Regression Sample Problem

Calculate the standard deviation for the X variable and the Y variable using the methodology presented earlier in the book. The standard deviation for X= 2.24 and the standard deviation for Y= 2.87. 

Calculate the term a for the equation of a line using the using the methodology presented in this chapter. 

Using this equation, it is now possible to predict future values of Y (number of traffic accidents) given a value for X (number trips to Big City, USA). For example, if there were 20 trips scheduled to Big City, USA next month, how many traffic accidents would one expect to be reported for the month  

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In the above formula, L represents a matrix with dimension r X p, /B represents the p X 1 regression parameters estimation and Vs is the corresponding estimated variance-covariance matrix. is an estimated positive scalar to prevent inverting a highly singular matrix. (Guo et al., 2003) However, the limitation of this method is that it is only suitable for one-sample problem and using the asymptotic theory will not be suitable for small number of replicates. 

This GLS estimator is akin to inverse variance-weighted regression discussed in Section 8.2.3. Again there is a limitation V can be inverted only when the number of calibration samples is larger than the number of predictor variables, i.e. spectral wavelengths. Thus, one either has to work with a limited set of selected wavelengths or one must apply other solutions which have been proposed for tackling this problem [5]. 

Several approaches have been investigated recently to achieve this multivariate calibration transfer. All of these require that a small set of transfer samples is measured on all instruments involved. Usually, this is a small subset of the larger calibration set that has been measured on the parent instrument A. Let Z indicate the set of spectra for the transfer set, X the full set of spectra measured on the parent instrument and a suffix Aor B the instrument on which the spectra were obtained. The oldest approach to the calibration transfer problem is to apply the calibration model, b, developed for the parent instrument A using a large calibration set (X ), to the spectra of the transfer set obtained on each instrument, i.e. and Zg. One then regresses the predictions (=Z b ) obtained for the parent instrument on those for the child instrument yg (=Z b ), giving... 

Instrument calibration is done during the analysis of samples by interspersing standards among the samples. Following completion of the samples and standards, a linear calibration curve is estimated from the response of the standards using standard linear regression techniques. The calibration constants obtained from each run are used only for the samples quantitated in that run. Drastic changes or lack of linearity may indicate a problem with the detector. 

Having the smoothed values of the state variables at each sampling point and the derivatives, q we have essentially transformed the problem to a "usual" linear regression problem. The parameter vector is obtained by minimizing the following LS objective function... 

In the calibration problem two related quantities, X and Y, are investigated where Y, the response variable, is relatively easy to measure while X, the amount or concentration variable, is relatively difficult to measure in terms of cost or effort Furthermore, the measurement error for X is small compared with that of Y The experimenter observes a calibration set of N pairs of values (x, y ), i l,...,N, of the quantities X and Y, x being the known standard amount or concentration values and y the chromatographic response from the known standard The calibration graph is determined from this set of calibration samples using regression techniques Additional values of the dependent variable Y, say y., j l,, M, where M is arbitrary, are also observed whose corresponding X values, x. are the unknown quantities of interest The statistical literature on the calibration problem considers the estimation of these unknown values, x, from the observed and the... 

The main advantage of PCR over inverse MLR is that it accounts for covariance between different x variables, thus avoiding any potential problems in the model computation mathematics, and removing the burden on the user to choose variables that are sufficiently independent of one another. From a practical viewpoint, it is much more flexible than CLS, because it can be used to develop regression models for any property, whether it is a concentration property or otherwise. Furthermore, one needs only the values of the property of interest for the calibration samples (and not the concentrations of aU components in the samples). 

A y-sample outlier, based on the sample s property value(s) (for regression problems only). 

Haaland and coworkers (5) discussed other problems with classical least-squares (CLS) and its performance relative to partial least-squares (PLS) and factor analysis (in the form of principal component regression). One of the disadvantages of CLS is that interferences from overlapping spectra are not handled well, and all the components in a sample must be included for a good analysis. For a material such as coal LTA, this is a significant limitation. 

Regression analysis often is used to assess differences in costs, in part because the sample size needed to detect economic differences may be larger than the sample needed to detect clinical differences (i.e., to overcome power problems). Traditionally, ordinary least-squares regression has been used to predict costs (or their log) as a function of the treatment group while controlling for covariables such as... 

The PLS approach to multivariate linear regression modeling is relatively new and not yet fully investigated from a theoretical point of view. The results with calibrating complex samples in food analysis 122,123) j y jnfj-ared reflectance spectroscopy, suggest that PLS could solve the general calibration problem in analytical chemistry. 

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