Abstract The problem is considered of approximately solving a system of univariate polynomials with one or more common roots and its coefficients corrupted by noise. The goal is to estimate the underlying common roots from the noisy system. Symbolic algebra methods are not suitable for this. New Rayleigh quotient methods are proposed and evaluated for estimating the common roots. Using tensor algebra, reasonable starting values for the Rayleigh quotient methods can be computed. The new methods are compared to Gauss–Newton, solving an eigenvalue problem obtained from the generalized Sylvester matrix, and finding a cluster among the roots of all polynomials. In a simulation study it is shown that Gauss–Newton and a new Rayleigh quotient method perform best, where the latter is more accurate when other roots than the true common roots are close together.

Background: Antidepressants are commonly used to treat depression, but only part of the patients show a treatment response. Unfortunately, it has proven hard to identify the patients for whom antidepressants are truly beneficial. This could be partly due to the fact that personal differences in how different symptoms respond to antidepressants are insufficiently understood. Therefore, this study aimed to capture and investigate these dynamic variations using Three-mode Principal Component Analysis (3MPCA). Methods: Data came from a group of depressed patients who all received Citalopram as part of phase 1 of STAR*D. The Quick Inventory of Depressive Symptomatology (QIDS) was administered at weeks 0, 2, 4, 6, 9 and 12. The resulting data were decomposed with 3MPCA into person-, symptomand timemode components that were associated with patient characteristics, short-term remission and long-term depression severity. Results: The optimal 3MPCA model (81% explained variance) had two symptom-mode (‘cognitive’ and ‘somatic’), two time-mode (‘first measurements’, ‘last measurements’) and three person-mode components (‘negative/suicidal thoughts’, ‘physical dysfunction’ and ‘general improvement’). ‘Negative/suicidal thoughts’ scores were correlated with suicidal thoughts (r=0.30-0.38) and negatively associated with remission. ‘Physical dysfunction’ was correlated with work/social difficulties (r=0.47), lower quality of life (r=-0.40) and side-effects (r=0.34-0.37), negatively associated with remission and positively associated with long-term severity. ‘General improvement’ scores were negatively correlated with quality of life (r=-0.38) but were positively associated with remission. Conclusions: 3MPCA offers an insightful longitudinal description of the most important sources of variation in antidepressant treatment response, providing interesting leads for clinical research into empirically-based antidepressant prescription. Introduction Antidepressants are commonly used for Major Depressive Disorder (MDD) treatment [1]. Although treatment studies have shown similar efficacy for different antidepressants [2-4], treatment responses are known to differ strongly across patients [5]. Given the large impact of MDD on both patients and society [6,7], optimizing antidepressant treatment is crucial. However, the heterogeneity of the diagnosis and course trajectories of MDD [8,9] has hindered this process. To obtain more precise prediction of antidepressant treatment response, patient subgroups based on baseline characteristics, such as symptom severity [3, 10, 11], gender [12] and comorbidity [13] have been studied. However, even within such strata, studies have been shown to have diverse treatment outcomes [14, 15]. A reason for this may be that the used outcomes to quantify treatment response are usually based on clinical cut-offs or minimal clinically-relevant change scores on severity scales, whereas it could very well be that the naturally occurring variations in treatment response are not captured very well in that way. As an alternative, data-driven approaches have been employed to describe the heterogeneity of treatment response. For example, longitudinal Latent Class Analysis has been used to find more homogeneous treatment response trajectories [16], growth mixture models have been used to find patient subgroups with similar treatment responses [17-19] and mixed-effects models have been used to identify the severity trajectories of respondents and non-respondents [20]. Although the latter studies have contributed considerably to our understanding of the existing variations in antidepressant treatment responses, they have not incorporated all relevant sources of heterogeneity (e.g. personsymptomand timeheterogeneity) into a single model [21]. For instance, traditional methods provide no insight into how patients can vary in terms of their trajectories on different subdomains of depressive symptomatology. To tackle this, more flexible methods, such as two-mode K-spectral centroid analysis [22] and Three-mode Principal Component Analysis (3MPCA [23-29]) can be used. The former approach offer insight into how temporal changes vary as a function of both symptoms and persons, by identification of patient subgroups with different trajectories on distinct symptom clusters [22]. 3MPCA provides an even more flexible approach to explain heterogeneity at the person-, symptomand timelevel in an integrated dimensional model. Provided with a dataset consisting of symptoms that were assessed repeatedly in a group of patients, 3MPCA can decompose symptoms, time points and persons into symptom-, timeand personmode components, respectively. Moreover, 3MPCA captures the interactions between the different modes’ components. As such, 3MPCA can describe how persons differ in their trajectories on different symptom-domains in response to 126 Chapter 5 | Decompose heterogeneity of antidepressant treatment outcome using 3MPCA 14468-Rei_BNW_DEF.indd 126 12-04-17 08:54

A new model for simultaneous component analysis (SCA) is introduced that contains the existing SCA models with common loading matrix as special cases. The new SCA-T3 model is a multi-set generalization of the Tucker3 model for component analysis of three-way data. For each mode (observational units, variables, sets) a different number of components can be chosen and the obtained solution can be rotated without loss of fit to facilitate interpretation. SCA-T3 can be fitted on centered multi-set data and also on the corresponding covariance matrices. For this purpose, alternating least squares algorithms are derived. SCA-T3 is evaluated in a simulation study, and its practical merits are demonstrated for several benchmark datasets.

Let $\sigma_n$ denote the largest mode-$n$ multilinear singular value of an $I_1\times\dots \times I_N$ tensor $\mathcal T$. We prove that $ \sigma_1^2+\dots+\sigma_{n-1}^2+\sigma_{n+1}^2+\dots+\sigma_{N}^2\leq (N-2)\|\mathcal T\|^2 + \sigma_n^2,\ n=1,\dots,N, $ where $\|\cdot\|$ denotes the Frobenius norm. We also show that at least for third-order cubic tensors the inverse problem always has a solution. Namely, for each $\sigma_1$, $\sigma_2$, and $\sigma_3$ that satisfy $ \sigma_1^2+\sigma_2^2\leq \|\mathcal T\|^2 + \sigma_3^2,\ \sigma_1^2+\sigma_3^2\leq \|\mathcal T\|^2 + \sigma_2^2,\ \sigma_2^2+\sigma_3^2\leq \|\mathcal T\|^2 + \sigma_1^2,$ and the trivial inequalities $\sigma_1\geq \frac{1}{\sqrt{n}}\|\mathcal T\|$, $\sigma_2\geq \frac{1}{\sqrt{n}}\|\mathcal T\|$, $\sigma_3\geq \frac{1}{\sqrt{n}}\|\mathcal T\|$, there always exists an $n\times n\times n$ tensor whose largest multilinear singular values are equal to $\sigma_1$, $\sigma_2$, and $\sigma_3$. We also show that if the equality $\sigma_1^2...

The perceived possibility of movement between groups, referred to as permeability of group boundaries, is considered a key factor in explaining intergroup relations. However, so far, permeability has been conceptualized in different ways and there exists no validated measure. Integrating different conceptualizations, we developed a scale distinguishing membership permeability (e.g., a person changing from one sport team to another) versus status permeability (e.g., a person acquiring a higher social status). Scale validation occurred across samples representing five lower status groups (older adults, women, obese, lower educated, ethnic minorities). Our scale was related to central indicators of intergroup relations such as self-reported intergroup attitudes (e.g., identification) and endorsement of behavioral strategies (individual mobility, collective action). Moreover, it distinguished permeability characteristics of different types of social groups. The scale provides a novel theoretical conceptualization of permeability and can be used to examine levels and correlates of permeability perceptions across social groups.

We consider multi-set data consisting of Nk observations, k = 1,…, K (e.g., subject scores), on J variables in K different samples. We introduce a factor model for the J × J covariance matrices Σk, k = 1,…, K, where the common part is modelled by Parafac2 and the unique variances Uk, k = 1,…, K, are diagonal. The Parafac2 model implies a common loadings matrix that is rescaled for each k, and a common factor correlation matrix. We estimate the unique variances Uk by minimum rank factor analysis on Σk for each k. The factors can be chosen orthogonal or oblique. We present a novel algorithm to estimate the Parafac2 part and demonstrate its performance in a simulation study. Also, we fit our model to a data set in the literature. Our model is easy to estimate and interpret. The unique variances, the factor correlation matrix and the communalities are guaranteed to be proper, and a percentage of explained common variance can be computed for each k. Also, the Parafac2 part is rotationally unique under mild conditions.

In the common factor model the observed data is conceptually split into a common covariance producing part and an uncorrelated unique part. The common factor model is fitted to the data itself and a new method is introduced for the simultaneous estimation of loadings, unique variances, factor scores, and unique parts. The method is based on Minimum Rank Factor Analysis and allows for the percentage of explained common variance to be computed. Taking into account factor indeterminacy, an explicit description of the complete class of solutions for the factor scores and unique parts is given. The method is evaluated in a simulation study and fitted to a dataset in the literature.

Maximum likelihood estimation of the linear factor model for continuous items assumes normally distributed item scores. We consider deviations from normality by means of a skew-normally distributed factor model or a quadratic factor model. We show that the item distributions under a skew-normal factor are equivalent to those under a quadratic model up to third-order moments. The reverse only holds if the quadratic loadings are equal to each other and within certain bounds. We illustrate that observed data which follow any skew-normal factor model can be so well approximated with the quadratic factor model that the models are empirically indistinguishable, and that the reverse does not hold in general. The choice between the two models to account for deviations of normality is illustrated by an empirical example from clinical psychology.

We consider multi-set data consisting of observations, k  = 1,…, K (e.g., subject scores), on J variables in K different samples. We introduce a factor model for the J  ×  J covariance matrices , k  = 1,…, K , where the common part is modelled by Parafac2 and the unique variances , k  = 1,…, K , are diagonal. The Parafac2 model implies a common loadings matrix that is rescaled for each k , and a common factor correlation matrix. We estimate the unique variances by minimum rank factor analysis on for each k . The factors can be chosen orthogonal or oblique. We present a novel algorithm to estimate the Parafac2 part and demonstrate its performance in a simulation study. Also, we fit our model to a data set in the literature. Our model is easy to estimate and interpret. The unique variances, the factor correlation matrix and the communalities are guaranteed to be proper, and a percentage of explained common variance can be computed for each k . Also, the Parafac2 part is rotationally unique under mild conditions.

Several conjectures and partial proofs have been formulated on the (non)existence of a best low-rank approximation of real-valued IxJx2 arrays. We analyze this problem using the Generalized Schur Decomposition and prove (non)existence of a best rank-R approximation for generic IxJx2 arrays, for all values of I,J,R. Moreover, for cases where a best rank-R approximation exists on a set of positive volume only, we provide easy-to-check necessary and sufficient conditions for the existence of a best rank-R approximation.

It is well known that a best rank-R approximation of order-3 tensors may not exist for R >= 2. A best rank-(R, R, R) approximation always exists, however, and is also a best rank-R approximation when it has rank (at most) R. For R = 2 and real order-3 tensors it is shown that a best rank-2 approximation is also a local minimum of the best rank-(2,2,2) approximation problem. This implies that if all rank-(2,2,2) minima have rank larger than 2, then a best rank-2 approximation does not exist. This provides an easy-to-check criterion for existence of a best rank-2 approximation. The result is illustrated by means of simulations.

Maximum likelihood estimation of the linear factor model for continuous items assumes normally distributed item scores. We consider deviations from normality by means of a skew-normally distributed factor model or a quadratic factor model. We show that the item distributions under a skew-normal factor are equivalent to those under a quadratic model up to third-order moments. The reverse only holds if the quadratic loadings are equal to each other and within certain bounds. We illustrate that observed data which follow any skew-normal factor model can be so well approximated with the quadratic factor model that the models are empirically indistinguishable, and that the reverse does not hold in general. The choice between the two models to account for deviations of normality is illustrated by an empirical example from clinical psychology.

Although heterogeneity of depression hinders research and clinical practice, attempts to reduce it with latent variable models have yielded inconsistent results, probably because these techniques cannot account for all interacting sources of heterogeneity at the same time. Therefore, to simultaneously decompose depression heterogeneity on the person-, symptom and time-level, three-mode Principal Component Analysis (3MPCA) was applied to data of 219 Major Depression patients, who provided Beck Depression Inventory assessments every three months for two years. The resulting person-level components were correlated with external baseline clinical and demographic variables. The 3MPCA extracted two symptom-level components (‘cognitive’, ‘somatic-affective’), two time-level components (‘improving’, ‘persisting’) and three person-level components, characterized by different interaction-patterns between the symptomand time-components (‘severe non-persisting’, ‘somatic depression’ and ‘cognitive depression’). This model explained 28% of the total variance and 65% when also incorporating the general trend in the data). Correlations with external variables illustrated the content differentiation between the person-components. Severe non-persisting depression was positively correlated with psychopathology (r=0.60) and negatively with quality of life (r=-0.50). Somatic depression was negatively correlated with physical functioning (r=-0.45). Cognitive depression was positively correlated with neuroticism (r=0.38) and negatively with self-esteem (r=-0.47). In conclusion, 3MPCA decomposes depression into homogeneous entities, while accounting for the interactions between different sources of heterogeneity, which shows the utility of the technique to investigate the PLOSONE | DOI:10.1371/journal.pone.0132765 July 15, 2015 1 / 16 OPEN ACCESS Citation: Monden R, Wardenaar KJ, Stegeman A, Conradi HJ, de Jonge P (2015) Simultaneous Decomposition of Depression Heterogeneity on the Person-, Symptomand Time-Level: The Use of Three-Mode Principal Component Analysis. PLoS ONE 10(7): e0132765. doi:10.1371/journal. pone.0132765 Editor: Daoqiang Zhang, Nanjing University of Aeronautic and Astronautics, CHINA Received: November 5, 2014 Accepted: June 18, 2015 Published: July 15, 2015 Copyright: © 2015 Monden et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the paper and its Supporting Information files. Funding: The current study was supported by a VICI grant (no: 91812607) received by Peter de Jonge from the Netherlands Organization for Scientific Research (NWO-ZonMW). The trial from which the data were sourced was supported by grants from NWO, Medical Sciences Program and Chronic Diseases Program, Research Foundations of Health Insurance Company ‘Het Groene Land’, Regional Health Insurance Company (RZG), National Fund underlying structure of complex psychopathology data and could help future development of better empirical depression subtypes.

Although heterogeneity of depression hinders research and clinical practice, attempts to reduce it with latent variable models have yielded inconsistent results, probably because these techniques cannot account for all interacting sources of heterogeneity at the same time. Therefore, to simultaneously decompose depression heterogeneity on the person-, symptom and time-level, three-mode Principal Component Analysis (3MPCA) was applied to data of 219 Major Depression patients, who provided Beck Depression Inventory assessments every three months for two years. The resulting person-level components were correlated with external baseline clinical and demographic variables. The 3MPCA extracted two symptom-level components (‘cognitive’, ‘somatic-affective’), two time-level components (‘improving’, ‘persisting’) and three person-level components, characterized by different interaction-patterns between the symptom- and time-components (‘severe non-persisting’, ‘somatic depression’ and ‘cognitive depression’). This model explained 28% of the total variance and 65% when also incorporating the general trend in the data). Correlations with external variables illustrated the content differentiation between the person-components. Severe non-persisting depression was positively correlated with psychopathology (r=0.60) and negatively with quality of life (r=-0.50). Somatic depression was negatively correlated with physical functioning (r=-0.45). Cognitive depression was positively correlated with neuroticism (r=0.38) and negatively with self-esteem (r=-0.47). In conclusion, 3MPCA decomposes depression into homogeneous entities, while accounting for the interactions between different sources of heterogeneity, which shows the utility of the technique to investigate the underlying structure of complex psychopathology data and could help future development of better empirical depression subtypes.

People’s Belief in a Just World (BJW) plays an important role in coping with misfortune and unfairness. This paper demonstrates that understanding of the BJW concept, and its consequences for behavior, is enhanced if we specify what (or who) the source of justice might be. We introduce a new scale, the 5-Dimensional Belief in a Just Treatment Scale (BJT5), which distinguishes five causal dimensions of BJW (God, Nature, Other People, Self, Chance). We confirm the 5-factor structure of the BJT5. We then address whether the BJW should be considered a uni- and/or multi-dimensional construct and find support for our multi-dimensional approach. Finally, we demonstrate convergent and discriminant validity with respect to important correlates of BJW as well as action in response to important negative life events and societal attitudes. This work illustrates the importance of distinguishing causal dimensions with regard to who distributes justice.

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximation generally does not decrease tensor rank. In this paper, we provide a mathematical treatment of this property for real-valued 2 × 2 × 2 tensors, with symmetric tensors as a special case. Regardless of the symmetry, we show that for generic 2×2×2 tensors (which have rank 2 or 3), subtracting a best rank-1 approximation results in a tensor that has rank 3 and lies on the boundary between the rank-2 and rank-3 sets. Hence, for a typical tensor of rank 2, subtracting a best rank-1 approximation increases the tensor rank.

BACKGROUND:Depression heterogeneity has hampered development of adequate prognostic models. Therefore, more homogeneous clinical entities (e.g. dimensions, subtypes) have been developed, but their differentiating potential is limited because neither captures all relevant variation across persons, symptoms and time. To address this, three-mode Principal Component Analysis (3MPCA) was previously applied to capture person-, symptom- and time-level variation in a single model (Monden et al., 2015). This study evaluated the added prognostic value of such an integrated model for longer-term depression outcomes. METHODS:The Beck Depression Inventory (BDI) was administered quarterly for two years to major depressive disorder outpatients participating in a randomized controlled trial. A previously developed 3MPCA model decomposed the data into 2 symptom-components ('somatic-affective', 'cognitive'), 2 time-components ('recovering', 'persisting') and 3 person-components ('severe non-persisting depression', 'somatic depression' and 'cognitive depression'). The predictive value of the 3MPCA model for BDI scores at 3-year (n=136) and 11-year follow-up (n=145) was compared with traditional latent variable models and traditional prognostic factors (e.g. baseline BDI component scores, personality). RESULTS:3MPCA components predicted 41% and 36% of the BDI variance at 3- and 11-year follow-up, respectively. A latent class model, growth mixture model and other known prognostic variables predicted 4-32% and 3-24% of the BDI variance at 3- and 11-year follow-up, respectively. LIMITATIONS:Only primary care patients were included. There was no independent validation sample. CONCLUSION:Accounting for depression heterogeneity at the person-, symptom- and time-level improves longer-term predictions of depression severity, underlining the potential of this approach for developing better prognostic models.

Three-way Candecomp/Parafac (CP) is a three-way generalization of principal component analysis (PCA) for matrices. Contrary to PCA, a CP decomposition is rotationally unique under mild conditions. However, a CP analysis may be hampered by the non-existence of a best-fitting CP decomposition with R>=2 components. In this case, fitting CP to a three-way data array results in diverging CP components. Recently, it has been shown that this can be solved by fitting a decomposition with several interaction terms, using initial values obtained from the diverging CP decomposition. The new decomposition is called CP"l"i"m"i"t, since it is the limit of the diverging CP decomposition. The practical merits of this procedure are demonstrated for a well-known three-way dataset of TV-ratings. CP"l"i"m"i"t finds main components with the same interpretation as Tucker models or when imposing orthogonality in CP. However, CP"l"i"m"i"t has higher joint fit of the main components than Tucker models, contains only one small interaction term, and does not impose the unnatural constraint of orthogonality. The uniqueness properties of the CP"l"i"m"i"t decomposition are discussed in detail.

A three-mode covariance matrix contains covariances of N observations (e.g., subject scores) on J variables for K different occasions or conditions. We model such an JK×JK covariance matrix as the sum of a (common) covariance matrix having Candecomp/Parafac form, and a diagonal matrix of unique variances. The Candecomp/Parafac form is a generalization of the two-mode case under the assumption of parallel factors. We estimate the unique variances by Minimum Rank Factor Analysis. The factors can be chosen oblique or orthogonal. Our approach yields a model that is easy to estimate and easy to interpret. Moreover, the unique variances, the factor covariance matrix, and the communalities are guaranteed to be proper, a percentage of explained common variance can be obtained for each variable-condition combination, and the estimated model is rotationally unique under mild conditions. We apply our model to several datasets in the literature, and demonstrate our estimation procedure in a simulation study.

A best rank-$R$ approximation of an order-3 tensor or three-way array may not exist due to the fact that the set of three-way arrays with rank at most $R$ is not closed. In this case, we are trying to compute the approximation results in diverging rank-1 terms. We show that this phenomenon can be seen as a three-way generalization of approximate diagonalization of a nondiagonalizable (real) matrix. Moreover, we show that, analogous to the matrix case, the limit point of the approximating rank-$R$ sequence satisfies a three-way generalization of the real Jordan canonical form. Recently, it was shown how to obtain the limit point and its three-way Jordan form for $R\le\min(I,J,K)$ and groups of two or three diverging rank-1 terms, where $I\times J\times K$ is the size of the array. We extend this to groups of four diverging rank-1 terms and show that $R>\min(I,J,K)$ is possible as long as no groups of more than $\min(I,J,K)$ diverging rank-1 terms occur. We demonstrate our results by means of numerical experiments.