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We assume a spatial blind source separation model in which the observed multivariate spatial data is a linear mixture of latent spatially uncorrelated random fields containing a number of pure white noise components. We propose a test on the number of white noise components and obtain the asymptotic distribution of its statistic for a general domain. We also demonstrate how computations can be facilitated in the case of gridded observation locations. Based on this test, we obtain a consistent estimator of the true dimension. Simulation studies and an environmental application in the Supplemental Material demonstrate that our test is at least comparable to and often outperforms bootstrap-based techniques, which are also introduced in this paper. peerReviewed

In environmental sciences, it is common to collect and analyze spatio-temporal multivariate data concerning several variables which are measured in time over a spatial domain. The spatio-temporal data are usually sparce in space, due to the high cost of the equipment, and temporal dense since the required variables are regularly sampled in time.In the literature different methods have been proposed for the analysis of such spatio-temporal data which exhibit a correlation in space and time as well as in-between variables. Among them it is worth recalling the generalization of Blind Source Separation technique for multivariate space-time random field (stBSS) and the space-time linear coregionalization model (ST-LCM). These methods are useful to simplify the spatio-temporal multivariate analysis since by a linear transformation of the original observations only the independent components which exhibit a spatio-temporal correlation are retained (lower than the number of observed variables) and modelled.In this paper a multivariate study regarding seven environmental variables (evapotranspiration level, minimum and maximum temperature, minimum and maximum humidity, wind speed and precipitation) measured between 2000 and 2022 in Veneto region (Italy) will be proposed. Both the stBSS and the joint diagonalization of the empirical covariance matrix approach will be used to identify the hidden components, and properly chosen spatio-temporal models will be fitted to the latent components. Note that for the first approach a BSS model for the multivariate random field will be assumed, whereas for second one a space-time linear coregionalization model (ST-LCM) for the independent components will be fitted to the matrix-valued covariance function estimated for seven relevant environmental variables.Finally, the fitted models have been used to predict evapotranspiration levels and a comparison of the values obtained by using the two different techniques will be provided.

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.

BACKGROUND Loyalty card data automatically collected by retailers provide an excellent source for evaluating health-related purchase behaviour of customers. The data comprise information on every grocery purchase, including expenditures on products and time of purchase for each customer. Such data where customers have an expenditure value for every product for each time can be formulated as three-dimensional tensorial data. OBJECTIVE The aim of the study was to use the modern tensorial principal component analysis (tensorial PCA) method to disclose characteristics of health-related purchase patterns from loyalty card data. A further aim was to identify card holders with distinct purchase patterns. We also considered the interpretation as well as advantages and challenges of tensorial PCA compared with standard PCA. METHODS Loyalty card programme members from the largest retailer in Finland were invited to participate in the study. Our data consist of the purchases of 7251 card holders who consented to the use of their data from the year 2016. The purchases were re-classified into 55 products and aggregated across 52 weeks. The data were then analysed by tensorial PCA, allowing us to effectively reduce the time and product-wise dimensions simultaneously. The augmentation method was utilized for selecting the suitable number of principal components for the analysis. RESULTS By tensorial PCA, we were able to systematically search for typical food purchasing patterns across time and product groups as well as detect atypical behaviour across products and time, and differing purchasing behaviours across groups of card holders. For example, we identified customers who purchaseds large amounts of meat products and separated them further into groups based on time profiles, i.e. customers whose purchases of meat stayed stable, increased, or decreased throughout the year, or varied between seasons of the year. CONCLUSIONS With tensorial PCA, we can effectively examine customers’ purchasing behaviour in more detail than with traditional methods because it can handle time and product dimensions simultaneously. When interpreting the results, both time and product dimensions must be considered. In further analysis, these time and product based patterns can directly be associated with other attributes of customers. This enables us to identify what kinds of people have specific purchasing patterns, which can help in the development of ways in which consumers can be steered towards making healthier food choices.

Many modern multivariate time series datasets contain a large amount of noise, and the first step of the data analysis is to separate the noise channels from the signals of interest. A crucial part of this dimension reduction is determining the number of signals. In this paper we approach this problem by considering a noisy latent variable time series model which comprises many popular blind source separation models. We propose a general framework for the estimation of the signal dimension that is based on testing for sub-sphericity and give examples of different tests suitable for time series settings. In the inference we rely on bootstrap null distributions. Several simulation studies are used to demonstrate the performances of the tests in different time series settings. peerReviewed

Multivariate functional anomaly detection has received a large amount of attention recently. Accounting both the time dimension and the correlations between variables is challenging due to the existence of different types of outliers and the dimension of the data. In the context of predictive maintenance and quality control, data sets often contain a large number of functional variables. However, most of the existing methods focus on a small number of functional variables. Moreover, in fields that have high reliability standards, detecting a small number of potential multivariate functional outliers with as few false positives as possible is crucial. In such a context, the adaptation of the Invariant Coordinate Selection (ICS) method from the multivariate to the multivariate functional case is of particular interest. Two extensions of ICS are proposed: point-wise and global. For both methods, the choice of the relevant components together with outlier identification and interpretation are discussed. A comparison is made on a predictive maintenance example from the avionics field and a quality control example from the microelectronics field. It appears that in such a context, point-wise and global ICS with a small number of selected components can be recommended.

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.

Scatter matrices generalize the covariance matrix and are useful in many multivariate data analysis methods, including well-known principal component analysis (PCA), which is based on the diagonalization of the covariance matrix. The simultaneous diagonalization of two or more scatter matrices goes beyond PCA and is used more and more often. In this paper, we offer an overview of many methods that are based on a joint diagonalization. These methods range from the unsupervised context with invariant coordinate selection and blind source separation, which includes independent component analysis, to the supervised context with discriminant analysis and sliced inverse regression. They also encompass methods that handle dependent data such as time series or spatial data. peerReviewed

Most linear dimension reduction methods proposed in the literature can be formulated using an appropriate pair of scatter matrices, see e.g. Ye and Weiss (2003), Tyler et al. (2009), Bura and Yang (2011), Liski et al. (2014) and Luo and Li (2016). The eigen-decomposition of one scatter matrix with respect to another is then often used to determine the dimension of the signal subspace and to separate signal and noise parts of the data. Three popular dimension reduction methods, namely principal component analysis (PCA), fourth order blind identification (FOBI) and sliced inverse regression (SIR) are considered in detail and the first two moments of subsets of the eigenvalues are used to test for the dimension of the signal space. The limiting null distributions of the test statistics are discussed and novel bootstrap strategies are suggested for the small sample cases. In all three cases, consistent test-based estimates of the signal subspace dimension are introduced as well. The asymptotic and bootstrap tests are compared in simulations and illustrated in real data examples.

To build a forecasting tool for the state of eutrophication in the Archipelago Sea, we fitted a Generalized Additive Mixed Model (GAMM) to marine environmental monitoring data, which were collected over the years 2011–2019 by an automated profiling buoy at the Seili ODAS-station. The resulting “Seili-index” can be used to predict the chlorophyll-α (chl-a) concentration in the seawater a number of days ahead by using the temperature forecast as a covariate. An array of test predictions with two separate models on the 2019 data set showed that the index is adept at predicting the amount of chl-a especially in the upper water layer. The visualization with 10 days of chl-a level predictions is presented online at https://saaristomeri.utu.fi/seili-index/. We also applied GAMMs to predict abrupt blooms of cyanobacteria on the basis of temperature and wind conditions and found the model to be feasible for short-term predictions. The use of automated monitoring data and the presented GAMM model in assessing the effects of natural resource management and pollution risks is discussed.

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 ModelsTime Series ModelsStatistical and Graphical Methods of Data AnalysisDimension ReductionData: Types and StructureTime Series, Stochastic Processes, and Functional Data.

Sliced inverse regression is one of the most popular sufficient dimension reduction methods. Originally, it was designed for independent and identically distributed data and recently extend to the case of serially and spatially dependent data. In this work we extend it to the case of spatially dependent data where the response might depend also on neighbouring covariates when the observations are taken on a grid-like structure as it is often the case in econometric spatial regression applications. We suggest guidelines on how to decide upon the dimension of the subspace of interest and also which spatial lag might be of interest when modeling the response. These guidelines are supported by a conducted simulation study.

We study the estimation of the linear discriminant with projection pursuit, a method that is unsupervised in the sense that it does not use the class labels in the estimation. Our viewpoint is asymptotic and, as our main contribution, we derive central limit theorems for estimators based on three different projection indices, skewness, kurtosis, and their convex combination. The results show that in each case the limiting covariance matrix is proportional to that of linear discriminant analysis (LDA), a supervised estimator of the discriminant. An extensive comparative study between the asymptotic variances reveals that projection pursuit gets arbitrarily close in efficiency to LDA when the distance between the groups is large enough and their proportions are reasonably balanced. Additionally, we show that consistent unsupervised estimation of the linear discriminant can be achieved also in high-dimensional regimes where the dimension grows at a suitable rate to the sample size, for example, pn=o(n1∕3) is sufficient under skewness-based projection pursuit. We conclude with a real data example and a simulation study investigating the validity of the obtained asymptotic formulas for finite samples. peerReviewed

Regional data analysis is concerned with the analysis and modeling of measurements that are spatially separated by specifically accounting for typical features of such data. Namely, measurements in close proximity tend to be more similar than the ones further separated. This might hold also true for cross-dependencies when multivariate spatial data is considered. Often, scientists are interested in linear transformations of such data which are easy to interpret and might be used as dimension reduction. Recently, for that purpose spatial blind source separation (SBSS) was introduced which assumes that the observed data are formed by a linear mixture of uncorrelated, weakly stationary random fields. However, in practical applications, it is well-known that when the spatial domain increases in size the weak stationarity assumptions can be violated in the sense that the second order dependency is varying over the domain which leads to non-stationary analysis. In our work we extend the SBSS model to adjust for these stationarity violations, present three novel estimators and establish the identifiability and affine equivariance property of the unmixing matrix functionals defining these estimators. In an extensive simulation study, we investigate the performance of our estimators and also show their use in the analysis of a geochemical dataset which is derived from the GEMAS geochemical mapping project. peerReviewed