## Stheno

Stheno is an implementation of Gaussian process modelling in Python. See also Stheno.jl.

## Nonlinear Regression in 20 Seconds

>>> import numpy as np

>>> from stheno import GP, EQ

>>> x = np.linspace(0, 2, 10)    # Points to predict at

>>> y = x ** 2                   # Observations

>>> (GP(EQ()) | (x, y))(3).mean  # Go GP!
array([[8.48258669]])

## Installation

Before installing the package, please ensure that gcc and gfortran are
available.
On OS X, these are both installed with brew install gcc;
users of Anaconda may want to instead consider conda install gcc.
Then simply

pip install stheno

## Manual

Note: here is a nicely rendered and more

### Kernel and Mean Design

Inputs to kernels, means, and GPs, henceforth referred to simply as inputs,
must be of one of the following three forms:

• If the input x is a rank 0 tensor, i.e. a scalar, then x refers to a
single input location. For example, 0 simply refers to the sole input
location 0.

• If the input x is a rank 1 tensor, then every element of x is
interpreted as a separate input location. For example, np.linspace(0, 1, 10)
generates 10 different input locations ranging from 0 to 1.

• If the input x is a rank 2 tensor, then every row of x is
interpreted as a separate input location. In this case inputs are
multi-dimensional, and the columns correspond to the various input dimensions.

If k is a kernel, say k = EQ(), then k(x, y) constructs the kernel
matrix
for all pairs of points between x and y. k(x) is shorthand for
k(x, x). Furthermore, k.elwise(x, y) constructs the kernel vector pairing
the points in x and y element wise, which will be a rank 2 column vector.

Example:

>>> EQ()(np.linspace(0, 1, 3))
array([[1.        , 0.8824969 , 0.60653066],
[0.8824969 , 1.        , 0.8824969 ],
[0.60653066, 0.8824969 , 1.        ]])

>>> EQ().elwise(np.linspace(0, 1, 3), 0)
array([[1.        ],
[0.8824969 ],
[0.60653066]])

Finally, mean functions output a rank 2 column vector.

#### Available Kernels

Constants function as constant kernels. Besides that, the following kernels are
available:

$$k(x, y) = \exp\left( -\frac{1}{2}|x - y|^2 \right);$$

$$k(x, y) = \left( 1 + \frac{|x - y|^2}{2 \alpha} \right)^{-\alpha};$$

• Exp() or Matern12(), the exponential kernel:

$$k(x, y) = \exp\left( -|x - y| \right);$$

• Matern32(), the Matern–3/2 kernel:

$$k(x, y) = \left( 1 + \sqrt{3}|x - y| \right)\exp\left(-\sqrt{3}|x - y|\right);$$

• Matern52(), the Matern–5/2 kernel:

$$k(x, y) = \left( 1 + \sqrt{5}|x - y| + \frac{5}{3} |x - y|^2 \right)\exp\left(-\sqrt{3}|x - y|\right);$$

• Delta(), the Kronecker delta kernel:

$$k(x, y) = \begin{cases} 1 & \text{if } x = y, \ 0 & \text{otherwise}; \end{cases}$$

• DecayingKernel(alpha, beta):

$$k(x, y) = \frac{|\beta|^\alpha}{|x + y + \beta|^\alpha};$$

• TensorProductKernel(f):

$$k(x, y) = f(x)f(y).$$

Adding or multiplying a FunctionType f to or with a kernel will
automatically translate f to TensorProductKernel(f). For example,
f * k will translate to TensorProductKernel(f) * k, and f + k will
translate to TensorProductKernel(f) + k.

#### Available Means

Constants function as constant means. Besides that, the following means are
available:

• TensorProductMean(f):

$$m(x) = f(x).$$

Adding or multiplying a FunctionType f to or with a mean will
automatically translate f to TensorProductMean(f). For example,
f * m will translate to TensorProductMean(f) * m, and f + m will
translate to TensorProductMean(f) + m.

#### Compositional Design

• Add and subtract kernels and means.

Example:

>>> EQ() + Exp()
EQ() + Exp()

>>> EQ() + EQ()
2 * EQ()

>>> EQ() + 1
EQ() + 1

>>> EQ() + 0
EQ()

>>> EQ() - Exp()
EQ() - Exp()

>>> EQ() - EQ()
0

• Multiply kernels and means.

Example:

>>> EQ() * Exp()
EQ() * Exp()

>>> 2 * EQ()
2 * EQ()

>>> 0 * EQ()
0

• Shift kernels and means:

Definition:

k.shift(c)(x, y) == k(x - c, y - c)

k.shift(c1, c2)(x, y) == k(x - c1, y - c2)

Example:

>>> Linear().shift(1)
Linear() shift 1

>>> EQ().shift(1, 2)
EQ() shift (1, 2)

• Stretch kernels and means.

Definition:

k.stretch(c)(x, y) == k(x / c, y / c)

k.stretch(c1, c2)(x, y) == k(x / c1, y / c2)

Example:

>>> EQ().stretch(2)
EQ() > 2

>>> EQ().stretch(2, 3)
EQ() > (2, 3)

The > operator is implemented to provide a shorthand for stretching:

>>> EQ() > 2
EQ() > 2

• Select particular input dimensions of kernels and means.

Definition:

k.select([0])(x, y) == k(x[:, 0], y[:, 0])

k.select([0, 1])(x, y) == k(x[:, [0, 1]], y[:, [0, 1]])

k.select([0], [1])(x, y) == k(x[:, 0], y[:, 1])

k.select(None, [1])(x, y) == k(x, y[:, 1])

Example:

>>> EQ().select([0])
EQ() : [0]

>>> EQ().select([0, 1])
EQ() : [0, 1]

>>> EQ().select([0], [1])
EQ() : ([0], [1])

>>> EQ().select(None, [1])
EQ() : (None, [1])

• Transform the inputs of kernels and means.

Definition:

k.transform(f)(x, y) == k(f(x), f(y))

k.transform(f1, f2)(x, y) == k(f1(x), f2(y))

k.transform(None, f)(x, y) == k(x, f(y))

Example:

>>> EQ().transform(f)
EQ() transform f

>>> EQ().transform(f1, f2)
EQ() transform (f1, f2)

>>> EQ().transform(None, f)
EQ() transform (None, f)

• Numerically, but efficiently, take derivatives of kernels and means.
This currently only works in TensorFlow.

Definition:

k.diff(0)(x, y) == d/d(x[:, 0]) d/d(y[:, 0]) k(x, y)

k.diff(0, 1)(x, y) == d/d(x[:, 0]) d/d(y[:, 1]) k(x, y)

k.diff(None, 1)(x, y) == d/d(y[:, 1]) k(x, y)

Example:

>>> EQ().diff(0)
d(0) EQ()

>>> EQ().diff(0, 1)
d(0, 1) EQ()

>>> EQ().diff(None, 1)
d(None, 1) EQ()

• Make kernels periodic, but not means.

Definition:

k.periodic(2 pi / w)(x, y) == k((sin(w * x), cos(w * x)), (sin(w * y), cos(w * y)))

Example:

>>> EQ().periodic(1)
EQ() per 1

• Reverse the arguments of kernels, but not means.

Definition:

reversed(k)(x, y) == k(y, x)

Example:

>>> reversed(Linear())
Reversed(Linear())

• Extract terms and factors from sums and products respectively of kernels and
means
.

Example:

>>> (EQ() + RQ(0.1) + Linear()).term(1)
RQ(0.1)

>>> (2 * EQ() * Linear).factor(0)
2

Kernels and means "wrapping" others can be "unwrapped" by indexing k[0]
or m[0].

Example:

>>> reversed(Linear())
Reversed(Linear())

>>> reversed(Linear())[0]
Linear()

>>> EQ().periodic(1)
EQ() per 1

>>> EQ().periodic(1)[0]
EQ()

#### Displaying Kernels and Means

Kernels and means have a display method.
The display method accepts a callable formatter that will be applied before any value is printed.
This comes in handy when pretty printing kernels, or when kernels contain TensorFlow objects.

Example:

>>> print((2.12345 * EQ()).display(lambda x: '{:.2f}'.format(x)))
2.12 * EQ(), 0

>>> tf.constant(1) * EQ()
Tensor("Const_1:0", shape=(), dtype=int32) * EQ()

>>> print((tf.constant(2) * EQ()).display(tf.Session().run))
2 * EQ()

#### Properties of Kernels

The stationarity of a kernel k can always be determined by querying
k.stationary. In many cases, the variance k.var, length scale
k.length_scale, and period k.period can also be determined.

Example of querying the stationarity:

>>> EQ().stationary
True

>>> (EQ() + Linear()).stationary
False

Example of querying the variance:

>>> EQ().var
1

>>> (2 * EQ()).var
2

Example of querying the length scale:

>>> EQ().length_scale
1

>>> (EQ() + EQ().stretch(2)).length_scale
1.5

Example of querying the period:

>>> EQ().periodic(1).period
1

>>> EQ().periodic(1).stretch(2).period
2

### Model Design

The basic building block of a model is a GP(kernel, mean=0, graph=model),
which necessarily takes in a kernel, and optionally a mean and a graph.
GPs can be combined into new GPs, and the graph is the thing that keeps
track of all of these objects.
If the graph is left unspecified, new GPs are appended to a provided default
graph model, which is exported by Stheno:

from stheno import model

Here's an example model:

>>> f1 = GP(EQ(), lambda x: x ** 2)

>>> f1
GP(EQ(), <lambda>)

>>> f2 = GP(Linear())

>>> f_sum = f1 + f2

>>> f_sum
GP(EQ() + Linear(), <lambda>)

#### Compositional Design

• Add and subtract GPs and other objects.

Example:

>>> GP(EQ()) + GP(Exp())
GP(EQ() + Exp(), 0)

>>> GP(EQ()) + GP(EQ())
GP(2 * EQ(), 0)

>>> GP(EQ()) + 1
GP(EQ(), 1)

>>> GP(EQ()) + 0
GP(EQ(), 0)

>>> GP(EQ()) + (lambda x: x ** 2)
GP(EQ(), <lambda>)

>>> GP(EQ(), 2) - GP(EQ(), 1)
GP(2 * EQ(), 1)

• Multiply GPs by other objects.

Example:

>>> 2 * GP(EQ())
GP(2 * EQ(), 0)

>>> 0 * GP(EQ())
GP(0, 0)

>>> (lambda x: x) * GP(EQ())
GP(<lambda> * EQ(), 0)

• Shift GPs.

Example:

>>> GP(EQ()).shift(1)
GP(EQ() shift 1, 0)

• Stretch GPs.

Example:

>>> GP(EQ()).stretch(2)
GP(EQ() > 2, 0)

The > operator is implemented to provide a shorthand for stretching:

>>> GP(EQ()) > 2
GP(EQ() > 2, 0)

• Select particular input dimensions.

Example:

>>> GP(EQ()).select(1, 3)
GP(EQ() : [1, 3], 0)

Indexing is implemented to provide a a shorthand for selecting input
dimensions:

>>> GP(EQ())[1, 3]
GP(EQ() : [1, 3], 0)

• Transform the input.

Example:

>>> GP(EQ()).transform(f)
GP(EQ() transform f, 0)

• Numerically take the derivative of a GP.
The argument specifies which dimension to take the derivative with respect
to.

Example:

>>> GP(EQ()).diff(1)
GP(d(1) EQ(), 0)

• Construct a finite difference estimate of the derivative of a GP.
See stheno.graph.Graph.diff_approx for a description of the arguments.

Example:

>>> GP(EQ()).diff_approx(deriv=1, order=2)
GP(50000000.0 * (0.5 * EQ() + 0.5 * ((-0.5 * (EQ() shift (0.0001414213562373095, 0))) shift (0, -0.0001414213562373095)) + 0.5 * ((-0.5 * (EQ() shift (0, 0.0001414213562373095))) shift (-0.0001414213562373095, 0))), 0)

• Construct the Cartesian product of a collection of GPs.

Example:

>>> model = Graph()

>>> f1, f2 = GP(EQ(), graph=model), GP(EQ(), graph=model)

>>> model.cross(f1, f2)
GP(MultiOutputKernel(EQ(), EQ()), MultiOutputMean(0, 0))

#### Displaying GPs

Like kernels and means, GPs have a display method that accepts a formatter.

Example:

>>> print(GP(2.12345 * EQ()).display(lambda x: '{:.2f}'.format(x)))
GP(2.12 * EQ(), 0)

#### Properties of GPs

Properties of kernels can be queried on GPs directly.

Example:

>>> GP(EQ()).stationary
True

>>> GP(RQ(1e-1)).length_scale
1

### Naming GPs

It is possible to give a name to GPs.
Names must be strings.
A graph then behaves like a two-way dictionary between GPs and their names.

Example:

>>> p = GP(EQ(), name='prior')

>>> p.name
'prior'

>>> p.name = 'alternative_prior'

>>> model['alternative_prior']
GP(EQ(), 0)

>>> model[p]
'alternative_prior'

### Finite-Dimensional Distributions, Inference, and Sampling

Simply call a GP to construct its finite-dimensional distribution:

>>> type(f(x))
stheno.random.Normal

>>> f(x).mean
array([[0.],
[0.],
[0.]])

>>> f(x).var
array([[1.        , 0.8824969 , 0.60653066],
[0.8824969 , 1.        , 0.8824969 ],
[0.60653066, 0.8824969 , 1.        ]])

>>> f(x).sample(1)
array([[-0.47676132],
[-0.51696144],
[-0.77643117]])

>>> y1 = f(x).sample(1)

>>> f(x).logpdf(y1)
-1.348196150807441

>>> y2 = f(x).sample(2)

>>> f(x).logpdf(y2)
array([-1.00581476, -1.67625465])

If you wish to compute the evidence of multiple observations,
then Graph.logpdf can be used.

Definition:

Graph.logpdf(f(x), y)

Graph.logpdf((f1(x1), y1), (f2(x2), y2), ...)

Furthermore, use f(x).marginals() to efficiently compute the means and
the marginal lower and upper 95% central credible region bounds:

>>> f(x).marginals()
(array([0., 0., 0.]), array([-2., -2., -2.]), array([2., 2., 2.]))

To condition on observations, use Graph.condition or GP.condition.
Syntax is much like the math:
compare f1_posterior = f1 | (f2(x), y) with $f_1 ,|, f_2(x) = y$.

Definition, where f* and g* are GPs:

f_posterior = f | (x, y)

f_posterior = f | (g1(x), y)

f_posterior = f | ((g1(x1), y1), (g2(x2), y2), ...)

f1_posterior, f2_posterior, ... = (f1, f2, ...) | Obs(g(x), y)

f1_posterior, f2_posterior, ... = (f1, f2, ...) | Obs((g1(x1), y1), (g2(x2), y2), ...)

Finally, Graph.sample can be used to get samples from multiple processes
jointly:

>>> model.sample(f(x), (2 * f)(x))
[array([[-0.35226314],
[-0.15521219],
[ 0.0752406 ]]),
array([[-0.70452827],
[-0.31042226],
[ 0.15048168]])]

### Inducing Points

Stheno supports sparse approximations of posterior distributions. To construct
a sparse approximation, use SparseObs instead of Obs.

Definition:

obs = SparseObs(u(z),  # Locations of inducing points.
e,     # Independent, additive noise process.
f(x),  # Locations of observations _without_ the noise
y)     # Observations.

obs = SparseObs(u(z), e, f(x), y)

obs = SparseObs(u(z), (e1, f1(x1), y1), (e2, f2(x2), y2), ...)

obs = SparseObs((u1(z1), u2(z2), ...), e, f(x), y)

obs = SparseObs(u(z), (e1, f1(x1), y1), (e2, f2(x2), y2), ...)

obs = SparseObs((u1(z1), u2(z2), ...), (e1, f1(x1), y1), (e2, f2(x2), y2), ...)

SparseObs will also compute the value of the ELBO in obs.elbo, which can be
optimised to select hyperparameters and locations of the inducing points.

### NumPy, TensorFlow, or PyTorch?

from stheno import GP, EQ
from stheno.tensorflow import GP, EQ
from stheno.torch import GP, EQ

### Undiscussed Features

• stheno.mokernel and stheno.momean offer multi-output kernels and means.

Example:

>>> model = Graph()

>>> f1, f2 = GP(EQ(), graph=model), GP(EQ(), graph=model)

>>> f = model.cross(f1, f2)

>>> f
GP(MultiOutputKernel(EQ(), EQ()), MultiOutputMean(0, 0))

>>> f(0).sample()
array([[ 1.1725799 ],
[-1.15642448]])

• stheno.eis offers kernels on an extended input space that allows one to
design kernels in an alternative, flexible way.

Example:

>>> p = GP(NoisyKernel(EQ(), Delta()))

>>> prediction = p.condition(Observed(x), y)(Latent(x)).marginals()

• stheno.normal offers an efficient implementation Normal of the normal
distribution, and a convenience constructor Normal1D for 1-dimensional normal
distributions.

• stheno.matrix offers structured representations of matrices and efficient
operations thereon.

• Approximate multiplication between GPs is implemented. This is an
experimental feature.

Example:

>>> GP(EQ(), 1) * GP(EQ(), 1)
GP(<lambda> * EQ() + <lambda> * EQ() + EQ() * EQ(), <lambda> + <lambda> + -1 * 1)

## Examples

### Simple Regression

import matplotlib.pyplot as plt
import numpy as np

from stheno import GP, EQ, Delta, model

# Define points to predict at.
x = np.linspace(0, 10, 100)
x_obs = np.linspace(0, 7, 20)

# Construct a prior.
f = GP(EQ().periodic(5.))  # Latent function.
e = GP(Delta())  # Noise.
y = f + .5 * e

# Sample a true, underlying function and observations.
f_true, y_obs = model.sample(f(x), y(x_obs))

# Now condition on the observations to make predictions.
mean, lower, upper = (f | (y(x_obs), y_obs))(x).marginals()

# Plot result.
plt.plot(x, f_true, label='True', c='tab:blue')
plt.scatter(x_obs, y_obs, label='Observations', c='tab:red')
plt.plot(x, mean, label='Prediction', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()
plt.show()

### Decomposition of Prediction

import matplotlib.pyplot as plt
import numpy as np

from stheno import GP, model, EQ, RQ, Linear, Delta, Exp, Obs, B

B.epsilon = 1e-10

# Define points to predict at.
x = np.linspace(0, 10, 200)
x_obs = np.linspace(0, 7, 50)

# Construct a latent function consisting of four different components.
f_smooth = GP(EQ())
f_wiggly = GP(RQ(1e-1).stretch(.5))
f_periodic = GP(EQ().periodic(1.))
f_linear = GP(Linear())

f = f_smooth + f_wiggly + f_periodic + .2 * f_linear

# Let the observation noise consist of a bit of exponential noise.
e_indep = GP(Delta())
e_exp = GP(Exp())

e = e_indep + .3 * e_exp

# Sum the latent function and observation noise to get a model for the
# observations.
y = f + .5 * e

# Sample a true, underlying function and observations.
f_true_smooth, f_true_wiggly, f_true_periodic, f_true_linear, f_true, y_obs = \
model.sample(f_smooth(x),
f_wiggly(x),
f_periodic(x),
f_linear(x),
f(x),
y(x_obs))

# Now condition on the observations and make predictions for the latent
# function and its various components.
f_smooth, f_wiggly, f_periodic, f_linear, f = \
(f_smooth, f_wiggly, f_periodic, f_linear, f) | Obs(y(x_obs), y_obs)

pred_smooth = f_smooth(x).marginals()
pred_wiggly = f_wiggly(x).marginals()
pred_periodic = f_periodic(x).marginals()
pred_linear = f_linear(x).marginals()
pred_f = f(x).marginals()

# Plot results.
def plot_prediction(x, f, pred, x_obs=None, y_obs=None):
plt.plot(x, f, label='True', c='tab:blue')
if x_obs is not None:
plt.scatter(x_obs, y_obs, label='Observations', c='tab:red')
mean, lower, upper = pred
plt.plot(x, mean, label='Prediction', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()

plt.figure(figsize=(10, 6))

plt.subplot(3, 1, 1)
plt.title('Prediction')
plot_prediction(x, f_true, pred_f, x_obs, y_obs)

plt.subplot(3, 2, 3)
plt.title('Smooth Component')
plot_prediction(x, f_true_smooth, pred_smooth)

plt.subplot(3, 2, 4)
plt.title('Wiggly Component')
plot_prediction(x, f_true_wiggly, pred_wiggly)

plt.subplot(3, 2, 5)
plt.title('Periodic Component')
plot_prediction(x, f_true_periodic, pred_periodic)

plt.subplot(3, 2, 6)
plt.title('Linear Component')
plot_prediction(x, f_true_linear, pred_linear)

plt.show()

### Learn a Function, Incorporating Prior Knowledge About Its Form

import matplotlib.pyplot as plt
import numpy as np
import tensorflow as tf
from tensorflow.contrib.opt import ScipyOptimizerInterface as SOI
from wbml import vars64 as vs

from stheno.tensorflow import GP, EQ, Delta

s = tf.Session()

# Define points to predict at.
x = np.linspace(0, 5, 100)
x_obs = np.linspace(0, 3, 20)

# Construct the model.
u = GP(vs.pos(.5) * EQ().stretch(vs.pos(1.)))
e = GP(vs.pos(.5) * Delta())
alpha = vs.pos(1.2)
vs.init(s)

f = u + (lambda x: x ** alpha)
y = f + e

# Sample a true, underlying function and observations.
f_true = x ** 1.8
y_obs = s.run((y | (f(x), f_true))(x_obs).sample())

# Learn.
lml = y(x_obs).logpdf(y_obs)
SOI(-lml).minimize(s)

# Print the learned parameters.
print('alpha', s.run(alpha))
print('prior', y.display(s.run))

# Condition on the observations to make predictions.
mean, lower, upper = s.run((f | (y(x_obs), y_obs))(x).marginals())

# Plot result.
plt.plot(x, f_true.squeeze(), label='True', c='tab:blue')
plt.scatter(x_obs, y_obs.squeeze(), label='Observations', c='tab:red')
plt.plot(x, mean, label='Prediction', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()
plt.show()

### Multi-Ouput Regression

import matplotlib.pyplot as plt
import numpy as np
from plum import Dispatcher, Referentiable, Self

from stheno import GP, EQ, Delta, model, Kernel, Obs

class VGP(Referentiable):
"""A vector-valued GP.

Args:
dim (int): Dimensionality.
kernel (instance of :class:stheno.kernel.Kernel): Kernel.
"""
dispatch = Dispatcher(in_class=Self)

@dispatch(int, Kernel)
def __init__(self, dim, kernel):
self.ps = [GP(kernel) for _ in range(dim)]

@dispatch([GP])
def __init__(self, *ps):
self.ps = ps

@dispatch(Self)
return VGP(*[f + g for f, g in zip(self.ps, other.ps)])

@dispatch(np.ndarray)
def lmatmul(self, A):
m, n = A.shape
ps = [0 for _ in range(m)]
for i in range(m):
for j in range(n):
ps[i] += A[i, j] * self.ps[j]
return VGP(*ps)

def sample(self, x):
return model.sample(*(p(x) for p in self.ps))

def __or__(self, obs):
return VGP(*(p | obs for p in self.ps))

def obs(self, x, ys):
return Obs(*((p(x), y) for p, y in zip(self.ps, ys)))

def marginals(self, x):
return [p(x).marginals() for p in self.ps]

# Define points to predict at.
x = np.linspace(0, 10, 100)
x_obs = np.linspace(0, 10, 10)

# Model parameters:
m = 2
p = 4
H = np.random.randn(p, m)

# Construct latent functions
us = VGP(m, EQ())
fs = us.lmatmul(H)

# Construct noise.
e = VGP(p, 0.5 * Delta())

# Construct observation model.
ys = e + fs

# Sample a true, underlying function and observations.
fs_true = fs.sample(x)
ys_obs = (ys | fs.obs(x, fs_true)).sample(x_obs)

# Condition the model on the observations to make predictions.
preds = (fs | ys.obs(x_obs, ys_obs)).marginals(x)

# Plot results.
def plot_prediction(x, f, pred, x_obs=None, y_obs=None):
plt.plot(x, f, label='True', c='tab:blue')
if x_obs is not None:
plt.scatter(x_obs, y_obs, label='Observations', c='tab:red')
mean, lower, upper = pred
plt.plot(x, mean, label='Prediction', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()

plt.figure(figsize=(10, 6))

for i in range(p):
plt.subplot(int(p ** .5), int(p ** .5), i + 1)
plt.title('Output {}'.format(i + 1))
plot_prediction(x, fs_true[i], preds[i], x_obs, ys_obs[i])

plt.show()

### Approximate Integration

import matplotlib.pyplot as plt
import numpy as np

from stheno import GP, EQ, Delta, Obs

# Define points to predict at.
x = np.linspace(0, 10, 200)
x_obs = np.linspace(0, 10, 10)

# Construct the model.
f = 0.7 * GP(EQ()).stretch(1.5)
e = 0.2 * GP(Delta())

# Construct derivatives via finite differences.
df = f.diff_approx(1)
ddf = f.diff_approx(2)
dddf = f.diff_approx(3) + e

# Fix the integration constants.
f, df, ddf, dddf = (f, df, ddf, dddf) | Obs((f(0), 1), (df(0), 0), (ddf(0), -1))

# Sample observations.
y_obs = np.sin(x_obs) + 0.2 * np.random.randn(*x_obs.shape)

# Condition on the observations to make predictions.
f, df, ddf, dddf = (f, df, ddf, dddf) | Obs(dddf(x_obs), y_obs)

# And make predictions.
pred_iiif = f(x).marginals()
pred_iif = df(x).marginals()
pred_if = ddf(x).marginals()
pred_f = dddf(x).marginals()

# Plot result.
def plot_prediction(x, f, pred, x_obs=None, y_obs=None):
plt.plot(x, f, label='True', c='tab:blue')
if x_obs is not None:
plt.scatter(x_obs, y_obs, label='Observations', c='tab:red')
mean, lower, upper = pred
plt.plot(x, mean, label='Prediction', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()

plt.figure(figsize=(10, 6))

plt.subplot(2, 2, 1)
plt.title('Function')
plot_prediction(x, np.sin(x), pred_f, x_obs=x_obs, y_obs=y_obs)
plt.legend()

plt.subplot(2, 2, 2)
plt.title('Integral of Function')
plot_prediction(x, -np.cos(x), pred_if)
plt.legend()

plt.subplot(2, 2, 3)
plt.title('Second Integral of Function')
plot_prediction(x, -np.sin(x), pred_iif)
plt.legend()

plt.subplot(2, 2, 4)
plt.title('Third Integral of Function')
plot_prediction(x, np.cos(x), pred_iiif)
plt.legend()

plt.show()

### Bayesian Linear Regression

import matplotlib.pyplot as plt
import numpy as np

from stheno import GP, Delta, model, Obs, dense

# Define points to predict at.
x = np.linspace(0, 10, 200)
x_obs = np.linspace(0, 10, 10)

# Construct the model.
slope = GP(1)
intercept = GP(5)
f = slope * (lambda x: x) + intercept

e = 0.2 * GP(Delta())  # Noise model

y = f + e  # Observation model

# Sample a slope, intercept, underlying function, and observations.
true_slope, true_intercept, f_true, y_obs = \
model.sample(slope(0), intercept(0), f(x), y(x_obs))

# Condition on the observations to make predictions.
slope, intercept, f = (slope, intercept, f) | Obs(y(x_obs), y_obs)
mean, lower, upper = f(x).marginals()

print('true slope', true_slope)
print('predicted slope', slope(0).mean)
print('true intercept', true_intercept)
print('predicted intercept', intercept(0).mean)

# Plot result.
plt.plot(x, f_true, label='True', c='tab:blue')
plt.scatter(x_obs, y_obs, label='Observations', c='tab:red')
plt.plot(x, mean, label='Prediction', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()
plt.show()

### GPAR

import matplotlib.pyplot as plt
import numpy as np
import tensorflow as tf
from tensorflow.contrib.opt import ScipyOptimizerInterface as SOI
from wbml import Vars

from stheno.tensorflow import GP, Delta, EQ, Graph, B

s = tf.Session()

# Define points to predict at.
x = np.linspace(0, 10, 200)
x_obs1 = np.linspace(0, 10, 30)
inds2 = np.random.permutation(len(x_obs1))[:10]
x_obs2 = x_obs1[inds2]

# Construct variable storages.
vs1 = Vars(np.float64)
vs2 = Vars(np.float64)

# Construct a model for each output.
m1 = Graph()
m2 = Graph()
f1 = vs1.pos(1.) * GP(EQ(), graph=m1).stretch(vs1.pos(1.))
f2 = vs2.pos(1.) * GP(EQ(), graph=m2).stretch(vs2.pos([1., .5]))
sig1 = vs1.pos(0.1)
sig2 = vs2.pos(0.1)

# Initialise variables.
vs1.init(s)
vs2.init(s)

# Noise models:
e1 = sig1 * GP(Delta(), graph=m1)
e2 = sig2 * GP(Delta(), graph=m2)

# Observation models:
y1 = f1 + e1
y2 = f2 + e2

# Construction functions to predict and observations.
f1_true = np.sin(x)
f2_true = np.sin(x) ** 2

y1_obs = np.sin(x_obs1) + 0.1 * np.random.randn(*x_obs1.shape)
y2_obs = np.sin(x_obs2) ** 2 + 0.1 * np.random.randn(*x_obs2.shape)

# Learn.
lml1 = y1(x_obs1).logpdf(y1_obs)
SOI(-lml1, var_list=vs1.vars).minimize(s)

lml2 = y2(np.stack((x_obs2, y1_obs[inds2]), axis=1)).logpdf(y2_obs)
SOI(-lml2, var_list=vs2.vars).minimize(s)

# Predict first output.
f1 = f1 | (y1(x_obs1), y1_obs)
mean1, lower1, upper1 = s.run(f1(x).marginals())

# Predict second output with Monte Carlo.
f2 = f2 | (y2(np.stack((x_obs2, y1_obs[inds2]), axis=1)), y2_obs)
sample = f2(B.concat([x[:, None], f1(x).sample()], axis=1)).sample()
samples = [s.run(sample).squeeze() for _ in range(100)]
mean2 = np.mean(samples, axis=0)
lower2 = np.percentile(samples, 2.5, axis=0)
upper2 = np.percentile(samples, 100 - 2.5, axis=0)

# Plot result.
plt.figure()

plt.subplot(2, 1, 1)
plt.title('Output 1')
plt.plot(x, f1_true, label='True', c='tab:blue')
plt.scatter(x_obs1, y1_obs, label='Observations', c='tab:red')
plt.plot(x, mean1, label='Prediction', c='tab:green')
plt.plot(x, lower1, ls='--', c='tab:green')
plt.plot(x, upper1, ls='--', c='tab:green')
plt.legend()

plt.subplot(2, 1, 2)
plt.title('Output 2')
plt.plot(x, f2_true, label='True', c='tab:blue')
plt.scatter(x_obs2, y2_obs, label='Observations', c='tab:red')
plt.plot(x, mean2, label='Prediction', c='tab:green')
plt.plot(x, lower2, ls='--', c='tab:green')
plt.plot(x, upper2, ls='--', c='tab:green')
plt.legend()

plt.show()

### A GP–RNN Model

import matplotlib.pyplot as plt
import numpy as np
import tensorflow as tf
from wbml import Vars, rnn as rnn_constructor

from stheno.tensorflow import GP, Delta, EQ, RQ, Obs

# Construct variable storages.
vs_gp = Vars(np.float32)
vs_rnn = Vars(np.float32)

# Construct a 1-layer RNN with GRUs.
f_rnn = rnn_constructor(1, 1, (10,))
f_rnn.initialise(vs_rnn)

# Wrap the RNN to be compatible with Stheno.
def rnn(x):
return f_rnn(x[:, :, None])[:, :, 0]

# Construct session.
s = tf.Session()

# Construct points which to predict at.
x = np.linspace(0, 1, 100, dtype=np.float32)
inds_obs = np.arange(0, int(0.75 * len(x)))  # Train on the first 75% only.
x_obs = x[inds_obs]

# Construct function and observations.
#   Draw a random fluctuation.
k_u = .2 * RQ(1e-1).stretch(0.05)
u = s.run(GP(k_u)(np.array(x, dtype=np.float64)).sample()).squeeze()
#   Construct the true, underlying function.
f_true = np.sin(2 * np.pi * 7 * x) + np.array(u, dtype=np.float32)
y_true = f_true + 0.2 * np.array(np.random.randn(*x.shape), dtype=np.float32)

# Normalise and split.
f_true = (f_true - np.mean(y_true)) / np.std(y_true)
y_true = (y_true - np.mean(y_true)) / np.std(y_true)
y_obs = y_true[inds_obs]

# Construct the model.
a = 0.1 * GP(EQ()).stretch(vs_gp.pos(0.1))
b = 0.1 * GP(EQ()).stretch(vs_gp.pos(0.1))
e = vs_gp.pos(0.1) * GP(Delta())

# RNN-only model:
y_rnn = rnn + e

# GP-RNN model:
f_gp_rnn = (1 + a) * rnn + b
y_gp_rnn = f_gp_rnn + e

# Construct evidences.
lml_rnn = y_rnn(x_obs).logpdf(y_obs)
lml_gp_rnn = y_gp_rnn(x_obs).logpdf(y_obs)

# Construct optimisers and initialise.
-lml_rnn, var_list=vs_rnn.vars
)
-lml_gp_rnn, var_list=vs_rnn.vars + vs_gp.vars
)
s.run(tf.global_variables_initializer())

# Nudge the RNN into the right direction.
for i in range(2000):
_, val = s.run([opt_rnn, lml_rnn])
if i % 100 == 0:
print(i, val)

# Jointly train the RNN and GPs.
for i in range(5000):
_, val = s.run([opt_jointly, lml_gp_rnn])
if i % 100 == 0:
print(i, val)

# Condition.
f_gp_rnn, a, b = (f_gp_rnn, a, b) | Obs(y_gp_rnn(x_obs), y_obs)

# Predict and plot results.
plt.figure(figsize=(10, 6))

plt.subplot(2, 1, 1)
plt.title('$(1 + a) \\cdot$ RNN ${}+b$')
plt.plot(x, f_true, label='True', c='tab:blue')
plt.scatter(x_obs, y_obs, label='Observations', c='tab:red')
mean, lower, upper = s.run(f_gp_rnn(x).marginals())
plt.plot(x, mean, label='Prediction', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()

plt.subplot(2, 2, 3)
plt.title('$a$')
mean, lower, upper = s.run(a(x).marginals())
plt.plot(x, mean, label='Prediction', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()

plt.subplot(2, 2, 4)
plt.title('$b$')
mean, lower, upper = s.run(b(x).marginals())
plt.plot(x, mean, label='Prediction', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()

plt.show()

### Approximate Multiplication Between GPs

import matplotlib.pyplot as plt
import numpy as np

from stheno import GP, EQ, model, Obs

# Define points to predict at.
x = np.linspace(0, 10, 100)

# Construct a prior.
f1 = GP(EQ(), 3)
f2 = GP(EQ(), 3)

# Compute the approximate product.
f_prod = f1 * f2

# Sample two functions.
s1, s2 = model.sample(f1(x), f2(x))

# Predict.
mean, lower, upper = (f_prod | ((f1(x), s1), (f2(x), s2)))(x).marginals()

# Plot result.
plt.plot(x, s1, label='Sample 1', c='tab:red')
plt.plot(x, s2, label='Sample 2', c='tab:blue')
plt.plot(x, s1 * s2, label='True product', c='tab:orange')
plt.plot(x, mean, label='Approximate posterior', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()
plt.show()

### Sparse Regression

import matplotlib.pyplot as plt
import numpy as np

from stheno import GP, EQ, Delta, SparseObs

# Define points to predict at.
x = np.linspace(0, 10, 100)
x_obs = np.linspace(0, 7, 50_000)
x_ind = np.linspace(0, 10, 20)

# Construct a prior.
f = GP(EQ().periodic(2 * np.pi))  # Latent function.
e = GP(Delta())  # Noise.
y = f + .5 * e

# Sample a true, underlying function and observations.
f_true = np.sin(x)
y_obs = np.sin(x_obs) + .5 * np.random.randn(*x_obs.shape)

# Now condition on the observations to make predictions.
obs = SparseObs(f(x_ind),  # Inducing points.
.5 * e,  # Noise process.
# Observations _without_ the noise process added on.
f(x_obs), y_obs)
print('elbo', obs.elbo)
mean, lower, upper = (f | obs)(x).marginals()

# Plot result.
plt.plot(x, f_true, label='True', c='tab:blue')
plt.scatter(x_obs, y_obs, label='Observations', c='tab:red')
plt.scatter(x_ind, 0 * x_ind, label='Inducing Points', c='black')
plt.plot(x, mean, label='Prediction', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()
plt.show()

### Smoothing with Nonparametric Basis Functions

import matplotlib.pyplot as plt
import numpy as np

from stheno import GP, EQ, Delta, model, Obs

# Define points to predict at.
x = np.linspace(0, 10, 100)
x_obs = np.linspace(0, 10, 20)

# Constuct a prior:
w = lambda x: np.exp(-x ** 2 / 0.5)  # Window
b = [(GP(EQ()) * w).shift(xi) for xi in x_obs]  # Weighted basis functions
f = sum(b)  # Latent function
e = GP(Delta())  # Noise
y = f + 0.2 * e  # Observation model

# Sample a true, underlying function and observations.
f_true, y_obs = model.sample(f(x), y(x_obs))

# Condition on the observations to make predictions.
obs = Obs(y(x_obs), y_obs)
f, b = (f | obs, b | obs)

# Plot result.
for i, bi in enumerate(b):
mean, lower, upper = bi(x).marginals()
kw_args = {'label': 'Basis functions'} if i == 0 else {}
plt.plot(x, mean, c='tab:orange', **kw_args)
plt.plot(x, f_true, label='True', c='tab:blue')
plt.scatter(x_obs, y_obs, label='Observations', c='tab:red')
mean, lower, upper = f(x).marginals()
plt.plot(x, mean, label='Prediction', c='tab:green')
plt.plot(x, lower, ls='--', c='tab:green')
plt.plot(x, upper, ls='--', c='tab:green')
plt.legend()
plt.show()