Least squares residue

One straightforward choice for the X-ray target function is the least square residual that represent the discrepancy between the observed and model-predicted structure factors ... 

Table II Reflned structural parameters for Sia, based on the rotational constants in Table I. For the set of five isotopomers, maximum least-squares residuals for the fits to all three constants...

For the model in Exercise 1, suppose, is normally distributed with mean zero and variance g2(1 + (yx)2). Show that g2 and y2 can be consistently estimated by a regression of the least squares residuals on a constant and x2. Is this estimator efficient ... 

Fit this model. If the software does not allow constraints on the coefficients, you can still do this with a two step least squares procedure, using the least squares residuals from the fust step. What do you find ... 

Derive the disturbance covariance matrix for the model y, = P x, + st, e, = ps, i + ut - A.u, i. What parameter is estimated by the regression of the ordinary least squares residuals on theft lagged values ... 

Ideally, experimental data (intensity) should be randomly distributed both above and below the calculated intensity profile. If there are multiple sequences with all observed points above or all below the calculated intensity values, it is said that serial correlation occurs. In other words, the 4-statistic reflects correlation between adjacent least-squares residuals and it can be used as an indicator that refined parameters are unbiased. 

In addition, in least-squares residual plots of S versus Xp the slope of the regression line of Si against Xj can be expected to be zero. In contrast, the regression of r against Xj should have a slope equal to the coefficient of Xj when the full model is fitted. This property of partial residuals makes these plots useful in assessing the extent of possible nonlinearity in a certain predictor (25). If the slope of the plot of r against Xj approximately equals the coefficient obtained from a ht of the full model, the specihcation of Xj in the regression model can be assumed to be correct. 

Harrison, M.J. and McCabe, B.P.M. A test for heteroscedas-ticity based on ordinary least squares residuals. Journal of the American Statistical Association 1979 74 494—499. 

Show that and can be consistently estimated by a regression of the least squares residuals on a constant and X. Is this estimator efficient ... 

Temperature Pulse Decay Technique. As described in Sec. 2.4 under Temperature Pulse Decay (TPD) Technique, local blood perfusion rate can be derived from the comparison between the dieoretically predicted and experimentally measured temperature decay of a thermistor bead probe. The details of the measurement mechanism have been described in that section. The temperature pulse decay technique has been used to measure the in vivo blood perfusion rates of different physical or physiological conditions in varimis tissues (Xu et al., 1991 1998). The advantages of this technique are that it is fast and induces little trauma. Using the Pennes bioheat transfer equation, the intrinsic thermal conductivity and blood perfusion rate can be simultaneously measured. In some of the applications, a two-parameter least-square residual fit was first performed to obtain the intrinsic therm conductivity of the tissue. This calculated value of thermal conductivity was then used to perform a one-parameter curve fit for the TPD measurements to obtain the local blood perfusion... 

Contributions to the Least Squares Residue and Residual Index... 

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