Application of a Multiple Regression Model for the Simultaneous Measurement of Refractive Index and Temperature Based on an Interferometric Optical System
Algorithm 1
Stage 1: to propose the first functions of the model, a basis of power functions, , is used. Each time a power function is added, the next steps are carried out:
(a) A set of values of is proposed
(b) For an value, the data matrix that contains the explanatory variables values is transformed, by using a power function (), into .
(c) The data matrix is standardized by using , where is the mean of the values of the -th column. Here, it is important to mention the matrix = (see Eq. (3))
(d) The -fold crossvalidation technique is applied in order to find the optimal value of that will be used to find the regression coefficients of the equation (Eq. (2)). Afterwards, with the estimated values obtained with the regression equation (Eq. (5)), the related to the value of is calculated
(e) Steps (b)–(d) are repeated for the entire set of values of proposed in the step (a)
(f) The exponent of the power function will be the value of for which the value of is the smallest ()
It is important to note that with respect to step (a), for the first power function, the dimensions of the transformed dataset are unchanged, , where is the data number and is the variable number. After this first function, features are added to the matrix, so its dimensions increases. Besides, when the matrix is transformed with the next added function, the optimal value of found for the previous function is fixed, and the steps to find the optimal value of of the added function are performed. Besides, each time a power function is added to transform the data matrix, a value of is calculated. Additionally, the total number of power functions, , to be used in the model will be chosen taking into account a criterion related to all the values. In this stage, the transformed matrix is expressed as
Stage 2: It is started with a transformed dataset with a basis of power functions proposed in stage 1. The purpose of this stage is to add features based on a function other than the power function that improves the estimation of the values of the response variable. In this way, each one of the remaining functions () is added separately to the proposed model in order to find the best next function. Here, each time a function is added, the steps (a)–(e) of the stage 1 are implemented. As step (f), the function that is chosen to be added to the model proposed in stage 1 will be the one with the smallest (). Here, the transformed matrix is expressed as
Stage 3: It is started with a transformed dataset with a basis of functions proposed in stages 1 and 2. The purpose of this stage is to add a function different from the power function and to the one proposed in stage 2 that improves the estimation of the values of the response variable. Here, each one of the remaining 3 functions is added separately to the proposed model in stage 2, and the steps (a)–(e) and step (f) described in stage 2 are again implemented. In this stage, the transformed matrix is expressed as
Further stages: to add another function, the steps of stage 3 are repeated. The total number of stages that can be carried out is five, and this implies that by using the power functions of stage 1 and the other four functions (), the transformed matrix can be expressed as Finally, the number of functions that will be taken into account is determined based on the a criterion of the smallest obtained with the added functions in each one of the stages
