Principal covariates regression

Principal covariates regression (PCovR) is a technique that recently has been put forward as a more flexible alternative to PLS regression [17]. Like CCA, RRR, PCR and PLS it extracts factors t from X that are used to estimate Y. These factors are chosen by a weighted least-squares criterion, viz. to fit both Y and X. By requiring the factors to be predictive not only for Y but also to represent X adequately, one introduces a preference towards the directions of the stable principal components of X. 

S. de Jong and H.A.L. Kiers, Principal covariates regression. Chemom. Intell. Lab. Syst., 14 (1992) 155-164. 

De Jong S, Kiers HAL, Principal covariates regression. Part 1. Theory, Chemometrics and Intelligent Laboratory Systems, 1992, 14, 155-164. 

We have seen that PLS regression (covariance criterion) forms a compromise between ordinary least squares regression (OLS, correlation criterion) and principal components regression (variance criterion). This has inspired Stone and Brooks [15] to devise a method in such a way that a continuum of models can be generated embracing OLS, PLS and PCR. To this end the PLS covariance criterion, cov(t,y) = s, s. r, is modified into a criterion T = r. (For... 

We will see that CLS and ILS calibration modelling have limited applicability, especially when dealing with complex situations, such as highly correlated predictors (spectra), presence of chemical or physical interferents (uncontrolled and undesired covariates that affect the measurements), less samples than variables, etc. More recently, methods such as principal components regression (PCR, Section 17.8) and partial least squares regression (PLS, Section 35.7) have been... 

What is special for the PLSR compared to, for example, the more well-known statistical technique of principal component regression (PCR) is that the y variable is used actively in determining how the regression factors ta,a= 1,2,..., A are computed from the spectra X. Each PLSR factor t<, is defined so that it describes as much as possible of the covariance between X and y remaining after the previous a - I factors have been estimated and subtracted. 

Problems with the inversion of the covariance matrix can be overcome by a preceding principal component analysis. Instead of (correlated) features a set of principal component scores is used as independent variables in the regression equation (principal component regression, PCR). Remember that PCA scores are uncorrelated by definition. 

Partial least squares regression A multivariate regression method that uses an algorithm that maximises the covariance between a number of independent variables (X-variables e.g., MIR absorbance values) and one or several dependent variables (Y-variables e.g., lake-water pH). This regression method summarises all X-variables containing the same information into latent variables, partial least square components (similar to principal components). 

Factor returns, hereafter called STB returns, are computed by regressing government bond s returns or LIBOR/swap key rate returns onto the shift, twist, and butterfly principal components. STB returns and other factor returns then go into the computation of the covariance matrix of all common factors. The shift, twist, and butterfly shapes are stable over time and only need to be reestimated periodically. 

Figure 6 Comparison of the regression between two x-variables and one y-variable by using latent variables from PCA, PCR and MLR. ftpci, direction of first principal component (maximum variance in x-space) Arlsi, direction of first PLS component (maximum covariance with y) Amlr. direction of steepest ascent in y-plane (maximum correlation coefficient with y). Three plots at the bottom show the different correlation coefficients r between the scores of these latent variables and y...

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