# Multivariate t-distribution of students with python

To generate samples with multivariate t-distribution, I use this function:

``````def multivariatet(mu,Sigma,N,M):
'''
Output:
Produce M samples of d-dimensional multivariate t distribution
Input:
mu = mean (d dimensional numpy array or scalar)
Sigma = scale matrix (dxd numpy array)
N = degrees of freedom
M = # of samples to produce
'''
d = len(Sigma)
g = np.tile(np.random.gamma(N/2.,2./N,M),(d,1)).T
Z = np.random.multivariate_normal(np.zeros(d),Sigma,M)
return mu + Z/np.sqrt(g)
```

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but now i am looking for student's multivariate t-distribution so i can calculate density of elements where `dimension > 1`

.

It would be something like `stats.t.pdf(x, df, loc, scale)`

a scipy package , but in a multidimensional space.

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4 answers

I coded the density myself:

``````import numpy as np
from math import *

def multivariate_t_distribution(x,mu,Sigma,df,d):
'''
Multivariate t-student density:
output:
the density of the given element
input:
x = parameter (d dimensional numpy array or scalar)
mu = mean (d dimensional numpy array or scalar)
Sigma = scale matrix (dxd numpy array)
df = degrees of freedom
d: dimension
'''
Num = gamma(1. * (d+df)/2)
Denom = ( gamma(1.*df/2) * pow(df*pi,1.*d/2) * pow(np.linalg.det(Sigma),1./2) * pow(1 + (1./df)*np.dot(np.dot((x - mu),np.linalg.inv(Sigma)), (x - mu)),1.* (d+df)/2))
d = 1. * Num / Denom
return d
```

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This estimates the log pdf of the student-T multivariate distribution for the nth data matrix X:

``````from scipy.special import gamma
from numpy.linalg import slogdet

def multivariate_student_t(X, mu, Sigma, df):
#multivariate student T distribution

[n,d] = X.shape
Xm = X-mu
V = df * Sigma
V_inv = np.linalg.inv(V)
(sign, logdet) = slogdet(np.pi * V)

logz = -gamma(df/2.0 + d/2.0) + gamma(df/2.0) + 0.5*logdet
logp = -0.5*(df+d)*np.log(1+ np.sum(np.dot(Xm,V_inv)*Xm,axis=1))

logp = logp - logz

return logp
```

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I summarized @farhawa's code to allow multiple entries in `x`

(I found that I wanted to query multiple points at once).

``````import numpy as np
from math import gamma

def multivariate_t_distribution(x, mu, Sigma, df):
'''
Multivariate t-student density. Returns the density
of the function at points specified by x.

input:
x = parameter (n-d numpy array; will be forced to 2d)
mu = mean (d dimensional numpy array)
Sigma = scale matrix (dxd numpy array)
df = degrees of freedom

Edited from: http://stackoverflow.com/a/29804411/3521179
'''

x = np.atleast_2d(x) # requires x as 2d
nD = Sigma.shape # dimensionality

numerator = gamma(1.0 * (nD + df) / 2.0)

denominator = (
gamma(1.0 * df / 2.0) *
np.power(df * np.pi, 1.0 * nD / 2.0) *
np.power(np.linalg.det(Sigma), 1.0 / 2.0) *
np.power(
1.0 + (1.0 / df) *
np.diagonal(
np.dot( np.dot(x - mu, np.linalg.inv(Sigma)), (x - mu).T)
),
1.0 * (nD + df) / 2.0
)
)

return 1.0 * numerator / denominator
```

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I tried the above answers and got different results from each, but I'm not sure why / what might be wrong. The following, which I based on scikit-learn code for Gaussian mixtures, I think works (for arbitrary size of numpy input arrays X and c t-distribution with parameters in lists and covars):

``````import numpy as np
from scipy import linalg
try:  # SciPy >= 0.19
from scipy.special import gammaln as sp_gammaln
except ImportError:
from scipy.misc import gammaln as sp_gammaln

def log_multivariate_t_density(X, means, covars, nu = 1):
n_samples, n_dim = X.shape
nmix = len(means)
log_prob = np.empty((n_samples, nmix))
for c, (mu, cv) in enumerate(zip(means, covars)):
try:
cv_chol = linalg.cholesky(cv, lower=True)
except linalg.LinAlgError:

try:
cv_chol = linalg.cholesky(cv + min_covar * np.eye(n_dim),
lower=True)
except linalg.LinAlgError:
raise ValueError("'covars' must be symmetric, "
"positive-definite")

cv_log_det = 2 * np.sum(np.log(np.diagonal(cv_chol)))
cv_sol = linalg.solve_triangular(cv_chol, (X - mu).T, lower=True).T

norm = (sp_gammaln((nu + n_dim) / 2.) - sp_gammaln(nu / 2.)
- 0.5 * n_dim * np.log(nu * np.pi))
inner = - (nu + n_dim) / 2. * np.log1p(np.sum(cv_sol ** 2, axis=1) / nu)
log_prob[:, c] = norm + inner - cv_log_det

return log_prob
```

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