# gaussian process regression explained

Since we are unable to completely remove uncertainty from the universe we best have a good way of dealing with it. To overcome this challenge, learning specialized kernel functions from the underlying data, for example by using deep learning, is an area of … Let’s consider that we’ve never heard of Barack Obama (bear with me), or at least we have no idea what his height is. On the left each line is a sample from the distribution of functions and our lack of knowledge is reflected in the wide range of possible functions and diverse function shapes on display. Also note how things start to go a bit wild again to the right of our last training point$x = 1$— that won’t get reined in until we observe some data over there. $$And we would like now to use our model and this regression feature of Gaussian Process to actually retrieve the full deformation field that fits to the observed data and still obeys to the properties of our model. The simplest example of this is linear regression, where we learn the slope and intercept of a line so we can predict the vertical position of points from their horizontal position. Given any set of N points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian. Anything other than 0 in the top right would be mirrored in the bottom left and would indicate a correlation between the variables. By the end of this maths-free, high-level post I aim to have given you an intuitive idea for what a Gaussian process is and what makes them unique among other algorithms. the square root of our covariance matrix. Gaussian Processes (GPs) are the natural next step in that journey as they provide an alternative approach to regression problems. In the next video, we will use Gaussian processes for Bayesian optimization. The probability distribution shown still reflects the small chance that Obama is average height and everyone else in the photo is unusually short. x_1 \\ Make learning your daily ritual. This lets you shape your fitted function in many different ways. This covariance matrix, along with a mean function to output the expected value of  f(x)  defines a Gaussian Process. \begin{pmatrix} Every finite set of the Gaussian process distribution is a multivariate Gaussian. See how the training points (the blue squares) have “reined in” the set of possible functions: the ones we have sampled from the posterior all go through those points. ∙ Penn State University ∙ 26 ∙ share . When you’re using a GP to model your problem you can shape your prior belief via the choice of kernel (a full explanation of these is beyond the scope of this post). This means going from a set of possible outcomes to just one real outcome — rolling the dice in this example. As we have seen, Gaussian processes offer a flexible framework for regression and several extensions exist that make them even more versatile. Bayesian inference might be an intimidating phrase but it boils down to just a method for updating our beliefs about the world based on evidence that we observe. Constructing Posterior Density We consider the regression model y = f(x) + ", where "˘N(0;˙2). Gaussian processes (GPs) provide a powerful probabilistic learning framework, including a marginal likelihood which represents the probability of data given only kernel hyperparameters. \end{pmatrix} So let’s put some constraints on it. This means not only that the training data has to be kept at inference time but also means that the computational cost of predictions scales (cubically!) It’s just that we’re not just talking about the joint probability of two variables, as in the bivariate case, but the joint probability of the values of  f(x)  for all the  x  values we’re looking at, e.g. \mu_1 \\ 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. At any rate, what we end up with are the mean,\mu_{*}and covariance matrix\Sigma_{*}that define our distribution f_{*} \sim \mathcal{N}{\left(\mu_{*}, \Sigma_{*}\right) }. Let’s run through an illustrative example of Bayesian inference — we are going to adjust our beliefs about the height of Barack Obama based on some evidence. Instead of updating our belief about Obama’s height based on photos we’ll update our belief about an unknown function given some samples from that function. Our prior belief about the the unknown function is visualized below. 0. \begin{pmatrix} Machine learning is an extension of linear regression in a few ways. a one in six chance of any particular face. In this method, a 'big' covariance is constructed, which describes the correlations between all the input and output variables taken in N points in the desired domain. Don’t Start With Machine Learning. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. We can see that Obama is definitely taller than average, coming slightly above several other world leaders, however we can’t be quite sure how tall exactly. If you use GPstuff, please use the reference (available online):Jarno Vanhatalo, Jaakko Riihimäki, Jouni Hartikainen, Pasi Jylänki, Ville Tolvanen, and Aki Vehtari (2013). Gaussian Process Regression Analysis for Functional Data presents nonparametric statistical methods for functional regression analysis, specifically the methods based on a Gaussian process prior in a functional space. with the number of training samples.$$, From both sides now: the math of linear regression, Machine Learning: A Probabilistic Perspective, Nando de Freitas’ UBC Machine Learning lectures. Now we can sample from this distribution. However we do know he’s a male human being resident in the USA. Gaussian processes know what they don’t know. Bayesian statistics provides us the tools to update our beliefs (represented as probability distributions) based on new data. 05/24/2020 ∙ by Junjie Liang, et al. Summary. Gaussian processes (GPs) are very widely used for modeling of unknown functions or surfaces in applications ranging from regression to classification to spatial processes. Bayesian linear regression provides a probabilistic approach to this by finding a distribution over the parameters that gets updated whenever new data points are observed. The models are fully probabilistic so uncertainty bounds are baked in with the model. Instead of observing some photos of Obama we will instead observe some outputs of the unknown function at various points. 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. There are some points$x_{*}$for which we would like to estimate$f(x_{*})$(denoted above as$f_{*}$). \right)} That’s what non-parametric means: it’s not that there aren’t parameters, it’s that there are infinitely many parameters. Take a look, Zillow house price prediction competition. But of course we need a prior before we’ve seen any data. Gaussian Process A Gaussian process (GP) is a generalization of a multivariate Gaussian distribution to infinitely many variables, thus functions Def: A stochastic process is Gaussian iff for every finite set of indices x 1, ..., x n in the index set is a vector-valued Gaussian random variable Their greatest practical advantage is that they can give a reliable estimate of their own uncertainty. AI, Machine Learning, Data Science, Language, Source: The Kernel Cookbook by David Duvenaud. Recall that when you have a univariate distribution$x \sim \mathcal{N}{\left(\mu, \sigma^2\right)}$you can express this in relation to standard normals, i.e. The most obvious example of a probability distribution is that of the outcome of rolling a fair 6-sided dice i.e. In statistics, originally in geostatistics, kriging or Gaussian process regression is a method of interpolation for which the interpolated values are modeled by a Gaussian process governed by prior covariances.Under suitable assumptions on the priors, kriging gives the best linear unbiased prediction of the intermediate values. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. By the end of this maths-free, high-level post I aim to have given you an intuitive idea for what a Gaussian process is and what makes them unique among other algorithms. First of all, we’re only interested in a specific domain — let’s say our x values only go from -5 to 5. That’s when I began the journey I described in my last post, From both sides now: the math of linear regression. The world around us is filled with uncertainty — we do not know exactly how long our commute will take or precisely what the weather will be at noon tomorrow. Want to Be a Data Scientist? In Bayesian inference our beliefs about the world are typically represented as probability distributions and Bayes’ rule tells us how to update these probability distributions. x_2 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 General Bounds on Bayes Errors for Regression with Gaussian Processes 303 2 Regression with Gaussian processes To explain the Gaussian process scenario for regression problems [4J, we assume that observations Y E R at input points x E RD are corrupted values of a function 8(x) by an independent Gaussian noise with variance u2 . \end{pmatrix} What might that look like? These documents show the start-to-finish process of quantitative analysis on the buy-side to produce a forecasting model. the similarity of each observed$x$to each other observed$x$.$K_{*}$gets us the similarity of the training values to the test values whose output values we’re trying to estimate.$K_{**}$gives the similarity of the test values to each other. This is an example of a discrete probability distributions as there are a finite number of possible outcomes. \sigma_{11} & \sigma_{12}\\ The authors focus on problems involving functional response variables and mixed covariates of functional and scalar variables. Since Gaussian processes let us describe probability distributions over functions we can use Bayes’ rule to update our distribution of functions by observing training data. To reinforce this intuition I’ll run through an example of Bayesian inference with Gaussian processes which is exactly analogous to the example in the previous section. We’d like to consider every possible function that matches our data, with however many parameters are involved. For Gaussian processes our evidence is the training data. 1.7.1. The key idea is that if $$x_i$$ and $$x_j$$ are deemed by the kernel to be similar, then we expect the output of the function at those points to be similar, too. 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. If we imagine looking at the bell from above and we see a perfect circle, this means these are two independent normally distributed variables — their covariance is 0. real numbers between -5 and 5. We also define the kernel function which uses the Squared Exponential, a.k.a Gaussian, a.k.a. Above we can see the classification functions learned by different methods on a simple task of separating blue and red dots. \sim \mathcal{N}{\left( Gaussian processes are a powerful algorithm for both regression and classification. as$x \sim \mu + \sigma(\mathcal{N}{\left(0, 1\right)})$. The shape of the bell is determined by the covariance matrix. ARMA models used in time series analysis and spline smoothing (e.g. Gaussian processes let you incorporate expert knowledge. The actual function generating the$y$values from our$x$values, unbeknownst to our model, is the$sin$function. Note that the K_ss variable here corresponds to$K_{**}$in the equation above for the joint probability. Now that we know how to represent uncertainty over numeric values such as height or the outcome of a dice roll we are ready to learn what a Gaussian process is. As with all Bayesian methods it begins with a prior distribution and updates this as data points are observed, producing the posterior distribution over functions. Note that two commonly used and powerful methods maintain high certainty of their predictions far from the training data — this could be linked to the phenomenon of adversarial examples where powerful classifiers give very wrong predictions for strange reasons. Probability distributions are exactly that and it turns out that these are the key to understanding Gaussian processes. Now let’s pretend that Wikipedia doesn’t exist so we can’t just look up Obama’s height and instead observe some evidence in the form of a photo. So we are trying to get the probability distribution$p(f_{*} | x_{*},x,f)$and we are assuming that $f$and$f_{*}$together are jointly Gaussian as defined above. On the right is the mean and standard deviation of our Gaussian process — we don’t have any knowledge about the function so the best guess for our mean is in the middle of the real numbers i.e. \sigma_{21} & \sigma_{22}\\ For this, the prior of the GP needs to be specified. I’m well aware that things may be getting hard to follow at this point, so it’s worth reiterating what we’re actually trying to do here. \end{pmatrix} This is shown below, the training data are the blue points and the learnt function is the red line. And generating standard normals is something any decent mathematical programming language can do (incidently, there’s a very neat trick involved whereby uniform random variables are projected on to the CDF of a normal distribution, but I digress…) We need the equivalent way to express our multivariate normal distribution in terms of standard normals:$f_{*} \sim \mu + B\mathcal{N}{(0, I)}$, where B is the matrix such that$BB^T = \Sigma_{*}$, i.e. In the discrete case a probability distribution is just a list of possible outcomes and the chance of them occurring. Here’s an example of a very wiggly function: There’s a way to specify that smoothness: we use a covariance matrix to ensure that values that are close together in input space will produce output values that are close together. We consider the problem of learning predictive models from longitudinal data, consisting of irregularly repeated, sparse observations from a set of individuals over time. Sampling from a Gaussian process is like rolling a dice but each time you get a different function, and there are an infinite number of possible functions that could result. The dotted red line shows the mean output and the grey area shows 2 standard deviations from the mean. Now we’ll observe some data. \mu_2 \begin{pmatrix} It always amazes me how I can hear a statement uttered in the space of a few seconds about some aspect of machine learning that then takes me countless hours to understand. 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. Longitudinal Deep Kernel Gaussian Process Regression. This approach was elaborated in detail for the matrix-valued Gaussian processes and generalised to processes with 'heavier tails' like Student-t processes. 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. We focus on regression problems, where the goal is to learn a mapping from some input space X= Rn of n-dimensional vectors to an output space Y= R of real-valued targets. If we have the joint probability of variables $x_1$ and $x_2$ as follows: it is possible to get the conditional probability of one of the variables given the other, and this is how, in a GP, we can derive the posterior from the prior and our observations. You’d really like a curved line: instead of just 2 parameters $\theta_0$ and $\theta_1$ for the function $\hat{y} = \theta_0 + \theta_1x$ it looks like a quadratic function would do the trick, i.e. \sim \mathcal{N}{\left( $y = \theta_0 + \theta_1x + \epsilon$. Image Source: Gaussian Processes for Machine Learning, C. E. Rasmussen & C. K. I. Williams Recall that in the simple linear regression setting, we have a dependent variable y that we assume can be modeled as a function of an independent variable x, i.e. Our updated belief (posterior in Bayesian terms) looks something like this. The goal of this example is to learn this function using Gaussian processes. This means that after they are trained the cost of making predictions is dependent only on the number of parameters. Gaussian processes are computationally expensive. A key benefit is that the uncertainty of a fitted GP increases away from the training data — this is a direct consequence of GPs roots in probability and Bayesian inference. The updated Gaussian process is constrained to the possible functions that fit our training data —the mean of our function intercepts all training points and so does every sampled function. Now we can say that within that domain we’d like to sample functions that produce an output whose mean is, say, 0 and that are not too wiggly. GPstuff - Gaussian process models for Bayesian analysis 4.7. Wahba, 1990 and earlier references therein) correspond to Gaussian process prediction with 1 We call the hyperparameters as they correspond closely to hyperparameters in neural I'm looking into GP regression, but I'm getting some behaviour that I do not understand. Gaussian processes are another of these methods and their primary distinction is their relation to uncertainty. The code presented here borrows heavily from two main sources: Nando de Freitas’ UBC Machine Learning lectures (code for GPs can be found here) and the PMTK3 toolkit, which is the companion code to Kevin Murphy’s textbook Machine Learning: A Probabilistic Perspective. A Gaussian process is a distribution over functions fully specified by a mean and covariance function. However, (Rasmussen & Williams, 2006) provide an efficient algorithm (Algorithm $2.1$ in their textbook) for fitting and predicting with a Gaussian process regressor. The world of Gaussian processes will remain exciting for the foreseeable as research is being done to bring their probabilistic benefits to problems currently dominated by deep learning — sparse and minibatch Gaussian processes increase their scalability to large datasets while deep and convolutional Gaussian processes put high-dimensional and image data within reach. \begin{pmatrix} \begin{pmatrix} How the Bayesian approach works is by specifying a prior distribution, p(w), on the parameter, w, and relocating probabilities based on evidence (i.e.observed data) using Bayes’ Rule: The updated distri… Radial Basis Function kernel. I created my own YouTube algorithm (to stop me wasting time), All Machine Learning Algorithms You Should Know in 2021, 5 Reasons You Don’t Need to Learn Machine Learning, 7 Things I Learned during My First Big Project as an ML Engineer, Building Simulations in Python — A Step by Step Walkthrough. Gaussian processes (O’Hagan, 1978; Neal, 1997) have provided a promising non-parametric Bayesian approach to metric regression (Williams and Rasmussen, 1996) and classiﬁcation prob-lems (Williams and Barber, 1998).