Straight-line model

To determine the reaction order we plot ln(%p-methoxyphenylacetylene) versus time for a first-order reaction, and (%p-methoxyphenylacetylene) versus time for a second-order reaction (Figure A5.1). Because the straight-line for the first-order plot fits the data nicely, we conclude that the reaction is first-order in p-methoxyphenylacetylene. Note that when plotted using the equation for a second-order reaction, the data show curvature that does not fit the straight-line model. 

Before we introduce the Kalman filter, we reformulate the least-squares algorithm discussed in Chapter 8 in a recursive way. By way of illustration, we consider a simple straight line model which is estimated by recursive regression. Firstly, the measurement model has to be specified, which describes the relationship between the independent variable x, e.g., the concentrations of a series of standard solutions, and the dependent variable, y, the measured response. If we assume a straight line model, any response is described by ... 

After each new observation, the estimates of the model parameters are updated (= new estimate of the parameters). In all equations below we treat the general case of a measurement model with p parameters. For the straight line model p = 2. An estimate of the parameters b based ony - 1 measurements is indicated by b(/ - 1). Let us assume that the parameters are recursively estimated and that an estimate h(j - 1) of the model parameters is available from y - 1 measurements. The next measurement y(j) is then performed at x(j), followed by the updating of the model parameters to b(/). 

In a well-behaved calibration model, residuals will have a Normal (i.e., Gaussian) distribution. In fact, as we have previously discussed, least-squares regression analysis is also a Maximum Likelihood method, but only when the errors are Normally distributed. If the data does not follow the straight line model, then there will be an excessive number of residuals with too-large values, and the residuals will then not follow the Normal distribution. It follows, then, that a test for Normality of residuals will also detect nonlinearity. 

The straight line model constrained to pass through the origin... 

Figure 5.1 shows the unconstrained straight line model y, = Po + PiX, passing... 

For the data used in this section, the unconstrained straight line model y, = Po + Pi- i, + "i, fits exactly. Although the constrained model y, = p,x, + r, does not fit exactly, it does explain the observed data better than the simple model yi, = Po + r,/ (sj values of 1/5 and 2, respectively). Thus, the response of the system would seem to increase as the level of the factor x, increases. 

Each element of the X X) matrix is a summation of products (see Appendix A). A common algebraic representation of the X X) matrix for the straight-line model y, - Po + PiXi,. + r, is... 

This is the straight line model constrained to pass through the origin (see Section 5.3). The fitted model is... 

A common algebraic representation of the (X X) matrix for the straight-line model >U = 0 PlXU rli S... 

This model is called simple since there is only one independent variable (X), and linear because it is linear in the parameters. This means that no parameters appear as exponents or are multiplied or divided by another parameter (i.e., Xp). The term linear can cause some confusion in nonstatistical literature since second or higher order polynomial models are also called linear models (but in the terminology of linear in the parameters and not as a straight line). To avoid any confusion, here the term straight-line model is used for the above-described simple linear model. For an in-depth discussion of linear models, the reader is referred to an appropriate statistical manual [8]. 

The residual standard error of a straight-line model is given by Sy =... 

A straight-line model is the most used, but also the most misused, model in analytical chemistry. The analytical chemist should check five basic assumptions during method validation before deciding whether to use a straight-line regression model for calibration purposes. These five assumptions are described in detail by MacTaggart and Farwell [6] and basically are linearity, error-free independent variable, random and homogeneous error, uncorrelated errors, and normal distribution of the error. The evaluation of these assumptions and the remedial actions are discussed hereafter. 

The linearity of (a part of) the range should be evaluated to check the appropriateness of the straight-line model. This can be achieved by a graphical evaluation of the residual plots or by using statistical tests. It is strongly recommended to use the residual plots in addition to the statistical tests. Mostly, the lack-of-fit test and Mandel s fitting test are used to evaluate the linearity of the regression line [8, 10]. The ISO 8466 describes in detail the statistical evaluation of the linear calibration function [11]. 

A much better way to evaluate the fitness of the regression model is by evaluating the residual plots. The residuals (e,) are plotted versus X or versus Y. Both graphs provide equivalent information for straight-line models. 

Consider a linear calibration experiment, for example measuring the peak height in electronic absorption spectroscopy as a function of concentration, at five different concentrations, illustrated in Figure 2.5. A chemist may wish to fit a straight line model to the experiment of the form... 

Figure 5.7 Evolution of DMS (a) and ME (b) during the period 1998-2005 in a straight line model for varietal wines from grape Chardonnay (C), Merlot (M), Marzemino (Ma) and Teroldego (T) four wines per vintage were considered...

Equation (9.15) can be written for a straight line model (see equation (9.9) with as the... 

The general straight-line model is often used as a starting point for model development. The confidence interval often widens at the ends (see Section 3.5). More levels of the factor act to reduce the confidence interval. Equally spaced levels are often tested. The number of levels depends on the complexity of the model or on the number of additional terms to be considered. Enough levels should be provided for left-out terms. The number of levels should exceed the number of parameters expected. Even if a straight line is likely, more than three levels should be considered, plus some replication. 

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