Degrees of freedom for the residuals of a principal component analysis — A clarification

Various conflicting proposals for degrees of freedom associated with the residuals of a principal component analysis have been published in the chemometrics-oriented literature. Here, a detailed derivation is given of the ‘standard’ formula from statistics. This derivation intends to be more accessible to chemometricians than, for example, the impeccable, but condensed proof that was published by John Mandel in a relatively unknown paper (J. Res. Nat. Bur. Stand., 74B (1970) 149–154). The derivation is presented in the form of a two-stage recipe that also appears to apply to more complex multiway models like the ones considered by Ceulemans and Kiers (Br. J. Math. Stat. Psych., 59 (2006) 133–150).     

Residual phase variance in partial correction: application to the estimate of the light intensity statistics

Although the wave-front correction provided by an adaptive optics system should be as complete as possible, only a partial compensation is attainable in the visible. An estimate of the residual phase variance in the compensated wave front can be used to calibrate system performance, but it is not a simple task when errors affect the compensation process. We propose a simple method for estimation of the residual phase variance that requires only the measurement of the Strehl ratio value. It provides good results over the whole range of compensation degrees. The estimate of the effective residual phase variance is useful not only for system calibration but also for determining the light intensity statistics to be expected in the image as a function of the degree of compensation introduced.     

An error estimate in the EFG method

In this paper, local and global error estimates for the element-free Galerkin (EFG) method are proposed. The essence of proposed error estimates is to use the difference between the values of the projected stress and these given directly by the EFG solution. The stress projection can be obtained simply by taking product of shape function based on a different domain of influence with the stresses at nodes. In this study, it was found that the effectivity index is optimized if the domain of influence in stress projection procedure is the smallest that retains regularity of the matrices in EFG. Numerical tests are shown for various 1D and 2D examples illustrating the good effectiveness of the proposed error estimator in the global energy norm and in the local error estimates.     

Postgrouped sampling method of estimation

A postgrouped sampling is considered for estimating the (finite or finite), population mean. Double sampling and an empirical-weighted estimator is used. Unbiasedness, variance and efficiency are considered. Its properties are discussed allowing the simple random sampling with replacement (SRSWR) design in the first phase, and in each stratum for the second phase. It is shown that for a fixed sample size in each postgroup, the variance of the proposed estimator with less prior information is asymptotically equivalent to the usual stratified estimator for fixed allocation. Some examples are provided for natural populations., The method is also extended to simple random sampling without replacement (SRSWOR) design in the first phase, and in each stratum for the second phase. Unbiased variance estimation is provided for both types of sampling designs.     

Cosmic radiation measurements on-board aircraft with the variance method

The microdosimetric variance-covariance method has been used for cosmic radiation measurements on-board aircraft. Two independent methods of data analysis are presented; the first based on a high energy neutron calibration and the second on identification of single high LET events in the measured multiple event spectrum. Reduced dose levels at high geomagnetic latitudes are observed on one flight in a period of enhanced solar activity as indicated by a reduced ground-level neutron fluence rate. It is shown that with a reduced 137Cs-calibration factor, a Geiger-Mueller tube can be used as a low LET monitor, and that the wall thickness of a tissue-equivalent proportional counter is not crucial for flight measurements. No covariance is observed on any flight indicating it is sufficient to base dose determinations on variance measurements with only one detector. The uncertainties involved are also discussed in some detail.     

结构非平稳随机响应方差数值计算方法

对受非平稳随机激励的工程结构,建立了在任意调制的演变随机白噪声和有色噪声激励下,结构响应方差矩阵的计算方法.提出了矩阵微分方程的数值解法,首先将矩阵微分方程转换为普通的代数微分方程,再应用常规的ODE求解器进行求解.该方法可以处理任意调制的演变随机白噪声或有色噪声,原理简明,易于掌握和应用.文中介绍了相应的关键计算步骤.数值仿真算例显示了该方法的正确性和可行性.     

An estimate of the truncation error in the asynchronous sampling of ac signals containing inter- and subharmonics

The estimation of the truncation error in the asynchronous sampling of an ac signal containing sub- and interharmonic components is considered. The expressions derived enable one to choose the optimum method and the necessary number of samples of a complex input signal. The limits of the error of the method of signal processing are determined taking the effect of sampling into account. Mathematical modeling of the procedure for estimating the possible truncation error is carried out.     

Optimal cluster selection probabilities to estimate the finite population distribution function under PPS cluster sampling

The estimation of the finite population distribution function under several sampling strategies based on a PPS cluster sampling, i.e., with cluster selection probabilities proportional to size, is studied. For the estimation of population means and totals, it is well-known that this type of strategies gives good results if the cluster selection probabilities are proportional to the total of the variable under study or to a related auxiliary variable over the cluster. It is proved that, for the estimation of the distribution function using cluster sampling, this solution is not good in general and, under an appropriate criteria, the optimal cluster selection probabilities that minimize the variance of the estimation, is obtained. This methodology is applied to two classical PPS sampling strategies: sampling with replacement, with the Hansen-Hurwitz estimator, and random groups sampling with the Rao-Hartley-Cochran estimator. Finally a small simulation to compare the efficiency of this approach with other methods is presented.     

Features of the sampling distribution of the ability estimate in computerized adaptive testing according to two stopping rules

Whether paper and pencil or computerized adaptive, tests are usually described by a set of rules managing how they are administered: which item will be first, which should follow any given item, when to administer the last one. This article focus on the latter and looks at the effect of two stopping rules on the estimated sampling distribution of the ability estimate in a CAT: the number of items administered and the a priori determined size of the standard error of the ability estimate.     

An estimate of the error of measuring sensitivity by the SINAD method

The limits of applicability of the SINAD method when measuring the signal-to-noise ratio, often used to estimate the sensitivity of receivers, are investigated. It is shown that if the systematic component of the error in measuring the harmonic coefficients of the generator being measured – the signal source – and of the high-frequency part of the receiver is ignored, there will be an unlimited increase in the error in measuring the sensitivity when the signal-to-noise ratio approaches the limiting value. The approach described enables the contribution to this error of components arising from distortions and noise of different origin to be separated.     

A CUSUM scheme with variable sample sizes and sampling intervals for monitoring the process mean and variance

The adaptive control feature and CUSUM chart are two monitoring schemes that are much more effective than the traditional static Shewhart chart in detecting process shifts in mean and variance. However, the designs and analyses of the adaptive CUSUM chart are mathematically intractable and the operation is very laborious. This article proposes a VSSI WLC scheme, which is a weighted‐loss‐function‐based CUSUM (WLC) scheme using variable sample sizes and sampling intervals (VSSI). This scheme detects the two‐sided mean shift and increasing standard deviation shift based on a single statistic WL (the weighted loss function). Most importantly, the VSSI WLC scheme is much easier to operate and design than a VSSI CCC scheme which comprises three individual CUSUM charts (two of them monitoring the increasing and decreasing mean shifts and one monitoring the increasing variance shift). Overall, the VSSI WLC scheme is much more effective than the static &S charts (by 72.36%), the VSSI &S charts (by 30.97%) and the static WLC scheme (by 50.94%) for detection. It is even more effective than the complicated VSSI CCC scheme for most cases. Copyright © 2006 John Wiley & Sons, Ltd.     

Monitoring the process mean and variance using a weighted loss function CUSUM scheme with variable sampling intervals

Recent research has shown that adaptive control charts and the CUmulative SUM (CUSUM) schemes are quicker in detecting process shifts than traditional static Shewhart charts. This article proposes a weighted loss function CUSUM (WLC) scheme with Variable Sampling Intervals (VSI). It simultaneously monitors both mean shifts and an increasing variance shift by manipulating a single CUSUM chart. Most importantly, this VSI WLC scheme is much easier to operate and design than a VSI CCC scheme which comprises of three CUSUM charts (two of them monitoring the increasing and decreasing mean shifts and one monitoring the increasing variance shift). In terms of detection efficiency, the VSI WLC scheme is a much more powerful tool than the static X&S chart, the VSI X&S chart and the static WLC scheme. It is even more powerful than the VSI CCC scheme for many different combinations of mean and increasing variance shifts.     

A note on calculation of the average fraction inspected for a continuous sampling plan

Ad important characteristic of a continuous sampling plan is the average fraction inspected. This paper presents a computationally efficient and easily programmable solution for calculation of the average fraction inspected.     

High-resolution digital holography utilized by the subpixel sampling method

A novel (to our knowledge) approach for resolution improvement in digital holography is presented in this paper. The proposed method is based on recording the incoming interference field on a complementary metal-oxide semiconductor (CMOS) camera with subpixel resolution. The method takes advantage of the small pixel size of the CMOS sensor, while overcoming the reduced fill factor. This paper describes the experimental and numerical procedures. The improvement of the obtainable optical resolution, image quality, and phase measurement accuracy are demonstrated within this paper.     

Reduction of the degrees of freedom in solving dynamic problems by the finite element method

The method of lowering the order of a matrix equation which describes the dynamics of a finite element model is presented. The finite element with a ‘truncated’ mass matrix is obtained for calculating thin plate vibrations. Such an element has one vibrational degree of freedom at each nodal point. In the case of uniform systems the accuracy provided by the suggested element is no less than that provided by the non-conforming elements, which have three vibrational degrees of freedom at each nodal point, and in some cases it is greater. The finite elements with a ‘truncated’ mass matrix have essential advantages in the study of non-uniform systems.     

A continuous sampling plan using sums of conforming run‐lengths

A continuous sampling plan, CSP‐SUM, is proposed based on the use of sums of run‐lengths of conforming items for deciding when to switch between the phases of sampling inspection and 100% inspection. The conventional measures of performance of CSPs such as the average outgoing quality, average fraction inspected, and the proportion passed under sampling inspection are evaluated for CSP‐SUM. Using these and other measures, comparisons with some standard CSPs are provided and indicate better performance for CSP‐SUM. Comparisons are also made with a recently‐proposed CSP utilizing CUSUMs, termed CSP‐CUSUM, and indicate that the performance of CSP‐SUM can be close to that of CSP‐CUSUM. A table is provided to aid the choice of parameters for the operation of CSP‐SUM. It is recommended that a conforming run‐length control chart be maintained in parallel with CSP‐SUM to detect significant upward shifts in the fraction defective of the process. Copyright © 2003 John Wiley & Sons, Ltd.     

A method to estimate the octanol-air partition coefficient of semivolatile organic compounds

A multicolumn method for the estimation of the octanol-air partition coefficient (K(OA)) of semivolatile compounds is described. The method is based on the retention time of a compound on gas chromatographic columns of different selectivity and polarity. Log K(OA) values of polychlorinated biphenyls (PCB) were estimated with the novel multicolumn method. The results were in excellent agreement with previously published data using a generator column method. Therefore, we used our multicolumn method to estimate K(OA) for more than 100 PCB congeners.