Linear least-squares fitting

Chapter 4, Model-Based Analyses, is essentially an introduction into least-squares fitting. It is crucial to clearly distinguish between linear and nonlinear least-squares fitting linear problems have explicit solutions while non-linear problems need to be solved iteratively. Linear regression forms the base for just about everything and thus requires particular consideration. 

There are two types of least-squares fit, linear and nonlinear. Linear least-squares fit is based on a function /(t) of the form... 

Least squared fit Linear correlation Linear regression Linkage distance LOD LOQ... 

Figure B2.4.6. Results of an offset-saturation expermient for measuring the spin-spin relaxation time, T. In this experiment, the signal is irradiated at some offset from resonance until a steady state is achieved. The partially saturated z magnetization is then measured with a kH pulse. This figure shows a plot of the z magnetization as a fiinction of the offset of the saturating field from resonance. Circles represent measured data the line is a non-linear least-squares fit. The signal is nonnal when the saturation is far away, and dips to a minimum on resonance. The width of this dip gives T, independent of magnetic field inliomogeneity.

Figure B2.4.8. Relaxation of two of tlie exchanging methyl groups in the TEMPO derivative in figure B2.4.7. The dotted lines show the relaxation of the two methyl signals after a non-selective inversion pulse (a typical experunent). The heavy solid line shows the recovery after the selective inversion of one of the methyl signals. The inverted signal (circles) recovers more quickly, under the combined influence of relaxation and exchange with the non-inverted peak. The signal that was not inverted (squares) shows a characteristic transient. The lines represent a non-linear least-squares fit to the data.

Once the least-squares fits to Slater functions with orbital exponents e = 1.0 are available, fits to Slater function s with oth er orbital expon cn ts can be obtained by siin ply m ii Itiplyin g th e cc s in th e above three equations by It remains to be determined what Slater orbital exponents to use in electronic structure calculation s. The two possibilities may be to use the "best atom" exponents (e = 1. f) for II. for exam pie) or to opiim i/e exponents in each calculation. The "best atom expon en ts m igh t be a rather poor ch oicc for mo lecular en viron men ts, and optirn i/.at ion of non linear exponents is not practical for large molecules, where the dimension of the space to be searched is very large.. 4 com prom isc is to use a set of standard exponents where the average values of expon en ts are optirn i/ed for a set of sin all rn olecules, fh e recom -mended STO-3G exponents are... 

Expand the three detemiinants D, Dt, and for the least squares fit to a linear function not passing through the origin so as to obtain explicit algebraic expressions for b and m, the y-intercept and the slope of the best straight line representing the experimental data. 

It is usually advisable to plot the observed pairs of y versus r, to support the linearity assumption and to detect potential outhers. Suspected outliers can be omitted from the least-squares Tit and then subsequently tested on the basis of the least-squares fit. 

To extract the agglomeration kernels from PSD data, the inverse problem mentioned above has to be solved. The population balance is therefore solved for different values of the agglomeration kernel, the results are compared with the experimental distributions and the sums of non-linear least squares are calculated. The calculated distribution with the minimum sum of least squares fits the experimental distribution best. 

It can be seen from Fig. 7 that V is a linear function of the qf This qV relation was pointed out and discussed at some length in the papers in ref. 6. It is not simple electrostatics in that it would not exist for an arbitrary set of charges on the sites, even if the potentials are calculated exactly. The charges must be the result of a self-consistent LDA calculation. The linearity of the relation and fie closeness of the points to the line is demonstrated by doing a least squares fit to the points. The sums that define the potentials V do not converge at all rapidly, as can be seen by calculating the Coulomb potential from the standard formula for one nn-shell after another. The qV relation leads to a special form for the interatomic Coulomb energy of the alloy... 

The hydrolysis of triphenylmethyl chloride follows first-order kinetics. The left panel shows the growth of (H J, a product (filled squares), and the exponential decrease in (PhiCCI], (open squares). The second and third panels show linearized forms, with [Ph,CCl], and [Ph CCl]o/tPhiCCI], being displayed on logarithmic scales. Each line is the least-squares fit to the indicated function. 

Various displays of data that follow the rate equation -d[A]/di = Jfc[AJ2. The panels display [A] 1AA] and In [A], versus time and [A], versus [A],r[A]0. Three of them show a line that is the least-squares fit to the appropriate form. The display of In [A], versus time is not linear because the reaction follows second-order, not first-order, kinetics. 

From this, the values of [B], follow from Eq. (2-17). This equation can also be used to fit the data with a nonlinear least-squares routine. Table 2-2 gives an example of data for a reaction that follows mixed second-order kinetics.3 Figure 2-3 displays the linear variation of ln([B],/tA],) with time as well as [A], itself against time. Both show a line corresponding to the least-squares fit of the function given. 

Substitution of the usual relation between concentration and property, Eq. (2-23), yields one form for linearized graphical analysis and another for least-squares fitting ... 

The choice of the y-variable is also important. If one records a series of concentrations, or a quantity proportional to them, then this set is a valid quantity to be fitted by linear least squares. On the other hand, if the equation is rearranged to a form that can be displayed in a linear graph, then the new variable may not be so suitable. Consider the equations for second-order kinetics. The correct form for least-squares fitting is... 

A plot of In Y, - Ye versus time will be linear. Its slope gives kt, = k + k- as before. Least-squares fitting to Eq. (3-16) is preferable, and Ye can be floated or fixed, as the system requires. Of course, Ye symbolizes the end point reading at equilibrium, not when the reaction has been drawn entirely to the right. 

Multiple Linear Least-Squares Fits with a Common Intercept Determination of the Intrinsic Viscosity of Macromolecules in Solution. Journal of Chemical Education 80(9), 1036-1038. 

The Monod kinetic parametos were evaluated by least squares fitting procedures, for tiie single and multiple substrate systems with/without mutual inhibition, and were indicated in Table 1 [6]. The value of indicates the linear decomposition rate. It is dear that the decomposition rate for prc iionic acid is significantly lower than those for acetic add and butyric acid. 

There have also been attempts to describe the temporal aspects of perception from first principles, the model including the effects of adaptation and integration of perceived stimuli. The parameters in the specific analytical model derived were estimated using non-linear regression [14]. Another recent development is to describe each individual TI-curve,/j(r), i = 1, 2,..., n, as derived from a prototype curve, S t). Each individual Tl-curve can be obtained from the prototype curve by shrinking or stretching the (horizontal) time axis and the (vertical) intensity axis, i.e. fff) = a, 5(b, t). The least squares fit is found in an iterative procedure, alternately adapting the parameter sets (a, Zi, for 1=1,2,..., n and the shape of the prototype curve [15],... 

Figure 4. Fits of lattice strain model to experimental mineral-melt partition coefficients for (a) plagioclase (run 90-6 of Blundy and Wood 1994) and (b) elinopyroxene (ran DC23 of Blundy and Dalton 2000). Different valence cations, entering the large cation site of each mineral, are denoted by different symbols. The curves are non-linear least squares fits of Equation (1) to the data for each valence. Errors bars, when larger than symbol, are 1 s.d. Ionic radii in Vlll-fold coordination are taken from Shannon (1976).

A central concept of statistical analysis is variance,105 which is simply the average squared difference of deviations from the mean, or the square of the standard deviation. Since the analyst can only take a limited number n of samples, the variance is estimated as the squared difference of deviations from the mean, divided by n - 1. Analysis of variance asks the question whether groups of samples are drawn from the same overall population or from different populations.105 The simplest example of analysis of variance is the F-test (and the closely related t-test) in which one takes the ratio of two variances and compares the result with tabular values to decide whether it is probable that the two samples came from the same population. Linear regression is also a form of analysis of variance, since one is asking the question whether the variance around the mean is equivalent to the variance around the least squares fit. 

This so-called measurement gain is really the slope of a calibration curve—an idea that we are familiar with. We do a least squares fit if this curve is linear, and find the tangent at the operating point if the curve is nonlinear. 

The impedance data were fitted to candidate electrical circuits using the non-linear weighted least-squares fitting program "EQIVCT" developed by Boukamp ( ). Graphical analysis was utilized to furnish reasonable first guesses of the circuit parameters for input to EQIVCT. 

Linear Least Squares Fitting in Case of Errors in Both Variables... 

The maximum search function is designed to locate intensity maxima within a limited area of x apace. Such information is important in order to ensure that the specimen is correctly aligned. The user must supply an initial estimate of the peak location and the boundary of the region of interest. Points surrounding this estimate are sampled in a systematic pattern to form a new estimate of the peak position. Several iterations are performed until the statistical uncertainties in the peak location parameters, as determined by a linearized least squares fit to the intensity data, are within bounds that are consistent with their estimated errors. 

We have found that data interpretation other than simple plotting from texture studies requires a sophisticated array of software tools. These include least squares curve resolving, non-linear least squares fitting, 2-dimensional data smoothing, numerical quadrature, and high-speed interactive graphics, to mention only a few. 

Any measure of the coordinate correlation is arbitrary. Here, the linear correlation coefficient r is used, largely because it is familiar. It measures the quality of a least-squares fit of a line to coordinates, with a magnitude that varies from 0 (for uncorrelated random coordinates) to 1 (for fully correlated coordinates lying on the line). For the correlated coordinates in Fig. 3.1 b, d, and f, theoretical relationships for r, that is, r =/(p1 p2) can be derived as shown in Appendix 3B. They show that r lies between the inclusive bounds of 1/2 and 1 for WEG, and between the inclusive bounds, 0 and 1, for FAN and PAR. 

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