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# gaussian process regression tutorial

Usually we have little prior knowledge about $\pmb{\theta}$, and so the prior distribution $p(\pmb{\theta})$ can be assumed flat. Any kernel function is valid so long it constructs a valid covariance matrix i.e. posterior distribution 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 $\mathbf{z}=\mathbf{m} + L\mathbf{u}$. realizations Note in the plots that the variance $\sigma_{2|1}^2$ at the observations is no longer 0, and that the functions sampled don't necessarily have to go through these observational points anymore. Gaussian Processes regression: basic introductory example¶ A simple one-dimensional regression example computed in two different ways: A noise-free case. With increasing data complexity, models with a higher number of parameters are usually needed to explain data reasonably well. In terms of implementation, we already computed $\mathbf{\alpha} = \left[K(X, X) + \sigma_n^2\right]^{-1}\mathbf{y}$ when dealing with the posterior distribution. The aim is to find $f(\mathbf{x})$, such that given some new test point $\mathbf{x}_*$, we can accurately estimate the corresponding $y_*$. For example the kind of functions that can be modelled with a Squared Exponential kernel with a characteristic length scale of 10 are completely different (much flatter) than those that can be modelled with the same kernel but a characteristic length scale of 1. """, # Fill the cost matrix for each combination of weights, Calculate the posterior mean and covariance matrix for y2. This might not mean much at this moment so lets dig a bit deeper in its meaning. The next figure on the left visualizes the 2D distribution for $X = [0, 0.2]$ where the covariance $k(0, 0.2) = 0.98$. \Sigma_{12} & = k(X_1,X_2) = k_{21}^\top \quad (n_1 \times n_2) typically describe systems randomly changing over time. We can now compute the $\pmb{\theta}_{MAP}$ for our Squared Exponential GP. a second post demonstrating how to fit a Gaussian process kernel Its mean and covariance are defined by a Gaussian process regression (GPR) is an even ﬁner approach than this. As always, I’m doing this in R and if you search CRAN, you will find a specific package for Gaussian process regression: gptk. By applying our linear model now on $\phi(x)$ rather than directly on the inputs $x$, we would implicitly be performing polynomial regression in the input space. a higher dimensional feature space). Increasing the noise variance allows the function values to deviate more from the observations, as can be verified by changing the $\texttt{noise}\_\texttt{var}$ parameter and re-running the code. Gaussian Processes (GPs) are the natural next step in that journey as they provide an alternative approach to regression problems. 1.7.1. The element-wise computations could be implemented with simple for loops over the rows of $X1$ and $X2$, but this is inefficient. \textit{Linear}: \quad &k(\mathbf{x}_i, \mathbf{x}_j) = \sigma_f^2\mathbf{x}_i^T \mathbf{x}_j \\ without any observed data. The plots should make clear that each sample drawn is an $n_*$-dimensional vector, containing the function values at each of the $n_*$ input points (there is one colored line for each sample). covariance If we assume that $f(\mathbf{x})$ is linear, then we can simply use the least-squares method to draw a line-of-best-fit and thus arrive at our estimate for $y_*$. For this, the prior of the GP needs to be specified. function . To implement this sampling operation we proceed as follows. 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. Gaussian processes are flexible probabilistic models that can be used to perform Bayesian regression analysis without having to provide pre-specified functional relationships between the variables. It returns the modelled models. Note that I could have parameterised each of these functions more to control other aspects of their character e.g. Gaussian Processes for regression: a tutorial José Melo Faculty of Engineering, University of Porto FEUP - Department of Electrical and Computer Engineering Rua Dr. Roberto Frias, s/n 4200-465 Porto, PORTUGAL jose.melo@fe.up.pt Abstract Gaussian processes are a powerful, non-parametric tool that can be be used in supervised learning, namely in re- is generated from an Python notebook file. 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. In this post we will model the covariance with the By Bayes' theorem, the posterior distribution over the kernel parameters $\pmb{\theta}$ is given by: $$p(\pmb{\theta}|\mathbf{y}, X) = \frac{p(\mathbf{y}|X, \pmb{\theta}) p(\pmb{\theta})}{p(\mathbf{y}|X)}.$$. and write the GP as However each realized function can be different due to the randomness of the stochastic process. solve There are some great resources out there to learn about them - Rasmussen and Williams, mathematicalmonk's youtube series, Mark Ebden's high level introduction and scikit-learn's implementations - but no single resource I found providing: This post aims to fix that by drawing the best bits from each of the resources mentioned above, but filling in the gaps, and structuring, explaining and implementing it all in a way that I think makes most sense. Gaussian process history Prediction with GPs: • Time series: Wiener, Kolmogorov 1940’s • Geostatistics: kriging 1970’s — naturally only two or three dimensional input spaces • Spatial statistics in general: see Cressie  for overview • General regression: O’Hagan  • Computer experiments (noise free): Sacks et al. We can treat the Gaussian process as a prior defined by the kernel function and create a 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 Before we can explore Gaussian processes, we need to understand the mathematical concepts they are based on. domain In other words we need to form the GP posterior. A noisy case with known noise-level per datapoint. In both cases, the kernel’s parameters are estimated using the maximum likelihood principle. In particular we first pre-compute the quantities $\mathbf{\alpha} = \left[K(X, X) + \sigma_n^2\right]^{-1}\mathbf{y} = L^T \backslash(L \backslash \mathbf{y})$ and $\mathbf{v} = L^T [K(X, X) + \sigma_n^2]^{-1}K(X, X_*) = L \backslash K(X, X_*)$, The prediction interval is computed from the standard deviation $\sigma_{2|1}$, which is the square root of the diagonal of the covariance matrix. Rather than claiming relates to some speciﬁc models (e.g. method below. The $\texttt{theta}$ parameter for the $\texttt{Linear}$ kernel (representing $\sigma_f^2$ in the linear kernel function formula above) controls the variance of the function gradients: small values give a narrow distribution of gradients around zero, and larger values the opposite. More formally, for any index set $\mathcal{S}$, a GP on $\mathcal{S}$ is a set of random variables $\{z_s: s \in \mathcal{S}\}$ s.t. Notice in the figure above that the stochastic process can lead to different paths, also known as Gaussian processes for regression ¶ Since Gaussian processes model distributions over functions we can use them to build regression models. \end{split}$$. prior$$\mathbf{y} \sim \mathcal{N}\left(\mathbf{0}, K(X, X) + \sigma_n^2I\right).$$. The \_\_\texttt{call}\_\_ function of the class constructs the full covariance matrix K(X1, X2) \in \mathbb{R}^{n_1 \times n_2} by applying the kernel function element-wise between the rows of X1 \in \mathbb{R}^{n_1 \times D} and X2 \in \mathbb{R}^{n_2 \times D}. function (a Gaussian process).$$\begin{bmatrix} \mathbf{y} \\ \mathbf{f}_* \end{bmatrix} = \mathcal{N}\left(\mathbf{0}, \begin{bmatrix} K(X, X) + \sigma_n^2I && K(X, X_*) \\ K(X_*, X) && K(X_*, X_*)\end{bmatrix}\right)., The GP posterior is found by conditioning the joint G.P prior distribution on the observations Here, and below, we use X \in \mathbb{R}^{n \times D} to denote the matrix of input points (one row for each input point). Examples of different kernels are given in a Note that \Sigma_{11} is independent of \Sigma_{22} and vice versa. It can be seen as a continuous This post explores some of the concepts behind Gaussian processes such as stochastic processes and the kernel function. jointly Gaussian In practice we can't just sample a full function evaluation f from a Gaussian process distribution since that would mean evaluating m(x) and k(x,x') at an infinite number of points since x can have an infinite \bar{\mathbf{f}}_* = K(X, X_*)^T\mathbf{\alpha} and \text{cov}(\mathbf{f}_*) = K(X_*, X_*) - \mathbf{v}^T\mathbf{v}. 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. It took me a while to truly get my head around Gaussian Processes (GPs). The position d(t) at time t evolves as d(t + \Delta t) = d(t) + \Delta d. between each pair in x_a and x_b. . This tutorial will introduce new users to specifying, fitting and validating Gaussian process models in Python. Have a look at normal distribution positive definite The notebook can be executed at. Note that the exponentiated quadratic covariance decreases exponentially the further away the function values x are from eachother. domain An example covariance matrix from the exponentiated quadratic covariance function is plotted in the figure below on the left. Gaussian Processes Tutorial Regression Machine Learning A.I Probabilistic Modelling Bayesian Python, You can modify those links in your config file. The bottom figure shows 5 realizations (sampled functions) from this distribution. Now we know what a GP is, we'll now explore how they can be used to solve regression tasks. Note that the distrubtion is quite confident of the points predicted around the observations (X_1,\mathbf{y}_1), and that the prediction interval gets larger the further away it is from these points. \Sigma_{22} & = k(X_2,X_2) \quad (n_2 \times n_2) \\ Machine Learning Tutorial at Imperial College London: Gaussian Processes Richard Turner (University of Cambridge) November 23, 2016 Let’s assume a linear function: y=wx+ϵ. We can get get a feel for the positions of any other local maxima that may exist by plotting the contours of the log marginal likelihood as a function of \pmb{\theta}. k(\mathbf{x}_n, \mathbf{x}_1) & \ldots & k(\mathbf{x}_n, \mathbf{x}_n) \end{bmatrix}. \begin{align*} Methods that use models with a fixed number of parameters are called parametric methods. 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. First we build the covariance matrix K(X_*, X_*) by calling the GPs kernel on X_*. The non-linearity is because the kernel can be interpreted as implicitly computing the inner product in a different space than the original input space (e.g. We can however sample function evaluations \mathbf{y} of a function f drawn from a Gaussian process at a finite, but arbitrary, set of points X: \mathbf{y} = f(X). Note that we have chosen the mean function m(\mathbf{x}) of our G.P prior to be 0, which is why the mean vector in the f.d.d above is the zero vector \mathbf{0}. Since functions can have an infinite input domain, the Gaussian process can be interpreted as an infinite dimensional Gaussian random variable. Here are 3 possibilities for the kernel function: \begin{align*} k(\mathbf{x}_i, \mathbf{x}_j) &= \mathbb{E}[(f(\mathbf{x}_i) - m(\mathbf{x}_i))(f(\mathbf{x}_j) -m(\mathbf{x}_j))], \end{align*} This posterior distribution can then be used to predict the expected value and probability of the output variable \mathbf{y} given input variables X. covariance function k(x,x'), with x the function values and (x,x') all possible pairs in the input "positive definite matrix. Observe that points close together in the input domain of x are strongly correlated (y_1 is close to y_2), while points further away from eachother are almost independent. \textit{Periodic}: \quad &k(\mathbf{x}_i, \mathbf{x}_j) = \text{exp}\left(-\sin(2\pi f(\mathbf{x}_i - \mathbf{x}_j))^T \sin(2\pi f(\mathbf{x}_i - \mathbf{x}_j))\right) marginal distribution This tutorial introduces the reader to Gaussian process regression as an expressive tool to model, actively explore and exploit unknown functions. Still, a principled probabilistic approach to classification tasks is a very attractive prospect, especially if they can be scaled to high-dimensional image classification, where currently we are largely reliant on the point estimates of Deep Learning models. # Instantiate GPs using each of these kernels. Note that the noise only changes kernel values on the diagonal (white noise is independently distributed). # Generate posterior samples, saving the posterior mean and covariance too. You can prove for yourself that each of these kernel functions is valid i.e. We can sample a realization of a function from a stochastic process. In the figure below we will sample 5 different function realisations from a Gaussian process with exponentiated quadratic prior We have some observed data \mathcal{D} = [(\mathbf{x}_1, y_1) \dots (\mathbf{x}_n, y_n)] with \mathbf{x} \in \mathbb{R}^D and y \in \mathbb{R}. Gaussian processes are a powerful algorithm for both regression and classification. For example, the f.d.d over \mathbf{f} = (f_{\mathbf{x}_1}, \dots f_{\mathbf{x}_n}) would be  \mathbf{f} \sim \mathcal{N}(\bar{\mathbf{f}}, K(X, X)), with. one that is positive semi-definite. information on this distribution. This is because these marginals come from a Gaussian process with as prior the exponentiated quadratic covariance which adds prior information that points close to eachother in the input space X must be close to eachother in the output space y. This is because the noise variance of the GP was set to it's default value of 10^{-8} during instantiation. The code demonstrates the use of Gaussian processes in a dynamic linear regression. L-BFGS. # Draw samples from the prior at our data points. \textit{Squared Exponential}: \quad &k(\mathbf{x}_i, \mathbf{x}_j) = \text{exp} \left(\frac{-1}{2l^2} (\mathbf{x}_i - \mathbf{x}_j)^T (\mathbf{x}_i - \mathbf{x}_j)\right) \\ Each kernel class has an attribute \texttt{theta}, which stores the parameter value of its associated kernel function (\sigma_f^2, l and f for the linear, squared exponential and periodic kernels respectively), as well as a \texttt{bounds} attribute to specify a valid range of values for this parameter. Jie Wang, Offroad Robotics, Queen's University, Kingston, Canada. GPyTorch Regression Tutorial¶ Introduction¶ In this notebook, we demonstrate many of the design features of GPyTorch using the simplest example, training an RBF kernel Gaussian process on a simple function. A finite dimensional subset of the Gaussian process distribution results in a The results are plotted below. The term marginal refers to the marginalisation over the function values \mathbf{f}. Whilst a multivariate Gaussian distribution is completely specified by a single finite dimensional mean vector and a single finite dimensional covariance matrix, in a GP this is not possible, since the f.d.ds in terms of which it is defined can have any number of dimensions. Gaussian Process Regression (GPR)¶ The GaussianProcessRegressor implements Gaussian processes (GP) for regression purposes. Gaussian process regression. 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 maximum a posteriori (MAP) estimate for \pmb{\theta}, \pmb{\theta}_{MAP}, occurs when p(\pmb{\theta}|\mathbf{y}, X) is greatest. The variance \sigma_2^2 of these predictions is then the diagonal of the covariance matrix \Sigma_{2|1}. In our case the index set \mathcal{S} = \mathcal{X} is the set of all possible input points \mathbf{x}, and the random variables z_s are the function values f_\mathbf{x} \overset{\Delta}{=} f(\mathbf{x}) corresponding to all possible input points \mathbf{x} \in \mathcal{X}. As the name suggests, the Gaussian distribution (which is often also referred to as normal distribution) is the basic building block of Gaussian processes. ⁽¹⁾ 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. The f.d.d of the observations \mathbf{y} \sim \mathbb{R}^n defined under the GP prior is: You can train a GPR model using the fitrgp function. By experimenting with the parameter \texttt{theta} for each of the different kernels, we can can change the characteristics of the sampled functions. The red cross marks the position of \pmb{\theta}_{MAP} for our G.P with fixed noised variance of 10^{-8}. To do this we can simply plug the above expression into a multivariate optimizer of our choosing, e.g. symmetric We know to place less trust in the model's predictions at these locations. Urtasun and Lawrence () Session 1: GP and Regression CVPR Tutorial 14 / 74 For general Bayesian inference need multivariate priors. Luckily, Bayes' theorem provides us a principled way to pick the optimal parameters. Note that X1 and X2 are identical when constructing the covariance matrices of the GP f.d.ds introduced above, but in general we allow them to be different to facilitate what follows. ). Once again Chapter 5 of Rasmussen and Williams outlines how to do this. Gaussian process regression (GPR) is an even ﬁner approach than this. 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}: We recently ran into these approaches in our robotics project that having multiple robots to generate environment models with minimum number of samples. conditional distribution Gaussian Process Regression Gaussian Processes: Deﬁnition A Gaussian process is a collection of random variables, any ﬁnite number of which have a joint Gaussian distribution. We do this by drawing correlated samples from a 41-dimensional Gaussian \mathcal{N}(0, k(X, X)) with X = [X_1, \ldots, X_{41}]. . We assume that each observation y can be related to an underlying function f(\mathbf{x}) through a Gaussian noise model:y = f(\mathbf{x}) + \mathcal{N}(0, \sigma_n^2)$$. If needed we can also infer a full posterior distribution p(θ|X,y) instead of a point estimate ˆθ. due to the uncertainty in the system. Of course the reliability of our predictions is dependent on a judicious choice of kernel function. Gaussian process regression (GPR) models are nonparametric kernel-based probabilistic models. If you played with the kernel parameters above you would have seen how much influence they have. Every finite set of the Gaussian process distribution is a multivariate Gaussian. multivariate Gaussian : We can write these as follows (Note here that \Sigma_{11} = \Sigma_{11}^{\top} since it's You can read In non-parametric methods, … Consistency: If the GP speciﬁes y(1),y(2) ∼ N(µ,Σ), then it must also specify y(1) ∼ N(µ 1,Σ 11): A GP is completely speciﬁed by a mean function and a For example, the covariance matrix associated with the linear kernel is simply \sigma_f^2XX^T, which is indeed symmetric positive semi-definite.$$\begin{split} Gaussian Processes Tutorial - Regression¶ It took me a while to truly get my head around Gaussian Processes (GPs). \mu_{2} & = m(X_2) \quad (n_2 \times 1) \\ The Gaussian processes regression is then described in an accessible way by balancing showing unnecessary mathematical derivation steps and missing key conclusive results. the periodic kernel could also be given a characteristic length scale parameter to control the co-variance of function values within each periodic element. By choosing a specific kernel function $k$ it is possible to set Unlike many popular supervised machine learning algorithms that learn exact values for every parameter in a function, the Bayesian approach infers a probability distribution over all possible values. Wiener process # in general these can be > 1d, hence the extra axis. It is often necessary for numerical reasons to add a small number to the diagonal elements of $K$ before the Cholesky factorisation. Let's define the methods to compute and optimize the log marginal likelihood in this way. $k(x_a, x_b)$ models the joint variability of the Gaussian process random variables. This tutorial was generated from an IPython notebook that can be downloaded here. We want to make predictions $\mathbf{y}_2 = f(X_2)$ for $n_2$ new samples, and we want to make these predictions based on our Gaussian process prior and $n_1$ previously observed data points $(X_1,\mathbf{y}_1)$. To sample functions from the Gaussian process we need to define the mean and covariance functions. exponentiated quadratic In this case $\pmb{\theta}_{MAP}$ can be found by maximising the marginal likelihood, $p(\mathbf{y}|X, \pmb{\theta}) = \mathcal{N}(\mathbf{0}, K(X, X) + \sigma_n^2I)$, which is just the f.d.d of our observations under the GP prior (see above). 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. This gradient will only exist if the kernel function is differentiable within the bounds of theta, which is true for the Squared Exponential kernel (but may not be for other more exotic kernels). \Sigma_{11} & = k(X_1,X_1) \quad (n_1 \times n_1) \\ with mean $0$ and variance $\Delta t$. $$\mathbf{f}_* | X_*, X, \mathbf{y} \sim \mathcal{N}\left(\bar{\mathbf{f}}_*, \text{cov}(\mathbf{f}_*)\right),$$, where A good high level exposition of what GPs actually are. # Compute L and alpha for this K (theta). This post aims to present the essentials of GPs without going too far down the various rabbit holes into which they can lead you (e.g. Try increasing", # First define input test points at which to evaluate the sampled functions. We will use simple visual examples throughout in order to demonstrate what's going on. This corresponds to sampling from the G.P prior, since we have not yet taken into account any observed data, only our prior belief (via the kernel function) as to which loose family of functions our target function belongs: $$\mathbf{f}_* \sim \mathcal{N}\left(\mathbf{0}, K(X_*, X_*)\right).$$. Each input to this function is a variable correlated with the other variables in the input domain, as defined by the covariance function. Introduction. For observations, we'll use samples from the prior. Try setting different initial value of theta.'. 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.