Least squares method estimated standard deviation

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. 

The structural parameters (r, u, sometimes k, and possibly other parameters) are then obtained by least-squares fitting, usually on the intensity curves, but occasionally on the RD curves (c/. p. 45). If the intensity data are used, the data from all the nozzle-to-plate distances are not necessarily combined into one composite curve an individual scale factor may be refined for each set of data. Since the observed data are considerably correlated, i.e. each point cannot be regarded as an independent observation, a number of problems are encountered in the application of the least-squares method. This problem is most important for the estimation of the standard deviations, since the parameters may be obtained fairly satis-factorally by conventional least-squares refinement. (A more detailed discussion is set out on p. 45.)... 

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... 

The response of many instruments is linear as a function of the measured variable, if variations due to experimental conditions or the instrument are taken into account. The objective is to determine the parameters of the linear equation that best represents the observations. The primary hypothesis in using the method of least squares is that one of the two variables should be without error while the second one is subject to random errors. This is the most frequently applied method. The coefficients a and b of the linear equation y = ax + b, as well as the standard deviation on a and on the estimation of y have been obtained in the past using a variety of similar equations. The choice of which formula to use depended on whether calculations were carried out manually, with calculator or using a spreadsheet. However, appropriate computer software is now widely used. 

The method of least squares is used to determine the equation of the best straight line through experimental data points. Equations 4-16 to 4-18 and 4-20 to 4-22 provide the least-squares slope and intercept and their standard deviations. Equation 4-27 estimates the uncertainty in x from a measured value of y with a calibration curve. A spreadsheet greatly simplifies least-squares calculations. 

The standard method of least-square fitting the model parameters from experimental data is only applicable if the deviation of estimated parameter-based model results from measured data can be explicitly calculated. To obtain the matrix of model error changes with the individual changes of each parameter for series of time dependent data points as measured in a... 

In surface tension measurements using the maximum bubble pressure method several sources of error may occur. As mentioned above, the exact machining of the capillary orifice is very important. A deviation from a circular orifice may cause an error of 0.3%. The determination of the immersion depth with an accuracy of 0.01 mm introduces an error of 0.3%. The accuracy of 1 Pa in the pressure measurement causes an additional error of 0.4%. The sum of all these errors gives an estimated total error of approximately 1%. Using the above-described apparatus, the standard deviations of the experimental data based on the least-squares statistical analysis were in the range 0.5% < sd > 1%. 

To conclude this discussion on IDLs, it is useful to compare Eqs. (2.19) and (2.29). Both equations relate Xp to a ratio of a standard deviation to the slope of the least squares regression line multiplied by a factor. In the simplified case, Eq. (2.19), the standard deviation refers to detector noise found in the baseline of a blank reference standard, whereas the standard deviation in the statistical case, Eq. (2.29), refers to the uncertainty in the least squares regression itself. This is an important conceptual difference in estimating IDLs. This difference should be understood and incorporated into all future methods. 

It may be necessary to use weighted least squares regression calculations in the method evaluation to obtain the Oji value, corresponding to the standard deviation at the conventional true value pti having unit weight. A valid estimate of aj,- at the concentration p-x,- of the control sample can be computed using the equation [12] ... 

The fitting procedure In this block the differences between experimental and theoretical data are minimized. A weighted least squares cost function is formulated. The Gauss-Newton and Levenberg-Marquardt method are implemented to minimize this cost fxmction and eventually provide the parameter values which best describe the data. Moreover, the standard deviations of the estimated parameters are also calculated. 

---