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RESEARCH PAPERS

# Bayesian and Sensitivity Analysis Approaches to Modeling the Direct Solar Irradiance

[+] Author and Article Information
Philippe Lauret

Faculté des Sciences et Technologies, Laboratoire de Génie Industriel, University of Reunion Island, Equipe Génie Civil et Thermique de l’Habitat, BP 7151, 15 avenue René Cassin, 97 715 Saint-Denis, Francelauret@univ-reunion.fr

Mathieu David

Faculté des Sciences et Technologies, Laboratoire de Génie Industriel, University of Reunion Island, Equipe Génie Civil et Thermique de l’Habitat, BP 7151, 15 avenue René Cassin, 97 715 Saint-Denis, Francemathieu.david@univ-reunion.fr

Eric Fock

Faculté des Sciences et Technologies, Laboratoire de Génie Industriel, University of Reunion Island, Equipe Génie Civil et Thermique de l’Habitat, BP 7151, 15 avenue René Cassin, 97 715 Saint-Denis, Franceeric.fock@univ-reunion.fr

Alain Bastide

Faculté des Sciences et Technologies, Laboratoire de Génie Industriel, University of Reunion Island, Equipe Génie Civil et Thermique de l’Habitat, BP 7151, 15 avenue René Cassin, 97 715 Saint-Denis, Francealain.bastide@univ-reunion.fr

Carine Riviere

Faculté des Sciences et Technologies, Laboratoire de Génie Industriel, University of Reunion Island, Equipe Génie Civil et Thermique de l’Habitat, BP 7151, 15 avenue René Cassin, 97 715 Saint-Denis, Francecarine.riviere@univ-reunion.fr

As an illustration, consider a polynomial model whose complexity is controlled by the number of coefficients. A too low-order polynomial will be unable to capture the underlying trends in the data while a too high-order polynomial will model the noise on the data.

In order to avoid any confusion with the hyperparameter $α$, we chose to name this variable $ϕ$ instead of the classical Greek letter $α$.

Note, however, that Bayesian inference may be also implemented by Monte Carlo sampling through the use of sophisticated methods like numerical techniques such as Markov Chain Monte Carlo (MCMC).

J. Sol. Energy Eng 128(3), 394-405 (Jan 19, 2006) (12 pages) doi:10.1115/1.2210495 History: Received August 23, 2005; Revised January 19, 2006

## Abstract

In this paper, emphasis is put on the design of a neural network (NN) to model the direct solar irradiance. Since, unfortunately, a neural network is not a statistician-in-a-box, building a NN for a particular problem is a nontrivial task. As a consequence, we argue that in order to properly model the direct solar irradiance, a systematic methodology must be employed. For this purpose, we propose a two-step approach to building the NN model. The first step deals with a probabilistic interpretation of the NN learning by using Bayesian techniques. The Bayesian approach to modeling offers significant advantages over the classical NN learning process. Among others, one can cite (i) automatic complexity control of the NN using all the available data and (ii) selection of the most important input variables. The second step consists of using a new sensitivity analysis-based pruning method in order to infer the optimal NN structure. We show that the combination of the two approaches makes the practical implementation of the Bayesian techniques more reliable.

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## Figures

Figure 1

Sketch of a MLP with d inputs and h hidden units. Note that biases are not represented and, in our case, d=8.

Figure 2

The EFAST method. The total variance is apportioned to the various input factors, as shown by the pie diagram.

Figure 3

The EFAST method applied to pruning of input units

Figure 4

Measured versus estimated direct irradiance (model 1): (a) training set and (b) test set. The solid line denotes the line 1:1 of the perfect fit.

Figure 5

Bayesian NN model 2: (a) training set and (b) test set

Figure 6

ARD technique: Relevance of the eight input variables. Variance of the fourth variable is equal to 3.41×10−13.

Figure 7

EFAST method: Sensitivity indices of the eight input variables

Figure 8

Bayesian NN model 3: (a) training set and (b) test set

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