Polynomial regression over-fitting

Insertion of the ambient temperatures in Eq. 1 results R0 in and a. Since the ambient temperature is calculated from a polynomial funtion (eq. 2) it is necessary to determine the polynomial coefficients by a least square fit regression over the ambient temperature range 0-50°C. Together with S0,. Si, b, and c, these values are written by the calibration computer to the non volatile memory of the microcontroller. As a calibration check two additional blackbody readings are performed at a third ambient temperature (see Fig. 3.48). 

The true model parameters (ft, ftj(...) are partial derivatives of the response function / and cannot be measured directly. It is, however, possible to otain estimates, ft, bV], bVl, of these parameters by multiple regression methods in which the polynomial model is fitted to known experimental results obtained by varying the settings of xr. These variations will then define an experimental design and are conveniently displayed as a design matrix, D, in which the rows describe the settings in the individual experiments and the columns describe the variations of the experimental variables over the series of experiments. 

Another possible source of variation within slides is known as spatial bias. This describes any effect that influences the intensity of a spot based on its position on a slide, such as variations in hybridization efficiency over the slide surface, variations in slide flatness, and scanner focal depth across a slide as well as print-tip-dependent effects. Currently, two methods are used to deal with these The first is to divide the slide into subsections, typically subgrids from individual pins, and normalize each subset separately. Or, when the variation is not restricted to a regular subgrid, such as with slide flatness problems, a two-dimensional Lowess technique can be used (see the Lorenze Wernisch website in the appendix). This is a more complex form of the Lowess regression, where a polynomial surface is fitted to the data based on slide coordinates and the surface is then smoothed. The procedure can be run on either the log ratio, the signal intensity, or the background intensities. 

Usually, the corrections Tc are smooth over the whole range. They may then be fitted as a function of Tctd by polynomial regression. Otherwise, Tc is linearly interpolated from the calibration table. 

Table 14 shows results obtained for every formula development according to MODDE 4.0 software. The collected experimental data were fitted by a multilinear regression (MLR) model with which several responses can be dealt with simultaneously to provide an overview of how all the factors affect all the responses. The responses of the model, R2 and Q2 values, were over 0.99 and 0.93 for tm% and 0.98 and 0.89 for /30%, respectively, implying that the data fitted well with the model. Here, R2 is the fraction of the variation of the response that can be modeled and Q2 is the fraction of the variation of the response that can be predicted by the model. The relationship between a response y and the variables xh xh... can be described by the polynomial ... 

The parabola (second degree polynomial) can only be fitted over a limited range (Fig. 15.11). For case a, the parabola is fitted between the log dilutions of 3 and 4.5 (regression formula in Fig. 15.11), whereas in case b the regression curve is fitted between the log dilutions of 1 and 5 (regression formula A = 0.6754 — 0.4417 (log D) -t- 0.0599 (log oy). Cubic, quartic, or quintic polynomials were calculated by the methods discussed in great detail by Snedecor and Cochran (1967). 

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