Measurement covariance

Hagan M. M., Havel P. J., Seeley R. J. et al (1999). Cerebrospinal fluid and plasma leptin measurements covariability with dopamine and cortisol in fasting humans. J. Clin. Endocrinol. Metah. 84(10), 3579-85. 

Furthermore, under symplectic transformations, it is relatively easy to show, using the Hessian formula for calculating the Fisher information matrix, that the measurement covariance matrix transforms as... 

For this library the corresponding measurement covariance matrices are given by (14) with... 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

The matrix gp, represents the components of a covariant second-order tensor called the metric tensor , because it defines distance measurement with respect to coordinates To illustrate the application of this definition in the... 

Measurement noise covariance matrix R The main problem with the instrumentation system was the randomness of the infrared absorption moisture eontent analyser. A number of measurements were taken from the analyser and eompared with samples taken simultaneously by work laboratory staff. The errors eould be approximated to a normal distribution with a standard deviation of 2.73%, or a varianee of 7.46. 

Measurement noise covariance matrix %Disturbance matrix... 

As expecfed the optimal estimator is indeed a function of both the covariance of the atmospheric turbulence and the measurement noise. A matrix identity can be used to derive an equivalent form of Eq. 16 (Law and Lane, 1996)... 

The term inverted in Eq. 17 is the covariance of the measurements and consequently directly measurable,... 

The relationship between the noise and atmospheric covariances is also evident in Eq. 17. If the noise on the measurements is large the N term dominates the inverse which means only the large eigenvalues of C contribute to the inverse. Consequently only the low order modes are compensated and a smooth reconstruction results. When the data is very noisy then 1 and hence a tend to zero. If the data is very noisy, then no estimate of the basis coefficients is made. 

Another approach requires the use of Wilks lambda. This is a measure of the quality of the separation, computed as the determinant of the pooled within-class covariance matrix divided by the determinant of the covariance matrix for the whole set of samples. The smaller this is, the better and one selects variables in a stepwise way by including those that achieve the highest decrease of the criterion. 

We will see that CLS and ILS calibration modelling have limited applicability, especially when dealing with complex situations, such as highly correlated predictors (spectra), presence of chemical or physical interferents (uncontrolled and undesired covariates that affect the measurements), less samples than variables, etc. More recently, methods such as principal components regression (PCR, Section 17.8) and partial least squares regression (PLS, Section 35.7) have been... 

One expects that during the measurement-prediction cycle the confidence in the parameters improves. Thus, the variance-covariance matrix needs also to be updated in each measurement-prediction cycle. This is done as follows [1] ... 

The sequence of the innovation, gain vector, variance-covariance matrix and estimated parameters of the calibration lines is shown in Figs. 41.1-41.4. We can clearly see that after four measurements the innovation is stabilized at the measurement error, which is 0.005 absorbance units. The gain vector decreases monotonously and the estimates of the two parameters stabilize after four measurements. It should be remarked that the design of the measurements fully defines the variance-covariance matrix and the gain vector in eqs. (41.3) and (41.4), as is the case in ordinary regression. Thus, once the design of the experiments is chosen... 

Equations (41.15) and (41.19) for the extrapolation and update of system states form the so-called state-space model. The solution of the state-space model has been derived by Kalman and is known as the Kalman filter. Assumptions are that the measurement noise v(j) and the system noise w(/) are random and independent, normally distributed, white and uncorrelated. This leads to the general formulation of a Kalman filter given in Table 41.10. Equations (41.15) and (41.19) account for the time dependence of the system. Eq. (41.15) is the system equation which tells us how the system behaves in time (here in j units). Equation (41.16) expresses how the uncertainty in the system state grows as a function of time (here in j units) if no observations would be made. Q(j - 1) is the variance-covariance matrix of the system noise which contains the variance of w. 

At this point let us assume that the covariance matrices (E,) of the measured responses (and hence of the error terms) during each experiment are known precisely. Obviously, in such a case the ML parameter estimates are obtained by minimizing the following objective function... 

If we assume that the residuals in Equation 2.35 (e,) are normally distributed, their covariance matrix ( ,) can be related to the covariance matrix of the measured variables (COV(sy.,)= LyJ through the error propagation law. Hence, if for example we consider the case of independent measurements with a constant variance, i.e. 

Let us now consider models that have only more than one measured variable (w>l). The previously described model adequacy tests have multivariate extensions that can be found in several advanced statistics textbooks. For example, the book Introduction to Applied Multivariate Statistics by Srivastava and Carter (1983) presents several tests on covariance matrices. 

Step 6. Based on the additional measurement of the response variables, estimate the parameter vector and its covariance matrix. 

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