Introduction

The Gaussian Process is an incredibly interesting object.

Paraphrasing from Rasmussen & Williams 2006, a Gaussian Process is a stochastic process (a collection of random variables, indexed on some set) where any finite number of them have a multivariate normal distribution.

GPs can represent functions and perhaps function spaces well (by having a covariance interpretation and also a basis function interpretation I guess). By contrast, a neural network can’t fit functions as easily - it needs a lot more data.

An MVN is completely characterized by its mean and covariance, and in the case of a GP, we can use the covariance to make magic. For example, to create a continuous function, let $y_d$ be a vector of points corresponding to domain points $x_{n,d}$. To enforce continuity, all we need is:

$Cor(y_i, y_j) \to 0 \; as \;||x_{i} - x_j||_m \to 0_.$

Covariance between points is usually enforced using kernels, so for example, this is the RBF kernel:

$Cov(y_i, y_j) = f(||x_{i} - x_j||) = \exp \left[ -\left( \sum_m (x_{m, i} - x_{m, i})^p \right)^{1/p} \right]$

This can be scaled (hence scaling the function along the y-axis), and a scale (the length-scale) can be applied to the domain variables (by “normalizing” them), which has the effect of scaling along the x-axis. The function enforced by this kernel is infinitely differentiable.

Knowing a few (“training”) points, one can predict all points between the training points (i.e. smoothing) by using the fact that, if $x_2$ is the set of training points, the conditional distribution of the predicted points is also multivariate normal:

$\begin{bmatrix}x_1\\x_2\end{bmatrix} \sim \mathcal N \left(\begin{bmatrix}\mu_1\\\mu_2\end{bmatrix}, \begin{bmatrix}\Sigma_{11} & \Sigma_{12}\\\Sigma_{21} & \Sigma_{22}\end{bmatrix} \right)$ $\Rightarrow x_1|x_2 \sim \mathcal N \left( \mu_1 + \Sigma_{12}\Sigma_{22}^{-1}(x_2 - \mu_2), \Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21} \right)$

In this case, $\Sigma_{ij} = k({\bf x_i}, {\bf x_j})$, but interestingly, due to the linearity of the MVN, a derivative of a GP is also a GP assuming that the mean and covariance are differentiable, in which case, the kernels are:

$\begin{bmatrix}x\\\frac{dx}{dt} \end{bmatrix} \sim \mathcal N \left(\begin{bmatrix}\mu\\ \frac{\partial \mu}{\partial t} \end{bmatrix}, \begin{bmatrix} K(t_i, t_j) & \frac{\partial K(t_i, t_j)}{\partial t_j} \\ \frac{\partial K(t_i, t_j)}{\partial t_i} & \frac{\partial^2 K(t_i, t_j)}{\partial t_i \partial t_j} \end{bmatrix} \right)$

Simulations

Prior Draw from a Gaussian Process

GP Prior Draw Code

# python / sklearn used to calculate the covariance matrix for efficiency

library(ggplot2)
library(mvtnorm)
library(reticulate) # let's one use python within R

k <- import("sklearn.gaussian_process.kernels")

x <- seq(-1, 1, length.out = 100)
dim(x) <- c(100, 1) # necessary to specify that the dimension of each x is 1

S <- k$RBF()(x, x) # RBF kernel with default length scale, 1., evaluated at x

y <- rmvnorm(1, sigma = S) # sample

x <- as.numeric(x) # restore the dimension (100,) for plotting
y <- as.numeric(y) # restore the dimension (100,) for plotting

qplot(x, y)

Posterior of a Gaussian Process

We fit a GP with an RBF kernel and lengthscale 1.0 to the points: $ \begin{bmatrix} [-0.5, -1] \ [0, 0.5] \ [0.5, 0] \end{bmatrix} $.

GP Posterior Code

# python / sklearn used to calculate the covariance matrix for efficiency

library(ggplot2)
library(mvtnorm)
library(reticulate) # let's one use python within R

k <- import("sklearn.gaussian_process.kernels")

x_test <- seq(-1, 1, length.out = 97) # prediction points
x_train <- c(-0.5, 0, 0.5)             # training points

x <- c(x_test, x_train) # concatenate them according to the convention above

dim(x) <- c(length(x_test) + length(x_train), 1) # tell sklearn that the dimension of each x is 1

S <- k$RBF()(x, x) # efficient application of the kernel function to our 100x100 matrix

# following the convention of the formulae above:

S11 <- S[1:length(x_test), 1:length(x_test)]
S12 <- S[1:length(x_test), (length(x_test) + 1):length(x)]
S21 <- S[(length(x_test) + 1):length(x), 1:length(x_test)]
S22 <- S[(length(x_test) + 1):length(x), (length(x_test) + 1):length(x)]

y_obs <- matrix(c(-1, 0.5, 0), 3, 1) # this is x_2 in the equation above

mu <- 0 + S12 %*% solve(S22) %*% (y_obs - 0)
sigma <- S11 - S12 %*% solve(S22) %*% S21

mu <- as.numeric(mu)
sigma <- diag(sigma) # obtain point-wise posterior variances

qplot(x = x_test, y = mu, geom = "line", color = "predicted") +
	geom_point(aes(x, y, color = "training"), data.frame(x = x_train, y = y_obs)) +
	geom_ribbon(ymin = mu - 1.96*sigma, ymax = mu + 1.96*sigma, alpha = 0.25) +
	labs(x = "x", y = "y")

Derivatives of the GP

I’ve written a piece to calculate the derivative kernels efficiently, but I’m still testing this:

GP Derivative Covariances (1-D)

#!/usr/bin/env python3

from autograd import numpy as np
from autograd import elementwise_grad as egrad

def K(xi, xj):
	""" Calculates the kernel matrix

	The domain values needn't be the same.
	Currently, the only supported dimension of each value of x is one.

	Args:
		xi: Sequence of domain values corresponding to the points changing over rows.
		xj: Sequence of domain values corresponding to the points changing over columns.

	Returns:
		numpy.array corresponding to the kernel values applied over the grid (xi, xj).
	"""
	if type(xi) is not float and type(xj) is not float: # if vectorization is possible:
		xi = xi.reshape(-1, 1) # reshape as a column vector
		xj = xj.reshape(1, -1) # reshape as a row vector
	return np.exp(-np.square(xi - xj)) # return kernel calculation

def dK(xi, xj):
	""" Returns the first kernel derivative matrix w.r.t. the second index

	Args:
		xi: Sequence of domain values corresponding to the points changing over rows.
		xj: Sequence of domain values corresponding to the points changing over columns.

	Returns:
		numpy.array corresponding to the kernel derivative values applied over the grid (xi, xj).		
	"""
	gradient = np.vectorize(egrad(K, 1)) # element-wise gradient along the second index
	xi = xi.reshape(-1, 1) # xi must be a column vector
	xj = xj.reshape(1, -1) # xj must be a row vector
	return gradient(xi, xj)

def dKt(xi, xj):
	""" Returns the transpose of Kernel Derivative Matrix by explicit computation

	Args:
		xi: Sequence of domain values corresponding to the points changing over rows.
		xj: Sequence of domain values corresponding to the points changing over columns.

	Returns:
		numpy.array corresponding to the kernel derivative values applied over the grid (xj, xi).
	
	"""
	gradient = np.vectorize(egrad(K, 0)) # element-wise gradient along the first index
	xi = xi.reshape(-1, 1) # xi must be a column vector
	xj = xj.reshape(1, -1) # xj must be a row vector
	return gradient(xi, xj)

def d2K(xi, xj):
	""" Returns the second kernel derivative matrix

	Args:
		xi: Sequence of domain values corresponding to the points changing over rows.
		xj: Sequence of domain values corresponding to the points changing over columns.

	Returns:
		numpy.array corresponding to the second kernel derivative applied over the grid (xi, xj).
	
	"""
	gradient = np.vectorize(egrad(egrad(K, 1), 0)) # two derivatives are taken
	xi = xi.reshape(-1, 1) # xi must be a column vector
	xj = xj.reshape(1, -1) # xj must be a row vector
	return gradient(xi, xj)

This seems to work and I’ve used (previous versions of it, coupled with Tensorflow’s Adam and Stan’s HMC/l_BFGS) to solve differential equations:

$\frac{dy}{dx} = sin(xy);$ $y(0) = 1$

Modeling SDE equivalents of GPs

There’s some great literature out there about modeling GPs as solutions of differential equations with a random component, but before I encountered that, the following was a brute-force attempt to model the functions $\mu(X_t), \sigma(X_t)$ where $X_t$ is a continuous time stochastic process and $W_t$ is the standard Weiner process:

$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$

When $X_t$ is a Gaussian Process, equating the Euler-Maruyama representation of the SDE above with the GP expressed as $LZ$ where $L$ is the Cholesky-decomposition of the covariance matrix, results in the random normal vector $Z$ being exactly equal to the random part of the SDE: $dW_t = \sqrt{\Delta t} Z$.

Hence modeling the functions $\mu, \sigma$ and minimizing the distance between $Z, dW_t$ is a way to obtain those functions without solving the SDE. When this is mathematically infeasible, the algorithm fails.

Simo Särkkä has shown that GPs can be written as state space models, which are models of the form:

$\dot {\bf x} = A {\bf x} + B {\bf u} \\ f = C {\bf x} + D {\bf u}$

For the centered Matern GP:

$A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -\lambda^3 & -3 \lambda^2 & -3 \lambda \end{bmatrix}, B = \begin{bmatrix} 0 \\0 \\ 1 \end{bmatrix} \\ C = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}, D = \begin{bmatrix} 0 \end{bmatrix} \\ u \sim \mathcal N, \; \nu = 2.5, \; \lambda = \sqrt{2\nu}/l$

Python code for Matern SSM-GP Simulation

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import StateSpace

plt.ion(); plt.style.use("ggplot")

##### MATERN GAUSSIAN PROCESS PARAMETERS

l = np.sqrt(2 * 2.5) / 1.0

a = np.array([[0, 1, 0], [0, 0, 1], [-l**3, -3*l**2, -3*l]])
b = np.array([[0], [0], [1]])
c = np.array([[1, 0, 0]])
d = np.array([[0]])

#### STATE SPACE MODEL INPUTS

u = np.random.normal(size = 1000)
t = np.linspace(0, 10, 1000)
x0 = np.random.normal()

#### SIMULATE SSM

model = StateSpace(a, b, c, d)
t, f, x = model.output(u, t)

#### PLOT

plt.plot(t, x)

I’ve got the suspicion that the RBF GP can’t be represented this way (easily) because it’s infinitely differentiable; the Matern GP isn’t, so a derivative of a high enough order would appear to be random (which is set to $u(t)$ and integrating that many times over would lead to a nice function. So, it’s probably unsurprising that the RBF needs to be written as its infinite series representation and then represented as a SSM.

Gaussian Processes

Introduction

Simulations

Prior Draw from a Gaussian Process

Posterior of a Gaussian Process

Derivatives of the GP

Modeling SDE equivalents of GPs

2019

Random Stuff

Random Stuff

Stochastic Bernoulli Probabilities

Random Projects

Random Projects

Musings on Statistics

Musings on Statistics

Measures & Defining Probability

Probability & Sigma Algebras

Sparse Gaussian Process Examples

A Minimal Working Example

2018

Gaussian Processes

Introduction

The Log-Likelihood & Dimensionality

Introduction

Non-Monotonic Transforms

Exact Densities for Non-Monotonic Transformations (draft)