Polynomial second order

Many systems that cannot be represented by a first-order empirical model can be described by a full second-order polynomial equation, such as that for two factors. 

Timoshenko et al (1967) recommended running a set of experiments in a CSTR on feed composition (now called feed-forward study), and then statistically correlating the discharge concentrations and rates with feed conditions by second order polynomials. In the second stage, mathematical experiments are executed on the previous empirical correlation to find the form and constants for the rate expressions. An example is presented for the dehydrogenation of butane. 

Simpson s method then uses second order polynomials to connect sets of three ordinates representing segment pairs to determine the composite area. The total area. A, under the curve y = f x) between the limits is then approximated by the following equation ... 

Development of the Nomograph. Tw o main sources of data were used to develop the nomograph McAuliffe and Price. The hydrocarbons were divided into 14 homologous series as listed in Table 1. Solubilities at 25°C were then regressed with the carbon numbers of the hydrocarbons in order to obtain the best fit for each homologous series. A second order polynomial equation fits the data very well ... 

Subsequently, the difference Ad(U0) (in A) is calculated from the theoretical value and the measured d value of the (1 1 0) reflection. Using the empirical calibration curve (a second- order polynomial, Eq.(6)), the pyrolusite concentration can be calculated or it may be taken from a diagram, as shown in Fig. 5. 

The effect of the number of finite element triangles is shown in Table III. These simulations were performed using 16 marching steps and second order polynomials. 

We will not prove it—hardly any text does anymore. Nonetheless, we use two general polynomials to illustrate some simple properties. First, consider a second order polynomial with the leading coefficient a2 = 1. If the polynomial has two real poles p, and p2, it can be factored as... 

Screening designs are mainly used in the intial exploratory phase to identify the most important variables governing the system performance. Once all the important parameters have been identified and it is anticipated that the linear model in Eqn (2) is inadequate to model the experimental data, then second-order polynomials are commonly used to extend the linear model. These models take the form of Eqn (3), where (3j are the coefficients for the squared terms in the model and 3-way and higher-order interactions are excluded. 

The MO concentrations versus time profiles were fitted to second order polynomial equations and the parameters estimated by nonlinear regression analysis. The initial rates of reactions were obtained by taking the derivative at t=0. The reaction is first order with respect to hydrogen pressure changing to zero order dependence above about 3.45 MPa hydrogen pressure. This was attributed to saturation of the catalyst sites. Experiments were conducted in which HPLC grade MIBK was added to the initial reactant mixture, there was no evidence of product inhibition. 

The ambient temperature sensor in the IRT 3000 is a KTY type spreading resistance sensor. The resistance of the sensor can be written as a second order polynomial function... 

The reorganization of the solvent molecules can be expressed through the change in the slow polarization. Consider a small volume element AC of the solvent in the vicinity of the reactant it has a dipole moment m = Ps AC caused by the slow polarization, and its energy of interaction with the external field Eex caused by the reacting ion is —Ps Eex AC = —Ps D AC/eo, since Eex = D/eo- We take the polarization Ps as the relevant outer-sphere coordinate, and require an expression for the contribution AU of the volume element to the potential energy of the system. In the harmonic approximation this must be a second-order polynomial in Ps, and the linear term is the interaction with the external field, so that the equilibrium values of Ps in the absence of a field vanishes ... 

For instance, if we fit a second-order polynomial through a window of five points... 

FIGURE 7.2 First derivatives of the seven NIR spectra from Figure 7.1. The Savitzky-Golay method was applied with a second-order polynomial for seven points. 

A characteristic measurement includes the determination of the microhotplate temperature as a function of its power consumption. The curves can be fitted by a second-order polynomial using the coefficients tjo and t]i. 

The discrete microhotplates were packaged and bonded in a DIL-28 package for temperature sensor cahbration. A Pt-lOO-temperature sensor was attached to the chip package in close vicinity to the sensors. The chips were then caHbrated in an oven at temperatures up to 325 °C with the help of the Pt-100 resistor. A second-order polynomial was extracted from the measurements for each temperature sensor providing the temperature coefficients i and a2. ... 

The relationship between the temperature difference, AT, and the input power is shown in Fig. 4.5 for microhotplate simulations and measurements. The simulated values are plotted together with the mean value of the experimental data for a set of three hotplates of the same wafer. The experimental curve was fitted with a second-order polynomial according to Eq. (3.24). As a result of the curve fit, the thermal resistance at room temperature, tjo, is 5.8 °C/mW with a standard deviation of 0.2 °C/mW, which is mainly due to variations in the etching process. 

In reference 88, response surfaces from optimization were used to obtain an initial idea about the method robustness and about the interval of the factors to be examined in a later robustness test. In the latter, regression analysis was applied and a full quadratic model was fitted to the data for each response. The method was considered robust concerning its quantitative aspect, since no statistically significant coefficients occurred. However, for qualitative responses, e.g., resolution, significant factors were found and the results were further used to calculate system suitability values. In reference 89, first a second-order polynomial model was fitted to the data and validated. Then response surfaces were drawn for... 

Anticipating a later section on canonical analysis of second-order polynomial models, we will show that the first-order term can be made to equal zero if we code the model using the stationary point as the center of the symmetrical design. For this new system of coding, c, = 10 2/3 and (see Section 8.5). 

One of the most useful models for approximating a region of a multifactor response surface is the full second-order polynomial model. For two factors, the model is of the form... 

In general, if k is the number of factors being investigated, the full second-order polynomial model contains V2 k -t- 1)(A -h 2) parameters. A rationalization for the widespread use of full second-order polynomial models is that they represent a truncated Taylor series expansion of any continuous function, and such models would therefore be expected to provide a reasonably good approximation of the true response surface over a local region of experiment space. 

Efficiency of full second-order polynomial models fit to data from central composite designs without replication. 

Full second-order polynomial models used with central composite experimental designs are very powerful tools for approximating the true behavior of many systems. However, the interpretation of the large number of estimated parameters in multifactor systems is not always straightforward. As an example, the parameter estimates of the coded and uncoded models in the previous section are quite different, even though the two models describe essentially the same response surface (see Equations 12.63 and 12.64). It is difficult to see this similarity by simple inspection of the two equations. Fortunately, canonical analysis is a mathematical technique that can be applied to full second-order polynomial models to reveal the essential features of the response surface and allow a simpler understanding of the factor effects and their interactions. 

To find the coordinates of the stationary point, we first differentiate the full second-order polynomial model with respect to each of the factors and set each derivative equal to zero. For two-factor models we obtain... 

The corresponding matrix least squares treatment for the full second-order polynomial model proceeds as follows. 

Figure 12.32 Mixture designs and fitted full second-order polynomial response surfaces. See text for details.

Write full second-order polynomial models for 1, 2, 3, 4, and 5 factors. 

Show that the full two-factor second-order polynomial model may be written y, - Po + Do d sd where Dq = [x, Xj,]. Show that this may be extended to full three-factor second-order polynomial models. 

Confidence Intervals for Full Second-Order Polynomial Models... 

In this chapter we investigate the interaction between experimental design and information quality in two-factor systems. However, instead of looking again at the uncertainty of parameter estimates, we will focus attention on uncertainty in the response surface itself. Although the examples are somewhat specific (i.e., limited to two factors and to full second-order polynomial models), the concepts are general and can be extended to other dimensional factor spaces and to other models. 

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