The USAF Multivariate Accommodation Method

Genome-wide association studies (GWAS) are susceptible to bias due to population stratification (PS). The most widely used method to correct bias due to PS is principal components (PCs) analysis (PCA), but there is no objective method to guide which PCs to include as covariates. Often, the ten PCs with the highest eigenvalues are included to adjust for PS. This selection is arbitrary, and patterns of local linkage disequilibrium may affect PCA corrections. To address these limitations, we estimate genomic propensity scores based on all statistically significant PCs selected by the Tracy-Widom (TW) statistic. We compare a principal components and propensity scores (PCAPS) approach to PCA and EMMAX using simulated GWAS data under no, moderate, and severe PS. PCAPS reduced spurious genetic associations regardless of the degree of PS, resulting in odds ratio (OR) estimates closer to the true OR. We illustrate our PCAPS method using GWAS data from a study of testicular germ cell tumors. PCAPS provided a more conservative adjustment than PCA. Advantages of the PCAPS approach include reduction of bias compared to PCA, consistent selection of propensity scores to adjust for PS, the potential ability to handle outliers, and ease of implementation using existing software packages.

Aim Bergmann's rule, one of the most studied and controversial ecogeographical generalizations, has rarely been tested with observations from high latitudes. We tested the rule using cranial measurements of the muskox [Ovibos moschatus (Zimmerman)], a homeotherm with an extremely northern distribution. We also used these data to describe geographical patterns in the species' dental architecture, an extension of the framework developed from interspecific comparisons.Location Specimens were compiled from arctic Canada, Alaska and Greenland, a latitudinal range of 60° N−83° N.Methods Body size was estimated from principal components analysis (PCA) of five cranial characters from 128 specimens. Mean scores on the first principal component from each locality were regressed against latitude and mean temperature to identify geographical variation in body size; scores on the second principal component were regressed against latitude to assess patterns in dental architecture. Regression analyses of the individual characters were performed as a complement to PCA.Results No latitudinal or climatic trend in body size was observed in either sex. On the other hand, for males, significant latitudinal variation was found for the second PCA axis (r = −0.434), and the feature which loaded most heavily on it, maxillary tooth row length (r = 0.429). For females, this dental structure also tended to increase with latitude (r = 0.423), but the trend was only marginally significant (P=0.12), perhaps owing to a smaller sample size.Main conclusions The geographically invariant body size of muskoxen failed to support current hypotheses of size variation. Behavioural and physiological adaptations may exempt the muskox from selective pressures underlying these hypotheses. We interpret latitudinal variation in dental architecture as a reflection of a cline in diet, dominated by graminoids at the expense of willows at higher latitudes. This intraspecific geographical trend is a recapitulation of the interspecific framework for large mammalian herbivores.

Children begin to train for the competitive swimming specifically from early age, therefore they can attain high performance at early childhood. In their background, coaches tend to impose hard training for children without considering physical maturity. The consideration for the physical maturity is necessary to impose the adequate training. Furthermore, revealing factors affecting swimming performance and developmental pattern of the children is essential. PURPOSE This study aims to quantify the effective factors to swimming performance at various stages of physical maturity, and then clarifies their developmental patterns in young swimmers. METHODS The participants were 175 boys and 213 girls aged 8 to 18 years old having regular training (3 times or more per week) at their swimming clubs. They were classified into three groups: earlier adolescence, later adolescence, and young adulthood according to their velocity curves of stature. We hypothesize that body size, muscle strength per body weight, flexibility, body fat and stroke technique strongly affect the swimming performance. The swimming performance defined as personal best record of 50m. A principal component analysis was applied to 11 measurements, and principal component score related to body size, muscle strength per body weight and flexibility was extracted. We applied simultaneous analysis of multiple groups to the multiple regression models of body fat (%fat), stroke technique (stroke index) and to the 3 variables from principal component analysis for each sex and maturity stage. RESULTS In the sexes, stroke technique was the most effective parameter to explain the swimming performance throughout adolescence, but its effect decreased in young adult. In contrast, body size had low contribution to the swimming performance in earlier adolescence, and increased with growth. The effects of other factors were low. CONCLUSIONS The swimming performance was affected by increasing body size, on the other hand, decreasing stroke technique with growth. Developments of fat free mass and absolute muscle strength are the accountable reason for these results.

Principal components analysis was carried out on 13 morphological dimensions collected in the first phase of the Québec Family Study (weight, height, fat mass (FM), fat-free mass (FFM), body surface area, six skinfolds, arm and calf girths). The first four principal components (PCs) account for 85.9% of the total phenotypic variance in the anthropometric dimensions. The first PC has almost equal loadings on the 13 variables, indicating its role as an overall body size and adiposity measure. This interpretation is supported by its high intraindividual correlations with other composite indices such as the BMI and sum of six skinfolds (SF6). An interesting finding compared to other studies dealing with PCs of only skinfolds is that the second component appears to contrast measures of body fat with those that are fat free or that represent overall body size, thus indexing a different dimension to the fat patterning. It explains 17% of the total phenotypic variance. The commonly observed component for the trunk-extremity contrast is represented as the fourth principal component. The third component primarily indexes height, although an extremity-trunk contrast is also weakly represented. Together, the 3rd and 4th components explain ∼10% of the total variance. The intraindividual correlation of PC3 with the trunk-extremity skinfold ratio (TER) is moderately negative (-0.307), whereas that of PC4 is highly positive (0.764). Analysis of familial correlations (parent-offspring and sibling) shows that the familial effect (transmissibility) ranges 0.43-0.75 for the first four PCs. The lack of significant spouse correlation for most of the PCs suggests that these familial effects may be primarily genetic. Am. J. Hum. Biol. 9:725-733, 1997. © 1997 Wiley-Liss, Inc.

Principal Component Analysis (PCA), also known as Proper Orthogonal Decomposition (POD), is one of the prevailing Reduced Order Modeling (ROM) techniques for aerodynamic data analysis. Along with the PCA, its variant for missing data, gappy POD, has recently found its application by transforming problems into missing data problems. In this paper, an alternative based on probability theory, i.e., Probabilistic Principal Component Analysis (PPCA), is employed for aerodynamic data reconstruction problems in lieu of the PCA and the gappy POD. Unlike the PCA, the PPCA generates spanning vectors of low-dimensional space iteratively with an Expectation Maximization (EM) algorithm. The combined algorithm, termed as Expectation-Maximization Probabilistic Principal Component Analysis (EM-PCA), can deal with not only complete data but also incomplete data sets, since the EM algorithm naturally incorporates missing data in its formulation. Therefore, the EM-PCA has potential to be a substitute for both PCA and gappy POD. For illustration, the EM-PCA is applied to complete as well as incomplete snapshot ensembles compiled from aerodynamic simulations, and acquired modes and restored data are verified with those obtained by the PCA and the gappy POD.

Groundwater is considered as one of the most important sources for water supply in Iran. The Fasa Plain in Fars Province, Southern Iran is one of the major areas of wheat production using groundwater for irrigation. A large population also uses local groundwater for drinking purposes. Therefore, in this study, this plain was selected to assess the spatial variability of groundwater quality and also to identify main parameters affecting the water quality using multivariate statistical techniques such as Cluster Analysis (CA), Discriminant Analysis (DA), and Principal Component Analysis (PCA). Water quality data was monitored at 22 different wells, for five years (2009-2014) with 10 water quality parameters. By using cluster analysis, the sampling wells were grouped into two clusters with distinct water qualities at different locations. The Lasso Discriminant Analysis (LDA) technique was used to assess the spatial variability of water quality. Based on the results, all of the variables except sodium absorption ratio (SAR) are effective in the LDA model with all variables affording 92.80% correct assignation to discriminate between the clusters from the primary 10 variables. Principal component (PC) analysis and factor analysis reduced the complex data matrix into two main components, accounting for more than 95.93% of the total variance. The first PC contained the parameters of TH, Ca2+, and Mg2+. Therefore, the first dominant factor was hardness. In the second PC, Cl-, SAR, and Na+ were the dominant parameters, which may indicate salinity. The originally acquired factors illustrate natural (existence of geological formations) and anthropogenic (improper disposal of domestic and agricultural wastes) factors which affect the groundwater quality.

Principal component analysis and hierarchical cluster analysis were applied to investigate physicochemical and instrumental textural properties of fresh kashar cheese. Four different principal components sufficiently explained the variability in the cheese samples. In addition, hierarchical cluster analysis was performed to group the kashar cheese samples regarding physicochemical and instrumental textural properties. Instrumental textural properties indicated greater variability than chemical composition of cheese samples. Principal component analysis revealed that color parameters were positively correlated with textural and chemical parameters. The results of this study revealed that other parameters rather than chemical composition would be effective on the instrumental textural properties. It was proved that principal component analysis was a very effective statistical tool to determine quality of cheese samples. According to the principal component analysis and hierarchical cluster analysis results, the attributes defining the kashar cheese samples were determined to be primarily the texture profile analysis parameters.

This study integrated conventional statistical tools with neighbourhood linkage to propose the statistical diagnosis approach. Fourteen monitoring wells in Kaohsiung Science Park were selected as study case, and lab data of routine groundwater analysis including pH, EC, hardness, TDS, TOC, ammonia, nitrate, nitrite, chloride, sulphate, fluoride, phenols, Fe, Mn, As, and temperature were subjected to principal component and cluster analysis. Principal component analysis (PCA) was utilized to reflect those chemical data with the greatest correlation, and PCA results identified five major principal components (PCs) representing 74.6% of cumulative variance. Based on the monitoring data between 2005 and 2008, the extracted information from the PCA mirrored the potential sources of groundwater contamination as acid leakage, arsenic dissolution, salinization, mineralization, and fluoride release. Cluster analysis (CA) was used to evaluate the similarities of water quality in groundwater samples, and five clusters were assigned in two-step clustering for corresponding with the number of PCs, i.e. the potential sources of groundwater contamination. The interpreted facts from CA illustrated that the classified monitoring wells in each cluster properly match up with the identified processes. With the aid of neighbourhood linkage, the domain of groundwater contamination can be spatially outlined by mapping the neighbouring wells within the identical cluster. Therefore, the nature of underlying processes affecting groundwater quality was explored by statistical diagnosis.

The aim of the present article was to reconsider several conclusions by Velicer and Jackson (1990a) in their review of issues that arise when comparing common factor analysis and principal component analysis. Specifically, the three conclusions by Velicer and Jackson that are considered in the present article are: (a) that common factor and principal component solutions are similar, (b) that differences between common factor and principal component solutions appear only when too many dimensions are extracted, and (c) that common factor and principal component parameters are equally generalizable. In contrast, Snook and Gorsuch (1989) argued recently that principal component analysis and common factor analysis led to different, dissimilar estimates of pattern loadings, terming the principal component loadings biased and the common factor loadings unbiased. In the present article, after replicating the Snook and Gorsuch results, an extension demonstrated that the difference between common factor and principal component pattern loadings is inversely related to the number of indicators per factor, not to the total number of observed variables in the analysis, countering claims by both Snook and Gorsuch and Velicer and Jackson. Considering the more general case of oblique factors, one concomitant of overrepresentation of pattern loadings is an underrepresentation of intercorrelations among dimensions represented by principal component analysis, whereas comparable values obtained using factor analysis are accurate. Differences in parameters deriving from principal component analysis and common factor analysis were explored in relation to several additional aspects of population data, such as variation in the level of communality of variables on a given factor and the moving of a variable from one battery of measures to another. The results suggest that principal component analysis should not be used if a researcher wishes to obtain parameters reflecting latent constructs or factors.

The aims of this study were twofold: first, to determine the impact of variance in dose‐volume histograms (DVH) on patient‐specific toxicity after 2 high‐dose fractions in a sample of 22 men with prostate cancer; and second, to compare the effectiveness of traditional DVH analysis and principal component analysis (PCA) in predicting rectum and urethra toxicity. A series of 22 patients diagnosed with prostate adenocarcinoma was treated with 45 Gy external beam and 20 Gy dose rate brachytherapy. Principal component analysis was applied to model the shapes of the rectum and urethra dose‐volume histograms. We used logistic regression to measure the correlations between the principal components and the incidence of rectal bleeding and urethra stricture. We also calculated the equivalent uniform dose (EUD) and normal tissue complication probability (NTCP) for the urethra and rectum, and tumor control probability (TCP) for the prostate using BioSuite software. We evaluated their correlations with rectal and urethra toxicity. The rectum DVHs are well described by one principal component (PC1), which accounts for 93.5% of the variance in their shapes. The urethra DVHs are described by two principal components, PC1 and PC2, which account for 94.98% and 3.15% of the variance, respectively. Multivariate exact logistic regression suggests that urethra PC2 is a good predictor of stricture, with Nagelkerke's R2 estimated at 0.798 and a Wald criterion of 5.421 (p<0.021). The average NTCPs were 0.06%±0.04% and 1.25%±0.22% for the rectum and urethra, respectively. The average TCP was 85.29%±2.28%. This study suggests that principal component analysis can be used to identify the shape variation in dose‐volume histograms, and that the principal components can be correlated with the toxicity of a treatment plan based on multivariate analysis. The principal components are also correlated with traditional dosimetric parameters.PACS number: 3.6.96.0

Principal component analysis (PCA) and minor component analysis (MCA) are two important statistical tools which have many applications in the fields of signal processing and data analysis. PCA and MCA neural networks (NNs) can be used to online extract principal component and minor component from input data. It is interesting to develop generalized learning algorithms of PCA and MCA NNs. Some novel generalized PCA and MCA learning algorithms are proposed in this paper. Convergence of PCA and MCA learning algorithms is an essential issue in practical applications. Traditionally, the convergence is studied via deterministic continuous-time (DCT) method. The DCT method requires the learning rate of the algorithms to approach to zero, which is not realistic in many practical applications. In this paper, deterministic discrete-time (DDT) method is used to study the dynamical behaviors of the proposed algorithms. The DDT method is more reasonable for the convergence analysis since it does not require constraints as that of the DCT method. It is proven that under some mild conditions, the weight vector in these proposed algorithms will converge exponentially to principal or minor component. Simulation results are further used to illustrate the theoretical results.

In standard Principal Component Analysis (PCA) comes one problem in addressing the set of incomplete data. The standard PCA procedure on incomplete data is to eliminate (listwise deletion procedure) or using the mean of the variable, this procedure may result in loss information from these observations. Another method used is to integrate Expectation Maximization (EM) to the method of Probabilistic Principal Component Analysis (PPCA). But PPCA can produce overfitting response prediction. In this study discussed the Variational Bayesian Principal Component Analysis (VBPCA) which is a method of development of PPCA method by incorporating prior information from the distribution of the principal components of the model parameters. From the simulation studies by eliminating the data through the concept of missing at random (MAR), obtained results that the value of the correlation scores principal components complete data with the principal component score predicted results PPCA method is superior when compared with VBPCA, as well as to the value of the correlation scores for the various percentages are generally incomplete data. However, judging from the size of a match between the response to predictions by the size normalized root mean square error of prediction (NRMSEP) VBPCA method produces better than PPCA.

Ocean bottom pressure-gauge (OBP) records play an essential role in seafloor geodesy. Oceanographic fluctuations in OBP data, however, pose as a significant noise source in seafloor transient crustal deformation observations, including slow slip events (SSEs), making it crucial to evaluate them quantitatively. To extract the significant fluctuation phenomena common to multiple observation networks, including oceanographic fluctuations and tectonic signals, we applied principal component analysis (PCA) to the 3-year Dense Oceanfloor Network System for Earthquakes and Tsunamis (DONET) OBP time series for 40 stations during 2016–2019. PCA could separate several oceanographic signals based on the characteristics of their spatial distributions, although evident transient tectonic signals could not be confirmed from the observed pressure records during this observed period. The spatial distribution of the first four principal components (PCs) reflected the common component, inclined component along sea depth, longitude component, and parabola-like pattern, respectively. By subtracting each PC (in particular, PC-2 and PC-4) from the time series, we could significantly reduce the sea depth dependence of OBP records, which has been highlighted in several previous studies and is also evident in this region. We interpreted PCs 2–4 as the reflection of the strength and meandering of ocean geostrophic currents based on a comparison with the PC spatial distribution of the numerical oceanographic models. In addition, to evaluate the ability of PCA to separate transient tectonic signal from OBP time series, including oceanographic fluctuations, we conducted a synthetic ramp assuming an SSE by rectangular fault and then applied PCA. The assumed synthetic tectonic signal could be separated from the oceanographic signals and included in the principal component independently depending on its amplitude, suggesting that the spatial distribution of each PC would change if the amplitude of the synthetic signal were sufficiently large. We propose a transient event-detection method based on the spatial distribution difference of a specific PC with or without a tectonic signal. We used the normalized inner product (NIP) between these PCs as the indicator of their similarities. This method can detect transient tectonic signals more significantly than the moment-magnitude scale of 5.9 from OBP records.Graphical

Objective To explore the principal component analysis (PCA)-Logistic regression model in predicting hospital-acquired pneumonia (HAP) in patients with craniocerebral injury, and find the influencing factors of mortality and HAP occurrence in patients with craniocerebral injury. Methods One hundred and eight patients with craniocerebral injury, admitted to our hospital from December 2011 to November 2017, were chosen in our hospital. Clinical diagnoses, 36 treatment indicators and laboratory results were constituted the original data set; 12 principal components with cumulative contribution>2/3 were extracted as independent variables, and mortality and HAP occurrence were as dependent variables to establish PCA-Logistic regression model. Receiver operating characteristic (ROC) curve was applied to forecast performance of PCA-R model. Results PCA-Logistic regression showed that 4 principal components had significant influence in mortality, and 5 principal components had significant influence in HAP outcomes. Open craniocerebral injury and coagulation changes were the clinical indexes with the highest coefficient when mortality was the outcome index. Gender and parenteral nutrition were the clinical indexes with the highest coefficient when HAP was outcome index. PCA-R model was able to identify the risk factors and forecast the clinical outcomes (HAP, sensitivity: 83.9%, specificity: 94.8%, area under the curve [AUC]: 0.949; mortality, sensitivity: 92.3%, specificity: 93.7%, AUC: 0.983). Conclusion PCA-Logistic regression model can effectively mine the clinical variables of patients with craniocerebral injury; insufficiency of blood perfusion after severe craniocerebral injury is an important factor affecting the survival of patients, and abnormal nutritional support may be an important clinical factor affecting the occurrence of HAP in patients. Key words: Hospital-acquired pneumonia; Severe traumatic brain injury; Principal component analysis; Logistic regression

This chapter explains the theory of Principal component analysis (PCA) in detail and presents practical implementation issues along with various application examples. It introduces the mathematical concepts behind PCA such as mean value, covariance, eigenvalues, and eigenvectors. The principal components are ordered in way that the principal components retain information in descending order. In many cases, PCA is implemented using the singular value decomposition (SVD) of the covariance matrix. The standard PCA algorithm can be extended to support nonlinear principal components using nonlinear kernels. SVD is an important tool for factorizing an arbitrary real or complex matrix, with many applications in various research areas, such as signal processing and statistics. PCA can also be performed without eigenanalysis. This can be achieved by using neural networks that can be trained to extract the principal components of the training set.