The Principle of Least Squares

Two types of error are generally recognized in discussions of experimental data. One is determinate or systematic error, that is, error which arises from the design of the experiment. The second is indeterminate or random error which arises because the experimental conditions cannot be perfectly controlled. It is assumed 

Determinate error usually results from experimental equipment which is faulty. Students usually first meet this concept in the analytical chemistry laboratory in determinations of weight and volume. The quality of the equipment used is reflected in the accuracy of the results obtained. Accuracy is a measure of how close the experimental result is to the truth. For example, if one wishes to make up a solution of accurately known concentration in a volumetric flask of 100 mL, both the flask and the balance used must be carefully calibrated. The flask is calibrated by weighing it empty and then filled with distilled water at a known temperature. On the basis of the weight of water, and the known density of the water, one may calculate an accurate volume for the flask when it is properly filled to the mark. Calibration of the balance is based on the use of standard weights which do not corrode and which cover the range in mass used in the experiment. The accuracy of the standard weights and the quality of the volumetric flask determine the accuracy of the concentration of the solution which is made. 

The operator s inability to control perfectly all the variables in an experiment leads to indeterminate error. For example, errors in the concentration of the solution made up in a volumetric flask can result from fluctuations in the room temperature. Changes in humidity in the room can also affect the weights involved in the experiment, for example, the weight of a solute which can adsorb water vapor. Indeterminate or random error affects the precision of the experiment. This type of error is treated by the laws of probability. If one determines the concentration of a solution each day on five successive days, the results are the same only within the precision of the experiment. On the basis of experience, the operator averages the results, and then estimates the standard deviation in order to have an indication of the precision of the experiment. 

When a given quantity x is observed an infinite number of times, the observed values follow a Gaussian or normal error distribution. Thus, the probability of observing a value x for the quantity being measured is given by 

This is the principle of least squares applied to the determination of a single quantity x. The sum of squares A is minimized, allowing x to have the optimum value. Thus, one has the condition 

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In this chapter, we shall use the principle of least squares to generate the equation of a unique curve for any given set of x-y pairs of data points. The ciu ve so obtained is the best fit to the points subject to... 

The principle of least squares is used to correlate Y with the X. values. The error e (or residual) is defined as... 

The purpose of the principle of least squares is to minimize the sum of the squares of the errors so that... 

The principle of least squares assumes that the best-fit line is the one for which the sum of the squares of the differences (d,) is a minimum. Note that this assumption considers all experimental error to be associated with v and none to be associated with v. In finding a best-fit line, therefore, it is important to let jc represent the variable that is known most accurately. The sum of the squares of the differences, taken for all values of / (from / = 1 up to and including / = n) is... 

While I was a graduate student in chemistry, I had several occasions to fit data to equations using the principle of least squares. However, I noticed that upon rearrangement... 

In all adjustments of observations, simple or complicated, the principle of least squares requires the minimization of the sum of the weighted squares of the residuals. This sum may be written... 

The weights are proportional to the reciprocal of the errors squared in the experimental values. The quantity Vx and Vy are the x and y residuals. The principle of least squares is the minimization of S. The method of least squares is a rule or set of rules for proceeding with the actual computation. [Chap 4, 36, p. ]... 

Curve fitting is applied when a dependent variable y is measured experimentally as a function of an independent variable x. The simplest relationship between these variables is a straight line. Under some circumstances, v and y do not come directly from experiment but are calculated from the experimental data on the basis of a theory which predicts that y should be linear in v. In the present context, the independent variable v is always that with a negligible level of error, whereas any significant experimental error is only associated with the dependent variable y. Under these circumstances the principle of least squares is applied to estimating the best straight line through the points with respect to the random error in y. 

The analysis presented for fitting a straight line to experimental data is easily extended to a curve or to a system with more than one independent variable on the basis of the principle of least squares. 

Three parameters, namely, a, b, and c are required to specify the relationship. The equation describing the principle of least squares is... 

To illustrate the reasonableness of the principle of least squares we may revert to an old regulation of the Belgian army in which the individual soores of the riflemen were formed by adding up the distances of each man s shots from the centre of the target. The... 

But 2 (a2) is the sum of the squares of the true errors. The true errors are unknown. By the principle of least squares, when measurements have an equal degree of confidence, the most probable value of the observed quantities are those which render the sum of the squares of the deviations of each observation from the mean, a minimum. Let 2(va) denote the sum of the squares of the deviations of each observation from the mean. If n is large, we may put 2(a 2) = S(v2) but if n is a limited number,... 

I. The best value to represent a number of observations of equal weight, is their arithmetical mean. If P denotes the most probable value of the observed magnitudes av a2f. . . an, then P - avP- a2,. . ., P - ant represent the several errors in the n observations. From the principle of least squares these errors will be a minimum when... 

VI. The principle of least squares for observations of different degrees of precision states that the most probable values of the observed quantities are those for which the sum of the weighted squares of the errors is a minimum, that is,... 

The above is based on the principle of least squares. A quicker method, not so exact, but accurate enough for most practical purposes, is due to Mayer. We can illustrate Mayer s method by equations (12). 

The modified Arrhenius parameters are determined from regression analysis with application of the principle of least squares. CTST describes the forward rate constant from reactant to the transition state (TS) s a function of the equilibrium between reactant and TS. ThermKin requires thermodynamic properties in the NASA polynomial format, needs to know whether the reaction is uni- or bimolecular, and either a two-parameter fit or a three-parameter fit is desired. Finally, the reaction to be calculated has to be given in the form ... 

The use of nonlinear least-squares analysis is ubiquitous in the analySK of fluorescence data, pairicululy time-domain and fiequency-domain data. A usefiil introduction to the principles of least-squares analysis is found in the compact but informative book by Bevington (1969). The applications of these concepts to diverse types of fluorescence data can be found in edited volumes (Brand and Johnson, 1992 Johnson and Brand, 1994). For more basic information about statistics and spectroscopy, one can examine several introductoiy texts (Ihylor, 1982 Mark and Workman, 1991). 

Method of Least Squares The best fit curve according to the principle of least squares is that curve for which the sum of squares of the residuals is a minimum. That is,... 

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