Unique variance

The main difference between factor analysis and principal component analysis is the way in which the variances of Eq. (8.20) are handled. Whereas the interest of FA is directed on the common variance var Xij)comm and both the other terms are summarized as unique variance... 

In the second study the same group evaluated the effects of prenatal cocaine exposure on child behavior in 506 African-American mother-child pairs (363). The mothers were identified as cocaine users and non-users during the initial prenatal visits with urine screen confirmation. Offspring behavior was assessed 6-7 years later using caregiver reports with the Achenbach Child Behavior Checklist (CBCL). Analyses stratified by sex and prenatal alcohol exposure showed that behaviors in girls without prenatal alcohol exposure but with prenatal cocaine exposure were adverse 6.5% of the unique variance in behavior was related to prenatal cocaine... 

Principal component analysis is used to reduce the information in many variables into a set of weighted linear combinations of those variables it does not differentiate between common and unique variance. If latent variables have to be determined, which contribute to the common variance in a set of measured variables, factor analysis (FA) is a valuable statistical method, since it attempts to exclude unique variance from the analysis. 

The a statistic, which measures the portion of variance of v which is identified as unique variance, i.e. variance not shared with other variables. 

Unique variance of CPT explained by CBLI metrics with increasing criterion BLLs found increasing variance from CBLI 20 (R = 5.8%,p<0.03)to CBLI 60 (R =23.3%,p < 0.005). Significant interaction between CBLI 60 and measure of ergonomic stess of upper extremities was foimd (R =... 

Safety precepts should allow for tailoring or options in the specific implementation, since different applications may have some unique variances. 

The Standard Error of Prediction (SEP) is supposed to refer uniquely to those situations when a calibration is generated with one data set and evaluated for its predictive performance with an independent data set. Unfortunately, there are times when the term SEP is wrongly applied to the errors in predicting y variables of the same data set which was used to generate the calibration. Thus, when we encounter the term SEP, it is important to examine the context in order to verify that the term is being used correctly. SEP is simply the square root of the Variance of Prediction, s2. The RMSEP (see below) is sometimes wrongly called the SEP. Fortunately, the difference between the two is usually negligible. 

Previously, in the case of constant detector noise, we then set Var(A s) and Var(A r) equal to the same value. This is the point at which must we now depart from the previous derivation, since in the case of Poisson-distributed noise the sample and reference noise levels will rarely, if ever, be the same. However, we are fortunate in this case that Poisson-distributed noise has a unique and very useful property that we have indeed previously made use of the variance of Poisson-distributed noise is equal to the mean signal value. Hence we can substitute Es for Var(A s) and Er for Var(A r) ... 

In a CFD calculation, one is usually interested in computing only the reacting-scalar means and (sometimes) the covariances. For binary mixing in the equilibrium-chemistry limit, these quantities are computed from (5.154) and (5.155), which contain the mixture-fraction PDF. However, since the presumed PDF is uniquely determined from the mixture-fraction mean and variance, (5.154) and (5.155) define mappings (or functions) from (I>- space ... 

The absorption spectra of the TIN(planar) and TIN(non-planar) forms can be resolved using the mathematical least-squares method of Principal Component Analysis (PCOMP) (26-28). In this analysis, the spectra of the two components can be calculated from a series of spectra containing different proportions of each component and no assumptions are made as to the shapes of the curves. A unique solution is obtained only if the two components have zero intensity at different wavelengths, otherwise a range of solutions is obtained. The unique component spectra obtained account for 99.99 percent of the variance of the original data. 

There is a problem with this approach, however - a problem with the residuals. The residuals are neither parameters of the model nor parameters associated with the uncertainty. They are quantities related to a parameter that expresses the variance of the residuals, The problem, then, is that the simultaneous equations approach in Equation 5.25 would attempt to uniquely calculate three items (P, r, and r,2) using only two experiments, clearly an impossible task. What is needed is an additional constraint to reduce the number of items that need to be estimated. A unique solution will then exist. 

In both the Ito and Stratonovich formulations, the randomness in a set of SDEs is generated by an auxiliary set of statistically independent Wiener processes [12,16]. The solution of an SDE is defined by a hmiting process (which is different in different interpretations) that yields a unique solution to any stochastic initial value problem for each possible reahzation of this underlying set of Wiener processes. A Wiener process W t) is a Gaussian Markov diffusion process for which the change in value W t) — W(t ) between any two times t and t has a mean and variance... 

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