Dependent estimates, estimated covariance between

Using this design, the covariances between the estimates of and P, and between the estimates of P, and P, are zero. This is confirmed in Figures 8.11 and 8.13. Thus, the estimation of Po does not depend on the estimated value of P, (and vice versa), and the estimated values of P, and P, do not depend on the estimated values of each other. 

One important practical aspect of PLS is that it takes into account errors both in the concentration estimates and spectra. A method such as PCR will assume that the concentration estimates are error free. Much traditional statistics rest on this assumption, that all errors are in the dependent variables (spectra). If in medicine it is decided to determine the concentration of a compound in the urine of patients as a function of age, it is assumed that age can be estimated exactly, the statistical variation being in the concentration of a compound and the nature of the urine sample. Yet in chemistry there are often significant errors in sample preparation, for example, accuracy of weighings and dilutions and so the independent variable (c) in itself also contains errors. With modem spectrometers, these are sometimes larger than spectroscopic errors. One way of overcoming this difficulty is to try to minimise the covariance between both types of variables, namely the x (spectroscopic) and c (concentration) variables. 

If the equity value is estimated by using the discounted cash flow method, the cost of capital assumes a particular relevance. Conventionally, the cost of capital is estimated through the capital asset pricing model (CAPM) that was introduced by Sharpe (1964) and subsequently improved by Lintner (1965). One of the most important variables is beta. Beta measures the sensitivity of the asset s or company returns to variation in the market or index returns. Therefore, according to CAPM theory, the risk assumed from an investor depends on the covariance (or correlation) between individual assets and market portfolio. Thus, if these singular assets do not have correlation, they will not add risk differently, if the correlation is positive they will add risk on market portfolio. 

The distance between object points is considered as an inverse similarity of the objects. This similarity depends on the variables used and on the distance measure applied. The distances between the objects can be collected in a distance matrk. Most used is the euclidean distance, which is the commonly used distance, extended to more than two or three dimensions. Other distance measures (city block distance, correlation coefficient) can be applied of special importance is the mahalanobis distance which considers the spatial distribution of the object points (the correlation between the variables). Based on the Mahalanobis distance, multivariate outliers can be identified. The Mahalanobis distance is based on the covariance matrix of X this matrix plays a central role in multivariate data analysis and should be estimated by appropriate methods—mostly robust methods are adequate. 

Another factor to be taken into account is the degree of over determination, or the ratio between the number of observations and the number of variable parameters in the least-squares problem. The number of observations depends on many factors, such as the X-ray wavelength, crystal quality and size, X-ray flux, temperature and experimental details like counting time, crystal alignment and detector characteristics. The number of parameters is likewise not fixed by the size of the asymmetric unit only and can be manipulated in many ways, like adding parameters to describe complicated modes of atomic displacements from their equilibrium positions. Estimated standard deviations on derived bond parameters are obtained from the least-squares covariance matrix as a measure of internal consistency. These quantities do not relate to the absolute values of bond lengths or angles since no physical factors feature in their derivation. 

Under Hanemann s (1984, 1989) well known linkage between random utility maximization and the functional form of econometric models with a binary dependent variable, a logit model has been estimated on SB-CVM data. It explains the log-odds ratio as a linear function of several household attributes (including income level as a covariate) and of the percentage premium price proposed (Franses and Paap, 2001 Gourieroux, 2000). Median WTP and truncated mean WTPs (both only at zero and between zero and 100%) have been calculated according to Hanemann and Kanninen... 

We should be aware that the estimations of the confidence intervals in the given way will only be valid if the parameters are independent of each other. All elements in the ofF-diagonals of the covariance matrix in Eq. (6.23) need to be zero. In the case of two parameters, the confidence intervals describe a square (cf. figure in the margin). If dependences exist between parameters, then an ellipse is obtained for the confidence interval of the parameters. The larger the elements in the ofF-diagonals in Eq. (6.23), the more pronounced deviations from the squared shape of the confidence intervals are to be expected. 

A determinant criterion is used to obtain least-squares estimates of model parameters. This entails minimizing the determinant of the matrix of cross products of the various residuals. The maximum likehhood estimates of the model parameters are thus obtained without knowledge of the variance-covariance matrix. The residuals e, , and correspond to the difference between predicted and actual values of the dependent variables at the different values of the Mth independent variable (m = to to u = tn), for the ith, 7th, and kth experiments (A, B, and C), respectively. It is possible to constmct an error covariance matrix with elements v,y ... 

One of the first concepts to increase the robustness of robust PLS was introduced by Gil and Romera [35]. They used robust covariance and crosscovariance between X and the dependent variable, y, adopting the Stahel-Donoho estimator of data scatter with Huber s weighing function [36] for this purpose. Another attempt was made by Cummins and Andrews [37]. They introduced an iterative approach based on down-weighting the influence of samples that have large residuals from the constructed PLS model. The sample weights are modified iteratively during the construction of the PLS model. 

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