algorithms.statistics.empirical_pvalue¶
Module: algorithms.statistics.empirical_pvalue¶
Inheritance diagram for nipy.algorithms.statistics.empirical_pvalue:
Routines to get corrected p-values estimates, based on the observations.
It implements 3 approaches:
Benjamini-Hochberg FDR: http://en.wikipedia.org/wiki/False_discovery_rate
a class that fits a Gaussian model to the central part of an histogram, following [1]
[1] Schwartzman A, Dougherty RF, Lee J, Ghahremani D, Taylor JE. Empirical null and false discovery rate analysis in neuroimaging. Neuroimage. 2009 Jan 1;44(1):71-82. PubMed PMID: 18547821. DOI: 10.1016/j.neuroimage.2008.04.182
This is typically necessary to estimate a FDR when one is not certain that the data behaves as a standard normal under H_0.
a model based on Gaussian mixture modelling ‘a la Oxford’
Author : Bertrand Thirion, Yaroslav Halchenko, 2008-2012
Class¶
NormalEmpiricalNull¶
-
class
nipy.algorithms.statistics.empirical_pvalue.NormalEmpiricalNull(x)[source]¶ Bases:
objectClass to compute the empirical null normal fit to the data.
The data which is used to estimate the FDR, assuming a Gaussian null from Schwartzmann et al., NeuroImage 44 (2009) 71–82
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__init__(x)[source]¶ Initialize an empirical null normal object.
- Parameters
x : 1D ndarray
The data used to estimate the empirical null.
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learn(left=0.2, right=0.8)[source]¶ Estimate the proportion, mean and variance of a Gaussian distribution for a fraction of the data
- Parameters
left: float, optional
Left cut parameter to prevent fitting non-gaussian data
right: float, optional
Right cut parameter to prevent fitting non-gaussian data
Notes
This method stores the following attributes:
mu = mu
p0 = min(1, np.exp(lp0))
sqsigma: variance of the estimated normal distribution
sigma: np.sqrt(sqsigma) : standard deviation of the estimated normal distribution
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threshold(alpha=0.05, verbose=0)[source]¶ Compute the threshold corresponding to an alpha-level FDR for x
- Parameters
alpha : float, optional
the chosen false discovery rate threshold.
verbose : boolean, optional
the verbosity level, if True a plot is generated.
- Returns
theta: float
the critical value associated with the provided FDR
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uncorrected_threshold(alpha=0.001, verbose=0)[source]¶ Compute the threshold corresponding to a specificity alpha for x
- Parameters
alpha : float, optional
the chosen false discovery rate (FDR) threshold.
verbose : boolean, optional
the verbosity level, if True a plot is generated.
- Returns
theta: float
the critical value associated with the provided p-value
-
fdr(theta)[source]¶ Given a threshold theta, find the estimated FDR
- Parameters
theta : float or array of shape (n_samples)
values to test
- Returns
afp : value of array of shape(n)
-
plot(efp=None, alpha=0.05, bar=1, mpaxes=None)[source]¶ Plot the histogram of x
- Parameters
efp : float, optional
The empirical FDR (corresponding to x) if efp==None, the false positive rate threshold plot is not drawn.
alpha : float, optional
The chosen FDR threshold
bar=1 : bool, optional
mpaxes=None: if not None, handle to an axes where the fig
will be drawn. Avoids creating unnecessarily new figures
-
Functions¶
-
nipy.algorithms.statistics.empirical_pvalue.check_p_values(p_values)[source]¶ Basic checks on the p_values array: values should be within [0,1]
Assures also that p_values are at least in 1d array. None of the checks is performed if p_values is None.
- Parameters
p_values : array of shape (n)
The sample p-values
- Returns
p_values : array of shape (n)
The sample p-values
-
nipy.algorithms.statistics.empirical_pvalue.fdr(p_values=None, verbose=0)[source]¶ Returns the FDR associated with each p value
- Parameters
p_values : ndarray of shape (n)
The samples p-value
- Returns
q : array of shape(n)
The corresponding fdr values
-
nipy.algorithms.statistics.empirical_pvalue.fdr_threshold(p_values, alpha=0.05)[source]¶ Return FDR threshold given p values
- Parameters
p_values : array of shape (n), optional
The samples p-value
alpha : float, optional
The desired FDR significance
- Returns
critical_p_value: float
The p value corresponding to the FDR alpha
-
nipy.algorithms.statistics.empirical_pvalue.gamma_gaussian_fit(x, test=None, verbose=0, mpaxes=False, bias=1, gaussian_mix=0, return_estimator=False)[source]¶ Computing some prior probabilities that the voxels of a certain map are in class disactivated, null or active using a gamma-Gaussian mixture
- Parameters
x: array of shape (nvox,)
the map to be analysed
test: array of shape (nbitems,), optional
the test values for which the p-value needs to be computed by default, test = x
verbose: 0, 1 or 2, optional
verbosity mode, 0 is quiet, and 2 calls matplotlib to display graphs.
mpaxes: matplotlib axes, optional
axes handle used to plot the figure in verbose mode if None, new axes are created if false, nothing is done
bias: float, optional
lower bound on the Gaussian variance (to avoid shrinkage)
gaussian_mix: float, optional
if nonzero, lower bound on the Gaussian mixing weight (to avoid shrinkage)
return_estimator: boolean, optional
if return_estimator is true, the estimator object is returned.
- Returns
bfp: array of shape (nbitems,3)
The probability of each component in the mixture model for each test value
estimator: nipy.labs.clustering.ggmixture.GGGM object
The estimator object, returned only if return_estimator is true.
-
nipy.algorithms.statistics.empirical_pvalue.gaussian_fdr(x)[source]¶ Return the FDR associated with each value assuming a Gaussian distribution
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nipy.algorithms.statistics.empirical_pvalue.gaussian_fdr_threshold(x, alpha=0.05)[source]¶ Return FDR threshold given normal variates
Given an array x of normal variates, this function returns the critical p-value associated with alpha. x is explicitly assumed to be normal distributed under H_0
- Parameters
x: ndarray
input data
alpha: float, optional
desired significance
- Returns
threshold : float
threshold, given as a Gaussian critical value
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nipy.algorithms.statistics.empirical_pvalue.smoothed_histogram_from_samples(x, bins=None, nbins=256, normalized=False)[source]¶ Smooth histogram corresponding to density underlying the samples in x
- Parameters
x: array of shape(n_samples)
input data
bins: array of shape(nbins+1), optional
the bins location
nbins: int, optional
the number of bins of the resulting histogram
normalized: bool, optional
if True, the result is returned as a density value
- Returns
h: array of shape (nbins)
the histogram
bins: array of shape(nbins+1),
the bins location
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nipy.algorithms.statistics.empirical_pvalue.three_classes_GMM_fit(x, test=None, alpha=0.01, prior_strength=100, verbose=0, fixed_scale=False, mpaxes=None, bias=0, theta=0, return_estimator=False)[source]¶ Fit the data with a 3-classes Gaussian Mixture Model, i.e. compute some probability that the voxels of a certain map are in class disactivated, null or active
- Parameters
x: array of shape (nvox,1)
The map to be analysed
test: array of shape(nbitems,1), optional
the test values for which the p-value needs to be computed by default (if None), test=x
alpha: float, optional
the prior weights of the positive and negative classes
prior_strength: float, optional
the confidence on the prior (should be compared to size(x))
verbose: int
verbosity mode
fixed_scale: bool, optional
boolean, variance parameterization. if True, the variance is locked to 1 otherwise, it is estimated from the data
mpaxes:
axes handle used to plot the figure in verbose mode if None, new axes are created
bias: bool
allows a rescaling of the posterior probability that takes into account the threshold theta. Not rigorous.
theta: float
the threshold used to correct the posterior p-values when bias=1; normally, it is such that test>theta note that if theta = -np.inf, the method has a standard behaviour
return_estimator: boolean, optional
If return_estimator is true, the estimator object is returned.
- Returns
bfp : array of shape (nbitems,3):
the posterior probability of each test item belonging to each component in the GMM (sum to 1 across the 3 classes) if np.size(test)==0, i.e. nbitem==0, None is returned
estimator : nipy.labs.clustering.GMM object
The estimator object, returned only if return_estimator is true.
Notes
Our convention is that:
class 1 represents the negative class
class 2 represents the null class
class 3 represents the positive class