# gaussian process regression tutorial

We will build up deeper understanding on how to implement Gaussian process regression from scratch on a toy example. We can treat the Gaussian process as a prior defined by the kernel function and create a This might not mean much at this moment so lets dig a bit deeper in its meaning. Instead we specify the GP in terms of an element-wise mean function $m:\mathbb{R}^D \mapsto \mathbb{R}$ and an element-wise covariance function (a.k.a kernel function) $k: \mathbb{R}^{D \times D} \mapsto \mathbb{R}$: This is common practice and isn't as much of a restriction as it sounds, since the mean of the posterior distribution is free to change depending on the observations it is conditioned on (see below). \end{split}$$. In other words, we can fit the data just as well (in fact better) if we increase the length scale but also increase the noise variance i.e. We can treat the Gaussian process as a prior defined by the kernel function and create a posterior distribution given some data. We can make predictions from noisy observations f(X_1) = \mathbf{y}_1 + \epsilon, by modelling the noise \epsilon as Gaussian noise with variance \sigma_\epsilon^2. First we build the covariance matrix K(X_*, X_*) by calling the GPs kernel on X_*. Gaussian Process Regression (GPR)¶ The GaussianProcessRegressor implements Gaussian processes (GP) for regression purposes. Chapter 4 of Rasmussen and Williams covers some other choices, and their potential use cases. Instead we use the simple vectorized form K(X1, X2) = \sigma_f^2X_1X_2^T for the linear kernel, and numpy's optimized methods \texttt{pdist} and \texttt{cdist} for the squared exponential and periodic kernels. This tutorial will introduce new users to specifying, fitting and validating Gaussian process models in Python. solve Gaussian processes are a powerful algorithm for both regression and classification. The multivariate Gaussian distribution is defined by a mean vector μ\muμ … # Instantiate GPs using each of these kernels. More generally, Gaussian processes can be used in nonlinear regressions in which the relationship between xs and ys is assumed to vary smoothly with respect to the values of the xs. # Plot poterior mean and 95% confidence interval. \Sigma_{22} & = k(X_2,X_2) \quad (n_2 \times n_2) \\ since they both come from the same multivariate distribution. By Bayes' theorem, the posterior distribution over the kernel parameters \pmb{\theta} is given by:$$ p(\pmb{\theta}|\mathbf{y}, X) = \frac{p(\mathbf{y}|X, \pmb{\theta}) p(\pmb{\theta})}{p(\mathbf{y}|X)}.. This post is part of series on Gaussian processes: In what follows we assume familiarity with basic probability and linear algebra especially in the context of multivariate Gaussian distributions. This tutorial aims to provide an accessible introduction to these techniques. Their greatest practical advantage is that they can give a reliable estimate of their own uncertainty. In other words we need to form the GP posterior. The main advantages of this method are the ability of GPs to provide uncertainty estimates and to learn the noise and smoothness parameters from training data. The posterior predictions of a Gaussian process are weighted averages of the observed data where the weighting is based on the coveriance and mean functions. By selecting alternative components (a.k.a basis functions) for $\phi(\mathbf{x})$ we can perform regression of more complex functions. We’ll be modeling the function \begin{align} y &= \sin(2\pi x) + \epsilon \\ \epsilon &\sim \mathcal{N}(0, 0.04) \end{align} The idea is that we wish to estimate an unknown function given noisy observations ${y_1, \ldots, y_N}$ of the function at a finite number of points ${x_1, \ldots x_N}.$ We imagine a generative process Perhaps the most important attribute of the GPR class is the $\texttt{kernel}$ attribute. For this we implement the following method: Finally, we use the fact that in order generate Gaussian samples $\mathbf{z} \sim \mathcal{N}(\mathbf{m}, K)$ where $K$ can be decomposed as $K=LL^T$, we can first draw $\mathbf{u} \sim \mathcal{N}(\mathbf{0}, I)$, then compute $\mathbf{z}=\mathbf{m} + L\mathbf{u}$. In fact, the Brownian motion process can be reformulated as a Gaussian process a second post demonstrating how to fit a Gaussian process kernel By experimenting with the parameter $\texttt{theta}$ for each of the different kernels, we can can change the characteristics of the sampled functions. This can be done with the help of the posterior distribution $p(\mathbf{y}_2 \mid \mathbf{y}_1,X_1,X_2)$. We will use simple visual examples throughout in order to demonstrate what's going on. But how do we choose the basis functions? A finite dimensional subset of the Gaussian process distribution results in a Using the marginalisation property of multivariate Gaussians, the joint distribution over the observations, $\mathbf{y}$, and test outputs $\mathbf{f_*}$ according to the GP prior is \begin{align*} marginal distribution $z$ has the desired distribution since $\mathbb{E}[\mathbf{z}] = \mathbf{m} + L\mathbb{E}[\mathbf{u}] = \mathbf{m}$ and $\text{cov}[\mathbf{z}] = L\mathbb{E}[\mathbf{u}\mathbf{u}^T]L^T = LL^T = K$. The $\_\_\texttt{call}\_\_$ function of the class constructs the full covariance matrix $K(X1, X2) \in \mathbb{R}^{n_1 \times n_2}$ by applying the kernel function element-wise between the rows of $X1 \in \mathbb{R}^{n_1 \times D}$ and $X2 \in \mathbb{R}^{n_2 \times D}$. That said, I have now worked through the basics of Gaussian process regression as described in Chapter 2 and I want to share my code with you here. Now we know what a GP is, we'll now explore how they can be used to solve regression tasks. Gaussian Processes regression: basic introductory example¶ A simple one-dimensional regression example computed in two different ways: A noise-free case. Any kernel function is valid so long it constructs a valid covariance matrix i.e. It took me a while to truly get my head around Gaussian Processes (GPs). This tutorial introduces the reader to Gaussian process regression as an expressive tool to model, actively explore and exploit unknown functions. Away from the observations the data lose their influence on the prior and the variance of the function values increases. # Create coordinates in parameter space at which to evaluate the lml. """, # Fill the cost matrix for each combination of weights, Calculate the posterior mean and covariance matrix for y2. this post The $\texttt{theta}$ parameter for the $\texttt{SquaredExponential}$ kernel (representing $l$ in the squared exponential kernel function formula above) is the characteristic length scale, roughly specifying how far apart two input points need to be before their corresponding function values can differ significantly: small values mean less 'co-variance' and so more quickly varying functions, whilst larger values mean more co-variance and so flatter functions. GP An example covariance matrix from the exponentiated quadratic covariance function is plotted in the figure below on the left. After a sequence of preliminary posts (Sampling from a Multivariate Normal Distribution and Regularized Bayesian Regression as a Gaussian Process), I want to explore a concrete example of a gaussian process regression.We continue following Gaussian Processes for Machine Learning, Ch 2.. Other recommended references are: based on the corresponding input X2, the observations (y1, X1), # Compute the posterior mean and covariance, # Define the true function that we want to regress on, # Number of points to condition on (training points), # Number of points in posterior (test points), # Number of functions that will be sampled from the posterior, # Sample observations (X1, y1) on the function, # Predict points at uniform spacing to capture function, # Compute the standard deviation at the test points to be plotted, # Plot the postior distribution and some samples, # Plot the distribution of the function (mean, covariance), 'Distribution of posterior and prior data. random walk $\bar{\mathbf{f}}_* = K(X, X_*)^T\mathbf{\alpha}$ and $\text{cov}(\mathbf{f}_*) = K(X_*, X_*) - \mathbf{v}^T\mathbf{v}$. In terms of basic understanding of Gaussian processes, the tutorial will cover the following topics: We will begin by an introduction to Gaussian processes starting with parametric models and generalized linear models. Still, $\pmb{\theta}_{MAP}$ is usually a good estimate, and in this case we can see that it is very close to the $\pmb{\theta}$ used to generate the data, which makes sense.