The Least Squares Method

Let b denote a / -element vector that is an estimate of the parameter vector p. We use this estimate to define a vector of residuals  

These residuals are the differences between the experimental observations F and the calculated values of Y using the estimated vector b. A common way for evaluation of the unknown vector b is the least squares method, which minimizes the sum of the squared residuals O  

In order to calculate the vector b, which minimizes O, we take the partial derivative of with respect to b and set it equal to zero  

We simplify this utilizing the matrix-vector identity A y = y A  

The above constitute a set of simultaneous linear algebraic equations, called the normal equations. The matrix X X) is a (/ x k) symmetric matrix. Assumption 4 made earlier guarantees that (X X) is nonsingular therefore, its inverse exists. Thus, the normal equations can be solved for the vector b  

As mentioned, the model parameters of the linear model can be effectively identified rrsing a least sqtrares method. The model eqiration for the different types of model can be written as a difference eqrratiorr, suehas, for example, the one shown inEqn. (24.4). When the meanvalnes of y and u are srrbtracted fiom the actual values of y and u, one may write this equation as  

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For a specified mean and standard deviation the number of degrees of freedom for a one-dimensional distribution (see sections on the least squares method and least squares minimization) of n data is (n — 1). This is because, given p and a, for n > 1 (say a half-dozen or more points), the first datum can have any value, the second datum can have any value, and so on, up to n — 1. When we come to find the... 

Example 10. Calculate by the least squares method the equation of the best straight line for the calibration curve given in the previous example. 

This system of i + 2 equations is nonlinear, and for this reason probably has not received attention in the least-squares method (207). We are able to give an explicit solution (163) for the particular case when Xy = xj and m,- = m for all values of i that is, when all reactions of the series are studied at a set of temperatures, not necessarily equidistant, but the same for all reactions. Let us introduce... 

This function depends on many parameters that will be refined using the least squared method. 

Similar procedures can be used to estimate coefficients in the van t Hoff equation. The least squares method for estimation of pre-exponential factor and activation energy is illustrated in Example 5.4.4.2b. 

The parameter values found by the two methods differ slightly owing to the different criteria used which were the least squares method for ESL and the maximum-likelihood method for SIMUSOLV and because the T=10 data point was included with the ESL run. The output curve is very similar and the parameters agree within the expected standard deviation. The quality of parameter estimation can also be judged from a contour plot as given in Fig. 2.41. 

Parameter estimation and identification are an essential step in the development of mathematical models that describe the behavior of physical processes (Seinfeld and Lapidus, 1974 Aris, 1994). The reader is strongly advised to consult the above references for discussions on what is a model, types of models, model formulation and evaluation. The paper by Plackett that presents the history on the discovery of the least squares method is also recommended (Plackett, 1972). 

Next, by trial and error an estimate value of K is sought such that the [IH + ] values obtained yield the best fit with the straight line for eqn. 3.77 (e.g., by means of the least-squares method) finally, K and Kh2o are calculated from the intercept and the slope of this line. 

Figure 1 shows the powder X-ray diffraction (XRD) pattern of the as-prepared Li(Nio.4Coo.2Mno.4)02 material. All of the peaks could be indexed based on the a-NaFeC>2 structure (R 3 m). The lattice parameters in hexagonal setting obtained by the least square method were a=2.868A and c=14.25A. Since no second-phase diffraction peaks were observed from the surface-coated materials and it is unlikely that the A1 ions were incorporated into the lattice at the low heat-treatment temperature (300°C), it is considered that the particle surface was coated with amorphous aluminum oxide. 

This is the situation we must handle. We cannot simply ignore one or more of these equations arbitrarily dealing with them properly has become known variously as the Least Squares method, Multiple Least Squares, or Multiple Linear Regression. 

Ryason and Russel measured the temperature dependence of the IR absorption band halfwidth for valence vibrations of hydroxyl groups on the silica surface.200 At T > 325 K, the least squares method permits a straight line to be drawn through experimental points of the dependence In Avv2 (Tl), the equation of the line appearing as follows 200... 

Use the least squares method to plot the best fit calibration curve. Comment upon its suitability for use. 

Fit the following data using the least squares method with the equation ... 

K and A being known, the solution of equation 13 allows the determination of the LSER parameters characterizing the sensors studied here. Equation 12 was solved with the least squares method. In Figure 8 the LSER parameters for each sensor are shown. 

Handling these equations is normally done through the least-square method just discussed on the right hand-side, the unknown vector will be the vector (a, b). The... 

The best fit to the Arrhenius plot can be found by the least squares method (applied to In t or log t) and extrapolated to find the time to the threshold value (tu) at a temperature of interest (Tu). To obtain an estimate of the maximum temperature of use, extrapolate the line to a specified reaction rate or time to reach a threshold value. 20,000 or 100,000 hours duration and 50% change as the threshold value are commonly used for establishing a general maximum temperature of use. 

The least-squares method is also widely applied to curve fitting in phase-modulation fluorometry the main difference with data analysis in pulse fluorometry is that no deconvolution is required curve fitting is indeed performed in the frequency domain, i.e. directly using the variations of the phase shift and the modulation ratio M as functions of the modulation frequency. Phase data and modulation data can be analyzed separately or simultaneously. In the latter case the reduced chi squared is given by... 

Returns statistics that describe a linear trend matching known data points, by fitting a straight line usiig the least squares method. 

Reflection intensity in the SAED negatives was measured with a microdensitometer. The refinement of the structure analysis was performed by the least square method over the intensity data (25 reflections) thus obtained. A PPX single-crystal is a mosaic crystal which gives an "N-pattem". Therefore we used the 1/d hko as the Lorentz correction factor [28], where d hko is the (hkO) spacing of the crystal. In this case, the reliability factor R was 31%, and the isotropic temperature factor B was 0.076nm. The molecular conformation of the P-form took after that of the P-form since R was minimized with this conformation benzene rings are perpendicular to the trans-zigzag plane of -CH2-CH2-. 

Therefore, uniaxially oriented samples should be prepared for this purpose, which give so-called fiber pattern in X-ray diffraction. The diffraction intensities from the PPX specimen of P-form, which had been elongated 6 times at 285°C, were measured by an ordinary photographic method. The reflections were indexed on the basis of the lattice constants a=ft=2.052nm, c(chain axis)=0.655nm, a=P=90°, and y=120°. Inseparable reflections were used in the lump in the computation by the least square method. 

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