Canonical Causal Analysis Between Multivariate Continuous Treatments and Outcomes

Chapter 11 - Canonical analysis

Canonical correlation analysis(CCA) is a well-known technique for simultaneously reducing two relevant data sets, and finding maximal correlation between them. However, it fails to preserve the local structure of each data set, as well as the global discriminant ability, which are important in real applications. In this paper, a new CCA model, called discriminant minimum class locality preserving canonical correlation analysis(called as DMPCCA) is proposed. The proposed method introduces locall structure information and global discriminant information into the classical CCA and considers a optimal combination of intra-class locality preserving, global discriminant ability and the maximal correlation between two sets. The experiments on data visualization, web image retrieval and face recognition validate the effectiveness of the proposed method.

Data integration through canonical correlation analysis and its application to OMICs research

The application of three multivariate analysis techniques (canonical discriminant analysis (CDA), principal component analysis (PCA), and canonical correlation analysis (CCA)) for evaluation of pasture botanical composition data is illustrated and discussed. CDA and PCA were used to describe differences in pasture botanical composition for different microsites within a pasture near Palmerston North, New Zealand. CCA could not be validly applied to this data set because a sampling strategy inappropriate for CCA had been used to collect the data. However, CCA is conceptually ideal for determining association between two groups of variables and CCA was used for a second data set from the Hawkes Bay region to establish association between differences in pasture botanical composition and differences in environmental variables. CCA identified a transition from white clover (Trifolium repens L.) to subterranean clover {Trifolium subterraneum L.) presence associated with decreasing rainfall, and a similar transition from ryegrass presence to browntop presence associated with altitude. There are practical difficulties in obtaining a suitable data set for canonical correlation, but with attention to sampling strategy a more precise definition of effect of environmental factors on pasture botanical composition would be expected.

The data of 60 male Chios lambs raised in western Turkey on a farm registered with the Sheep and Goat Breeders' Association were used. Two datasets were generated to determine the correlations between weaning (90 d) and six-month traits in lambs. A canonical analysis was performed to determine the correlation between body weight measured at the weaning period and six morphological traits —dataset X-weaning weight (WW), body length (BL), wither height (WH), heart girth (HG), chest width (CW), rump width (RW), and rump length (RL)— and body weight measured at six months and six morphological traits —dataset Y-body weight (LW_6), body length (BL_6), wither height (WH_6), heart girth (HG_6), chest width (CW_6), rump width (RW_6), and rump length (RL_6). Only the first (U1V1) of the seven pairs of canonical dimensions was statistically significant. The contribution of body weight, WH, and RW to the explanatory effect of canonical variables estimated from morphological traits of Chios lambs was greater at weaning age than other body measurements. Canonical correlation analysis and variables (U1 and V1) can be used as a generic measure of variables, X and Y, respectively. The cross-loadings indicated that body weight, RW, and WH in the weaning period were more effective in V1 determination than other traits; WH, RL, and LW_6 were more influential in U1 determination. This study is the first to use canonical correlation analysis to determine the correlation between live weight and body measurements in Chios lambs, presenting early selection criteria for Chios lamb breeding studies.

This paper gives results from joint analyses of dual polarimety synthetic aperture radar data from the Sentinel-1 mission and optical data from the Sentinel-2 mission. The analyses are carried out by means of traditional canonical correlation analysis (CCA) and canonical information analysis (CIA). Where CCA is based on maximising correlation between linear combinations of the two data sets, CIA maximises mutual information between the two. CIA is a conceptually more pleasing method for the analysis of data with very different modalities such as radar and optical data. Although a little inconclusive as far as the change detection aspect is concerned, results show that CIA analysis gives conspicuously less noisy appearing images of canonical variates (CVs) than CCA. Also, the 2D histogram of the mutual information based leading CVs clearly reveals much more structure than the correlation based one. This gives promise for potentially better change detection results with CIA than can be obtained by means of CCA.

This study explores various options available for choosing the number of principal coordinates m in the canonical analysis of principal coordinates ‘CAP’, a useful procedure that has wide-ranging application wherever multivariate data sets are collected or generated. Choosing too few coordinates (small m) in this constrained (i.e. hypothesis-based) ordination procedure may lead to inadequate separation of the groups (when used as a canonical discriminant analysis) or to inadequate correlation between explanatory and response variables (when used as a canonical correlations analysis), whereas choosing too many (large m) may lead to overparameterization, resulting in overfitting of the data and spurious relationships. It is shown here that the optimum number of principal coordinates is simply the one that results in the smallest P value in the canonical analysis carried out using permutations. For data in which more than one m value results in the same minimum P value, m should be chosen from that set to be the number of principal coordinates that minimizes the leave-one-out residual sum of squares. This choice of m provides suitable solutions for each of the 17 case studies investigated here (which yielded 17 canonical discriminant analyses and 7 canonical correlation analyses).

In the face of multicollinearity, researchers face challenges interpreting canonical correlation analysis (CCA) results. Although standardized function and structure coefficients provide insight into the canonical variates produced, they fall short when researchers want to fully report canonical effects. This article revisits the interpretation of CCA results, providing a tutorial and demonstrating canonical commonalty analysis. Commonality analysis fully explains the canonical effects produced by using the variables in a given canonical set to partition the variance of canonical variates produced from the other canonical set. Conducting canonical commonality analysis without the aid of software is laborious and may be untenable, depending on the number of noteworthy canonical functions and variables in either canonical set. Commonality analysis software is identified for the canonical correlation case and we demonstrate its use in facilitating model interpretation. Data from Holzinger and Swineford (1939) are employed to test a hypothetical theory that problem-solving skills are predicted by fundamental math ability.

We have endeavored in the preceding chapters to draw attention to the richness and flexibility of canonical analysis. The flexibility of the underlying model, in particular, is expressed in part by the number of apparently distinct methods which prove on examination to be special cases of canonical correlation analysis. Canonical variate analysis and dual scaling were both shown abstractly to be equivalent to canonical correlation analysis. The model’s flexibility is complemented by a correspondingly diverse range of application. While, in relation to the worked examples of Chaps. 6–12, it may be that the ability of canonical analysis to have parsimoniously summarized the dependence structures in question is to be attributed to the small scale of these investigations, within the class of small-scale investigations the applicability of canonical analysis was shown to be very wide. Altogether, it seems justifiable to conclude that canonical analysis offers an unusually versatile structure by means of which statistical theory, ecological theory and empirical data can be harnessed to provide insight into the organization of ecological systems.

There have been an increasing number of applications of sparse Canonical Correlation Analysis (sCCA) to genomic data during the past several years. Most of the research in this area has focused on the relationships between gene expression levels and phenotype variations. However, as multimodal omics data becomes available there is a need to integrate these data modalities into a framework that allows for simultaneous data analyses, thereby providing novel insight for various fields in the life sciences. The pioneering work of Schwartz et al. (2012) used the classical Canonical Correlation Analysis (CCA) to provide an integrative approach to the analysis of host gene expression and microbiota composition data from neonates with different feeding types. Although promising, the proposed approach has serious deficiencies. First, the statistical interpretation is problematic because the involved two-stage analysis makes the results sensitive to the variations of data and the original interpretation of CCA is lost. Second, the associated computational cost is tremendous, O(n 3 ) where n is the number of variables involved in the analysis. Thus, we developed a methodology based on the sCCA to overcome these problems. The performance of our approach is compared to that of Schwartz et al. (2012) and to the sparse Principal Component Analysis (sPCA) on a large synthetic data set with the subsequent application to a multimodal omics data (gene expression, microbiota composition, and metabolites) from neonates with two different feeding types.

A classification algorithm of an SSVEP brain-Computer interface based on CCA fusion wavelet coefficients

A number of measures of canonical correlation coefficient are now used in pattern recognition in the different literature. Some robust forms of classical canonical correlation coefficient are introduced recently to address the robustness issue of the canonical coefficient in the presence of outliers and departure from normality. Also a few number of kernels are used in canonical analysis to capture nonlinear relationship in data space, which is linear in some higher dimensional feature space. But not much work has been done to investigate their relative performances through simulation and also from the view point of sensitivity. In this paper an attempt has been made to compare performances of kernel canonical correlation coefficients (Gaussian, Laplacian and Polynomial) with that of classical and robust canonical correlation coefficient measures using simulation and influence function. We investigate the bias, standard error, MSE, qualitative robustness index, sensitivity curve of each estimator under a variety of situations and also employ boxplots and scatter plots of canonical variates to judge their performances. We observe that the class of kernel estimators perform better than the class of classical and robust estimators in general and the kernel estimator with Laplacian function has shown the best performance for large sample size.

Using a hypothetical data set, the authors provide concrete examples to illustrate that canonical correlation analysis is a general linear model, subsuming other parametric procedures as special cases. Specific statistical techniques included in the analysis are t tests, Pearson correlation, multiple regression, ANOVA, MANOVA, and discriminant analysis. The discussion is aided by an initial explanation of the logic of canonical analysis. Further, similarities between the canonical technique and other univariate and multivariate procedures are highlighted. The treatment is intended to reiterate a framework in which statistical concepts can be presented to students.

In recent years, the number of people with Alzheimer’s disease (AD) has grown worldwide. Since no cure has yet been found for the disease, detection and initiation of treatment is the best way to prevent the disease from progressing behavioral and physical symptoms in the patient. There are several methods to diagnose AD, and neuroimaging, as a noninvasive approach, reveals changes in the brain due to the disease. Diagnosis accuracy can be improved by fusion of various neuroimaging modalities. Canonical correlation analysis (CCA) and its extensions have been widely used for fusing multi-modal datasets, where healthy controls (HCs) are differentiated from patients by applying classification methods to canonical variables (CVs) resulting from CCA or its extensions. The goal of our study is to find an optimal method, from the perspective of accuracy and processing complexity, to diagnose patients with AD. This goal is achieved by fusing anatomical magnetic resonance imaging (MRI) and functional MRI (fMRI) data using CCA-based methods. Experimental results illustrate that HCs and AD patients can be classified using CVs obtained from structured and sparse CCA with greater than 90% accuracy.

BackgroundAlthough hypercaloric interventions are associated with nutritional, endocrine, metabolic, and cardiovascular disorders in obesity experiments, a rational distinction between the effects of excess adiposity and the individual roles of dietary macronutrients in relation to these disturbances has not previously been studied. This investigation analyzed the correlation between ingested macronutrients (including sucrose and saturated and unsaturated fatty acids) plus body adiposity and metabolic, hormonal, and cardiovascular effects in rats with diet-induced obesity.MethodsNormotensive Wistar-Kyoto rats were submitted to Control (CD; 3.2 Kcal/g) and Hypercaloric (HD; 4.6 Kcal/g) diets for 20 weeks followed by nutritional evaluation involving body weight and adiposity measurement. Metabolic and hormonal parameters included glycemia, insulin, insulin resistance, and leptin. Cardiovascular analysis included systolic blood pressure profile, echocardiography, morphometric study of myocardial morphology, and myosin heavy chain (MHC) protein expression. Canonical correlation analysis was used to evaluate the relationships between dietary macronutrients plus adiposity and metabolic, hormonal, and cardiovascular parameters.ResultsAlthough final group body weights did not differ, HD presented higher adiposity than CD. Diet induced hyperglycemia while insulin and leptin levels remained unchanged. In a cardiovascular context, systolic blood pressure increased with time only in HD. Additionally, in vivo echocardiography revealed cardiac hypertrophy and improved systolic performance in HD compared to CD; and while cardiomyocyte size was unchanged by diet, nuclear volume and collagen interstitial fraction both increased in HD. Also HD exhibited higher relative β-MHC content and β/α-MHC ratio than their Control counterparts. Importantly, body adiposity was weakly associated with cardiovascular effects, as saturated fatty acid intake was directly associated with most cardiac remodeling measurements while unsaturated lipid consumption was inversely correlated with these effects.ConclusionHypercaloric diet was associated with glycemic metabolism and systolic blood pressure disorders and cardiac remodeling. These effects directly and inversely correlated with saturated and unsaturated lipid consumption, respectively.