[1]:

import matplotlib.cm as cmap
import matplotlib.pyplot as plt
import numpy as np
import pymc3 as pm
import theano
import theano.tensor as tt

%matplotlib inline

np.random.seed(206)


# Mean and Covariance Functions¶

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.

## Mean functions¶

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¶

[2]:

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.

### Constant¶

gp.mean.Constant returns a vector whose value is provided.

[3]:

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.

[4]:

const_func_vec = pm.gp.mean.Constant(tt.ones(5))

print(const_func_vec(X).eval())

[1. 1. 1. 1. 1.]


### Linear¶

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).

[5]:

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())

[-0.36242716  2.20718024  0.41039522  2.50735445 -2.06177178]


## Defining a custom mean function¶

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.

## Covariance functions¶

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.

$k(x, x') = \mathrm{exp}\left[ -\frac{(x - x')^2}{2 \ell^2} \right]$
[6]:

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Two (and higher) Dimensional Inputs¶

#### Both dimensions active¶

It is easy to define kernels with higher dimensional inputs. Notice that the ls (lengthscale) parameter is an array of length 2. Lists of PyMC3 random variables can be used for automatic relevance determination (ARD).

[7]:

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])

K = theano.function([], cov(X2))()
m = plt.imshow(K, cmap="inferno", interpolation="none")
plt.colorbar(m);


#### One dimension active¶

[8]:

ls = 0.2

K = theano.function([], cov(X2))()
m = plt.imshow(K, cmap="inferno", interpolation="none")
plt.colorbar(m);


#### Product of covariances over different dimensions¶

Note that this is equivalent to using a two dimensional ExpQuad with separate lengthscale parameters for each dimension.

[9]:

ls1 = 0.2
ls2 = 1.0
cov = cov1 * cov2

K = theano.function([], cov(X2))()
m = plt.imshow(K, cmap="inferno", interpolation="none")
plt.colorbar(m);


### White Noise¶

$k(x, x') = \sigma^2 \mathrm{I}_{xx}$
[10]:

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Constant¶

$k(x, x') = c$
[11]:

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


$k(x, x') = \left(1 + \frac{(x - x')^2}{2\alpha\ell^2} \right)^{-\alpha}$
[12]:

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Exponential¶

$k(x, x') = \mathrm{exp}\left[ -\frac{||x - x'||}{2\ell^2} \right]$
[13]:

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Matern 5/2¶

$k(x, x') = \left(1 + \frac{\sqrt{5(x - x')^2}}{\ell} + \frac{5(x-x')^2}{3\ell^2}\right) \mathrm{exp}\left[ - \frac{\sqrt{5(x - x')^2}}{\ell} \right]$
[14]:

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Matern 3/2¶

$k(x, x') = \left(1 + \frac{\sqrt{3(x - x')^2}}{\ell}\right) \mathrm{exp}\left[ - \frac{\sqrt{3(x - x')^2}}{\ell} \right]$
[15]:

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Matern 1/2¶

$k(x, x') = \mathrm{exp}\left[ -\frac{(x - x')^2}{\ell} \right]$
[16]:

ls = 0.2
tau = 2.0
cov = tau * pm.gp.cov.Matern12(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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Cosine¶

$k(x, x') = \mathrm{cos}\left( 2 \pi \frac{||x - x'||}{ \ell^2} \right)$
[17]:

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Linear¶

$k(x, x') = (x - c)(x' - c)$
[18]:

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Polynomial¶

$k(x, x') = [(x - c)(x' - c) + \mathrm{offset}]^{d}$
[19]:

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Multiplication with a precomputed covariance matrix¶

A covariance function cov can be multiplied with numpy matrix, K_cos, as long as the shapes are appropriate.

[20]:

# 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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Applying an arbitary warping function on the inputs¶

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.

[21]:

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 = pm.gp.cov.WarpedInput(1, warp_func=warp_func, args=(a, b, c), cov_func=cov_exp)

X = np.linspace(0, 2, 400)[:, None]
wf = warp_func(X.flatten(), a, b, c).eval()

plt.figure(figsize=(14, 4))
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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");

/home/ferres/dev/pymc3/pymc3/gp/cov.py:90: UserWarning: Only 1 column(s) out of Subtensor{int64}.0 are being used to compute the covariance function. If this is not intended, increase 'input_dim' parameter to the number of columns to use. Ignore otherwise.
warnings.warn(f"Only {self.input_dim} column(s) out of {X.shape[-1]} are"


### Constructing 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.

[22]:

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 = 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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");

/home/ferres/dev/pymc3/pymc3/gp/cov.py:90: UserWarning: Only 2 column(s) out of Subtensor{int64}.0 are being used to compute the covariance function. If this is not intended, increase 'input_dim' parameter to the number of columns to use. Ignore otherwise.
warnings.warn(f"Only {self.input_dim} column(s) out of {X.shape[-1]} are"


### Periodic¶

There is no need to construct the periodic covariance this way every time. A more efficient implementation of this covariance function is built in.

[23]:

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)
for p in np.arange(0, 2, period):
plt.axvline(p, color="black")
plt.axhline(0, color="black")
plt.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Circular¶

Circular kernel is similar to Periodic one but has an additional nuisance parameter $$\tau$$

In the following paper, the Weinland function is used to solve the problem and ensures positive definite kernel on the circular domain (and not only).

$W_c(t) = \left(1 + \tau \frac{t}{c}\right)\left(1-\frac{t}{c}\right)_+^\tau$

where $$c$$ is maximum value for $$t$$ and $$\tau\ge 4$$ is some positive number

The kernel itself for geodesic distance (arc length) on a circle looks like

$k_g(x, y) = W_\pi(\text{dist}_{\mathit{geo}}(x, y))$

Briefly, you can think

• $$t$$ is time, it runs from $$0$$ to $$24$$ and then goes back to $$0$$

• $$c$$ is maximum distance between any timestamps, here it would be $$12$$

• $$\tau$$ controls for correlation strength, larger $$\tau$$ leads to less smooth functions

[24]:

period = 0.6
tau = 4
cov = pm.gp.cov.Circular(1, period=period, tau=tau)

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)
for p in np.arange(0, 2, period):
plt.axvline(p, color="black")
plt.axhline(0, color="black")
plt.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


We can see the effect of $$\tau$$, it adds more non-smooth patterns

[25]:

period = 0.6
tau = 40
cov = pm.gp.cov.Circular(1, period=period, tau=tau)

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)
for p in np.arange(0, 2, period):
plt.axvline(p, color="black")
plt.axhline(0, color="black")
plt.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Gibbs¶

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.

[26]:

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 = tanh_func(X, ls1, ls2, w, x0).eval()
plt.figure(figsize=(14, 4))
plt.plot(X, wf)
plt.ylabel("lengthscale")
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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Scaled Covariance¶

One can construct a new kernel or covariance function by multiplying some base kernel by a nonnegative function $$\phi(x)$$,

$k_{\mathrm{scaled}}(x, x') = \phi(x) k_{\mathrm{base}}(x, x') \phi(x') \,.$

This is useful for specifying covariance functions whose amplitude changes across the domain.

[27]:

def logistic(x, a, x0, c, d):
# a is the slope, x0 is the location
return d * pm.math.invlogit(a * (x - x0)) + c

a = 2.0
x0 = 5.0
c = 0.1
d = 2.0

cov = pm.gp.cov.ScaledCov(1, scaling_func=logistic, args=(a, x0, c, d), cov_func=cov_base)

X = np.linspace(0, 10, 400)[:, None]
lfunc = logistic(X.flatten(), a, b, c, d).eval()

plt.figure(figsize=(14, 4))
plt.plot(X, lfunc)
plt.xlabel("X")
plt.ylabel(r"$\phi(x)$")
plt.title("The scaling function")

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");

/home/ferres/dev/pymc3/pymc3/gp/cov.py:90: UserWarning: Only 1 column(s) out of Subtensor{int64}.0 are being used to compute the covariance function. If this is not intended, increase 'input_dim' parameter to the number of columns to use. Ignore otherwise.
warnings.warn(f"Only {self.input_dim} column(s) out of {X.shape[-1]} are"


### Constructing a Changepoint kernel using ScaledCov¶

The ScaledCov kernel can be used to create the Changepoint covariance. This covariance models a process that gradually transitions from one type of behavior to another.

The changepoint kernel is given by

$k(x, x') = \phi(x)k_{1}(x, x')\phi(x) + (1 - \phi(x))k_{2}(x, x')(1 - \phi(x'))$

where $$\phi(x)$$ is the logistic function.

[28]:

def logistic(x, a, x0):
# a is the slope, x0 is the location
return pm.math.invlogit(a * (x - x0))

a = 2.0
x0 = 5.0

cov1 = pm.gp.cov.ScaledCov(
1, scaling_func=logistic, args=(-a, x0), cov_func=pm.gp.cov.ExpQuad(1, 0.2)
)
cov2 = pm.gp.cov.ScaledCov(
1, scaling_func=logistic, args=(a, x0), cov_func=pm.gp.cov.Cosine(1, 0.5)
)
cov = cov1 + cov2

X = np.linspace(0, 10, 400)
plt.figure(figsize=(14, 4))
plt.fill_between(
X,
np.zeros(400),
logistic(X, -a, x0).eval(),
color="slateblue",
alpha=0.4,
)
plt.fill_between(
X, np.zeros(400), logistic(X, a, x0).eval(), label="Cosine region", color="firebrick", alpha=0.4
)
plt.legend()
plt.xlabel("X")
plt.ylabel(r"$\phi(x)$")
plt.title("The two scaling functions")

K = cov(X[:, None]).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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");

/home/ferres/dev/pymc3/pymc3/gp/cov.py:90: UserWarning: Only 1 column(s) out of Subtensor{int64}.0 are being used to compute the covariance function. If this is not intended, increase 'input_dim' parameter to the number of columns to use. Ignore otherwise.
warnings.warn(f"Only {self.input_dim} column(s) out of {X.shape[-1]} are"


### Combination of two or more Covariance functions¶

You can combine different covariance functions to model complex data.

In particular, you can perform the following operations on any covaraince functions:

• Add other covariance function with equal or broadcastable dimensions with first covariance function

• Multiply with a scalar or a covariance function with equal or broadcastable dimensions with first covariance function

• Exponentiate with a scalar.

[29]:

ls_1 = 0.1
tau_1 = 2.0
ls_2 = 0.5
tau_2 = 1.0
cov_1 = tau_1 * pm.gp.cov.ExpQuad(1, ls=ls_1)
cov_2 = tau_2 * pm.gp.cov.ExpQuad(1, ls=ls_2)

cov = cov_1 + cov_2

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


#### Multiplication¶

[30]:

ls_1 = 0.1
tau_1 = 2.0
ls_2 = 0.5
tau_2 = 1.0
cov_1 = tau_1 * pm.gp.cov.ExpQuad(1, ls=ls_1)
cov_2 = tau_2 * pm.gp.cov.ExpQuad(1, ls=ls_2)

cov = cov_1 * cov_2

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


#### Exponentiation¶

[31]:

ls_1 = 0.1
tau_1 = 2.0
power = 2
cov_1 = tau_1 * pm.gp.cov.ExpQuad(1, ls=ls_1)

cov = cov_1 ** power

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.title("Samples from the GP prior")
plt.ylabel("y")
plt.xlabel("X");


### Defining a custom covariance function¶

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!

[32]:

%load_ext watermark
%watermark -n -u -v -iv -w

numpy  1.19.0
theano 1.0.5
pymc3  3.9.3
last updated: Sat Sep 26 2020

CPython 3.8.5
IPython 7.17.0
watermark 2.0.2