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Gaussian Process Regression Models

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Gaussian process regression (GPR) models are nonparametric kernel-based probabilistic models. You can train a GPR model using the fitrgp function.

Consider the training set $$\{(x_i,y_i);i=1,2,\mathrm{...},n\}$$, where $$x_i\in {ℝ}^d$$ and $$y_i\in ℝ$$, drawn from an unknown distribution. A GPR model addresses the question of predicting the value of a response variable $$y_{new}$$, given the new input vector $$x_{new}$$, and the training data. A linear regression model is of the form

where $$\epsilon \sim N(0,{\sigma }^2\text{)}$$. The error variance σ² and the coefficients β are estimated from the data. A GPR model explains the response by introducing latent variables, $$f\left(x_i\right),i=1,2,\mathrm{...},n$$, from a Gaussian process (GP), and explicit basis functions, h. The covariance function of the latent variables captures the smoothness of the response and basis functions project the inputs $$x$$ into a p-dimensional feature space.

A GP is a set of random variables, such that any finite number of them have a joint Gaussian distribution. If $$\left\{f\left(x\right),x\in {ℝ}^d\right\}$$ is a GP, then given n observations $$x_1,x_2,\mathrm{...},x_n$$, the joint distribution of the random variables $$f(x_1),f(x_2),\mathrm{...},f(x_n)$$ is Gaussian. A GP is defined by its mean function $$m\left(x\right)$$ and covariance function, $$k\left(x,x^{\prime }\right)$$. That is, if $$\left\{f\left(x\right),x\in {ℝ}^d\right\}$$ is a Gaussian process, then $$E\left(f\left(x\right)\right)=m\left(x\right)$$ and $$Cov\left[f\left(x\right),f\left(x^{\prime }\right)\right]=E\left[\left\{f\left(x\right)-m\left(x\right)\right\}\left\{f\left(x^{\prime }\right)-m\left(x^{\prime }\right)\right\}\right]=k\left(x,x^{\prime }\right).$$

Now consider the following model.

where $$f\left(x\right)~GP\left(0,k\left(x,x^{\prime }\right)\right)$$, that is f(x) are from a zero mean GP with covariance function, $$k\left(x,x^{\prime }\right)$$. h(x) are a set of basis functions that transform the original feature vector x in R^d into a new feature vector h(x) in R^p. β is a p-by-1 vector of basis function coefficients. This model represents a GPR model. An instance of response y can be modeled as

Hence, a GPR model is a probabilistic model. There is a latent variable f(x_i) introduced for each observation $$x_i$$, which makes the GPR model nonparametric. In vector form, this model is equivalent to

where

$$X=\left(\begin{array}{c}{x}_1^T\\ x_2^T\\ ⋮\\ x_n^T\end{array}\right),y=\left(\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right),H=\left(\begin{array}{c}h\left(x_1^T\right)\\ h\left(x_2^T\right)\\ ⋮\\ h\left(x_n^T\right)\end{array}\right),f=\left(\begin{array}{c}f\left(x_1\right)\\ f\left(x_2\right)\\ ⋮\\ f\left(x_n\right)\end{array}\right).$$

The joint distribution of latent variables $$f\left(x_1\right),f\left(x_2\right),\mathrm{...},f\left(x_n\right)$$ in the GPR model is as follows:

close to a linear regression model, where $$K\left(X,X\right)$$ looks as follows:

$$K\left(X,X\right)=\left(\begin{array}{cccc}k\left(x_1,x_1\right)& k\left(x_1,x_2\right)& \cdots & k\left(x_1,x_n\right)\\ k\left(x_2,x_1\right)& k\left(x_2,x_2\right)& \cdots & k\left(x_2,x_n\right)\\ ⋮& ⋮& ⋮& ⋮\\ k\left(x_n,x_1\right)& k\left(x_n,x_2\right)& \cdots & k\left(x_n,x_n\right)\end{array}\right).$$

The covariance function $$k\left(x,x^{\prime }\right)$$ is usually parameterized by a set of kernel parameters or hyperparameters, $$\theta$$. Often $$k\left(x,x^{\prime }\right)$$ is written as $$k\left(x,x^{\prime }\theta \right)$$ to explicitly indicate the dependence on $$\theta$$.

fitrgp estimates the basis function coefficients, $$\beta$$, the noise variance, $${\sigma }^2$$, and the hyperparameters,$$\theta$$, of the kernel function from the data while training the GPR model. You can specify the basis function, the kernel (covariance) function, and the initial values for the parameters.

Because a GPR model is probabilistic, it is possible to compute the prediction intervals using the trained model (see predict and resubPredict).

You can also compute the regression error using the trained GPR model (see loss and resubLoss).

Compare Prediction Intervals of GPR Models

This example fits GPR models to a noise-free data set and a noisy data set. The example compares the predicted responses and prediction intervals of the two fitted GPR models.

Generate two observation data sets from the function $\mathit{g}\left(\mathit{x}\right)=\mathit{x}\cdot \sin \left(\mathit{x}\right)$.

The values in y_observed1 are noise free, and the values in y_observed2 include some random noise.

Fit GPR models to the observed data sets.

Compute the predicted responses and 95% prediction intervals using the fitted models.

Resize a figure to display two plots in one figure.

Create a 1-by-2 tiled chart layout.

For each tile, draw a scatter plot of observed data points and a function plot of $\mathit{x}\cdot \sin \left(\mathit{x}\right)$. Then add a plot of GP predicted responses and a patch of prediction intervals.

When the observations are noise free, the predicted responses of the GPR fit cross the observations. The standard deviation of the predicted response is almost zero. Therefore, the prediction intervals are very narrow. When observations include noise, the predicted responses do not cross the observations, and the prediction intervals become wide.

References

[1] Rasmussen, C. E. and C. K. I. Williams. Gaussian Processes for Machine Learning. MIT Press. Cambridge, Massachusetts, 2006.

See Also

fitrgp predict RegressionGP

Related Topics

From CFD-Wiki

Introduction

The K-epsilon model is one of the most common turbulence models, although it just doesn't perform well in cases of large adverse pressure gradients (Reference 4). It is a two equation model, that means, it includes two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy.

The first transported variable is turbulent kinetic energy, . The second transported variable in this case is the turbulent dissipation, . It is the variable that determines the scale of the turbulence, whereas the first variable, , determines the energy in the turbulence.

There are two major formulations of K-epsilon models (see References 2 and 3). That of Launder and Sharma is typically called the 'Standard' K-epsilon Model. The original impetus for the K-epsilon model was to improve the mixing-length model, as well as to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows.

As described in Reference 1, the K-epsilon model has been shown to be useful for free-shear layer flows with relatively small pressure gradients. Similarly, for wall-bounded and internal flows, the model gives good results only in cases where mean pressure gradients are small; accuracy has been shown experimentally to be reduced for flows containing large adverse pressure gradients. One might infer then, that the K-epsilon model would be an inappropriate choice for problems such as inlets and compressors.

To calculate boundary conditions for these models see turbulence free-stream boundary conditions.

Usual K-epsilon models

Miscellaneous

References

[1] Bardina, J.E., Huang, P.G., Coakley, T.J. (1997), 'Turbulence Modeling Validation, Testing, and Development', NASA Technical Memorandum 110446.

[2] Jones, W. P., and Launder, B. E. (1972), 'The Prediction of Laminarization with a Two-Equation Model of Turbulence', International Journal of Heat and Mass Transfer, vol. 15, 1972, pp. 301-314.

[3] Launder, B. E., and Sharma, B. I. (1974), 'Application of the Energy Dissipation Model of Turbulence to the Calculation of Flow Near a Spinning Disc', Letters in Heat and Mass Transfer, vol. 1, no. 2, pp. 131-138.

[4] Wilcox, David C (1998). 'Turbulence Modeling for CFD'. Second edition. Anaheim: DCW Industries, 1998. pp. 174.

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