Tensorial Principal Component Analysis in Detecting Temporal Trajectories of Purchase Patterns in Loyalty Card Data: Retrospective Cohort Study

BACKGROUND

Loyalty card data automatically collected by retailers provide an excellent source for evaluating health-related purchase behavior of customers. The data comprise information on every grocery purchase, including expenditures on product groups and the time of purchase for each customer. Such data where customers have an expenditure value for every product group for each time can be formulated as 3D tensorial data.

OBJECTIVE

This study aimed to use the modern tensorial principal component analysis (PCA) method to uncover the characteristics of health-related purchase patterns from loyalty card data. Another aim was to identify card holders with distinct purchase patterns. We also considered the interpretation, advantages, and challenges of tensorial PCA compared with standard PCA.

METHODS

Loyalty card program members from the largest retailer in Finland were invited to participate in this study. Our LoCard data consist of the purchases of 7251 card holders who consented to the use of their data from the year 2016. The purchases were reclassified into 55 product groups and aggregated across 52 weeks. The data were then analyzed using tensorial PCA, allowing us to effectively reduce the time and product group-wise dimensions simultaneously. The augmentation method was used for selecting the suitable number of principal components for the analysis.

RESULTS

Using tensorial PCA, we were able to systematically search for typical food purchasing patterns across time and product groups as well as detect different purchasing behaviors across groups of card holders. For example, we identified customers who purchased large amounts of meat products and separated them further into groups based on time profiles, that is, customers whose purchases of meat remained stable, increased, or decreased throughout the year or varied between seasons of the year.

CONCLUSIONS

Using tensorial PCA, we can effectively examine customers' purchasing behavior in more detail than with traditional methods because it can handle time and product group dimensions simultaneously. When interpreting the results, both time and product dimensions must be considered. In further analyses, these time and product groups can be directly associated with additional consumer characteristics such as socioeconomic and demographic predictors of dietary patterns. In addition, they can be linked to external factors that impact grocery purchases such as inflation and unexpected pandemics. This enables us to identify what types of people have specific purchasing patterns, which can help in the development of ways in which consumers can be steered toward making healthier food choices.

Data Type Agnostic Visual Sensitivity Analysis

Modern science and industry rely on computational models for simulation, prediction, and data analysis. Spatial blind source separation (SBSS) is a model used to analyze spatial data. Designed explicitly for spatial data analysis, it is superior to popular non-spatial methods, like PCA. However, a challenge to its practical use is setting two complex tuning parameters, which requires parameter space analysis. In this paper, we focus on sensitivity analysis (SA). SBSS parameters and outputs are spatial data, which makes SA difficult as few SA approaches in the literature assume such complex data on both sides of the model. Based on the requirements in our design study with statistics experts, we developed a visual analytics prototype for data type agnostic visual sensitivity analysis that fits SBSS and other contexts. The main advantage of our approach is that it requires only dissimilarity measures for parameter settings and outputs (Fig. 1). We evaluated the prototype heuristically with visualization experts and through interviews with two SBSS experts. In addition, we show the transferability of our approach by applying it to microclimate simulations. Study participants could confirm suspected and known parameter-output relations, find surprising associations, and identify parameter subspaces to examine in the future. During our design study and evaluation, we identified challenging future research opportunities.

Multivariate Independent Component Analysis Identifies Patients in Newborn Screening Equally to Adjusted Reference Ranges

Newborn screening (NBS) of inborn errors of metabolism (IEMs) is based on the reference ranges established on a healthy newborn population using quantile statistics of molar concentrations of biomarkers and their ratios. The aim of this paper is to investigate whether multivariate independent component analysis (ICA) is a useful tool for the analysis of NBS data, and also to address the structure of the calculated ICA scores. NBS data were obtained from a routine NBS program performed between 2013 and 2022. ICA was tested on 10,213/150 free-diseased controls and 77/20 patients (9/3 different IEMs) in the discovery/validation phases, respectively. The same model computed during the discovery phase was used in the validation phase to confirm its validity. The plots of ICA scores were constructed, and the results were evaluated based on 5sd levels. Patient samples from 7/3 different diseases were clearly identified as 5sd-outlying from control groups in both phases of the study. Two IEMs containing only one patient each were separated at the 3sd level in the discovery phase. Moreover, in one latent variable, the effect of neonatal birth weight was evident. The results strongly suggest that ICA, together with an interpretation derived from values of the "average member of the score structure", is generally applicable and has the potential to be included in the decision process in the NBS program.

Blind recovery of sources for multivariate space-time random fields

With advances in modern worlds technology, huge datasets that show dependencies in space as well as in time occur frequently in practice. As an example, several monitoring stations at different geographical locations track hourly concentration measurements of a number of air pollutants for several years. Such a dataset contains thousands of multivariate observations, thus, proper statistical analysis needs to account for dependencies in space and time between and among the different monitored variables. To simplify the consequent multivariate spatio-temporal statistical analysis it might be of interest to detect linear transformations of the original observations that result in straightforward interpretative, spatio-temporally uncorrelated processes that are also highly likely to have a real physical meaning. Blind source separation (BSS) represents a statistical methodology which has the aim to recover so-called latent processes, that exactly meet the former requirements. BSS was already successfully used in sole temporal and sole spatial applications with great success, but, it was not yet introduced for the spatio-temporal case. In this contribution, a reasonable and innovative generalization of BSS for multivariate space-time random fields (stBSS), under second-order stationarity, is proposed, together with two space-time extensions of the well-known algorithms for multiple unknown signals extraction (stAMUSE) and the second-order blind identification (stSOBI) which solve the formulated problem. Furthermore, symmetry and separability properties of the model are elaborated and connections to the space-time linear model of coregionalization and to the classical principal component analysis are drawn. Finally, the usefulness of the new methods is shown in a thorough simulation study and on a real environmental application.

A review of second-order blind identification methods

Second-order source separation (SOS) is a data analysis tool which can be used for revealing hidden structures in multivariate time series data or as a tool for dimension reduction. Such methods are nowadays increasingly important as more and more high-dimensional multivariate time series data are measured in numerous fields of applied science. Dimension reduction is crucial, as modeling such high-dimensional data with multivariate time series models is often impractical as the number of parameters describing dependencies between the component time series is usually too high. SOS methods have their roots in the signal processing literature, where they were first used to separate source signals from an observed signal mixture. The SOS model assumes that the observed time series (signals) is a linear mixture of latent time series (sources) with uncorrelated components. The methods make use of the second-order statistics-hence the name "second-order source separation." In this review, we discuss the classical SOS methods and their extensions to more complex settings. An example illustrates how SOS can be performed. This article is categorized under:Statistical Models > Time Series ModelsStatistical and Graphical Methods of Data Analysis > Dimension ReductionData: Types and Structure > Time Series, Stochastic Processes, and Functional Data.

Visual Parameter Selection for Spatial Blind Source Separation

Analysis of spatial multivariate data, i.e., measurements at irregularly-spaced locations, is a challenging topic in visualization and statistics alike. Such data are inteGral to many domains, e.g., indicators of valuable minerals are measured for mine prospecting. Popular analysis methods, like PCA, often by design do not account for the spatial nature of the data. Thus they, together with their spatial variants, must be employed very carefully. Clearly, it is preferable to use methods that were specifically designed for such data, like spatial blind source separation (SBSS). However, SBSS requires two tuning parameters, which are themselves complex spatial objects. Setting these parameters involves navigating two large and interdependent parameter spaces, while also taking into account prior knowledge of the physical reality represented by the data. To support analysts in this process, we developed a visual analytics prototype. We evaluated it with experts in visualization, SBSS, and geochemistry. Our evaluations show that our interactive prototype allows to define complex and realistic parameter settings efficiently, which was so far impractical. Settings identified by a non-expert led to remarkable and surprising insights for a domain expert. Therefore, this paper presents important first steps to enable the use of a promising analysis method for spatial multivariate data.

Fast tensorial JADE

We propose a novel method for tensorial-independent component analysis. Our approach is based on TJADE and k-JADE, two recently proposed generalizations of the classical JADE algorithm. Our novel method achieves the consistency and the limiting distribution of TJADE under mild assumptions and at the same time offers notable improvement in computational speed. Detailed mathematical proofs of the statistical properties of our method are given and, as a special case, a conjecture on the properties of k-JADE is resolved. Simulations and timing comparisons demonstrate remarkable gain in speed. Moreover, the desired efficiency is obtained approximately for finite samples. The method is applied successfully to large-scale video data, for which neither TJADE nor k-JADE is feasible. Finally, an experimental procedure is proposed to select the values of a set of tuning parameters. Supplementary material including the R-code for running the examples and the proofs of the theoretical results is available online.

On linear dimension reduction based on diagonalization of scatter matrices for bioinformatics downstream analyses

Dimension reduction is often a preliminary step in the analysis of data sets with a large number of variables. Most classical, both supervised and unsupervised, dimension reduction methods such as principal component analysis (PCA), independent component analysis (ICA) or sliced inverse regression (SIR) can be formulated using one, two or several different scatter matrix functionals. Scatter matrices can be seen as different measures of multivariate dispersion and might highlight different features of the data and when compared might reveal interesting structures. Such analysis then searches for a projection onto an interesting (signal) part of the data, and it is also important to know the correct dimension of the signal subspace. These approaches usually make either no model assumptions or work in wide classes of semiparametric models. Theoretical results in the literature are however limited to the case where the sample size exceeds the number of variables which is hardly ever true for data sets encountered in bioinformatics. In this paper, we briefly review the relevant literature and explore if the dimension reduction tools can be used to find relevant and interesting subspaces for small-n-large-p data sets. We illustrate the methods with a microarray dataset of prostate cancer patients and healthy controls.

Blind Source Separation for Compositional Time Series

Many geological phenomena are regularly measured over time to follow developments and changes. For many of these phenomena, the absolute values are not of interest, but rather the relative information, which means that the data are compositional time series. Thus, the serial nature and the compositional geometry should be considered when analyzing the data. Multivariate time series are already challenging, especially if they are higher dimensional, and latent variable models are a popular way to deal with this kind of data. Blind source separation techniques are well-established latent factor models for time series, with many variants covering quite different time series models. Here, several such methods and their assumptions are reviewed, and it is shown how they can be applied to high-dimensional compositional time series. Also, a novel blind source separation method is suggested which is quite flexible regarding the assumptions of the latent time series. The methodology is illustrated using simulations and in an application to light absorbance data from water samples taken from a small stream in Lower Austria.

Team-based "Get-a-Grip" lifestyle management programme in the treatment of obesity

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Team-based one-year lifestyle programme led to mean weight loss of 4.8 kg.

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Among the participants 44% lost ≥ 5%, while 21% lost ≥ 10% of their initial weight.

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Beneficial changes were detected in muscle mass, body fat, and visceral fat.

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Systolic and diastolic blood pressure was reduced significantly.

This study examined weight loss during an extensive 1-year lifestyle programme in primary care in Finland in overweight subjects (n = 134, age 18–69 years; BMI > 30, or BMI > 25 with a comorbidity that would benefit from weight loss) between 2009 and 2013 in a single arm design. The programme included four medical doctor visits, five sessions by a dietitian (advice on diet and on-location shopping behaviour), cooking classes, exercise supervised by personal trainer, and group discussions. A motivational interview method was applied.

Of the 134 participants, 92 (69%) completed the 1-year programme. Among the participants 44% lost ≥ 5%, while 21% lost ≥ 10% of their initial body weight. In intention-to-treat-analyses, the mean weight loss during one year was 4.8 kg (p < 0.001). Mean BMI decreased by 1.7 kg/m² (p < 0.001) and waist circumference by 5.6 cm (p < 0.001). Mean muscle mass increased by 3.3% (p < 0.001), and body fat decreased by 5.0% (p < 0.001). After the programme mean visceral fat content was reduced by 6.4%, systolic blood pressure by 8 mmHg (p < 0.001), and diastolic blood pressure by 6 mmHg (p < 0.001).

In conclusion, retention to the team-based lifestyle management programme resulted in moderate but significant weight loss with beneficial changes in body composition, and the trend to lose weight was maintained throughout the year.

Trial registration: Clinicaltrials.gov identifier NCT04003259.

Minimum Distance Index for BSS, Generalization, Interpretation and Asymptotics

We consider complex valued linear blind source separation, where the signal dimension might be smaller than the dimension of the observable data vector. In order to measure the success of the signal separation, we propose an extension of the minimum distance index and establish its properties. Interpretations for the index are derived through connections to signal-to-noise ratios and correlations. The interpretations are novel also for the real valued original case. In addition, we consider the asymptotic behavior of the extended minimum distance index. This paper is an invited extended version of the paper presented at the CDAM 2019 conference.

Extracting Conditionally Heteroskedastic Components using Independent Component Analysis

In the independent component model, the multivariate data are assumed to be a mixture of mutually independent latent components. The independent component analysis (ICA) then aims at estimating these latent components. In this article, we study an ICA method which combines the use of linear and quadratic autocorrelations to enable efficient estimation of various kinds of stationary time series. Statistical properties of the estimator are studied by finding its limiting distribution under general conditions, and the asymptotic variances are derived in the case of ARMA-GARCH model. We use the asymptotic results and a finite sample simulation study to compare different choices of a weight coefficient. As it is often of interest to identify all those components which exhibit stochastic volatility features we suggest a test statistic for this problem. We also show that a slightly modified version of the principal volatility component analysis can be seen as an ICA method. Finally, we apply the estimators in analysing a data set which consists of time series of exchange rates of seven currencies to US dollar. Supporting information including proofs of the theorems is available online.

Spatial blind source separation

Recently a blind source separation model was suggested for spatial data, along with an estimator based on the simultaneous diagonalization of two scatter matrices. The asymptotic properties of this estimator are derived here, and a new estimator based on the joint diagonalization of more than two scatter matrices is proposed. The asymptotic properties and merits of the novel estimator are verified in simulation studies. A real-data example illustrates application of the method.

Tensorial blind source separation for improved analysis of multi-omic data

There is an increased need for integrative analyses of multi-omic data. We present and benchmark a novel tensorial independent component analysis (tICA) algorithm against current state-of-the-art methods. We find that tICA outperforms competing methods in identifying biological sources of data variation at a reduced computational cost. On epigenetic data, tICA can identify methylation quantitative trait loci at high sensitivity. In the cancer context, tICA identifies gene modules whose expression variation across tumours is driven by copy-number or DNA methylation changes, but whose deregulation relative to normal tissue is independent of such alterations, a result we validate by direct analysis of individual data types.

On Independent Component Analysis with Stochastic Volatility Models

Consider a multivariate time series where each component series is assumed to be a linear mixture of latent mutually independent stationary time series. Classical independent component analysis (ICA) tools, such as fastICA, are often used to extract latent series, but they don't utilize any information on temporal dependence. Also financial time series often have periods of low and high volatility. In such settings second order source separation methods, such as SOBI, fail. We review here some classical methods used for time series with stochastic volatility, and suggest modifications of them by proposing a family of vSOBI estimators. These estimators use different nonlinearity functions to capture nonlinear autocorrelation of the time series and extract the independent components. Simulation study shows that the proposed method outperforms the existing methods when latent components follow GARCH and SV models. This paper is an invited extended version of the paper presented at the CDAM 2016 conference.

Subgroup detection in genotype data using invariant coordinate selection

BACKGROUND

The current gold standard in dimension reduction methods for high-throughput genotype data is the Principle Component Analysis (PCA). The presence of PCA is so dominant, that other methods usually cannot be found in the analyst's toolbox and hence are only rarely applied.

RESULTS

We present a modern dimension reduction method called 'Invariant Coordinate Selection' (ICS) and its application to high-throughput genotype data. The more commonly known Independent Component Analysis (ICA) is in this framework just a special case of ICS. We use ICS on both, a simulated and a real dataset to demonstrate first some deficiencies of PCA and how ICS is capable to recover the correct subgroups within the simulated data. Second, we apply the ICS method on a chicken dataset and also detect there two subgroups. These subgroups are then further investigated with respect to their genotype to provide further evidence of the biological relevance of the detected subgroup division. Further, we compare the performance of ICS also to five other popular dimension reduction methods.

CONCLUSION

The ICS method was able to detect subgroups in data where the PCA fails to detect anything. Hence, we promote the application of ICS to high-throughput genotype data in addition to the established PCA. Especially in statistical programming environments like e.g. R, its application does not add any computational burden to the analysis pipeline.

The type of the functional cardiovascular response to upright posture is associated with arterial stiffness: a cross-sectional study in 470 volunteers

BACKGROUND

In a cross-sectional study we examined whether the haemodynamic response to upright posture could be divided into different functional phenotypes, and whether the observed phenotypes were associated with known determinants of cardiovascular risk.

METHODS

Volunteers (n = 470) without medication with cardiovascular effects were examined using radial pulse wave analysis, whole-body impedance cardiography, and heart rate variability analysis. Based on the passive head-up tilt induced changes in systemic vascular resistance and cardiac output, the principal determinants of blood pressure, a cluster analysis was performed.

RESULTS

The haemodynamic response could be clustered into 3 categories: upright increase in vascular resistance and decrease in cardiac output were greatest in the first (+45 % and -27 %, respectively), smallest in the second (+2 % and -2 %, respectively), and intermediate (+22 % and -13 %, respectively) in the third group. These groups were named as 'constrictor' (n = 109), 'sustainer' (n = 222), and 'intermediate' (n = 139) phenotypes, respectively. The sustainers were characterized by male predominance, higher body mass index, blood pressure, and also by higher pulse wave velocity, an index of large arterial stiffness, than the other groups (p < 0.01 for all). Heart rate variability analysis showed higher supine and upright low frequency/high frequency (LF/HF) ratio in the sustainers than constrictors, indicating increased sympathovagal balance. Upright LF/HF ratio was also higher in the sustainer than intermediate group. In multivariate analysis, independent explanatory factors for higher pulse wave velocity were the sustainer (p < 0.022) and intermediate phenotypes (p < 0.046), age (p < 0.001), body mass index (p < 0.001), and hypertension (p < 0.001).

CONCLUSIONS

The response to upright posture could be clustered to 3 functional phenotypes. The sustainer phenotype, with smallest upright decrease in cardiac output and highest sympathovagal balance, was independently associated with increased large arterial stiffness. These results indicate an association of the functional haemodynamic phenotype with an acknowledged marker of cardiovascular risk.

TRIAL REGISTRATION

ClinicalTrials.gov NCT01742702.

Robust Independent Component Analysis Based on Two Scatter Matrices

Oja, Sirkiä, and Eriksson (2006) and Ollila, Oja, and Koivunen (2007) showed that, under general assumptions, any two scatter matrices with the so called independent components property can be used to estimate the unmixing matrix for the independent component analysis (ICA). The method is a generalization of Cardoso’s (Cardoso, 1989) FOBI estimate which uses the regular covariance matrix and a scatter matrix based on fourth moments. Different choices of the two scatter matrices are compared in a simulation study. Based on the study, we recommend always the use of two robust scatter matrices. For possible asymmetric independent components, symmetrized versions of the scatter matrix estimates should be used.

Vasopeptidase Inhibition Corrects the Structure and Function of the Small Arteries in Experimental Renal Insufficiency

BACKGROUND

We studied whether vasopeptidase inhibition corrects the structure and function of the small arteries in experimental chronic renal insufficiency (CRI).

METHODS

After 5/6 nephrectomy (NX) surgery was performed on rats, there was a 14-week follow-up, allowing CRI to become established. Omapatrilat (40 mg/kg/day in chow) was then given for 8 weeks, and the small mesenteric arterial rings were investigated in vitro using wire and pressure myographs.

RESULTS

Plasma and ventricular B-type natriuretic peptide (BNP) concentrations were increased 2- to 2.7-fold, while systolic blood pressure (BP) increased by 32 mm Hg after NX. Omapatrilat treatment normalized the BNP and reduced the BP by 45 mm Hg in the NX rats. Endothelium-dependent vasorelaxation was impaired but the response to acetylcholine was normalized after omapatrilat treatment. Vasorelaxations induced by nitroprusside, isoprenaline and levcromakalim were enhanced after omapatrilat, and the responses were even more pronounced than in untreated sham-operated rats. Arterial wall thickness and wall-to-lumen ratio were increased after NX, whereas omapatrilat normalized these structural features and improved the strain-stress relationship in the small arteries; this suggests improved arterial elastic properties.

CONCLUSION

Omapatrilat treatment reduced BP, normalized volume overload, improved vasorelaxation and corrected the dimensions and passive elastic properties of the small arteries in the NX rats. Therefore, we consider vasopeptidase inhibition to be an effective treatment for CRI-induced changes in the small arteries.

Reduced systemic vascular resistance in healthy volunteers with presyncopal symptoms during a nitrate-stimulated tilt-table test

WHAT IS ALREADY KNOWN ABOUT THIS SUBJECT

Nitrates may facilitate syncope through various pathways, but the precise mechanism of nitrate-induced syncope is still under debate. The purpose of the present study was to compare the underlying haemodynamic mechanisms in subjects without and with presyncopal symptoms during a nitroglycerin-stimulated tilt-table test.

WHAT THIS STUDY ADDS

A major decrease in systemic vascular resistance was documented in subjects with presyncope during 0.25 mg nitroglycerin-stimulated tilt-table test, in the absence of changes in cardiac output. These findings indicated that even a small dose of nitroglycerin significantly decreased arterial resistance and cardiac afterload. AIMS The mechanism of nitrate-induced syncope remains controversial. We examined the haemodynamic changes in healthy volunteers during nitroglycerin-stimulated tilt-table test.

METHODS

Continuous radial pulse wave analysis, whole-body impedance cardiography and plethysmographic finger blood pressure were recorded in a supine position and during head-up tilt in 21 subjects with presyncopal symptoms (6 male/15 female, age 43 ± 3 years) after 0.25 mg sublingual nitroglycerin and 21 control subjects (6 male/15 female, age 43 ± 2 years). The drug was administered in the supine position and a passive head-up tilt followed 5 min later. Additionally, nitroglycerin was only administered during head-up tilt in 19 subjects and the haemodynamics were recorded.

RESULTS

Supine and upright haemodynamics were similar before nitroglycerin administration in the two groups. During the nitroglycerin-stimulated tilt test, aortic and radial mean blood pressure decreased significantly more in the presyncope group when compared with the controls (P= 0.0006 and P= 0.0004, respectively). The decreases in systemic vascular resistance (P= 0.0008) and heart rate (P= 0.002), and increase in aortic reflection time (P= 0.0002) were greater in the presyncope group, while the change in cardiac index was not different between the groups (P= 0.14). If nitroglycerin was administered during the upright tilt and not in supine position, the haemodynamic changes were quite corresponding.

CONCLUSIONS

Presyncopal symptoms during nitrate-stimulated tilt test were explained by decreased systemic vascular resistance and increased aortic reflection time, while cardiac output remained unchanged. These findings indicated reduced arterial resistance in nitroglycerin-induced presyncope.