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Students T Processes

Student T Processes (STP) can be viewed as a generalization of Gaussian Processes, in GP models we use the multivariate normal distribution to model noisy observations of an unknown function. Likewise for STP models, we employ the multivariate student t distribution. Formally a student t process is a stochastic process where the finite dimensional distribution is multivariate t.

\begin{align} \mathbf{y} & \in \mathbb{R}^n \\ \mathbf{y} & \sim MVT_{n}(\nu, \phi, K) \\ p(\mathbf{y}) & = \frac{\Gamma(\frac{\nu + n}{2})}{(\nu \pi)^{n/2} \Gamma(\nu/2)} |K|^{-1/2} \\ & \times (1 + \frac{(\mathbf{y} - \phi)^T K^{-1} (\mathbf{y} - \phi)}{\nu})^{-\frac{\nu + n}{2}} \end{align}

It is known that as \nu \rightarrow \infty, the MVT_{n}(\nu, \phi, K) tends towards the multivariate normal distribution \mathcal{N}_{n}(\phi, K).

Regression with Student T Processes

The regression formulation for STP models is identical to the GP regression framework, to summarize the posterior predictive distribution takes the following form.

Suppose \mathbf{t} \sim MVT_{n_{tr} + n_t}(\nu, \mathbf{0}, K) is the process producing the data. Let [\mathbf{f_*}]_{n_{t} \times 1} represent the values of the function on the test inputs and [\mathbf{y}]_{n_{tr} \times 1} represent noisy observations made on the training data points.

\begin{align} & \mathbf{f_*}|X,\mathbf{y},X_* \sim MVT_{\nu + n_{tr}}(\mathbf{\bar{f_*}}, \frac{\nu + \beta - 2}{\nu + n_{tr} - 2} \times cov(\mathbf{f_*})) \label{eq:posterior}\\ & \beta = \mathbf{y}^T K^{-1} \mathbf{y} \\ & \mathbf{\bar{f_*}} \overset{\triangle}{=} \mathbb{E}[\mathbf{f_*}|X,y,X_*] = K(X_*,X)[K(X,X) + \sigma^{2}_n \it{I}]^{-1} \mathbf{y} \label{eq:posterior:mean} \\ & cov(\mathbf{f_*}) = K(X_*,X_*) - K(X_*,X)[K(X,X) + \sigma^{2}_n \it{I}]^{-1}K(X,X_*) \end{align}

STP models for a single output

For univariate GP models (single output), use the StudentTRegressionModel class (an extension of AbstractSTPRegressionModel). To construct a STP regression model you would need:

  • The degrees of freedom \nu
  • Kernel/covariance instance to model correlation between values of the latent function at each pair of input features.
  • Kernel instance to model the correlation of the additive noise, generally the DiracKernel (white noise) is used.
  • Training data
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val trainingdata: Stream[(DenseVector[Double], Double)] = ...
val num_features = trainingdata.head._1.length

// Create an implicit vector field for the creation of the stationary
// radial basis function kernel
implicit val field = VectorField(num_features)

val kernel = new RBFKernel(2.5)
val noiseKernel = new DiracKernel(1.5)
val model = new StudentTRegression(1.5, kernel, noiseKernel, trainingData)

STP models for Multiple Outputs

You can use the MOStudentTRegression[I] class to create multi-output GP models.

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val trainingdata: Stream[(DenseVector[Double], DenseVector[Double])] = ...

val model = new MOStudentTRegression[DenseVector[Double]](
    sos_kernel, sos_noise, trainingdata,
    trainingdata.length, trainingdata.head._2.length)

Tip

Working with multi-output Student T models is similar to multi-output GP models. We need to create a kernel function over the combined index set (DenseVector[Double], Int). This can be done using the sum of separable kernel idea.

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val linearK = new PolynomialKernel(2, 1.0)
val tKernel = new TStudentKernel(0.2)
val d = new DiracKernel(0.037)

val mixedEffects = new MixedEffectRegularizer(0.5)
val coRegCauchyMatrix = new CoRegCauchyKernel(10.0)
val coRegDiracMatrix = new CoRegDiracKernel

val sos_kernel: CompositeCovariance[(DenseVector[Double], Int)] =
    (linearK :* mixedEffects)  + (tKernel :* coRegCauchyMatrix)

val sos_noise: CompositeCovariance[(DenseVector[Double], Int)] =
    d :* coRegDiracMatrix

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