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[ENH] Implement pure-Python ICA with mixture modeling #51
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8391f4b
Add nipy's mixture model.
tsalo 0351b88
Add MELODIC-less ICA pseudocode.
tsalo 796f842
Fix up some bugs and add test.
tsalo 0f646f4
Add tedana as a dependency.
tsalo 49d719b
Add nipy license information.
tsalo 3d03758
Fix tedana version.
tsalo 7b7739d
Fix up new approach.
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| Original file line number | Diff line number | Diff line change |
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| """ | ||
| This file was taken from the nipy Python library and reduced to only contain | ||
| code necessary to define and use the GGM class. Minor changes have been made | ||
| to the documentation, in order to follow project convention. | ||
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| License | ||
| ------- | ||
| Copyright (c) 2006-2018, NIPY Developers | ||
| All rights reserved. | ||
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| Redistribution and use in source and binary forms, with or without | ||
| modification, are permitted provided that the following conditions are | ||
| met: | ||
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| * Redistributions of source code must retain the above copyright | ||
| notice, this list of conditions and the following disclaimer. | ||
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| * Redistributions in binary form must reproduce the above | ||
| copyright notice, this list of conditions and the following | ||
| disclaimer in the documentation and/or other materials provided | ||
| with the distribution. | ||
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| * Neither the name of the NIPY Developers nor the names of any | ||
| contributors may be used to endorse or promote products derived | ||
| from this software without specific prior written permission. | ||
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| THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS | ||
| "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT | ||
| LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR | ||
| A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT | ||
| OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, | ||
| SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT | ||
| LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, | ||
| DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY | ||
| THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT | ||
| (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE | ||
| OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. | ||
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| Notes | ||
| ----- | ||
| Taken from the nipy version at commit c7ae00435ef1134cee28eb0d31d2f1a253328f6b. | ||
| """ | ||
| import numpy as np | ||
| import scipy.special as sp | ||
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| def _dichopsi_log(u, v, y, eps=0.00001): | ||
| """ Implements the dichotomic part of the solution of psi(c)-log(c)=y | ||
| """ | ||
| if u > v: | ||
| u, v = v, u | ||
| t = (u + v) / 2 | ||
| if np.absolute(u - v) < eps: | ||
| return t | ||
| else: | ||
| if sp.psi(t) - np.log(t) > y: | ||
| return _dichopsi_log(u, t, y, eps) | ||
| else: | ||
| return _dichopsi_log(t, v, y, eps) | ||
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| def _psi_solve(y, eps=0.00001): | ||
| """ Solve psi(c)-log(c)=y by dichotomy | ||
| """ | ||
| if y > 0: | ||
| print("y", y) | ||
| raise ValueError("y>0, the problem cannot be solved") | ||
| u = 1. | ||
| if y > sp.psi(u) - np.log(u): | ||
| while sp.psi(u) - np.log(u) < y: | ||
| u *= 2 | ||
| u /= 2 | ||
| else: | ||
| while sp.psi(u) - np.log(u) > y: | ||
| u /= 2 | ||
| return _dichopsi_log(u, 2 * u, y, eps) | ||
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| def _compute_c(x, z, eps=0.00001): | ||
| """ | ||
| this function returns the mle of the shape parameter if a 1D gamma | ||
| density | ||
| """ | ||
| eps = 1.e-7 | ||
| y = np.dot(z, np.log(x)) / np.sum(z) - np.log(np.dot(z, x) / np.sum(z)) | ||
| if y > - eps: | ||
| c = 10 | ||
| else: | ||
| c = _psi_solve(y, eps=0.00001) | ||
| return c | ||
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| def _gaus_dens(mean, var, x): | ||
| """ evaluate the gaussian density (mean,var) at points x | ||
| """ | ||
| Q = - (x - mean) ** 2 / (2 * var) | ||
| return 1. / np.sqrt(2 * np.pi * var) * np.exp(Q) | ||
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| def _gam_dens(shape, scale, x): | ||
| """evaluate the gamma density (shape,scale) at points x | ||
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| Notes | ||
| ----- | ||
| Returns 0 on negative subspace | ||
| """ | ||
| ng = np.zeros(np.size(x)) | ||
| cst = - shape * np.log(scale) - sp.gammaln(shape) | ||
| i = np.ravel(np.nonzero(x > 0)) | ||
| if np.size(i) > 0: | ||
| lz = cst + (shape - 1) * np.log(x[i]) - x[i] / scale | ||
| ng[i] = np.exp(lz) | ||
| return ng | ||
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| def _gam_param(x, z): | ||
| """ Compute the parameters of a gamma density from data weighted points | ||
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| Parameters | ||
| ---------- | ||
| x: array of shape(nbitem) the learning points | ||
| z: array of shape(nbitem), their membership within the class | ||
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| Notes | ||
| ----- | ||
| if no point is positive then the couple (1, 1) is returned | ||
| """ | ||
| eps = 1.e-5 | ||
| i = np.ravel(np.nonzero(x > 0)) | ||
| szi = np.sum(z[i]) | ||
| if szi > 0: | ||
| shape = _compute_c(x[i], z[i], eps) | ||
| scale = np.dot(x[i], z[i]) / (szi * shape) | ||
| else: | ||
| shape = 1 | ||
| scale = 1 | ||
| return shape, scale | ||
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| class GGM(object): | ||
| """ | ||
| This is the basic one dimensional Gaussian-Gamma Mixture estimation class | ||
| Note that it can work with positive or negative values, | ||
| as long as there is at least one positive value. | ||
| NB : The gamma distribution is defined only on positive values. | ||
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| 5 scalar members | ||
| - mean: gaussian mean | ||
| - var: gaussian variance (non-negative) | ||
| - shape: gamma shape (non-negative) | ||
| - scale: gamma scale (non-negative) | ||
| - mixt: mixture parameter (non-negative, weight of the gamma) | ||
| """ | ||
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| def __init__(self, shape=1, scale=1, mean=0, var=1, mixt=0.5): | ||
| self.shape = shape | ||
| self.scale = scale | ||
| self.mean = mean | ||
| self.var = var | ||
| self.mixt = mixt | ||
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| def parameters(self): | ||
| """ print the paramteres of self | ||
| """ | ||
| print("Gaussian: mean: ", self.mean, "variance: ", self.var) | ||
| print("Gamma: shape: ", self.shape, "scale: ", self.scale) | ||
| print("Mixture gamma: ", self.mixt, "Gaussian: ", 1 - self.mixt) | ||
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| def Mstep(self, x, z): | ||
| """ | ||
| Mstep of the model: maximum likelihood | ||
| estimation of the parameters of the model | ||
|
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||
| Parameters | ||
| ---------- | ||
| x : array of shape (nbitems,) | ||
| input data | ||
| z array of shape(nbitrems, 2) | ||
| the membership matrix | ||
| """ | ||
| # z[0,:] is the likelihood to be generated by the gamma | ||
| # z[1,:] is the likelihood to be generated by the gaussian | ||
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| tiny = 1.e-15 | ||
| sz = np.maximum(tiny, np.sum(z, 0)) | ||
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| self.shape, self.scale = _gam_param(x, z[:, 0]) | ||
| self.mean = np.dot(x, z[:, 1]) / sz[1] | ||
| self.var = np.dot((x - self.mean) ** 2, z[:, 1]) / sz[1] | ||
| self.mixt = sz[0] / np.size(x) | ||
|
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| def Estep(self, x): | ||
| """ | ||
| E step of the estimation: | ||
| Estimation of ata membsership | ||
|
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||
| Parameters | ||
| ---------- | ||
| x: array of shape (nbitems,) | ||
| input data | ||
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| Returns | ||
| ------- | ||
| z: array of shape (nbitems, 2) | ||
| the membership matrix | ||
| """ | ||
| eps = 1.e-15 | ||
| z = np.zeros((np.size(x), 2), 'd') | ||
| z[:, 0] = _gam_dens(self.shape, self.scale, x) | ||
| z[:, 1] = _gaus_dens(self.mean, self.var, x) | ||
| z = z * np.array([self.mixt, 1. - self.mixt]) | ||
| sz = np.maximum(np.sum(z, 1), eps) | ||
| L = np.sum(np.log(sz)) / np.size(x) | ||
| z = (z.T / sz).T | ||
| return z, L | ||
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| def estimate(self, x, niter=10, delta=0.0001, verbose=False): | ||
| """ Complete EM estimation procedure | ||
|
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| Parameters | ||
| ---------- | ||
| x : array of shape (nbitems,) | ||
| the data to be processed | ||
| niter : int, optional | ||
| max nb of iterations | ||
| delta : float, optional | ||
| criterion for convergence | ||
| verbose : bool, optional | ||
| If True, print values during iterations | ||
|
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| Returns | ||
| ------- | ||
| LL, float | ||
| average final log-likelihood | ||
| """ | ||
| if x.max() < 0: | ||
| # all the values are generated by the Gaussian | ||
| self.mean = np.mean(x) | ||
| self.var = np.var(x) | ||
| self.mixt = 0. | ||
| L = 0.5 * (1 + np.log(2 * np.pi * self.var)) | ||
| return L | ||
|
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| # proceed with standard estimate | ||
| z, L = self.Estep(x) | ||
| L0 = L - 2 * delta | ||
| for i in range(niter): | ||
| self.Mstep(x, z) | ||
| z, L = self.Estep(x) | ||
| if verbose: | ||
| print(i, L) | ||
| if (L < L0 + delta): | ||
| break | ||
| L0 = L | ||
| return L | ||
|
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| def show(self, x): | ||
| """ Visualization of the mm based on the empirical histogram of x | ||
|
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| Parameters | ||
| ---------- | ||
| x : array of shape (nbitems,) | ||
| the data to be processed | ||
| """ | ||
| step = 3.5 * np.std(x) / np.exp(np.log(np.size(x)) / 3) | ||
| bins = max(10, int((x.max() - x.min()) / step)) | ||
| h, c = np.histogram(x, bins) | ||
| h = h.astype(np.float) / np.size(x) | ||
| p = self.mixt | ||
|
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| dc = c[1] - c[0] | ||
| y = (1 - p) * _gaus_dens(self.mean, self.var, c) * dc | ||
| z = np.zeros(np.size(c)) | ||
| z = _gam_dens(self.shape, self.scale, c) * p * dc | ||
|
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| import matplotlib.pylab as mp | ||
| mp.figure() | ||
| mp.plot(0.5 * (c[1:] + c[:-1]), h) | ||
| mp.plot(c, y, 'r') | ||
| mp.plot(c, z, 'g') | ||
| mp.plot(c, z + y, 'k') | ||
| mp.title('Fit of the density with a Gamma-Gaussians mixture') | ||
| mp.legend(('data', 'gaussian acomponent', 'gamma component', | ||
| 'mixture distribution')) | ||
|
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| def posterior(self, x): | ||
| """Posterior probability of observing the data x for each component | ||
|
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| Parameters | ||
| ---------- | ||
| x: array of shape (nbitems,) | ||
| the data to be processed | ||
|
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| Returns | ||
| ------- | ||
| y, pg : arrays of shape (nbitem) | ||
| the posterior probability | ||
| """ | ||
| p = self.mixt | ||
| pg = p * _gam_dens(self.shape, self.scale, x) | ||
| y = (1 - p) * _gaus_dens(self.mean, self.var, x) | ||
| return y / (y + pg), pg / (y + pg) |
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More aggressive masking seems to be necessary. I noticed that the ICA components were showing up outside the brain with the old method.