def test_scalar_values():
np.random.seed(1234)
# When evaluated on scalar data, the pdf should return a scalar
x, mean, cov = 1.5, 1.7, 2.5
pdf = multivariate_normal.pdf(x, mean, cov)
assert_equal(pdf.ndim, 0)
# When evaluated on a single vector, the pdf should return a scalar
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5)) # Diagonal values for cov. matrix
pdf = multivariate_normal.pdf(x, mean, cov)
assert_equal(pdf.ndim, 0)
def test_scalar_values():
np.random.seed(1234)
# When evaluated on scalar data, the pdf should return a scalar
x, mean, cov = 1.5, 1.7, 2.5
pdf = multivariate_normal.pdf(x, mean, cov)
assert_equal(pdf.ndim, 0)
# When evaluated on a single vector, the pdf should return a scalar
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5)) # Diagonal values for cov. matrix
pdf = multivariate_normal.pdf(x, mean, cov)
assert_equal(pdf.ndim, 0)
def test_R_values():
# Compare the multivariate pdf with some values precomputed
# in R version 3.0.1 (2013-05-16) on Mac OS X 10.6.
# The values below were generated by the following R-script:
# > library(mnormt)
# > x <- seq(0, 2, length=5)
# > y <- 3*x - 2
# > z <- x + cos(y)
# > mu <- c(1, 3, 2)
# > Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
# > r_pdf <- dmnorm(cbind(x,y,z), mu, Sigma)
r_pdf = np.array(
[0.0002214706, 0.0013819953, 0.0049138692, 0.0103803050, 0.0140250800])
x = np.linspace(0, 2, 5)
y = 3 * x - 2
z = x + np.cos(y)
r = np.array([x, y, z]).T
mean = np.array([1, 3, 2], 'd')
cov = np.array([[1, 2, 0], [2, 5, .5], [0, .5, 3]], 'd')
pdf = multivariate_normal.pdf(r, mean, cov)
assert_allclose(pdf, r_pdf, atol=1e-10)
def test_R_values():
# Compare the multivariate pdf with some values precomputed
# in R version 3.0.1 (2013-05-16) on Mac OS X 10.6.
# The values below were generated by the following R-script:
# > library(mnormt)
# > x <- seq(0, 2, length=5)
# > y <- 3*x - 2
# > z <- x + cos(y)
# > mu <- c(1, 3, 2)
# > Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
# > r_pdf <- dmnorm(cbind(x,y,z), mu, Sigma)
r_pdf = np.array([0.0002214706, 0.0013819953, 0.0049138692,
0.0103803050, 0.0140250800])
x = np.linspace(0, 2, 5)
y = 3 * x - 2
z = x + np.cos(y)
r = np.array([x, y, z]).T
mean = np.array([1, 3, 2], 'd')
cov = np.array([[1, 2, 0], [2, 5, .5], [0, .5, 3]], 'd')
pdf = multivariate_normal.pdf(r, mean, cov)
assert_allclose(pdf, r_pdf, atol=1e-10)
def test_logpdf():
# Check that the log of the pdf is in fact the logpdf
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5))
d1 = multivariate_normal.logpdf(x, mean, cov)
d2 = multivariate_normal.pdf(x, mean, cov)
assert_allclose(d1, np.log(d2))
def test_normal_1D():
# The probability density function for a 1D normal variable should
# agree with the standard normal distribution in scipy.stats.distributions
x = np.linspace(0, 2, 10)
mean, cov = 1.2, 0.9
scale = cov**0.5
d1 = norm.pdf(x, mean, scale)
d2 = multivariate_normal.pdf(x, mean, cov)
assert_allclose(d1, d2)
def test_logpdf():
# Check that the log of the pdf is in fact the logpdf
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5))
d1 = multivariate_normal.logpdf(x, mean, cov)
d2 = multivariate_normal.pdf(x, mean, cov)
assert_allclose(d1, np.log(d2))
def test_normal_1D():
# The probability density function for a 1D normal variable should
# agree with the standard normal distribution in scipy.stats.distributions
x = np.linspace(0, 2, 10)
mean, cov = 1.2, 0.9
scale = cov**0.5
d1 = norm.pdf(x, mean, scale)
d2 = multivariate_normal.pdf(x, mean, cov)
assert_allclose(d1, d2)
def test_frozen():
# The frozen distribution should agree with the regular one
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5))
norm_frozen = multivariate_normal(mean, cov)
assert_allclose(norm_frozen.pdf(x), multivariate_normal.pdf(x, mean, cov))
assert_allclose(norm_frozen.logpdf(x),
multivariate_normal.logpdf(x, mean, cov))
def test_broadcasting():
np.random.seed(1234)
n = 4
# Construct a random covariance matrix.
data = np.random.randn(n, n)
cov = np.dot(data, data.T)
mean = np.random.randn(n)
# Construct an ndarray which can be interpreted as
# a 2x3 array whose elements are random data vectors.
X = np.random.randn(2, 3, n)
# Check that multiple data points can be evaluated at once.
for i in range(2):
for j in range(3):
actual = multivariate_normal.pdf(X[i, j], mean, cov)
desired = multivariate_normal.pdf(X, mean, cov)[i, j]
assert_allclose(actual, desired)
def test_frozen():
# The frozen distribution should agree with the regular one
np.random.seed(1234)
x = np.random.randn(5)
mean = np.random.randn(5)
cov = np.abs(np.random.randn(5))
norm_frozen = multivariate_normal(mean, cov)
assert_allclose(norm_frozen.pdf(x), multivariate_normal.pdf(x, mean, cov))
assert_allclose(norm_frozen.logpdf(x),
multivariate_normal.logpdf(x, mean, cov))
def test_broadcasting():
np.random.seed(1234)
n = 4
# Construct a random covariance matrix.
data = np.random.randn(n, n)
cov = np.dot(data, data.T)
mean = np.random.randn(n)
# Construct an ndarray which can be interpreted as
# a 2x3 array whose elements are random data vectors.
X = np.random.randn(2, 3, n)
# Check that multiple data points can be evaluated at once.
for i in range(2):
for j in range(3):
actual = multivariate_normal.pdf(X[i, j], mean, cov)
desired = multivariate_normal.pdf(X, mean, cov)[i, j]
assert_allclose(actual, desired)
def test_marginalization():
# Integrating out one of the variables of a 2D Gaussian should
# yield a 1D Gaussian
mean = np.array([2.5, 3.5])
cov = np.array([[.5, 0.2], [0.2, .6]])
n = 2**8 + 1 # Number of samples
delta = 6 / (n - 1) # Grid spacing
v = np.linspace(0, 6, n)
xv, yv = np.meshgrid(v, v)
pos = np.empty((n, n, 2))
pos[:, :, 0] = xv
pos[:, :, 1] = yv
pdf = multivariate_normal.pdf(pos, mean, cov)
# Marginalize over x and y axis
margin_x = romb(pdf, delta, axis=0)
margin_y = romb(pdf, delta, axis=1)
# Compare with standard normal distribution
gauss_x = norm.pdf(v, loc=mean[0], scale=cov[0, 0]**0.5)
gauss_y = norm.pdf(v, loc=mean[1], scale=cov[1, 1]**0.5)
assert_allclose(margin_x, gauss_x, rtol=1e-2, atol=1e-2)
assert_allclose(margin_y, gauss_y, rtol=1e-2, atol=1e-2)
def test_marginalization():
# Integrating out one of the variables of a 2D Gaussian should
# yield a 1D Gaussian
mean = np.array([2.5, 3.5])
cov = np.array([[.5, 0.2], [0.2, .6]])
n = 2 ** 8 + 1 # Number of samples
delta = 6 / (n - 1) # Grid spacing
v = np.linspace(0, 6, n)
xv, yv = np.meshgrid(v, v)
pos = np.empty((n, n, 2))
pos[:, :, 0] = xv
pos[:, :, 1] = yv
pdf = multivariate_normal.pdf(pos, mean, cov)
# Marginalize over x and y axis
margin_x = romb(pdf, delta, axis=0)
margin_y = romb(pdf, delta, axis=1)
# Compare with standard normal distribution
gauss_x = norm.pdf(v, loc=mean[0], scale=cov[0, 0] ** 0.5)
gauss_y = norm.pdf(v, loc=mean[1], scale=cov[1, 1] ** 0.5)
assert_allclose(margin_x, gauss_x, rtol=1e-2, atol=1e-2)
assert_allclose(margin_y, gauss_y, rtol=1e-2, atol=1e-2)
