam -- number of times to apply box filter (default = 25)

•   parallel -- run in parallel (default = F)

•   C -- scale balance parameter for combined method (default = None)

predict(newdata=None)
    This function performs prediction on regression model on new data if available or current stored data in object Usage: obj.predict() 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

ELASTIC FUNCTIONAL TOLERANCE BOUNDS

Functional Tolerance Bounds using SRSF

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

tolerance.bootTB(f, time, a=0.5, p=0.99, B=500, no=5, parallel=True)
    This function computes tolerance bounds for functional data containing phase and amplitude variation using bootstrap sampling

Parameters

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

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

•   a -- confidence level of tolerance bound (default = 0.05)

•   p -- coverage level of tolerance bound (default = 0.99)

•   B -- number of bootstrap samples (default = 500)

•   no -- number of principal components (default = 5)

•   parallel -- enable parallel processing (default = T)

Return type
    tuple of boxplot objects

Return amp
    amplitude tolerance bounds

Rtype out_med
    ampbox object

Return ph
    phase tolerance bounds

Rtype out_med
    phbox object

Return out_med
    alignment results

Rtype out_med
    fdawarp object

tolerance.mvtol_region(x, alpha, P, B)
    Computes tolerance factor for multivariate normal

    Krishnamoorthy, K. and Mondal, S. (2006), Improved Tolerance Factors for Multivariate Normal Distributions, Communications in Statistics - Simulation and Computation, 35, 461–478.

Parameters

•   x -- (M,N) matrix defining N variables of M samples

•   alpha -- confidence level

•   P -- coverage level

•   B -- number of bootstrap samples

Return type
    double

Return tol
    tolerance factor

tolerance.pcaTB(f, time, a=0.5, p=0.99, no=5, parallel=True)
    This function computes tolerance bounds for functional data containing phase and amplitude variation using fPCA

Parameters

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

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

•   a -- confidence level of tolerance bound (default = 0.05)

•   p -- coverage level of tolerance bound (default = 0.99)

•   no -- number of principal components (default = 5)

•   parallel -- enable parallel processing (default = T)

Return type
    tuple of boxplot objects

Return warp
    alignment data from time_warping

Return pca
    functional pca from jointFPCA

Return tol
    tolerance factor

tolerance.rwishart(df, p)
    Computes a random wishart matrix

Parameters

•   df -- degree of freedom

•   p -- number of dimensions

Return type
    double

Return R
    matrix

CURVE REGISTRATION

statistic calculation for SRVF (curves) open and closed using Karcher Mean and Variance

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

class curve_stats.fdacurve(beta, mode='O', N=200, scale=False)
    This class provides alignment methods for open and closed curves using the SRVF framework

    Usage: obj = fdacurve(beta, mode, N, scale) :param beta: numpy ndarray of shape (n, M, N) describing N curves in R^M :param mode: Open ('O') or closed curve ('C') (default 'O') :param N: resample curve to N points :param scale: scale curve to length 1 (true/false) :param q: (n,T,K) matrix defining n dimensional srvf on T samples with K srvfs :param betan: aligned curves :param qn: aligned srvfs :param basis: calculated basis :param beta_mean: karcher mean curve :param q_mean: karcher mean srvf :param gams: warping functions :param v: shooting vectors :param C: karcher covariance :param s: pca singular values :param U: pca singular vectors :param coef: pca coefficients :param qun: cost function :param samples: random samples :param gamr: random warping functions :param cent: center :param scale: scale :param len: length of curve :param len_q: length of srvf :param mean_scale mean length :param mean_scale_q mean length srvf :param E: energy

    Author : J. D. Tucker (JDT) <jdtuck AT sandia.gov> Date : 26-Aug-2020

karcher_cov()
    This calculates the mean of a set of curves

karcher_mean(parallel=False, cores=- 1, method='DP')
    This calculates the mean of a set of curves :param parallel: run in parallel (default = F) :param cores: number of cores for parallel (default = -1 (all)) :param method: method to apply optimization (default="DP") options are "DP" or "RBFGS"

plot()
    plot curve mean results

sample_shapes(no=3, numSamp=10)
    Computes sample shapes from mean and covariance

Parameters

•   no -- number of direction (default 3)

•   numSamp -- number of samples (default 10)

shape_pca(no=10)
    Computes principal direction of variation specified by no. N is Number of shapes away from mean. Creates 2*N+1 shape sequence

Parameters
    no -- number of direction (default 3)

srvf_align(parallel=False, cores=- 1, method='DP')
    This aligns a set of curves to the mean and computes mean if not computed :param parallel: run in parallel (default = F) :param cores: number of cores for parallel (default = -1 (all)) :param method: method to apply optimization (default="DP") options are "DP" or "RBFGS"

curve_stats.randn(d0, d1, ..., dn)
    Return a sample (or samples) from the "standard normal" distribution.

    NOTE: This is a convenience function for users porting code from Matlab, and wraps standard_normal. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

    NOTE: New code should use the standard_normal method of a default_rng() instance instead; please see the random-quick-start.

    If positive int_like arguments are provided, randn generates an array of shape (d0, d1, ..., dn), filled with random floats sampled from a univariate "normal" (Gaussian) distribution of mean 0 and variance 1. A single float randomly sampled from the distribution is returned if no argument is provided.

d0, d1, ..., dn
    int, optional The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

Z   ndarray or float A (d0, d1, ..., dn)-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied.

    standard_normal : Similar, but takes a tuple as its argument. normal : Also accepts mu and sigma arguments. Generator.standard_normal: which should be used for new code.

    For random samples from N(\mu, \sigma^2), use:

    sigma * np.random.randn(...) + mu

    >>> np.random.randn()
    2.1923875335537315  # random

    Two-by-four array of samples from N(3, 6.25):

    >>> 3 + 2.5 * np.random.randn(2, 4)
    array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
           [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random

SRVF GEODESIC COMPUTATION

geodesic calculation for SRVF (curves) open and closed

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

geodesic.back_parallel_transport(u1, alpha, basis, T=100, k=5)
    backwards parallel translates q1 and q2 along manifold

Parameters

•   u1 -- numpy ndarray of shape (2,M) of M samples

•   alpha -- numpy ndarray of shape (2,M) of M samples

•   basis -- list numpy ndarray of shape (2,M) of M samples

•   T -- Number of samples of curve (Default = 100)

•   k -- number of samples along path (Default = 5)

Return type
    numpy ndarray

Return utilde
    translated vector

geodesic.calc_alphadot(alpha, basis, T=100, k=5)
    calculates derivative along the path alpha

Parameters

•   alpha -- numpy ndarray of shape (2,M) of M samples

•   basis -- list of numpy ndarray of shape (2,M) of M samples

•   T -- Number of samples of curve (Default = 100)

•   k -- number of samples along path (Default = 5)

Return type
    numpy ndarray

Return alphadot
    derivative of alpha

geodesic.calculate_energy(alphadot, T=100, k=5)
    calculates energy along path

Parameters

•   alphadot -- numpy ndarray of shape (2,M) of M samples

•   T -- Number of samples of curve (Default = 100)

•   k -- number of samples along path (Default = 5)

Return type
    numpy scalar

Return E
    energy

geodesic.calculate_gradE(u, utilde, T=100, k=5)
    calculates gradient of energy along path

Parameters

•   u -- numpy ndarray of shape (2,M) of M samples

•   utilde -- numpy ndarray of shape (2,M) of M samples

•   T -- Number of samples of curve (Default = 100)

•   k -- number of samples along path (Default = 5)

Return type
    numpy scalar

Return gradE
    gradient of energy

Return normgradE
    norm of gradient of energy

geodesic.cov_integral(alpha, alphadot, basis, T=100, k=5)
    Calculates covariance along path alpha

Parameters

•   alpha -- numpy ndarray of shape (2,M) of M samples (first curve)

•   alphadot -- numpy ndarray of shape (2,M) of M samples

•   basis -- list numpy ndarray of shape (2,M) of M samples

•   T -- Number of samples of curve (Default = 100)

•   k -- number of samples along path (Default = 5)

Return type
    numpy ndarray

Return u
    covariance

geodesic.find_basis_normal_path(alpha, k=5)
    computes orthonormalized basis vectors to the normal space at each of the k points (q-functions) of the path alpha

Parameters

•   alpha -- numpy ndarray of shape (2,M) of M samples (path)

•   k -- number of samples along path (Default = 5)

Return type
    numpy ndarray

Return basis
    basis vectors along the path

geodesic.geod_dist_path_strt(beta, k=5)
    calculate geodisc distance for path straightening

Parameters

•   beta -- numpy ndarray of shape (2,M) of M samples

•   k -- number of samples along path (Default = 5)

Return type
    numpy scalar

Return dist
    geodesic distance

geodesic.geod_sphere(beta1, beta2, k=5, scale=False)
    This function calculates the geodesics between open curves beta1 and beta2 with k steps along path

Parameters

•   beta1 -- numpy ndarray of shape (2,M) of M samples

•   beta2 -- numpy ndarray of shape (2,M) of M samples

•   k -- number of samples along path (Default = 5)

•   scale -- include length (Default = False)

Return type
    numpy ndarray

Return dist
    geodesic distance

Return path
    geodesic path

Return O
    rotation matrix

geodesic.init_path_geod(beta1, beta2, T=100, k=5)
    Initializes a path in \cal{C}. beta1, beta2 are already standardized curves. Creates a path from beta1 to beta2 in shape space, then projects to the closed shape manifold.

Parameters

•   beta1 -- numpy ndarray of shape (2,M) of M samples (first curve)

•   beta2 -- numpy ndarray of shape (2,M) of M samples (end curve)

•   T -- Number of samples of curve (Default = 100)

•   k -- number of samples along path (Default = 5)

Return type
    numpy ndarray

Return alpha
    a path between two q-functions

Return beta
    a path between two curves

Return O
    rotation matrix

geodesic.init_path_rand(beta1, beta_mid, beta2, T=100, k=5)
    Initializes a path in \cal{C}. beta1, beta_mid beta2 are already standardized curves. Creates a path from beta1 to beta_mid to beta2 in shape space, then projects to the closed shape manifold.

Parameters

•   beta1 -- numpy ndarray of shape (2,M) of M samples (first curve)

•   betamid -- numpy ndarray of shape (2,M) of M samples (mid curve)

•   beta2 -- numpy ndarray of shape (2,M) of M samples (end curve)

•   T -- Number of samples of curve (Default = 100)

•   k -- number of samples along path (Default = 5)

Return type
    numpy ndarray

Return alpha
    a path between two q-functions

Return beta
    a path between two curves

Return O
    rotation matrix

geodesic.path_straightening(beta1, beta2, betamid, init='rand', T=100, k=5)
    Perform path straightening to find geodesic between two shapes in either the space of closed curves or the space of affine standardized curves. This algorithm follows the steps outlined in section 4.6 of the manuscript.

Parameters

•   beta1 -- numpy ndarray of shape (2,M) of M samples (first curve)

•   beta2 -- numpy ndarray of shape (2,M) of M samples (end curve)

•   betamid -- numpy ndarray of shape (2,M) of M samples (mid curve Default = NULL, only needed for init "rand")

•   init -- initialize path geodesic or random (Default = "rand")

•   T -- Number of samples of curve (Default = 100)

•   k -- number of samples along path (Default = 5)

Return type
    numpy ndarray

Return dist
    geodesic distance

Return path
    geodesic path

Return pathsqnc
    geodesic path sequence

Return E
    energy

geodesic.update_path(alpha, beta, gradE, delta, T=100, k=5)
    Update the path along the direction -gradE

Parameters

•   alpha -- numpy ndarray of shape (2,M) of M samples

•   beta -- numpy ndarray of shape (2,M) of M samples

•   gradE -- numpy ndarray of shape (2,M) of M samples

•   delta -- gradient paramenter

•   T -- Number of samples of curve (Default = 100)

•   k -- number of samples along path (Default = 5)

Return type
    numpy scalar

Return alpha
    updated path of srvfs

Return beta
    updated path of curves

UTILITY FUNCTIONS

Utility functions for SRSF Manipulations

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

utility_functions.SqrtMean(gam)
    calculates the srsf of warping functions with corresponding shooting vectors

Parameters
    gam -- numpy ndarray of shape (M,N) of M warping functions with N samples

Return type
    2 numpy ndarray and vector

Return mu
    Karcher mean psi function

Return gam_mu
    vector of dim N which is the Karcher mean warping function

Return psi
    numpy ndarray of shape (M,N) of M SRSF of the warping functions

Return vec
    numpy ndarray of shape (M,N) of M shooting vectors

utility_functions.SqrtMeanInverse(gam)
    finds the inverse of the mean of the set of the diffeomorphisms gamma

Parameters
    gam -- numpy ndarray of shape (M,N) of M warping functions with N samples

Return type
    vector

Return gamI
    inverse of gam

utility_functions.SqrtMedian(gam)
    calculates the median srsf of warping functions with corresponding shooting vectors

Parameters
    gam -- numpy ndarray of shape (M,N) of M warping functions with N samples

Return type
    2 numpy ndarray and vector

Return gam_median
    Karcher median warping function

Return psi_meidan
    vector of dim N which is the Karcher median srsf function

Return psi
    numpy ndarray of shape (M,N) of M SRSF of the warping functions

Return vec
    numpy ndarray of shape (M,N) of M shooting vectors

utility_functions.cumtrapzmid(x, y, c, mid)
    cumulative trapezoidal numerical integration taken from midpoint

Parameters

•   x -- vector of size N describing the time samples

•   y -- vector of size N describing the function

•   c -- midpointtic

•   mid -- midpiont location

Return type
    vector

Return fa
    cumulative integration

utility_functions.diffop(n, binsize=1)
    Creates a second order differential operator

Parameters

•   n -- dimension

•   binsize -- dx (default = 1)

Return type
    numpy ndarray

Return m
    matrix describing differential operator

utility_functions.elastic_depth(f, time, method='DP2', lam=0.0, parallel=True)
    calculates the elastic depth between functions in matrix f

Parameters

•   f -- matrix of size MxN (M time points for N functions)

•   time -- vector of size M describing the sample points

•   method -- method to apply optimization (default="DP2") options are "DP","DP2","RBFGS"

•   lam -- controls the elasticity (default = 0.0)

Return type
    scalar

Return amp
    amplitude depth

Return phase
    phase depth

utility_functions.elastic_distance(f1, f2, time, method='DP2', lam=0.0)
    " calculates the distances between function, where f1 is aligned to f2. In other words calculates the elastic distances

Parameters

•   f1 -- vector of size N

•   f2 -- vector of size N

•   time -- vector of size N describing the sample points

•   method -- method to apply optimization (default="DP2") options are "DP","DP2","RBFGS"

•   lam -- controls the elasticity (default = 0.0)

Return type
    scalar

Return Dy
    amplitude distance

Return Dx
    phase distance

utility_functions.f_K_fold(Nobs, K=5)
    generates sample indices for K-fold cross validation

    :param Nobs number of observations :param K number of folds

Return type
    numpy ndarray

Return train
    train indexes (Nobs*(K-1)/K X K)

Return test
    test indexes (Nobs*(1/K) X K)

utility_functions.f_to_srsf(f, time, smooth=False)
    converts f to a square-root slope function (SRSF)

Parameters

•   f -- vector of size N samples

•   time -- vector of size N describing the sample points

Return type
    vector

Return q
    srsf of f

utility_functions.geigen(Amat, Bmat, Cmat)
    generalized eigenvalue problem of the form

    max tr L'AM / sqrt(tr L'BL tr M'CM) w.r.t. L and M

    :param Amat numpy ndarray of shape (M,N) :param Bmat numpy ndarray of shape (M,N) :param Bmat numpy ndarray of shape (M,N)

Return type
    numpy ndarray

Return values
    eigenvalues

Return Lmat
    left eigenvectors

Return Mmat
    right eigenvectors

utility_functions.gradient_spline(time, f, smooth=False)
    This function takes the gradient of f using b-spline smoothing

Parameters

•   time -- vector of size N describing the sample points

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

•   smooth -- smooth data (default = F)

Return type
    tuple of numpy ndarray

Return f0
    smoothed functions functions

Return g
    first derivative of each function

Return g2
    second derivative of each function

utility_functions.innerprod_q(time, q1, q2)
    calculates the innerproduct between two srsfs

    :param time vector descrbing time samples :param q1 vector of srsf 1 :param q2 vector of srsf 2

Return type
    scalar

Return val
    inner product value

utility_functions.invertGamma(gam)
    finds the inverse of the diffeomorphism gamma

Parameters
    gam -- vector describing the warping function

Return type
    vector

Return gamI
    inverse of gam

utility_functions.optimum_reparam(q1, time, q2, method='DP2', lam=0.0, grid_dim=7)
    calculates the warping to align srsf q2 to q1

Parameters

•   q1 -- vector of size N or array of NxM samples of first SRSF

•   time -- vector of size N describing the sample points

•   q2 -- vector of size N or array of NxM samples samples of second SRSF

•   method -- method to apply optimization (default="DP2") options are "DP","DP2","RBFGS"

•   lam -- controls the amount of elasticity (default = 0.0)

•   grid_dim -- size of the grid, for the DP2 method only (default = 7)

Return type
    vector

Return gam
    describing the warping function used to align q2 with q1

utility_functions.optimum_reparam_pair(q, time, q1, q2, lam=0.0)
    calculates the warping to align srsf pair q1 and q2 to q

Parameters

•   q -- vector of size N or array of NxM samples of first SRSF

•   time -- vector of size N describing the sample points

•   q1 -- vector of size N or array of NxM samples samples of second SRSF

•   q2 -- vector of size N or array of NxM samples samples of second SRSF

•   lam -- controls the amount of elasticity (default = 0.0)

Return type
    vector

Return gam
    describing the warping function used to align q2 with q1

utility_functions.outlier_detection(q, time, mq, k=1.5)
    calculates outlier's using geodesic distances of the SRSFs from the median

Parameters

•   q -- numpy ndarray of N x M of M SRS functions with N samples

•   time -- vector of size N describing the sample points

•   mq -- median calculated using time_warping.srsf_align()

•   k -- cutoff threshold (default = 1.5)

Returns
    q_outlier: outlier functions

utility_functions.randomGamma(gam, num)
    generates random warping functions

Parameters

•   gam -- numpy ndarray of N x M of M of warping functions

•   num -- number of random functions

Returns
    rgam: random warping functions

utility_functions.resamplefunction(x, n)
    resample function using n points

Parameters

•   x -- functions

•   n -- number of points

Return type
    numpy array

Return xn
    resampled function

utility_functions.rgam(N, sigma, num)
    Generates random warping functions

Parameters

•   N -- length of warping function

•   sigma -- variance of warping functions

•   num -- number of warping functions

Returns
    gam: numpy ndarray of warping functions

utility_functions.smooth_data(f, sparam)
    This function smooths a collection of functions using a box filter

Parameters

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

•   sparam -- Number of times to run box filter (default = 25)

Return type
    numpy ndarray

Return f
    smoothed functions functions

utility_functions.srsf_to_f(q, time, f0=0.0)
    converts q (srsf) to a function

Parameters

•   q -- vector of size N samples of srsf

•   time -- vector of size N describing time sample points

•   f0 -- initial value

Return type
    vector

Return f
    function

utility_functions.update_progress(progress)
    This function creates a progress bar

Parameters
    progress -- fraction of progress

utility_functions.warp_f_gamma(time, f, gam)
    warps a function f by gam

    :param time vector describing time samples :param q vector describing srsf :param gam vector describing warping function

Return type
    numpy ndarray

Return f_temp
    warped srsf

utility_functions.warp_q_gamma(time, q, gam)
    warps a srsf q by gam

    :param time vector describing time samples :param q vector describing srsf :param gam vector describing warping function

Return type
    numpy ndarray

Return q_temp
    warped srsf

utility_functions.zero_crossing(Y, q, bt, time, y_max, y_min, gmax, gmin)
    finds zero-crossing of optimal gamma, gam = s*gmax + (1-s)*gmin from elastic regression model

Parameters

•   Y -- response

•   q -- predicitve function

•   bt -- basis function

•   time -- time samples

•   y_max -- maximum repsonse for warping function gmax

•   y_min -- minimum response for warping function gmin

•   gmax -- max warping function

•   gmin -- min warping fucntion

Return type
    numpy array

Return gamma
    optimal warping function

CURVE FUNCTIONS

functions for SRVF curve manipulations

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

curve_functions.Basis_Normal_A(q)
    Find Normal Basis

Parameters
    q -- numpy ndarray (n,T) defining T points on n dimensional SRVF

    :rtype list :return delg: basis

curve_functions.calc_j(basis)
    Calculates Jacobian matrix from normal basis

Parameters
    basis -- list of numpy ndarray of shape (2,M) of M samples basis

Return type
    numpy ndarray

Return j
    Jacobian

curve_functions.calculate_variance(beta)
    This function calculates variance of curve beta

Parameters
    beta -- numpy ndarray of shape (2,M) of M samples

Return type
    numpy ndarray

Return variance
    variance

curve_functions.calculatecentroid(beta)
    This function calculates centroid of a parameterized curve

Parameters
    beta -- numpy ndarray of shape (2,M) of M samples

Return type
    numpy ndarray

Return centroid
    center coordinates

curve_functions.curve_to_q(beta, mode='O')
    This function converts curve beta to srvf q

Parameters

•   beta -- numpy ndarray of shape (2,M) of M samples

•   mode -- Open ('O') or closed curve ('C') (default 'O')

Return type
    numpy ndarray

Return q
    srvf of curve

Return lenb
    length of curve

Return lenq
    length of srvf

curve_functions.curve_zero_crossing(Y, beta, bt, y_max, y_min, gmax, gmin)
    finds zero-crossing of optimal gamma, gam = s*gmax + (1-s)*gmin from elastic curve regression model

Parameters

•   Y -- response

•   beta -- predicitve function

•   bt -- basis function

•   y_max -- maximum repsonse for warping function gmax

•   y_min -- minimum response for warping function gmin

•   gmax -- max warping function

•   gmin -- min warping fucntion

Return type
    numpy array

Return gamma
    optimal warping function

Return O_hat
    rotation matrix

curve_functions.elastic_distance_curve(beta1, beta2, closed=0, scale=False, method='DP')
    Calculates the two elastic distances between two curves :param beta1: numpy ndarray of shape (2,M) of M samples :param beta2: numpy ndarray of shape (2,M) of M samples :param closed: open (0) or closed (1) curve (default=0) :param scale: include scale (default=False) :param method: method to apply optimization (default="DP") options are "DP" or "RBFGS"

Return type
    scalar

Return dist
    shape distance

curve_functions.elastic_shooting(q1, v)
    Calculates shooting vector from v to q1

Parameters

•   q1 -- vector of srvf

•   v -- shooting vector

    :rtype numpy ndarray :return q2n: vector of srvf

curve_functions.find_basis_normal(q)
    Finds the basis normal to the srvf

Parameters
    q1 -- numpy ndarray of shape (2,M) of M samples

Return type
    list of numpy ndarray

Return basis
    list containing basis vectors

curve_functions.find_best_rotation(q1, q2)
    This function calculates the best rotation between two srvfs using procustes rigid alignment

Parameters

•   q1 -- numpy ndarray of shape (2,M) of M samples

•   q2 -- numpy ndarray of shape (2,M) of M samples

Return type
    numpy ndarray

Return q2new
    optimal rotated q2 to q1

Return R
    rotation matrix

curve_functions.find_rotation_and_seed_coord(beta1, beta2, closed=0, method='DP')
    This function returns a candidate list of optimally oriented and registered (seed) shapes w.r.t. beta1

Parameters

•   beta1 -- numpy ndarray of shape (2,M) of M samples

•   beta2 -- numpy ndarray of shape (2,M) of M samples

•   closed -- Open (0) or Closed (1)

•   method -- method to apply optimization (default="DP") options are "DP" or "RBFGS"

Return type
    numpy ndarray

Return beta2new
    optimal rotated beta2 to beta1

Return O
    rotation matrix

Return tau
    seed

curve_functions.find_rotation_and_seed_q(q1, q2)
    This function returns a candidate list of optimally oriented and registered (seed) shapes w.r.t. beta1

Parameters

•   q1 -- numpy ndarray of shape (2,M) of M samples

•   q2 -- numpy ndarray of shape (2,M) of M samples

Return type
    numpy ndarray

Return beta2new
    optimal rotated beta2 to beta1

Return O
    rotation matrix

Return tau
    seed

curve_functions.find_rotation_and_seed_unique(q1, q2, closed=0, method='DP')
    This function returns a candidate list of optimally oriented and registered (seed) shapes w.r.t. beta1

Parameters

•   beta1 -- numpy ndarray of shape (2,M) of M samples

•   beta2 -- numpy ndarray of shape (2,M) of M samples

•   closed -- Open (0) or Closed (1)

•   method -- method to apply optimization (default="DP") options are "DP" or "RBFGS"

Return type
    numpy ndarray

Return beta2new
    optimal rotated beta2 to beta1

Return O
    rotation matrix

Return tau
    seed

curve_functions.gram_schmidt(basis)
    Performs Gram Schmidt Orthogonlization of a basis_o

param basis
    list of numpy ndarray of shape (2,M) of M samples

rtype
    list of numpy ndarray

return basis_o
    orthogonlized basis

curve_functions.group_action_by_gamma(q, gamma)
    This function reparamerized srvf q by gamma

Parameters

•   f -- numpy ndarray of shape (2,M) of M samples

•   gamma -- numpy ndarray of shape (2,M) of M samples

Return type
    numpy ndarray

Return qn
    reparatermized srvf

curve_functions.group_action_by_gamma_coord(f, gamma)
    This function reparamerized curve f by gamma

Parameters

•   f -- numpy ndarray of shape (2,M) of M samples

•   gamma -- numpy ndarray of shape (2,M) of M samples

Return type
    numpy ndarray

Return fn
    reparatermized curve

curve_functions.innerprod_q2(q1, q2)
    This function calculates the inner product in srvf space

Parameters

•   q1 -- numpy ndarray of shape (2,M) of M samples

•   q2 -- numpy ndarray of shape (2,M) of M samples

Return type
    numpy ndarray

Return val
    inner product

curve_functions.inverse_exp(q1, q2, beta2)
    Calculate the inverse exponential to obtain a shooting vector from q1 to q2 in shape space of open curves

Parameters

•   q1 -- numpy ndarray of shape (2,M) of M samples

•   q2 -- numpy ndarray of shape (2,M) of M samples

•   beta2 -- numpy ndarray of shape (2,M) of M samples

Return type
    numpy ndarray

Return v
    shooting vectors

curve_functions.inverse_exp_coord(beta1, beta2, closed=0, method='DP')
    Calculate the inverse exponential to obtain a shooting vector from beta1 to beta2 in shape space of open curves

Parameters

•   beta1 -- numpy ndarray of shape (2,M) of M samples

•   beta2 -- numpy ndarray of shape (2,M) of M samples

•   closed -- open (0) or closed (1) curve

•   method -- method to apply optimization (default="DP") options are "DP" or "RBFGS"

Return type
    numpy ndarray

Return v
    shooting vectors

Return dist
    distance

curve_functions.optimum_reparam_curve(q1, q2, lam=0.0, method='DP')
    calculates the warping to align srsf q2 to q1

Parameters

•   q1 -- matrix of size nxN or array of NxM samples of first SRVF

•   time -- vector of size N describing the sample points

•   q2 -- matrix of size nxN or array of NxM samples samples of second SRVF

•   lam -- controls the amount of elasticity (default = 0.0)

•   method -- method to apply optimization (default="DP") options are "DP" or "RBFGS"

Return type
    vector

Return gam
    describing the warping function used to align q2 with q1

curve_functions.parallel_translate(w, q1, q2, basis, mode=0)
    parallel translates q1 and q2 along manifold

Parameters

•   w -- numpy ndarray of shape (2,M) of M samples

•   q1 -- numpy ndarray of shape (2,M) of M samples

•   q2 -- numpy ndarray of shape (2,M) of M samples

•   basis -- list of numpy ndarray of shape (2,M) of M samples

•   mode -- open 0 or closed curves 1 (default 0)

Return type
    numpy ndarray

Return wbar
    translated vector

curve_functions.pre_proc_curve(beta, T=100)
    This function prepcoessed a curve beta to set of closed curves

Parameters

•   beta -- numpy ndarray of shape (2,M) of M samples

•   T -- number of samples (default = 100)

Return type
    numpy ndarray

Return betanew
    projected beta

Return qnew
    projected srvf

Return A
    alignment matrix (not used currently)

curve_functions.project_curve(q)
    This function projects srvf q to set of close curves

Parameters
    q -- numpy ndarray of shape (2,M) of M samples

Return type
    numpy ndarray

Return qproj
    project srvf

curve_functions.project_tangent(w, q, basis)
    projects srvf to tangent space w using basis

Parameters

•   w -- numpy ndarray of shape (2,M) of M samples

•   q -- numpy ndarray of shape (2,M) of M samples

•   basis -- list of numpy ndarray of shape (2,M) of M samples

Return type
    numpy ndarray

Return wproj
    projected q

curve_functions.psi(x, a, q)
    This function formats variance output

Parameters

•   x -- numpy ndarray of shape (2,M) of M samples curve

•   a -- numpy ndarray of shape (2,1) mean

•   q -- numpy ndarray of shape (2,M) of M samples srvf

Return type
    numpy ndarray

Return psi1
    variance

Return psi2
    cross variance

Return psi3
    curve end

Return psi4
    curve end

curve_functions.q_to_curve(q, scale=1)
    This function converts srvf to beta

Parameters

•   q -- numpy ndarray of shape (n,M) of M samples

•   scale -- scale of curve

Return type
    numpy ndarray

Return beta
    parameterized curve

curve_functions.resamplecurve(x, N=100, mode='O')
    This function resamples a curve to have N samples

Parameters

•   x -- numpy ndarray of shape (2,M) of M samples

•   N -- Number of samples for new curve (default = 100)

•   mode -- Open ('O') or closed curve ('C') (default 'O')

Return type
    numpy ndarray

Return xn
    resampled curve

curve_functions.scale_curve(beta)
    scales curve to length 1

Parameters
    beta -- numpy ndarray of shape (2,M) of M samples

Return type
    numpy ndarray

Return beta_scaled
    scaled curve

Return scale
    scale factor used

curve_functions.shift_f(f, tau)
    shifts a curve f by tau

Parameters

•   f -- numpy ndarray of shape (2,M) of M samples

•   tau -- scalar

Return type
    numpy ndarray

Return fn
    shifted curve

UMAP EFDA METRICS

Distance metrics for functions and curves in R^n for use with UMAP (https://github.com/lmcinnes/umap)

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

umap_metric.efda_distance(q1, q2)
    " calculates the distances between two curves, where q2 is aligned to q1. In other words calculates the elastic distances/ This metric is set up for use with UMAP or t-sne from scikit-learn

Parameters

•   q1 -- vector of size N

•   q2 -- vector of size N

Return type
    scalar

Return dist
    amplitude distance

umap_metric.efda_distance_curve(beta1, beta2, closed)
    " calculates the distances between two curves, where beta2 is aligned to beta1. In other words calculates the elastic distance. This metric is set up for use with UMAP or t-sne from scikit-learn

Parameters

•   beta1 -- vector of size n*M

•   beta2 -- vector of size n*M

•   closed -- .INDENT 2.0

0.  if open curves and (1) if closed curves

Return type
    scalar

Return dist
    shape distance Tucker, J. D. 2014, Functional Component Analysis and Regression using Elastic Methods. Ph.D. Thesis, Florida State University.

    Robinson, D. T. 2012, Function Data Analysis and Partial Shape Matching in the Square Root Velocity Framework. Ph.D. Thesis, Florida State University.

    Huang, W. 2014, Optimization Algorithms on Riemannian Manifolds with Applications. Ph.D. Thesis, Florida State University.

    Srivastava, A., Wu, W., Kurtek, S., Klassen, E. and Marron, J. S. (2011). Registration of Functional Data Using Fisher-Rao Metric. arXiv:1103.3817v2 [math.ST].

    Tucker, J. D., Wu, W. and Srivastava, A. (2013). Generative models for functional data using phase and amplitude separation. Computational Statistics and Data Analysis 61, 50-66.

    J. D. Tucker, W. Wu, and A. Srivastava, "Phase-Amplitude Separation of Proteomics Data Using Extended Fisher-Rao Metric," Electronic Journal of Statistics, Vol 8, no. 2. pp 1724-1733, 2014.

    J. D. Tucker, W. Wu, and A. Srivastava, "Analysis of signals under compositional noise With applications to SONAR data," IEEE Journal of Oceanic Engineering, Vol 29, no. 2. pp 318-330, Apr 2014.

    Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

    S. Kurtek, A. Srivastava, and W. Wu. Signal estimation under random time-warpings and nonlinear signal alignment. In Proceedings of Neural Information Processing Systems (NIPS), 2011.

    Wen Huang, Kyle A. Gallivan, Anuj Srivastava, Pierre-Antoine Absil. "Riemannian Optimization for Elastic Shape Analysis", Short version, The 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014).

    Cheng, W., Dryden, I. L., and Huang, X. (2016). Bayesian registration of functions and curves. Bayesian Analysis, 11(2), 447-475.

    W. Xie, S. Kurtek, K. Bharath, and Y. Sun, A geometric approach to visualization of variability in functional data, Journal of American Statistical Association 112 (2017), pp. 979-993.

    Lu, Y., R. Herbei, and S. Kurtek, 2017: Bayesian registration of functions with a Gaussian process prior. Journal of Computational and Graphical Statistics, 26, no. 4, 894–904.

    Lee, S. and S. Jung, 2017: Combined analysis of amplitude and phase variations in functional data. arXiv:1603.01775 [stat.ME], 1–21.

    J. D. Tucker, J. R. Lewis, and A. Srivastava, “Elastic Functional Principal Component Regression,” Statistical Analysis and Data Mining, vol. 12, no. 2, pp. 101-115, 2019.

    J. D. Tucker, J. R. Lewis, C. King, and S. Kurtek, “A Geometric Approach for Computing Tolerance Bounds for Elastic Functional Data,” Journal of Applied Statistics, 10.1080/02664763.2019.1645818, 2019.

    T. Harris, J. D. Tucker, B. Li, and L. Shand, "Elastic depths for detecting shape anomalies in functional data," Technometrics, 10.1080/00401706.2020.1811156, 2020.

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AUTHOR
J. Derek Tucker

COPYRIGHT
2020, J. Derek Tucker