import matplotlib.pyplot as plt
import matplotlib.cm as cmap
%matplotlib inline
import numpy as np
np.random.seed(206)
import theano
import theano.tensor as tt
import pymc3 as pm
A large set of mean and covariance functions are available in PyMC3. It is relatively easy to define custom mean and covariance functions. Since PyMC3 uses Theano, their gradients do not need to be defined by the user.
The following mean functions are available in PyMC3.
gp.mean.Zero
gp.mean.Constant
gp.mean.Linear
All follow a similar usage pattern. First, the mean function is specified. Then it can be evaluated over some inputs. The first two mean functions are very simple. Regardless of the inputs, gp.mean.Zero
returns a vector of zeros with the same length as the number of input values.
zero_func = pm.gp.mean.Zero()
X = np.linspace(0, 1, 5)[:, None]
print(zero_func(X).eval())
[ 0. 0. 0. 0. 0.]
The default mean functions for all GP implementations in PyMC3 is Zero
.
gp.mean.Constant
returns a vector whose value is provided.
const_func = pm.gp.mean.Constant(25.2)
print(const_func(X).eval())
[ 25.2 25.2 25.2 25.2 25.2]
As long as the shape matches the input it will receive, gp.mean.Constant
can also accept a Theano tensor or vector of PyMC3 random variables.
const_func_vec = pm.gp.mean.Constant(tt.ones(5))
print(const_func_vec(X).eval())
[ 1. 1. 1. 1. 1.]
gp.mean.Linear
is a takes as input a matrix of coefficients and a vector of intercepts (or a slope and scalar intercept in one dimension).
beta = np.random.randn(3)
b = 0.0
lin_func = pm.gp.mean.Linear(coeffs=beta, intercept=b)
X = np.random.randn(5, 3)
print(lin_func(X).eval())
[-2.21615117 0.18483208 -0.13889476 -0.8641522 -1.89552731]
To define a custom mean function, subclass gp.mean.Mean
, and provide __call__
and __init__
methods. For example, the code for the Constant
mean function is
import theano.tensor as tt
class Constant(pm.gp.mean.Mean):
def __init__(self, c=0):
Mean.__init__(self)
self.c = c
def __call__(self, X):
return tt.alloc(1.0, X.shape[0]) * self.c
Remember that Theano must be used instead of NumPy.
PyMC3 contains a much larger suite of built-in covariance functions. The following shows functions drawn from a GP prior with a given covariance function, and demonstrates how composite covariance functions can be constructed with Python operators in a straightforward manner. Our goal was for our API to follow kernel algebra (see Ch.4 of Rassmussen + Williams) as closely as possible. See the main documentation page for an overview on their usage in PyMC3.
lengthscale = 0.2
eta = 2.0
cov = eta**2 * pm.gp.cov.ExpQuad(1, lengthscale)
X = np.linspace(0, 2, 200)[:,None]
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
x1, x2 = np.meshgrid(np.linspace(0,1,10), np.arange(1,4))
X2 = np.concatenate((x1.reshape((30,1)), x2.reshape((30,1))), axis=1)
ls = np.array([0.2, 1.0])
cov = pm.gp.cov.ExpQuad(input_dim=2, ls=ls)
K = theano.function([], cov(X2))()
m = plt.imshow(K, cmap="inferno", interpolation='none'); plt.colorbar(m);
ls = 0.2
cov = pm.gp.cov.ExpQuad(input_dim=2, ls=ls, active_dims=[0])
K = theano.function([], cov(X2))()
m = plt.imshow(K, cmap="inferno", interpolation='none'); plt.colorbar(m);
Note that this is equivalent to using a two dimensional ExpQuad
with separate lengthscale parameters for each dimension.
ls1 = 0.2
ls2 = 1.0
cov1 = pm.gp.cov.ExpQuad(2, ls1, active_dims=[0])
cov2 = pm.gp.cov.ExpQuad(2, ls2, active_dims=[1])
cov = cov1 * cov2
K = theano.function([], cov(X2))()
m = plt.imshow(K, cmap="inferno", interpolation='none'); plt.colorbar(m);
sigma = 2.0
cov = pm.gp.cov.WhiteNoise(sigma)
X = np.linspace(0, 2, 200)[:,None]
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
c = 2.0
cov = pm.gp.cov.Constant(c)
X = np.linspace(0, 2, 200)[:,None]
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
alpha = 0.1
ls = 0.2
tau = 2.0
cov = tau * pm.gp.cov.RatQuad(1, ls, alpha)
X = np.linspace(0, 2, 200)[:,None]
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
inverse_lengthscale = 5
cov = pm.gp.cov.Exponential(1, ls_inv=inverse_lengthscale)
X = np.linspace(0, 2, 200)[:,None]
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
ls = 0.2
tau = 2.0
cov = tau * pm.gp.cov.Matern52(1, ls)
X = np.linspace(0, 2, 200)[:,None]
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
ls = 0.2
tau = 2.0
cov = tau * pm.gp.cov.Matern32(1, ls)
X = np.linspace(0, 2, 200)[:,None]
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
period = 0.5
cov = pm.gp.cov.Cosine(1, period)
X = np.linspace(0, 2, 200)[:,None]
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
c = 1.0
tau = 2.0
cov = tau * pm.gp.cov.Linear(1, c)
X = np.linspace(0, 2, 200)[:,None]
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
c = 1.0
d = 3
offset = 1.0
tau = 0.1
cov = tau * pm.gp.cov.Polynomial(1, c=c, d=d, offset=offset)
X = np.linspace(0, 2, 200)[:,None]
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
A covariance function cov
can be multiplied with numpy matrix, K_cos
, as long as the shapes are appropriate.
# first evaluate a covariance function into a matrix
period = 0.2
cov_cos = pm.gp.cov.Cosine(1, period)
K_cos = theano.function([], cov_cos(X))()
# now multiply it with a covariance *function*
cov = pm.gp.cov.Matern32(1, 0.5) * K_cos
X = np.linspace(0, 2, 200)[:,None]
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
If $k(x, x')$ is a valid covariance function, then so is $k(w(x), w(x'))$.
The first argument of the warping function must be the input X
. The remaining arguments can be anything else, including random variables.
def warp_func(x, a, b, c):
return 1.0 + x + (a * tt.tanh(b * (x - c)))
a = 1.0
b = 5.0
c = 1.0
cov_m52 = pm.gp.cov.ExpQuad(1, 0.2)
cov = pm.gp.cov.WarpedInput(1, warp_func=warp_func, args=(a,b,c), cov_func=cov_m52)
X = np.linspace(0, 2, 400)[:,None]
wf = theano.function([], warp_func(X.flatten(), a,b,c))()
plt.plot(X, wf); plt.xlabel("X"); plt.ylabel("warp_func(X)");
plt.title("The warping function used");
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
Periodic
using WarpedInput
¶The WarpedInput
kernel can be used to create the Periodic
covariance. This covariance models functions that are periodic, but are not an exact sine wave (like the Cosine
kernel is).
The periodic kernel is given by
$$ k(x, x') = \exp\left( -\frac{2 \sin^{2}(\pi |x - x'|\frac{1}{T})}{\ell^2} \right) $$Where T is the period, and $\ell$ is the lengthscale. It can be derived by warping the input of an ExpQuad
kernel with the function $\mathbf{u}(x) = (\sin(2\pi x \frac{1}{T})\,, \cos(2 \pi x \frac{1}{T}))$. Here we use the WarpedInput
kernel to construct it.
The input X
, which is defined at the top of this page, is 2 "seconds" long. We use a period of $0.5$, which means that functions
drawn from this GP prior will repeat 4 times over 2 seconds.
def mapping(x, T):
c = 2.0 * np.pi * (1.0 / T)
u = tt.concatenate((tt.sin(c*x), tt.cos(c*x)), 1)
return u
T = 0.6
ls = 0.4
# note that the input of the covariance function taking
# the inputs is 2 dimensional
cov_exp = pm.gp.cov.ExpQuad(2, ls)
cov = pm.gp.cov.WarpedInput(1, cov_func=cov_exp,
warp_func=mapping, args=(T, ))
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
There is no need to construct the periodic covariance this way every time. A more efficient implementation of this covariance function is built in.
period = 0.6
ls = 0.4
cov = pm.gp.cov.Periodic(1, period=period, ls=ls)
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
The Gibbs covariance function applies a positive definite warping function to the lengthscale. Similarly to WarpedInput
, the lengthscale warping function can be specified with parameters that are either fixed or random variables.
def tanh_func(x, ls1, ls2, w, x0):
"""
ls1: left saturation value
ls2: right saturation value
w: transition width
x0: transition location.
"""
return (ls1 + ls2) / 2.0 - (ls1 - ls2) / 2.0 * tt.tanh((x - x0) / w)
ls1 = 0.05
ls2 = 0.6
w = 0.3
x0 = 1.0
cov = pm.gp.cov.Gibbs(1, tanh_func, args=(ls1, ls2, w, x0))
wf = theano.function([], tanh_func(X, ls1, ls2, w, x0))()
plt.plot(X, wf); plt.ylabel("tanh_func(X)"); plt.xlabel("X"); plt.title("Lengthscale as a function of X");
K = cov(X).eval()
plt.figure(figsize=(14,4))
plt.plot(X, pm.MvNormal.dist(mu=np.zeros(K.shape[0]), cov=K).random(size=3).T);
plt.xlabel("X");
Covariance function objects in PyMC3 need to implement the __init__
, diag
, and full
methods, and subclass gp.cov.Covariance
. diag
returns only the diagonal of the covariance matrix, and full
returns the full covariance matrix. The full
method has two inputs X
and Xs
. full(X)
returns the square covariance matrix, and full(X, Xs)
returns the cross-covariances between the two sets of inputs.
For example, here is the implementation of the WhiteNoise
covariance function:
class WhiteNoise(pm.gp.cov.Covariance):
def __init__(self, sigma):
super(WhiteNoise, self).__init__(1, None)
self.sigma = sigma
def diag(self, X):
return tt.alloc(tt.square(self.sigma), X.shape[0])
def full(self, X, Xs=None):
if Xs is None:
return tt.diag(self.diag(X))
else:
return tt.alloc(0.0, X.shape[0], Xs.shape[0])
If we have forgotten an important covariance or mean function, please feel free to submit a pull request!