Slope least-squares

Note that m and b do not have subscripts because there is only one slope and one intercept they are the minimization parameters for the least squares function. Now there are two minimization conditions... 

Procedure. Compute the slope of the function by a linear least squares procedure and obtain a value of Boltzmann s constant. How many particles do you expect to find 125 pm above the reference point Take the uncertainty you have calculated for the slope, as the uncertainty in k. Is the modem value of = 1.381 x 10 within these enor limits ... 

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. 

Dichromate-permanganate determination is an artificial problem because the matrix of coefficients can be obtained as the slopes of A vs. x from four univariate least squares regression treatments, one on solutions containing only at... 

Subtracting the slope matrix obtained by the multivariate least squares tieatment from that obtained by univariate least squares slope matiix yields the error mahix... 

A straight line may be fitted to the (X, Y) or (X, Y) pairs of data when plotted on log-log graph paper from which the slope N and the intercept log K with X = 1 may be read. Alternatively, the method of least squares may be used to estimate the values of K and N, giving the best fit to the available data. 

As an example, consider the polymerization of methyl methacrylate, initiated by a,a -azobisisobutyronitrile.4 The dependence of the initial rate on the concentration of the initiator is displayed in Fig. 1-1, which shows them on a double logarithmic scale. The points define a straight line with a least-squares slope of0.496. Clearly the reaction is half-order with respect to the concentration of the initiator. 

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. 

The buildup of [P] is biexponential. Nonlinear least-squares fitting will provide the best solution for k and k2 (and, if they are unknown, for [A ]o and [A2]o also). Another method is to plot ln([P] - [P],) against time. At long times, the exponent with the higher value of k will have fallen off, such that the slope will approximate the slower one. Calling this ks (it might be either k or k2), we perform a subtraction, emphasizing the data at shorter times ... 

The first of these reactions was carried out in 1,4-cyclohexadiene over a temperature range of 39 to 100 °C. It is fairly slow the half-times were 20 h and 3.4 min at the extremes. Reaction (7-11) is quite fast the second-order rate constant, kn, was evaluated over the range 6.4 to 47.5 °C. Values of feio and fen are presented in Table 7-1. The temperature profiles are depicted in Fig. 7-1 from their intercepts and slopes the activation parameters can be obtained. A nonlinear least-squares fit to Eq. (7-1) or... 

Linear least squares treatments of plots of the logarithm of the vapor pressure versus the reciprocal temperature were performed. The second-law enthalpy and entropy of sublimation at the median temperature are proportioned to the slope and... 

Leach, R. A., Carter, C. A., and Harris, J. M., Least-Square Polynomial Filters for Initial Point and Slope Estimation, Anal. Chem. 56, 1984, 2304-2307. 

Kahn, A., Procedure for Increasing the Accuracy of the Initial Data Point Slope Estimation by Least-Squares Polynomial Filters, Anal. Chem. 60, 1988,... 

The experimental points scatter uniformly on both sides of the line. Accordingly, it can be concluded that the tested rate equation should not be rejected. The slope, k, is 0.02 min. This is only a rough estimate of the rate constant because numerical and graphical differentiations are very inaccurate procedures. The slope was also calculated by the least squares technique minimizing the sum of squares... 

The data are also represented in Fig. 39.5a and have been replotted semi-logarithmically in Fig. 39.5b. Least squares linear regression of log Cp with respect to time t has been performed on the first nine data points. The last three points have been discarded as the corresponding concentration values are assumed to be close to the quantitation limit of the detection system and, hence, are endowed with a large relative error. We obtained the values of 1.701 and 0.005117 for the intercept log B and slope Sp, respectively. From these we derive the following pharmacokinetic quantities ... 

A least squares linear regression has been applied to the data pertaining to the p-phase, yielding the values of 1.745 and 0.005166 for the intercept log B and slope Sp, respectively. Using these results, we can compute the extrapolated plasma concentrations between 0 and 20 minutes. From the latter, we subtract the observed concentrations C which yields the concentrations of the a-phase C ... 

These data are also shown in the semilogarithmic plot in Fig. 39.13a, which clearly shows two distinct phases. A straight line has been fitted by least-squares regression through the data starting from the observation at 90 minutes down to the last one. This yields the values of 1.086 and -0.001380 for the intercept log and slope ip, respectively. From these results we have computed the extrapolated p-phase values between 2 and 60 minutes. These have been subtracted from the experimental Cp values in order to yield the a-phase concentrations C ... 

The residual a-phase concentrations C are shown in the semilogarithmic plot of Fig. 39.13b. Least-squares linear regression of log C upon time produced 1.524 and -0.02408 for the intercept log and the slope respectively. 

Ordinary least squares regression of MV upon MX produces a slope of 9.32 and an intercept of 2.36. From these we derive the parameters of the simple Michaelis-Menten reaction (eq. (39.116)) ... 

The amount of tebuconazole residue is calculated by using a least-squares fitting algorithm to generate the best line which can be used to calculate the corresponding concentration for a given peak area or peak height. Calculate the slope and the intercept of the standard calibration curve. 

One can apply a similar approach to samples drawn from a process over time to determine whether a process is in control (stable) or out of control (unstable). For both kinds of control chart, it may be desirable to obtain estimates of the mean and standard deviation over a range of concentrations. The precision of an HPLC method is frequently lower at concentrations much higher or lower than the midrange of measurement. The act of drawing the control chart often helps to identify variability in the method and, given that variability in the method is less than that of the process, the control chart can help to identify variability in the process. Trends can be observed as sequences of points above or below the mean, as a non-zero slope of the least squares fit of the mean vs. batch number, or by means of autocorrelation.106... 

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. 

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