Random covariance

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. 

Finally, an infinite set of random vectors is defined to be statistically independent if all finite subfamilies are statistically independent. Given an infinite family of identically distributed, statistically independent random vectors having finite means and covariances, we define their normalized sum to be the vector sfn, , sj where... 

In astronomy, we are interested in the optical effects of the turbulence. A wave with complex amplitude U(x) = exp[ irefractive index, resulting in a random phase structure by the time it reaches the telescope pupil. If the turbulence is weak enough, the effect of the aberrations can be approximated by summing their phase along a path (the weak phase screen approximation), then the covariance of the complex amplitude at the telescope can be shown to be... 

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. 

Just as in everyday life, in statistics a relation is a pair-wise interaction. Suppose we have two random variables, ga and gb (e.g., one can think of an axial S = 1/2 system with gN and g ). The g-value is a random variable and a function of two other random variables g = f(ga, gb). Each random variable is distributed according to its own, say, gaussian distribution with a mean and a standard deviation, for ga, for example, (g,) and oa. The standard deviation is a measure of how much a random variable can deviate from its mean, either in a positive or negative direction. The standard deviation itself is a positive number as it is defined as the square root of the variance ol. The extent to which two random variables are related, that is, how much their individual variation is intertwined, is then expressed in their covariance Cab ... 

If two random variables are nncorrelated, then both their covariance Cab and their correlation coefficient rab are equal to zero. If two random variables are fully correlated, then the absolute value of their covariance is C,J = cacb, and the absolute value of their correlation coefficient is unity rab = 1. A key point to note for our EPR linewidth theory to be developed is that two fully correlated variables can be fully positively correlated rab = 1, or fully negatively correlated rab = -1. Of course, if two random variables are correlated to some extent, then 0 < Cab < oacb, and 0 < IrJ < 1. 

Equation 41-A3 can be checked by expanding the last term, collecting terms and verifying that all the terms of equation 41-A2 are regenerated. The third term in equation 41-A3 is a quantity called the covariance between A and B. The covariance is a quantity related to the correlation coefficient. Since the differences from the mean are randomly positive and negative, the product of the two differences from their respective means is also randomly positive and negative, and tend to cancel when summed. Therefore, for independent random variables the covariance is zero, since the correlation coefficient is zero for uncorrelated variables. In fact, the mathematical definition of uncorrelated is that this sum-of-cross-products term is zero. Therefore, since A and B are random, uncorrelated variables ... 

First, we conjecture that each item is a separate marker of the taxon. Now there is a pool of indicators to work with. Second, we choose a random pair of indicators, for example, item 1 ( When I go shopping, I check several times to be sure I have my wallet/ purse with me ) and item 2 ( Before I leave my house, I check whether all windows are closed ), as the output indicators. Third, we sum scores on items 3 to 8, which makes a 7-point scale that ranges from 0 (none of the 6 checking behaviors are endorsed) to 6 (all of the 6 checking behaviors are endorsed) this is the input variable. Fourth, we calculate the covariance between items 1 and 2 in a subsample of individuals who scored 0 on the input variable, next we calculate the covariance for individuals who scored 1 on the input variable, and so forth. Fifth, we choose another pair of output indicators (e.g., items 1 and 3), and combine the other six items together to make a new input variable. This process is repeated 28 times until all possible pairs are drawn (1-2 and 2-1 are not considered different pairs). Next, we take 28 covariances from 0 subsamples and average them we do the same for all seven sets of numbers and plot the average covariances. SSMAXCOV plots look similar to the plots from the MAXCOV section and are interpreted the same way. 

Here 4 is the target state vector at time index k and Wg contains two random variables which describe the unknown process error, which is assumed to be a Gaussian random variable with expectation zero and covariance matrix Q. In addition to the target dynamic model, a measurement equation is needed to implement the Kalman filter. This measurement equation maps the state vector t. to the measurement domain. In the next section different measurement equations are considered to handle various types of association strategies. 

The vector nk describes the unknown additive measurement noise, which is assumed in accordance with Kalman filter theory to be a Gaussian random variable with zero mean and covariance matrix R. Instead of the additive noise term nj( in equation (20), the errors of the different measurement values are assumed to be statistically independent and identically Gaussian distributed, so... 

The adjustment of measurements to compensate for random errors involves the resolution of a constrained minimization problem, usually one of constrained least squares. Balance equations are included in the constraints these may be linear but are generally nonlinear. The objective function is usually quadratic with respect to the adjustment of measurements, and it has the covariance matrix of measurements errors as weights. Thus, this matrix is essential in the obtaining of reliable process knowledge. Some efforts have been made to estimate it from measurements (Almasy and Mah, 1984 Darouach et al., 1989 Keller et al., 1992 Chen et al., 1997). The difficulty in the estimation of this matrix is associated with the analysis of the serial and cross correlation of the data. 

Most techniques for process data reconciliation start with the assumption that the measurement errors are random variables obeying a known statistical distribution, and that the covariance matrix of measurement errors is given. In Chapter 10 direct and indirect approaches for estimating the variances of measurement errors are discussed, as well as a robust strategy for dealing with the presence of outliers in the data set. 

Assume that s is a random error vector with zero mean and covariance P, that is,... 

In the previous development it was assumed that only random, normally distributed measurement errors, with zero mean and known covariance, are present in the data. In practice, process data may also contain other types of errors, which are caused by nonrandom events. For instance, instruments may not be adequately compensated, measuring devices may malfunction, or process leaks may be present. These biases are usually referred as gross errors. The presence of gross errors invalidates the statistical basis of data reconciliation procedures. It is also impossible, for example, to prepare an adequate process model on the basis of erroneous measurements or to assess production accounting correctly. In order to avoid these shortcomings we need to check for the presence of gross systematic errors in the measurement data. 

Summarizing, the statistical characterisation of the random process (mean and covariance) can be projected through the interval tk < t < tk+1, and in this process there is an input noise that will increase the error, damaging the quality of the estimate. 

Cases 4 and 5 deserve some special consideration. They were performed under the same conditions in terms of noise and initial parameter value, but in case 5 the covariances (weights) of the temperature measurements were increased with respect to those in the remaining measurements. For case 4 it was noticed that, although a normal random distribution of the errors was considered in generating the measurements, some systematic errors occur, especially in measurement numbers 6, 8,... 

The estimation of means, variances, and covariances of random variables from the sample data is called point estimation, because one value for each parameter is obtained. By contrast, interval estimation establishes confidence intervals from sampling. 

However, care must be taken to avoid the singularity that occurs when C is not full rank. In general, the rank of C will be equal to the number of random variables needed to define the joint PDF. Likewise, its rank deficiency will be equal to the number of random variables that can be expressed as linear functions of other random variables. Thus, the covariance matrix can be used to decompose the composition vector into its linearly independent and linearly dependent components. The joint PDF of the linearly independent components can then be approximated by (5.332). 

The eigenvalue/eigenvector decomposition of the covariance matrix thus allows us to redefine the problem in terms of Nc independent, standard normal random variables 0in. 

For a vector X of n random variables with mean vector p and nxn symmetric covariance-matrix an m-point sample is a matrix X with n rows and m columns. 

Parallel to the case of a single random variable, the mean vector and covariance matrix of random variables involved in a measurement are usually unknown, suggesting the use of their sampling distributions instead. Let us assume that x is a vector of n normally distributed variables with mean n-column vector ft and covariance matrix L. A sample of m observations has a mean vector x and annxn covariance matrix S. The properties of the t-distribution are extended to n variables by stating that the scalar m(x—p)TS ( —p) is distributed as the Hotelling s-T2 distribution. The matrix S/m is simply the covariance matrix of the estimate x. There is no need to tabulate the T2 distribution since the statistic... 

A random n-vector X has a mean vector fi and an n x n covariance matrix . a is the diagonal matrix with standard deviations as diagonal terms and p the correlation matrix. Find the correlation matrix of the reduced vector given by... 

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