Gaussian calibration

Introduction

In this page, we present the method used in the GaussianLinearCalibration and GaussianNonLinearCalibration classes. Gaussian calibration is a particular case of Bayesian calibration (see Bayesian calibration) where the prior has a Gaussian distribution and the observation error has a Gaussian distribution. In this case, simplifications occur which lead to an efficient algorithm that requires fewer model evaluations compared to Monte-Carlo Markov Chains (MCMC).

We consider a computer model \vect{h} (i.e. a deterministic function) to calibrate:

\vect{z} = \vect{h}(\vect{x}, \vect{\theta}),

where

  • \vect{x} \in \Rset^{d_x} is the input vector;

  • \vect{z} \in \Rset^{d_z} is the output vector;

  • \vect{\theta} \in \Rset^{d_h} are the unknown parameters of \vect{h} to calibrate.

Let n \in \Nset be the number of observations. The standard hypothesis of the probabilistic calibration is:

\vect{y}^i = \vect{z}^i + \vect{\varepsilon}^i,

for i=1,...,n where \vect{\varepsilon}^i is a random measurement error.

The goal of Gaussian calibration is to estimate \vect{\theta}, based on observations of n inputs (\vect{x}^1, \ldots, \vect{x}^n) and the associated n observations of the output (\vect{y}^1, \ldots, \vect{y}^n). In other words, the calibration process reduces the discrepancy between the observations (\vect{y}^1, \ldots, \vect{y}^n) and the predictions \vect{h}(\vect{\theta}). Given that (\vect{y}^1, \ldots, \vect{y}^n) are realizations of a random variable, the estimate of \vect{\theta}, denoted by \hat{\vect{\theta}}, is also a random variable. Hence, the secondary goal of calibration is to estimate the distribution of \hat{\vect{\theta}} representing the uncertainty of the calibration process.

In the remaining of this section, the input \vect{x} is not involved anymore in the equations. This is why we simplify the equation into:

\vect{z} = \vect{h}(\vect{\theta}).

Bayesian calibration

The Bayesian calibration framework is based on two hypotheses. The first hypothesis is that the parameter \vect{\theta} has a known distribution, called the prior distribution, and denoted by p(\vect{\theta}). The second hypothesis is that the output observations (\vect{y}^1, \ldots, \vect{y}^n) are sampled from a known conditional distribution denoted by p(\vect{y} | \vect{\theta}). For any \vect{y}\in\Rset^{d_z} such that p(\vect{y})>0, the Bayes theorem implies that the conditional distribution of \vect{\theta} given \vect{y} is:

p(\vect{\theta} | \vect{y}) = \frac{p(\vect{y} | \vect{\theta}) p(\vect{\theta})}{p(\vect{y})}

for any \vect{\theta}\in\Rset^{d_h}. The denominator of the previous Bayes fraction is independent of \vect{\theta}, so that the posterior distribution is proportional to the numerator:

p(\vect{\theta} | \vect{y}) \propto p(\vect{y} | \vect{\theta}) p(\vect{\theta}).

for any \vect{\theta}\in\Rset^{d_h}.

In Gaussian calibration, the two previous distributions are assumed to be gaussian. More precisely, we make the hypothesis that the parameter \vect{\theta} follows the normal distribution:

\vect{\theta} \sim \mathcal{N}(\vect{\mu}, B),

where \vect{\mu}\in\Rset^{d_h} is the mean of the gaussian prior distribution, which is named the background and B\in\Rset^{d_h \times d_h} is the covariance matrix of the parameter. Secondly, we make the hypothesis that the output observations follow the conditional normal distribution:

\vect{y} | \vect{\theta} \sim \mathcal{N}(\vect{h}(\vect{\theta}), \; R),

where R\in\Rset^{d_z \times d_z} is the covariance matrix of the output observations.

Posterior distribution

Let \|\cdot\|_B be the Mahalanobis distance associated with the matrix B:

\|\vect{\theta} - \vect{\mu} \|^2_B = (\vect{\theta} - \vect{\mu} )^T B^{-1} (\vect{\theta} - \vect{\mu} ),

for any \vect{\theta},\vect{\mu} \in \Rset^{d_h}. Denote by \|\cdot\|_R the Mahalanobis distance associated with the matrix R:

\|\vect{y} - h(\vect{\theta})\|^2_R = (\vect{y} - h(\vect{\theta}))^T R^{-1} (\vect{y} - h(\vect{\theta})).

for any \vect{\theta} \in \Rset^{d_h} and any \vect{y} \in \Rset^{d_z}. Therefore, the posterior distribution of \vect{\theta} given the observations \vect{y} is:

p(\vect{\theta}|\vect{y}) \propto \exp\left( -\frac{1}{2} \left( \|\vect{y}-H(\vect{\theta})\|^2_R + \|\vect{\theta}-\vect{\mu} \|^2_B \right) \right)

for any \vect{\theta}\in\Rset^{d_h}.

MAP estimator

The maximum of the posterior distribution of \vect{\theta} given the observations \vect{y} is reached at :

\hat{\vect{\theta}}_{MAP} = \argmin_{\vect{\theta}\in\Rset^{d_h}} \frac{1}{2} \left( \|\vect{y} - H(\vect{\theta})\|^2_R + \|\vect{\theta}-\vect{\mu} \|^2_B \right).

It is called the maximum a posteriori posterior estimator or MAP estimator.

Regularity of solutions of the Gaussian Calibration

The gaussian calibration is a tradeoff, so that the second expression acts as a spring which pulls the parameter \vect{\theta} closer to the background \vect{\mu} (depending on the “spring constant” B) meanwhile getting as close a possible to the observations. Depending on the matrix B, the computation may have better regularity properties than the plain non linear least squares problem.

Non Linear Gaussian Calibration : 3DVAR

The cost function of the gaussian nonlinear calibration problem is :

c(\vect{\theta}) = \frac{1}{2}\|\vect{y} - h(\vect{\theta})\|^2_R + \frac{1}{2}\|\vect{\theta}-\vect{\mu} \|^2_B

for any \vect{\theta}\in\Rset^{d_h}. The goal of non-linear Gaussian calibration is to find the value of \vect{\theta} which minimizes the cost function C. In general, this involves using a nonlinear unconstrained optimization solver. Let J \in \Rset^{n \times d_h} be the Jacobian matrix made of the partial derivatives of \vect{h} with respect to \vect{\theta}:

J(\vect{\theta}) = \frac{\partial \vect{h}}{\partial \vect{\theta}}.

The Jacobian matrix of the cost function C can be expressed depending on the matrices R and B and the Jacobian matrix of the function h:

\frac{d }{d\vect{\theta}} c(\vect{\theta}) = B^{-1} (\vect{\theta}-\vect{\mu}) + J(\vect{\theta})^T R^{-1} (H(\vect{\theta}) - \vect{y})

for any \vect{\theta}\in\Rset^{d_h}. The Hessian matrix of the cost function is:

\frac{d^2 }{d\vect{\theta}^2} c(\vect{\theta}) = B^{-1} + J(\vect{\theta})^T R^{-1} J(\vect{\theta})

for any \vect{\theta}\in\Rset^{d_h}.

If the covariance matrix B is positive definite, then the Hessian matrix of the cost function is positive definite. Under this hypothesis, the solution of the nonlinear Gaussian calibration is unique.

Solving the Non Linear Gaussian Calibration Problem

The implementation of the resolution of the Gaussian non linear calibration problem involves the Cholesky decomposition of the covariance matrices B and R. This allows one to transform the sum of two Mahalanobis distances into a single Euclidean norm. This leads to a classical non linear least squares problem.

Linear Gaussian Calibration

In Gaussian linear calibration, we assume that the parametric function is linear with respect to the parameters. This method is known as the Kalman filter in data assimilation. We make the hypothesis that h is linear with respect to \vect{\theta}, i.e., for any \vect{\theta}\in\Rset^{d_h}, we have:

h(\vect{\theta}) = h(\vect{\mu}) + J(\vect{\theta}-\vect{\mu} ),

where J is the constant Jacobian matrix of h.

Let A be the matrix:

A^{-1} = B^{-1} + J^T R^{-1} J.

We denote by K the Kalman matrix:

K = A J^T R^{-1}.

The maximum of the posterior distribution of \vect{\theta} given the observations \vect{y} is:

\hat{\vect{\theta}}_{MAP} = \vect{\mu} + K (\vect{y} - h(\vect{\mu})).

It can be proved that:

p(\vect{\theta} | \vect{y}) \propto \exp\left(\frac{1}{2} (\vect{\theta} - \hat{\vect{\theta}})^T A^{-1} (\vect{\theta} - \hat{\vect{\theta}}) \right)

for any \vect{\theta}\in\Rset^{d_h}. This implies:

\vect{\theta} \; | \; \vect{y} \sim \mathcal{N}\left(\hat{\vect{\theta}}_{MAP}, \; A\right)

Bias of Linear Gaussian Calibration

Assume that there is a true value of the parameter \vect{\theta}^\star that generated the output observations \vect{y}. Consider the MAP estimator \hat{\vect{\theta}}_{MAP} as an estimator of \vect{\theta}^\star. This estimator is a random variable which depends on the random sample of output observations. The MAP estimator is biased and the bias is:

\operatorname{Bias}\left(\hat{\vect{\theta}}_{MAP}\right) = (KJ - \operatorname{I})\left(\vect{\theta}^\star - \vect{\mu}\right).

This bias is introduced as a consequence of the regularization properties of Gaussian calibration. Due to the trade-off between bias and variance of the estimator of the parameters, reducing the variance comes at the price of an increased bias.