arXiv subjects

Peter G. M. van der Heijden

Publications and source records attributed to Peter G. M. van der Heijden.

At least 19 recordsLinked to original sources

arXiv2026

Nonnegative matrix factorizations and related compositional models: Equivalence, identifiability, and an application on the grain-size analysis of sediments

Across fields such as machine learning, social science, and geology, considerable attention has been given to models that factorize a nonnegative matrix into the product of two or three matrices, subject to nonnegative or row-sum-to-1 constraints. Although these models are to a large extent similar or even equivalent, they are presented under different names, and their similarity is not well known. This paper highlights similarities among five models, latent budget analysis (LBA) and latent class analysis (LCA) from social science, end-member analysis (EMA) from geology, probabilistic latent semantic analysis (PLSA) and nonnegative matrix factorization (NMF) from machine learning. We focus on the identifiability of these models. We prove that the solution of LBA, EMA, LCA, PLSA is unique if and only if the solution of NMF is unique. Consequently, existing uniqueness theorems for NMF directly apply to LBA, EMA, LCA, PLSA, and vice versa. We also provide a brief review of algorithms for the estimation of these models. We illustrate NMF on a sedimentary grain-size distribution dataset from sedimentary geology, and end the paper with a discussion of closely related model: archetypal analysis.

stat.ML↗

arXiv2026

Maximum-distance nonnegative matrix factorization for unmixing highly mixed grain-size distribution data: A generalization of AnalySize

Nonnegative matrix factorization (NMF) decomposes a nonnegative matrix into the product of two nonnegative matrices. This property makes NMF well suited for unmixing grain-size distribution data, which are inherently nonnegative and have row sums equal to one. Previous studies have shown that AnalySize, an NMF-based method, performs well on poorly mixed grain-size distribution data but struggles when the data is highly mixed, where no observed samples are close to the true end members. To overcome this limitation, we introduce a maximum-distance NMF that encourages the estimated end members to be as distinct as possible and develop a hierarchical alternating least squares algorithm for optimization. The proposed formulation can be regarded as a generalization of AnalySize, where AnalySize minimizes the distance among end members while the proposed method maximizes it. Experimental results demonstrate that the method effectively decomposes highly mixed grain-size distribution data.

cs.LG↗

arXiv2026

Dual system estimation using mixed effects loglinear models

In official statistics, dual system estimation (DSE) is a well-known tool to estimate the size of a population. Two sources are linked, and the number of units that are missed by both sources is estimated. Often dual system estimation is carried out in each of the levels of a stratifying variable, such as region. DSE can be considered a loglinear independence model, and, with a stratifying variable, a loglinear conditional independence model. The standard approach is to estimate parameters for each level of the stratifying variable. Thus, when the number of levels of the stratifying variable is large, the number of parameters estimated is large as well. Mixed effects loglinear models, where sets of parameters involving the stratifying variable are replaced by a distribution parameterised by its mean and a variance, have also been proposed, and we investigate their properties through simulation. In our simulation studies the mixed effects loglinear model outperforms the fixed effects loglinear model although only to a small extent in terms of mean squared error. We show how mixed effects dual system estimation can be extended to multiple system estimation.

stat.ME↗

arXiv2026

Identification of NMF by choosing maximum-volume basis vectors

In nonnegative matrix factorization (NMF), minimum-volume-constrained NMF is a widely used framework for identifying the solution of NMF by making basis vectors as similar as possible. This typically induces sparsity in the coefficient matrix, with each row containing zero entries. Consequently, minimum-volume-constrained NMF may fail for highly mixed data, where such sparsity does not hold. Moreover, the estimated basis vectors in minimum-volume-constrained NMF may be difficult to interpret as they may be mixtures of the ground truth basis vectors. To address these limitations, in this paper we propose a new NMF framework, called maximum-volume-constrained NMF, which makes the basis vectors as distinct as possible. We further establish an identifiability theorem for maximum-volume-constrained NMF and provide an algorithm to estimate it. Experimental results demonstrate the effectiveness of the proposed method.

cs.LG↗

arXiv2026

Correspondence Analysis and PMI-Based Word Embeddings: A Comparative Study

Popular word embedding methods such as GloVe and Word2Vec are related to the factorization of the pointwise mutual information (PMI) matrix. In this paper, we establish a formal connection between correspondence analysis (CA) and PMI-based word embedding methods. CA is a dimensionality reduction method that uses singular value decomposition (SVD), and we show that CA is mathematically close to the weighted factorization of the PMI matrix. We further introduce variants of CA for word-context matrices, namely CA applied after a square-root transformation (ROOT-CA) and after a fourth-root transformation (ROOTROOT-CA). We analyze the performance of these methods and examine how their success or failure is influenced by extreme values in the decomposed matrix. Although our primary focus is on traditionalstatic word embedding methods, we also include a comparison with a transformer-based encoder (BERT) to situate the results relative to contextual embeddings. Empirical evaluations across multiple corpora and word-similarity benchmarks show that ROOT-CA and ROOTROOT-CA perform slightly better overall than standard PMI-based methods and achieve results competitive with BERT.

cs.CL↗

arXiv2026

Unmixing highly mixed grain size distribution data via maximum volume constrained end member analysis

End member analysis (EMA) unmixes grain size distribution (GSD) data into a mixture of end members (EMs), thus helping understand sediment provenance and depositional regimes and processes. In highly mixed data sets, however, many EMA algorithms find EMs which are still a mixture of true EMs. To overcome this, we propose maximum volume constrained EMA (MVC-EMA), which finds EMs as different as possible. We provide a uniqueness theorem and a quadratic programming algorithm for MVC-EMA. Experimental results show that MVC-EMA can effectively find true EMs in highly mixed data sets.

stat.ME↗

arXiv2024

The Extended Crosswise Model Adjusted for Random Answering

The Extended Crosswise Model is a popular randomized response design that employs a sensitive and a randomized innocuous statement, and asks respondents if one of these statements is true, or that none or both are true. The model has a degree of freedom to test for response biases, but is unable to detect random answering. In this paper, we propose two new methods to indirectly estimate and correct for random answering. One method uses a non-sensitive control statement and a quasi-randomized innocuous statement to which both answers are known to estimate the proportion of random respondents. The other method assigns less weight in the estimation procedure to respondents who complete the survey in an unrealistically short time. For four surveys among elite athletes, we use these methods to correct the prevalence estimates of doping use for random answering.

stat.ME↗

arXiv2024

Nowcasting in triple-system estimation

Multiple systems estimation uses samples that each cover part of a population to obtain a total population size estimate. Ideally, all the available samples are used, but if some samples are available (much) later, one may use only the samples that are available early. Under some regularity conditions, including sample independence, two samples is enough to obtain an asymptotically unbiased population size estimate. However, the assumption of sample independence may be unrealistic, especially when samples are derived from administrative sources. The sample independence assumption can be relaxed when three or more samples are used, which is therefore generally recommended. This may be a problem if the third sample is available much later than the first two samples. Therefore, in this paper we propose a new approach that deals with this issue by utilising older samples, using the so-called expectation maximisation algorithm. This leads to a population size nowcast estimate that is asymptotically unbiased under more relaxed assumptions than the estimate based on two samples. The resulting nowcasting model is applied to the problem of estimating the number of homeless people in The Netherlands, which leads to reasonably accurate nowcast estimates.

stat.ME↗

arXiv2024

Generating contingency tables with fixed marginal probabilities and dependence structures described by loglinear models

We present a method to generate contingency tables that follow loglinear models with prescribed marginal probabilities and dependence structures. We make use of (loglinear) Poisson regression, where the dependence structures, described using odds ratios, are implemented using an offset term. We apply this methodology to carry out simulation studies in the context of population size estimation using dual system and triple system estimators, popular in official statistics. These estimators use contingency tables that summarise the counts of elements enumerated or captured within lists that are linked. The simulation is used to investigate these estimators in the situation that the model assumptions are fulfilled, and the situation that the model assumptions are violated.

stat.ME↗

arXiv2024

Correspondence analysis: handling cell-wise outliers via the reconstitution algorithm

Correspondence analysis (CA) is a popular technique to visualize the relationship between two categorical variables. CA uses the data from a two-way contingency table and is affected by the presence of outliers. The supplementary points method is a popular method to handle outliers. Its disadvantage is that the information from entire rows or columns is removed. However, outliers can be caused by cells only. In this paper, a reconstitution algorithm is introduced to cope with such cells. This algorithm can reduce the contribution of cells in CA instead of deleting entire rows or columns. Thus the remaining information in the row and column involved can be used in the analysis. The reconstitution algorithm is compared with two alternative methods for handling outliers, the supplementary points method and MacroPCA. It is shown that the proposed strategy works well.

stat.ME↗

arXiv2024

Using Chao's Estimator as a Stopping Criterion for Technology-Assisted Review

Technology-Assisted Review (TAR) aims to reduce the human effort required for screening processes such as abstract screening for systematic literature reviews. Human reviewers label documents as relevant or irrelevant during this process, while the system incrementally updates a prediction model based on the reviewers' previous decisions. After each model update, the system proposes new documents it deems relevant, to prioritize relevant documentsover irrelevant ones. A stopping criterion is necessary to guide users in stopping the review process to minimize the number of missed relevant documents and the number of read irrelevant documents. In this paper, we propose and evaluate a new ensemble-based Active Learning strategy and a stopping criterion based on Chao's Population Size Estimator that estimates the prevalence of relevant documents in the dataset. Our simulation study demonstrates that this criterion performs well on several datasets and is compared to other methods presented in the literature.

cs.IR↗

arXiv2023

Bias correction in multiple-systems estimation

If part of a population is hidden but two or more sources are available that each cover parts of this population, dual- or multiple-system(s) estimation can be applied to estimate this population. For this it is common to use the log-linear model, estimated with maximum likelihood. These maximum likelihood estimates are based on a non-linear model and therefore suffer from finite-sample bias, which can be substantial in case of small samples or a small population size. This problem was recognised by Chapman, who derived an estimator with good small sample properties in case of two available sources. However, he did not derive an estimator for more than two sources. We propose an estimator that is an extension of Chapman's estimator to three or more sources and compare this estimator with other bias-reduced estimators in a simulation study. The proposed estimator performs well, and much better than the other estimators. A real data example on homelessness in the Netherlands shows that our proposed model can make a substantial difference.

stat.ME↗

arXiv2023

Improving information retrieval through correspondence analysis instead of latent semantic analysis

Both latent semantic analysis (LSA) and correspondence analysis (CA) are dimensionality reduction techniques that use singular value decomposition (SVD) for information retrieval. Theoretically, the results of LSA display both the association between documents and terms, and marginal effects; in comparison, CA only focuses on the associations between documents and terms. Marginal effects are usually not relevant for information retrieval, and therefore, from a theoretical perspective CA is more suitable for information retrieval. In this paper, we empirically compare LSA and CA. The elements of the raw document-term matrix are weighted, and the weighting exponent of singular values is adjusted to improve the performance of LSA. We explore whether these two weightings also improve the performance of CA. In addition, we compare the optimal singular value weighting exponents for LSA and CA to identify what the initial dimensions in LSA correspond to. The results for four empirical datasets show that CA always performs better than LSA. Weighting the elements of the raw data matrix can improve CA; however, it is data dependent and the improvement is small. Adjusting the singular value weighting exponent usually improves the performance of CA; however, the extent of the improved performance depends on the dataset and number of dimensions. In general, CA needs a larger singular value weighting exponent than LSA to obtain the optimal performance. This indicates that CA emphasizes initial dimensions more than LSA, and thus, margins play an important role in the initial dimensions in LSA.

cs.IR↗

arXiv2022

A comparison of latent semantic analysis and correspondence analysis of document-term matrices

Latent semantic analysis (LSA) and correspondence analysis (CA) are two techniques that use a singular value decomposition (SVD) for dimensionality reduction. LSA has been extensively used to obtain low-dimensional representations that capture relationships among documents and terms. In this article, we present a theoretical analysis and comparison of the two techniques in the context of document-term matrices. We show that CA has some attractive properties as compared to LSA, for instance that effects of margins, i.e. sums of row elements and column elements, arising from differing document-lengths and term-frequencies are effectively eliminated, so that the CA solution is optimally suited to focus on relationships among documents and terms. A unifying framework is proposed that includes both CA and LSA as special cases. We empirically compare CA to various LSA based methods on text categorization in English and authorship attribution on historical Dutch texts, and find that CA performs significantly better. We also apply CA to a long-standing question regarding the authorship of the Dutch national anthem Wilhelmus and provide further support that it can be attributed to the author Datheen, amongst several contenders.

cs.IR↗

arXiv2020

Multiple system estimation using covariates having missing values and measurement error: estimating the size of the Māori population in New Zealand

We investigate use of two or more linked registers, or lists, for both population size estimation and to investigate the relationship between variables appearing on all or only some registers. This relationship is usually not fully known because some individuals appear in only some registers, and some are not in any register. These two problems have been solved simultaneously using the EM algorithm. We extend this approach to estimate the size of the indigenous Māori population in New Zealand, leading to several innovations: (1) the approach is extended to four registers (including the population census), where the reporting of Māori status differs between registers; (2) some individuals in one or more registers have missing ethnicity, and we adapt the approach to handle this additional missingness; (3) some registers cover subsets of the population by design. We discuss under which assumptions such structural undercoverage can be ignored and provide a general result; (4) we treat the Māori indicator in each register as a variable measured with error, and embed a latent class model in the multiple system estimation to estimate the population size of a latent variable, interpreted as the true Māori status. Finally, we discuss estimating the Māori population size from administrative data only. Supplementary materials for our article are available online.

stat.AP↗

arXiv2019

Linkage Free Dual System Estimation

In this paper it is shown that under certain conditions there is a relationship between the parameter estimation of the Fellegi--Sunter probabilistic linkage model and dual system estimation. This relationship can be used as the basis of an approach to population size estimation. In this case it is sufficient to estimate the parameters of the linkage model in order to obtain the population size estimate. Neither classification of the record pairs into links and non-links, nor forcing the records into a series of 1-1 matches, nor clerical review of the potential links is required. The accuracy of the proposed estimator appears to be bounded by the accuracy of the dual system estimator with perfect linkage and it diminishes as the discriminatory power of the linkage variables decreases.

stat.ME↗

arXiv2012

People born in the Middle East but residing in the Netherlands: Invariant population size estimates and the role of active and passive covariates

Including covariates in loglinear models of population registers improves population size estimates for two reasons. First, it is possible to take heterogeneity of inclusion probabilities over the levels of a covariate into account; and second, it allows subdivision of the estimated population by the levels of the covariates, giving insight into characteristics of individuals that are not included in any of the registers. The issue of whether or not marginalizing the full table of registers by covariates over one or more covariates leaves the estimated population size estimate invariant is intimately related to collapsibility of contingency tables [Biometrika 70 (1983) 567-578]. We show that, with information from two registers, population size invariance is equivalent to the simultaneous collapsibility of each margin consisting of one register and the covariates. We give a short path characterization of the loglinear model which describes when marginalizing over a covariate leads to different population size estimates. Covariates that are collapsible are called passive, to distinguish them from covariates that are not collapsible and are termed active. We make the case that it can be useful to include passive covariates within the estimation model, because they allow a finer description of the population in terms of these covariates. As an example we discuss the estimation of the population size of people born in the Middle East but residing in the Netherlands.

stat.AP↗

arXiv2009

A covariate adjustment for zero-truncated approaches to estimating the size of hidden and elusive populations

In this paper we consider the estimation of population size from one-source capture--recapture data, that is, a list in which individuals can potentially be found repeatedly and where the question is how many individuals are missed by the list. As a typical example, we provide data from a drug user study in Bangkok from 2001 where the list consists of drug users who repeatedly contact treatment institutions. Drug users with 1, 2, 3$,...$ contacts occur, but drug users with zero contacts are not present, requiring the size of this group to be estimated. Statistically, these data can be considered as stemming from a zero-truncated count distribution. We revisit an estimator for the population size suggested by Zelterman that is known to be robust under potential unobserved heterogeneity. We demonstrate that the Zelterman estimator can be viewed as a maximum likelihood estimator for a locally truncated Poisson likelihood which is equivalent to a binomial likelihood. This result allows the extension of the Zelterman estimator by means of logistic regression to include observed heterogeneity in the form of covariates. We also review an estimator proposed by Chao and explain why we are not able to obtain similar results for this estimator. The Zelterman estimator is applied in two case studies, the first a drug user study from Bangkok, the second an illegal immigrant study in the Netherlands. Our results suggest the new estimator should be used, in particular, if substantial unobserved heterogeneity is present.

stat.AP↗