Polynomial multivariate least-squares regression for modeling nonlinear data applied to in-depth characterization of chromatographic resolution

Abstract

Well-established, linear multivariate calibration methods such as multivariate least-squares regression (MLR), principal component regression (PCR), or partial least squares (PLS) have two limitations: (i) measured data must be linearly related to the response variables and (ii) predictor variables x n=1,. . .,N cannot be coupled to each other. For evaluation of nonlinear data, however, these restrictions need to be overcome and thus polynomial multivariate least-squares regression (PMLR or "response surfaces") has been introduced here. PMLR is based on multivariate least squares but incorporates all combinations of predictor variables up to a user-selected polynomial order (e.g., x u i Áx v j including u or v=0). Because of the inclusion of such coupled terms and their powers, PMLR models are better adapted to model nonlinear data and can help to enhance the prediction step's accuracy and precision. PMLR has been based on MLR because it facilitates-unlike PCR or PLS-a physical and chemical interpretation of the predictors. Hence, the origins and the relevance of nonlinear and/or coupled predictors can be investigated. The details of the PMLR algorithm and its implementation are presented along with a method for model optimization utilizing gradients of response surfaces.Newly developed PMLR models up to quintic order have been applied to predict a chromatograph's peak resolution as a function of six-instrument parameters. It has been demonstrated that PMLR is better capable than MLR and PCR to describe these nonlinear and coupled instrument parameters. In addition, the novel software tool has been utilized for model optimization to determine instrument parameters, which result in the best chromatographic resolution.

Modeling nutrient impacts on microalgae cells via image analyses

“…Again, artifacts in the images from which these histograms had been derived required an additional background polynomial of order M R . Equation (3) has been introduced here to concurrently model cell shape R and concentration information and thus to gain insights into the concentration dependency of m(c).…”

“…Similarly, σ ( c ) = γ σ , 0 + γ σ , 1 · c 1 + γ σ , 2 · c 2 + ⋯ describes the concentration dependency of the size distribution's width and A ( c ) = γ A , 0 + γ A , 1 · c 1 + γ A , 2 · c 2 + ⋯ the concentration dependency of the normal distributions' maximum height. Altogether, the model function comprises [3 + (1 + M S )] · W model parameters γ with W = W ( M S , P ) . These need to be derived via nonlinear least‐squares regression.…”

A self‐guided search for good local minima of the sum‐of‐squared‐error in nonlinear least squares regression

“…(ii) The evaluation of (1) is performed on CUDA-enabled [20] video cards (GPUs). Both parallel-computation options can be easily realized in C-code 4 For the calculation of W, refer to [11]. Figure 5.…”

“…This is feasible because an explicit data model is rarely required for quantitative predictions or for sample classification. In soft-modeling procedures, nonlinear relations between measured data and the wanted chemical information can be empirically approximated for instance by a piecewise linear model [9] or by multivariate, polynomial models [10][11][12]. While successful for many quantitative calibrations, due to the absence of an interpretable model function, soft modeling cannot gain insights into the underlying chemical processes.…”

A Color Gamut Description Algorithm for Liquid Crystal Displays inCIELABSpace

“…There are actually several color conversion models, such as 3D interpolation [12–14], polynomial regression [15, 16], and neural network [17, 18]. In the paper, the polynomial regression method is used because of the model's precision and the quantity of sample colors, and the utilized polynomials are expressed as below: $\begin{array}{c}P\left(R,G,B\right)=c_0+c_{1}R+c_{2}G+c_{3}B+c_{4}RG+c_{5}GB\\ \hspace{1em}+c_{6}RB+c_7{R}^2+c_8{G}^2+c_9{B}^2+c_{10}RGB\\ \hspace{1em}+c_{11}{R}^{2}G+c_{12}{R}^{2}B+c_{13}{G}^{2}R+c_{14}{G}^{2}B+c_{15}{B}^{2}R\\ \hspace{1em}+c_{16}{B}^{2}G+c_{17}{R}^3+c_{18}{G}^3+c_{19}{B}^3,\end{array}$ where c i  ( i = 0,1,…, 19) are the coefficients which can be determined by the least square method [19, 20]. Take one constant-lightness boundary, for example; it is depicted in RGB and CIELAB spaces respectively, as shown in Figure 5.…”