Pith. sign in

REVIEW 3 major objections 6 minor 116 references

Bayesian nonparametric clustering for spatio-temporal data, with an application to air pollution

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Folding station coordinates into the clustering prior turns nine PM10 clusters into five.

desk verdict A competent review-and-application paper whose central five-versus-nine cluster comparison is not yet reproducible because the similarity-function hyperparameters are never reported. read the letter →

arxiv 2505.24694 v1 pith:OWIOUKAL submitted 2025-05-30 stat.ME stat.AP

Luca Aiello , Raffaele Argiento , Sirio Legramanti , Lucia Paci This is my paper
classification stat.MEstat.AP MSC 62F1562H3062M1062M30
keywords BayesiannonparametricclusteringspatialproductpartitionmodelsimilarityfunctionPM10airpollutionspatio-temporaldatatimeseriesDirichletprocess
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that swapping a plain product partition prior for a spatial one changes what clustering of air-quality time series reveals. On 2019 daily PM10 readings from 162 stations in Northern Italy, the authors cluster stations by the autoregressive coefficient and variance of a latent pollution process, and they show that a spatial Product Partition Model, which multiplies the partition prior by a similarity function rewarding stations that are close on the map, collapses nine clusters into five. The five-cluster partition is more parsimonious, reduces posterior uncertainty about which stations belong together, and divides the region into a high-persistence, high-variability Po Valley core surrounded by lower-persistence coastal and mountain zones. The point is not just that spatial information helps, but that a closed-form similarity function makes the added spatial structure cheap and routine to include.

What carries the argument

The engine of the analysis is the spatial Product Partition Model (sPPM), a Product Partition Model whose prior on the partition is weighted by a similarity function g(s*_k) that scores how spatially close the stations inside each cluster are. The paper uses the version obtained by treating station coordinates as Normal draws with a cluster-specific mean and covariance, integrating out those parameters under their conjugate Normal-Inverse-Wishart prior; this yields a closed-form similarity function, so adding spatial structure costs almost nothing in computation. Clustering targets the latent AR(1) parameters θ_i = (ϕ_i, τ²_i), with Dirichlet-process cohesion αΓ(n_k), and posterior sampling is made feasible at T = 365 by exploiting the tridiagonal structure of the precision matrix, cutting the cost of each density evaluation from O(T³) to O(T).

What would settle it

Re-estimate the sPPM on the same 162 stations with visibly different Normal-Inverse-Wishart settings, for example κ0 near zero versus κ0 large, or µ0 placed far from the observed station coordinates, and check whether the five-cluster partition and the three-region geography survive; separately, randomly permuting the station coordinates among locations should wipe out the spatial structure, so if the cluster count stays at five under permutation, the spatial term is not what is driving the parsimony.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that incorporating geographical coordinates into the prior over the partition, the one change separating the spatial PPM from the plain PPM, delivers a materially better summary of how PM10 dynamics vary across Northern Italy. Applying the sPPM with a Normal-Inverse-Wishart similarity function to the two parameters (ϕ, τ²) that drive an AR(1) latent process at each station produces a posterior point estimate with five clusters instead of the nine found without spatial covariates, with smaller co-clustering uncertainty. The estimated clusters trace a coherent geography: a central Po Valley cluster with high persistence and variability, an adjacent belt of moderate values, and coastal and mountain clusters with lower persistence; the cluster-specific time series remain as distinct as they were under nine clusters, so the reduction is a simplification rather than a loss.

Load-bearing premise

The load-bearing premise is that the unreported Normal-Inverse-Wishart hyperparameters that define the spatial similarity function are reasonable choices, and the five-versus-nine cluster comparison is computed under those values with no evidence about how the result would shift if they changed.

Editorial extensions

If this is right

  • Folding station coordinates into the partition prior reduces the estimated number of clusters from nine to five while shrinking posterior uncertainty about station membership.
  • The five-cluster partition separates the Po Valley into a high-persistence, high-variability core, a surrounding belt, and lower-persistence coastal and mountain zones, a geography that matches known pollution patterns.
  • Cluster-specific time-series bands stay distinct under fewer clusters, so the spatial prior simplifies the clustering without blurring the differences that matter.
  • The closed-form similarity function and the O(T) tridiagonal computations make the spatial model no harder to run than the plain model on daily data over a full year.
  • Because the whole procedure is Bayesian, each station's cluster assignment comes with an estimated co-clustering probability, not just a hard label.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct test the paper leaves implicit: the Normal-Inverse-Wishart hyperparameters that define the similarity function are not reported in the text, so the five-versus-nine comparison has not been shown to be stable across reasonable choices of those values.
  • Because predictive performance is flagged as an open gap, an obvious next test is whether the five-cluster partition improves forecasts of PM10 at unmonitored locations; the spatial similarity function already encodes distance, so kriging within clusters is a short step.
  • If the five-cluster partition is stable, it gives regulators a direct mapping from monitoring-station dynamics to regions, and comparing those clusters against external covariates such as emission inventories or land use would test whether the cluster boundaries carry substantive meaning beyond the PM10 series themselves.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This manuscript reviews Bayesian nonparametric clustering methods for spatio-temporal data, with a focus on spatial product partition models (sPPMs), and illustrates the methodology on daily PM10 concentrations from 162 monitoring stations in Northern Italy. The authors fit a dynamic linear model with station-specific AR(1) parameters and compare three specifications: no clustering, a product partition model (PPM) without spatial covariates, and an sPPM whose prior encourages spatially cohesive clusters via a Normal-Inverse-Wishart similarity function. The headline empirical result is that the sPPM yields five clusters versus nine for the PPM, with more interpretable spatial patterns.

Significance. If fully supported, the application would provide a valuable, reproducible demonstration of sPPM for environmental data, and the computational details in Algorithm 1 and Appendix A3 (exploiting tridiagonal precision matrices) are useful contributions. The methodological review sections are generally sound. However, the central parsimony comparison is not yet established, because two key prior inputs—the NIW hyperparameters of the similarity function and the two-stage value of α—are either unreported or fixed from the same data, and the posterior distribution of the number of clusters is not displayed. These are fixable issues, but they are load-bearing for the paper's main empirical claim.

major comments (3)
  1. [Section 5.2, Eq. (12), Appendix A2] The Normal-Inverse-Wishart hyperparameters (µ0, κ0, ν0, Λ0) that define the similarity function g3 in Eq. (12) are never reported. The spatial scale of the prior is entirely set by these values, and Λ0 in particular determines how strongly spatially dispersed clusters are penalized; for small Λ0 relative to inter-station distances, the sPPM may tend to produce spatially compact (and hence fewer) clusters almost by construction. The paper's central parsimony result (five versus nine clusters, Section 5.2.1, Figures 6–7) therefore cannot be evaluated from the manuscript alone. Please report the chosen hyperparameter values and provide a sensitivity analysis (e.g., varying κ0 and the scale of Λ0) showing the resulting number of clusters and partitions.
  2. [Section 5.2.1, Algorithm 1 lines 13–15] The sPPM sets α to the posterior mean of the PPM's DP concentration parameter estimated on the same data. Since α directly controls the prior tendency to create new clusters (Eq. (7)), the nine-to-five reduction is a comparison between a random-α PPM and a fixed-α sPPM, so the effect of the spatial similarity function is confounded with the choice of α. This is not by construction circular, but it is an avoidable two-stage estimation step. Please either estimate α jointly under the sPPM (e.g., with an approximation of the EPPF normalizing constant) or perform a sensitivity analysis over a range of α values and report how the number of clusters and the partition change.
  3. [Section 5.2.1, Figures 5–7] The parsimony claim rests solely on point estimates of the partition obtained by minimizing the VI loss. The posterior distribution of the number of clusters K under each model is not reported, so it is unclear whether the five-versus-nine gap is a systematic feature of the posteriors or an artifact of the loss function and the point estimate. Please report the posterior distribution of K (e.g., a histogram or credible interval) for both the PPM and the sPPM, as well as the uncertainty associated with the estimated partitions.
minor comments (6)
  1. [Figure 7 caption] The typo 'luster-specific' should read 'cluster-specific'.
  2. [Appendix A4, Figures A1–A2] 'Interquantile bands' should be 'interquartile bands' in the captions and text.
  3. [Figure 2] The x-axis labels are nearly unreadable in the current figure; please reformat the date ticks.
  4. [Section A1, after Eq. (A1)] The sentence beginning 'Where SM is...' should be cleaned up (lowercase 'where', and the simplex S_M should be defined precisely, with the expectation notation clarified).
  5. [Algorithm 1] The quantities K(−i) and Kaux are used in lines 6–8 but are not defined in the text; adding one sentence with definitions would improve reproducibility.
  6. [Section 5.2] The statement that aα=2, bα=0.5 gives a 'fairly diffuse' prior on α could be supported by reporting the implied prior mean and variance.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sPPM application uses established methodology, and the five-vs-nine cluster comparison is a posterior outcome, not a fitted input relabeled as a prediction.

full rationale

The paper does not derive predictions from fitted constants. Section 4 introduces the sPPM prior (Eq. 9) with similarity functions g1–g4 taken from Müller et al. (2011) and Page and Quintana (2016), which are external, not self, citations; the closed form for g3 in Appendix A2 is a standard Normal-Inverse-Wishart prior predictive computation, and neither the cohesion C(S_k)=αΓ(n_k) nor the similarity function is defined in terms of the PM10 outcome or the final partition. In Section 5.2.1, the sPPM fixes α at the PPM posterior mean, but this plug-in value does not by construction generate the five-cluster result: the PPM run with the same model class and its own posterior α yields nine clusters, so the reduction to five is an empirical effect of adding g3. The reported VI-loss partitions are posterior point estimates from MCMC, not out-of-sample predictions, and no statistical quantity in the paper is shown to equal its own input. The main concerns—the Normal-Inverse-Wishart hyperparameters (µ0, κ0, ν0, Λ0) in Eq. (12)/Appendix A2 are never stated in the text, and α is data-dependently fixed—are reproducibility and robustness issues, not circularity. Self-citations (Argiento and De Iorio 2022; Argiento et al. 2024; Paci et al. 2013) are background methodological references and are not load-bearing for the application. Hence score 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on several modeling assumptions: the AR(1) latent process with positivity constraint, Gaussian likelihood, the choice of similarity function with unspecified hyperparameters, and the two-stage estimation of α. No new entities are introduced.

free parameters (3)
  • DP concentration parameter α (PPM and sPPM) = PPM: posterior sample with Gamma(2,0.5) prior; sPPM: fixed to posterior mean from PPM
    In the PPM, α is assigned a Gamma(2,0.5) hyperprior and estimated; in the sPPM it is fixed to its posterior mean from the PPM (Section 5.2), which makes the comparison between methods depend on this data-derived value.
  • NIW hyperparameters for similarity function (µ0, κ0, ν0, Λ0) = not reported
    The similarity function g3(s_k*) in (12) uses a Normal-Inverse-Wishart prior on the cluster location parameters, but the hyperparameter values are not stated in the paper; they are chosen by hand in the code and influence spatial cohesion.
  • Hyperparameters for priors (aζ,bζ,aσ,bσ,aϕ,bϕ,aτ,bτ,aα,bα) = aζ=aσ=2, bζ=bσ=1, aϕ=bϕ=1, aτ=2, bτ=1, aα=2, bα=0.5
    These are chosen by hand to be vague or weakly informative (Section 5.2). They affect the prior on residual variances, AR coefficient, and α.
assumptions (6)
  • domain assumption The latent AR(1) process is stationary with positive autocorrelation, so ϕ is constrained to (0,1) via Beta(1,1).
    Used in Section 5.2, model (14) and prior (15). This rules out non-stationary or negatively autocorrelated behavior, which is reasonable for daily PM10 but still an assumption.
  • domain assumption The data likelihood is Gaussian after accounting for seasonal dummies and latent AR process.
    Model (14) assumes Gaussian noise for the observed PM10 and innovations, which may not hold for pollution data with occasional spikes; no residual diagnostics are provided.
  • domain assumption The similarity function g3 integrates spatial locations under a Normal-Inverse-Wishart prior predictive (Appendix A2), assuming the spatial distribution within each cluster is Gaussian.
    This is a modeling choice for spatial cohesion; the locations are not actually Gaussian, but the density is used as a similarity measure.
  • standard math The EPPF representations for finite and DP mixtures (equations (6), (7)) are correct and the PPM form (8) holds.
    These are standard results from Argiento and De Iorio (2022) and Ferguson (1973); the paper does not re-derive them fully but relies on the cited literature.
  • standard math Pitman's species sampling model representation ensures that a PPM with a valid EPPF corresponds to a mixture model (Section A1).
    Used to connect model (4) and (5); standard result.
  • ad hoc to paper Setting α in the sPPM to the posterior mean from the PPM is a valid basis for comparing the two models.
    This two-stage approach is not fully Bayesian and could favor one model; the paper does not conduct sensitivity analysis across α values.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bayesian nonparametric clustering for spatio-temporal data, with an application to air pollution." pith.science (2026). https://pith.science/paper/OWIOUKAL

@misc{pith2026250524694,
  author       = {Pith},
  title        = {Pith review of: Bayesian nonparametric clustering for spatio-temporal data, with an application to air pollution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OWIOUKAL}},
  note         = {Machine review of arXiv:2505.24694}
}
read the original abstract

Air pollution is a major global health hazard, with fine particulate matter (PM10) linked to severe respiratory and cardiovascular diseases. Hence, analyzing and clustering spatio-temporal air quality data is crucial for understanding pollution dynamics and guiding policy interventions. This work provides a review of Bayesian nonparametric clustering methods, with a particular focus on their application to spatio-temporal data, which are ubiquitous in environmental sciences. We first introduce key modeling approaches for point-referenced spatio-temporal data, highlighting their flexibility in capturing complex spatial and temporal dependencies. We then review recent advancements in Bayesian clustering, focusing on spatial product partition models, which incorporate spatial structure into the clustering process. We illustrate the proposed methods on PM10 monitoring data from Northern Italy, demonstrating their ability to identify meaningful pollution patterns. This review highlights the potential of Bayesian nonparametric methods for environmental risk assessment and offers insights into future research directions in spatio-temporal clustering for public health and environmental science.

Figures

Figures reproduced from arXiv: 2505.24694 by the authors.

Figure 1
Figure 1. Monitoring stations above the PM10 daily limit in Northern Italy in 2019. lower PM10 concentrations, often below 50 µg/m3 . The confidence bands in [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Time series of the PM10 concentrations at the 162 considered monitoring stations in Northern Italy along the year 2019. The average across stations is reported in red, while interquartile and 90% bands are reported in orange and yellow, respectively. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Mean (a) and standard deviation (b) of PM10 concentrations along year 2019 for each of the 162 considered monitoring stations in Northern Italy. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Posterior means of the autoregressive coefficient ϕi (left panel) and the vari￾ance parameter τ 2 i (right panel) under the model without clustering. the estimated posterior probability that any two locations belong to the same cluster, with darker entries indicating h…
Figure 5
Figure 5. Figure 5: Posterior co-clustering matrix (a) and monitoring stations colored according to the partition obtained from the PPM and the VI criterion (b). The legend displays the estimated cluster-specific values of θ = (ϕ, τ 2 ). 24 [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: Posterior co-clustering matrix (a) and monitoring stations colored according to the partition obtained from the sPPM and the VI criterion (b). The legend displays the estimated cluster-specific values of θ = (ϕ, τ 2 ). The results obtained after incorporating the simil…
Figure 7
Figure 7. Figure 7: Riverplot illustrating the difference between the VI-loss point estimates of the partitions obtained from the PPM (left) and the sPPM (right). The values in the boxes display the estimated luster-specific parameters θ ∗ k = (ϕ ∗ k , τ 2 k ∗ ) under the PPM and the sPPM…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

116 extracted references · 71 canonical work pages

  1. [1]

    and De Iorio, M

    Argiento, R. and De Iorio, M. (2022). Is infinity that far? A B ayesian nonparametric perspective of finite mixture models. The Annals of Statistics , 50(5):2641--2663

    2022

  2. [2]

    Argiento, R., Filippi-Mazzola, E., and Paci, L. (2024). Model-based clustering of categorical data based on the H amming distance. Journal of the American Statistical Association , doi:10.1080/01621459.2024.2402568:1--20

    arXiv 2024

  3. [3]

    P., and Gelfand, A

    Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2014). Hierarchical Modeling and Analysis for Spatial Data . CRC Press

    2014

  4. [4]

    Beraha, M., Argiento, R., Camerlenghi, F., and Guglielmi, A. (2025). Bayesian mixtures models with repulsive and attractive atoms. arXiv 2302.09034

    work page Pith review arXiv 2025

  5. [5]

    Berrocal, V. (2016). Identifying trends in the spatial errors of a regional climate model via clustering. Environmetrics , 27(2):90--102

    2016

  6. [6]

    C., Ehrlich, R., and Full, W

    Bezdek, J. C., Ehrlich, R., and Full, W. (1984). FCM: The fuzzy c-means clustering algorithm . Computers & geosciences , 10(2-3):191--203

    1984

  7. [7]

    Binder, D. A. (1978). Bayesian cluster analysis. Biometrika , 65(1):31--38

    1978

  8. [8]

    and Fr \"u hwirth-Schnatter, S

    Bitto, A. and Fr \"u hwirth-Schnatter, S. (2019). Achieving shrinkage in a time-varying parameter model framework. Journal of Econometrics , 210(1):75--97

    2019

Show all 116 references
  1. [9]

    B., and Raftery, A

    Bouveyron, C., Celeux, G., Murphy, T. B., and Raftery, A. E. (2019). Model-based clustering and classification for data science: with applications in R , volume 50. Cambridge University Press

    2019

  2. [10]

    Bucci, A., Ippoliti, L., Valentini, P., and Fontanella, S. (2022). Clustering spatio-temporal series of confirmed COVID -19 deaths in europe. Spatial Statistics , 49:100543

    2022

  3. [11]

    Cameletti, M., Ignaccolo, R., and Bande, S. (2011). Comparing spatio-temporal models for particulate matter in P iemonte. Environmetrics , 22(8):985--996

    2011

  4. [12]

    Cheam, A., Marbac, M., and McNicholas, P. (2017). Model-based clustering for spatiotemporal data on air quality monitoring. Environmetrics , 28(3):e2437

    2017

  5. [13]

    Cocchi, D., Greco, F., and Trivisano, C. (2007). Hierarchical space-time modelling of PM10 pollution. Atmospheric Environment , 41(3):532--542

    2007

  6. [14]

    and Wikle, C

    Cressie, N. and Wikle, C. K. (2015). Statistics for spatio-temporal data . John Wiley & Sons

    2015

  7. [15]

    B., Johnson, D

    Dahl, D. B., Johnson, D. J., and M \"u ller, P. (2022). Search algorithms and loss functions for B ayesian clustering. Journal of Computational and Graphical Statistics , 31(4):1189--1201

    2022

  8. [16]

    B., Johnson, D

    Dahl, D. B., Johnson, D. J., and Müller, P. (2021). salso: Search Algorithms and Loss Functions for B ayesian Clustering . R package version 0.2.22

    2021

  9. [17]

    and Raftery, A

    Dasgupta, A. and Raftery, A. E. (1998). Detecting features in spatial point processes with clutter via model-based clustering. Journal of the American statistical Association , 93(441):294--302

    1998

  10. [18]

    O., Hamm, N

    Datta, A., Banerjee, S., Finley, A. O., Hamm, N. A. S., and Schaap, M. (2016). Nonseparable dynamic nearest neighbor Gaussian process models for large spatio-temporal data with an application to particulate matter analysis . The Annals of Applied Statistics , 10(3):1286--1316

    2016

  11. [19]

    De Blasi, P., Favaro, S., Lijoi, A., Mena, R., Pr \"u nster, I., and Ruggiero, M. (2015). Are G ibbs-type priors the most natural generalization of the D irichlet process? IEEE Transactions on Pattern Analysis and Machine Intelligence , 37(2):212--229

    2015

  12. [20]

    A., Guindani, M., and Gelfand, A

    Duan, J. A., Guindani, M., and Gelfand, A. E. (2007). Generalized spatial D irichlet process models. Biometrika , 94(4):809--825

    2007

  13. [21]

    and Sanderson, C

    Eddelbuettel, D. and Sanderson, C. (2014). RcppArmadillo: Accelerating R with high-performance C++ linear algebra . Computational Statistics and Data Analysis , 71:1054--1063

    2014

  14. [22]

    Escobar, M. D. (1994). Estimating N ormal means with a D irichlet process prior. Journal of the American Statistical Association , 89(425):268--277

    1994

  15. [23]

    Escobar, M. D. and West, M. (1995). Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association , 90(430):577--588

    1995

  16. [24]

    Directive 2008/50/ec of the European Parliament and of the Council of 21 May 2008 on ambient air quality and cleaner air for Europe

    European Commission (2008). Directive 2008/50/ec of the European Parliament and of the Council of 21 May 2008 on ambient air quality and cleaner air for Europe . https://eur-lex.europa.eu/eli/dir/2008/50/2015-09-18

    2008

  17. [25]

    Ferguson, T. S. (1973). A B ayesian analysis of some nonparametric problems. The Annals of Statistics , 1(2):209--230

    1973

  18. [26]

    and Green, P

    Fern \'a ndez, C. and Green, P. J. (2002). Modelling spatially correlated data via mixtures: a B ayesian approach. Journal of the Royal Statistical Society Series B: Statistical Methodology , 64(4):805--826

    2002

  19. [27]

    and Raftery, A

    Fraley, C. and Raftery, A. E. (2002). Model-based clustering, discriminant analysis, and density estimation. Journal of the American statistical Association , 97(458):611--631

    2002

  20. [28]

    C., Browne, R

    Franczak, B. C., Browne, R. P., and McNicholas, P. D. (2013). Mixtures of shifted asymmetric L aplace distributions. IEEE Transactions on Pattern Analysis and Machine Intelligence , 36(6):1149--1157

    2013

  21. [29]

    Fr \"u hwirth-Schnatter, S., Celeux, G., and Robert, C. P. (2019). Handbook of mixture analysis . CRC press

    2019

  22. [30]

    and Kaufmann, S

    Fr \"u hwirth-Schnatter, S. and Kaufmann, S. (2008). Model-based clustering of multiple time series. Journal of Business & Economic Statistics , 26(1):78--89

    2008

  23. [31]

    u hwirth-Schnatter, S., Malsiner-Walli, G., and Gr \

    Fr \"u hwirth-Schnatter, S., Malsiner-Walli, G., and Gr \"u n, B. (2021). Generalized mixtures of finite mixtures and telescoping sampling. Bayesian Analysis , 16(4):1279--1307

    2021

  24. [32]

    and Pyne, S

    Fr \"u hwirth-Schnatter, S. and Pyne, S. (2010). Bayesian inference for finite mixtures of univariate and multivariate skew- N ormal and skew-t distributions. Biostatistics , 11(2):317--336

    2010

  25. [33]

    F \'u quene, J., Steel, M., and Rossell, D. (2019). On choosing mixture components via non-local priors. Journal of the Royal Statistical Society Series B: Statistical Methodology , 81(5):809--837

    2019

  26. [34]

    E., Kottas, A., and MacEachern, S

    Gelfand, A. E., Kottas, A., and MacEachern, S. N. (2005). Bayesian nonparametric spatial modeling with D irichlet process mixing. Journal of the American Statistical Association , 100(471):1021--1035

    2005

  27. [35]

    Gnedin, A. (2010). A species sampling model with finitely many types. Electronic Communications in Probability [electronic only] , 15:79--88

    2010

  28. [36]

    and Pitman, J

    Gnedin, A. and Pitman, J. (2006). Exchangeable G ibbs partitions and S tirling triangles. Journal of Mathematical sciences , 138:5674--5685

    2006

  29. [37]

    Goodman, L. A. (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika , 61(2):215--231

    1974

  30. [38]

    Grazian, C. (2023). A review on B ayesian model-based clustering. arXiv preprint arXiv:2303.17182

    2023 arXiv

  31. [39]

    Grazian, C., Villa, C., and Liseo, B. (2020). On a loss-based prior for the number of components in mixture models. Statistics & Probability Letters , 158:108656

    2020

  32. [40]

    Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and B ayesian model determination. Biometrika , 82(4):711--732

    1995

  33. [41]

    C., Engen, G

    Griffiths, R. C., Engen, G. F., and McCloskey, R. J. (1974). Properties of the two-parameter Poisson–Dirichlet distribution. Journal of Applied Probability , 11(2):319--338

    1974

  34. [42]

    Hamm, N., Finley, A., Schaap, M., and Stein, A. (2015). A spatially varying coefficient model for mapping PM10 air quality at the E uropean scale. Atmospheric Environment , 102:393 -- 405

    2015

  35. [43]

    Harrison, P. J. and Stevens, C. F. (1976). Bayesian forecasting. Journal of the Royal Statistical Society Series B: Statistical Methodology , 38(3):205--228

    1976

  36. [44]

    Hartigan, J. A. (1990). Partition M odels. Communications in statistics-Theory and methods , 19(8):2745--2756

    1990

  37. [45]

    J., Hooten, M

    Hefley, T. J., Hooten, M. B., Hanks, E. M., Russell, R. E., and Walsh, D. P. (2017). Dynamic spatio-temporal models for spatial data. Spatial statistics , 20:206--220

    2017

  38. [46]

    and James, L

    Ishwaran, H. and James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. Journal of the American statistical Association , 96(453):161--173

    2001

  39. [47]

    C., and Stephens, D

    Jasra, A., Holmes, C. C., and Stephens, D. A. (2005). Markov chain Monte Carlo methods and the label switching problem in B ayesian mixture modeling. Statistical Science , 20(1):50--67

    2005

  40. [48]

    E., and Walker, S

    Kalli, M., Griffin, J. E., and Walker, S. G. (2011). Slice sampling mixture models. Statistics and computing , 21:93--105

    2011

  41. [49]

    and Xekalaki, E

    Karlis, D. and Xekalaki, E. (2005). Mixed P oisson distributions. International Statistical Review/Revue Internationale de Statistique , pages 35--58

    2005

  42. [50]

    and Fr \"u hwirth-Schnatter, S

    Kastner, G. and Fr \"u hwirth-Schnatter, S. (2014). Ancillarity-sufficiency interweaving strategy ( ASIS ) for boosting mcmc estimation of stochastic volatility models. Computational Statistics & Data Analysis , 76:408--423

    2014

  43. [51]

    and Besag, J

    Knorr-Held, L. and Besag, J. (1998). Modelling risk from a disease in time and space. Statistics in medicine , 17(18):2045--2060

    1998

  44. [52]

    A., and Gelfand, A

    Kottas, A., Duan, J. A., and Gelfand, A. E. (2008). Modeling disease incidence data with spatial and spatio temporal D irichlet process mixtures. Biometrical Journal: Journal of Mathematical Methods in Biosciences , 50(1):29--42

    2008

  45. [53]

    Krnjaji \'c , M., Kottas, A., and Draper, D. (2008). Parametric and nonparametric B ayesian model specification: A case study involving models for count data. Computational Statistics & Data Analysis , 52(4):2110--2128

    2008

  46. [54]

    Larsen, B., Gilardoni, S., Stenstr \"o m, K., Niedzialek, J., Jimenez, J., and Belis, C. (2012). Sources for PM air pollution in the Po Plain, Italy: II. Probabilistic uncertainty characterization and sensitivity analysis of secondary and primary sources . Atmospheric Environm...

    2012

  47. [55]

    Laurini, M. P. (2019). A spatio-temporal approach to estimate patterns of climate change. Environmetrics , 30(1):e2542

    2019

  48. [56]

    E., Gangnon, R

    Lee, J., Kamenetsky, M. E., Gangnon, R. E., and Zhu, J. (2021). Clustered spatio-temporal varying coefficient regression model. Statistics in medicine , 40(2):465--480

    2021

  49. [57]

    and McLachlan, G

    Lee, S. and McLachlan, G. J. (2014). Finite mixtures of multivariate skew t-distributions: some recent and new results. Statistics and Computing , 24:181--202

    2014

  50. [58]

    Li, B., Zhang, X., and Smerdon, J. E. (2016). Comparison between spatio-temporal random processes and application to climate model data. Environmetrics , 27(5):267--279

    2016

  51. [59]

    Lijoi, A., Prunster, I., and Walker, S. G. (2008). Bayesian nonparametric estimators derived from conditional G ibbs structures. Annals of Applied Probability , 18(4):1519--1547

    2008

  52. [60]

    Liu, J., Wade, S., and Bochkina, N. (2024). Shared differential clustering across single-cell rna sequencing datasets with the hierarchical D irichlet process. Econometrics and Statistics

    2024

  53. [61]

    Lo, A. Y. (1984). On a class of B ayesian nonparametric estimates: I. density estimates. The Annals of Statistics , pages 351--357

    1984

  54. [62]

    MacEachern, S. N. (1994). Estimating N ormal means with a conjugate style D irichlet process prior. Communications in Statistics-Simulation and Computation , 23(3):727--741

    1994

  55. [63]

    MacEachern, S. N. (1998). Computational methods for mixture of D irichlet process models. In Practical nonparametric and semiparametric Bayesian statistics , pages 23--43. Springer

    1998

  56. [64]

    MacEachern, S. N. and M \"u ller, P. (1998). Estimating mixture of D irichlet process models. Journal of Computational and Graphical Statistics , 7(2):223--238

    1998

  57. [65]

    Mastrantonio, G., Grazian, C., Mancinelli, S., and Bibbona, E. (2019). New formulation of the logistic- G aussian process to analyze trajectory tracking data. The Annals of Applied Statistics , 13(4):2483--2508

    2019

  58. [66]

    J., Miller, S., and Pelletier, D

    McCausland, W. J., Miller, S., and Pelletier, D. (2011). Simulation smoothing for state--space models: A computational efficiency analysis. Computational Statistics & Data Analysis , 55(1):199--212

    2011

  59. [67]

    Meil a , M. (2007). Comparing clusterings--an information based distance. Journal of multivariate analysis , 98(5):873--895

    2007

  60. [68]

    Meurant, G. (1992). A review on the inverse of symmetric tridiagonal and block tridiagonal matrices. SIAM Journal on Matrix Analysis and Applications , 13(3):707--728

    1992

  61. [69]

    Miller, J. W. and Harrison, M. T. (2018). Mixture models with a prior on the number of components. Journal of the American Statistical Association , 113(521):340--356

    2018

  62. [70]

    Molitor, J., Papathomas, M., Jerrett, M., and Richardson, S. (2010). Bayesian profile regression with an application to the National Survey of Children's Health . Biostatistics , 11(3):484--498

    2010

  63. [71]

    M \"u ller, P., Quintana, F., and Rosner, G. L. (2011). A P roduct P artition M odel with regression on covariates. Journal of Computational and Graphical Statistics , 20(1):260--278

    2011

  64. [72]

    A., Jara, A., and Hanson, T

    M \"u ller, P., Quintana, F. A., Jara, A., and Hanson, T. (2015). Bayesian nonparametric data analysis , volume 1. Springer

    2015

  65. [73]

    M., Gaetan, C., and Girardi, P

    Musau, V. M., Gaetan, C., and Girardi, P. (2022). Clustering of bivariate satellite time series: A quantile approach. Environmetrics , 33(7):e2755

    2022

  66. [74]

    Neal, R. M. (2000). Markov chain sampling methods for D irichlet process mixture models. Journal of computational and graphical statistics , 9(2):249--265

    2000

  67. [75]

    and Gelfand, A

    Nguyen, X. and Gelfand, A. E. (2011). The D irichlet labeling process for clustering functional data. Statistica Sinica , pages 1249--1289

    2011

  68. [76]

    Nieto-Barajas, L. E. and Contreras-Crist \'a n, A. (2014). A B ayesian nonparametric approach for time series clustering. Bayesian Analysis , 9(1):147--170

    2014

  69. [77]

    Nobile, A. (2004). On the posterior distribution of the number of components in a finite mixture. Annals of Statistics , 32:2044--2073

    2004

  70. [78]

    B., Gormley, I

    O’Hagan, A., Murphy, T. B., Gormley, I. C., McNicholas, P. D., and Karlis, D. (2016). Clustering with the multivariate normal inverse G aussian distribution. Computational Statistics & Data Analysis , 93:18--30

    2016

  71. [79]

    E., and Holland, D

    Paci, L., Gelfand, A. E., and Holland, D. M. (2013). Spatio-temporal modeling for real-time ozone forecasting. Spatial Statistics , 4:79--93

    2013

  72. [80]

    Page, G. L. and Quintana, F. A. (2016). Spatial P roduct P artition M odels. Bayesian Analysis , 11(1):265--298

    2016

  73. [81]

    L., Quintana, F

    Page, G. L., Quintana, F. A., and Dahl, D. B. (2022). Dependent modeling of temporal sequences of random partitions. Journal of Computational and Graphical Statistics , 31(2):614--627

    2022

  74. [82]

    Palla, G., Der \'e nyi, I., Farkas, I., and Vicsek, T. (2005). Uncovering the overlapping community structure of complex networks in nature and society. Nature , 435(7043):814--818

    2005

  75. [83]

    and Roberts, G

    Papaspiliopoulos, O. and Roberts, G. O. (2008). Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models . Biometrika , 95(1):169--186

    2008

  76. [84]

    and McNicholas, P

    Paton, F. and McNicholas, P. D. (2020). Detecting British Columbia coastal rainfall patterns by clustering G aussian processes. Environmetrics , 31(8):e2631

    2020

  77. [85]

    J., Benvenuti, C., Cianella, R., Caputo, M

    Peluso, S., Mira, A., Rue, H., Tierney, N. J., Benvenuti, C., Cianella, R., Caputo, M. L., and Auricchio, A. (2020). A B ayesian spatiotemporal statistical analysis of out-of-hospital cardiac arrests. Biometrical Journal , 62(4):1105--1119

    2020

  78. [86]

    Petralia, F., Rao, V., and Dunson, D. (2012). Repulsive mixtures. Advances in neural information processing systems , 25

    2012

  79. [87]

    C., Demaria, G., Colombi, C., Cuccia, E., and Dal Santo, U

    Pietrogrande, M. C., Demaria, G., Colombi, C., Cuccia, E., and Dal Santo, U. (2022). Seasonal and spatial variations of PM 10 and PM 2.5 oxidative potential in five urban and rural sites across L ombardia R egion, I taly. International Journal of Environmental Research and Pub...

    2022

  80. [88]

    Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probability theory and related fields , 102(2):145--158

    1995

  81. [89]

    Pitman, J. (1996). Some developments of the Blackwell-Macqueen URN scheme. Lecture Notes-Monograph Series , 30:245--267

    1996

  82. [90]

    Quintana, F. A. (2006). A predictive view of B ayesian clustering. Journal of Statistical Planning and Inference , 136(8):2407--2429

    2006

  83. [91]

    and Maruotti, A

    Ranalli, M. and Maruotti, A. (2020). Model-based clustering for noisy longitudinal circular data, with application to animal movement. Environmetrics , 31(2):e2572

    2020

  84. [92]

    Rasmussen, C. E. and Williams, C. K. I. (2005). Gaussian Processes for Machine Learning . The MIT Press

    2005

  85. [93]

    Reich, B. J. and Fuentes, M. (2007). A multivariate semiparametric B ayesian spatial modeling framework for hurricane surface wind fields. The Annals of Applied Statistics , 1(1):249--264

    2007

  86. [94]

    and Green, P

    Richardson, S. and Green, P. J. (1997). On B ayesian analysis of mixtures with an unknown number of components (with discussion). Journal of the Royal Statistical Society Series B: Statistical Methodology , 59(4):731--792

    1997

  87. [95]

    and Dunson, D

    Rodr guez, A. and Dunson, D. B. (2011). Nonparametric B ayesian models through probit stick-breaking processes. Bayesian Analysis , 6(1):145--178

    2011

  88. [96]

    Rousseau, J., Grazian, C., and Lee, J. E. (2019). Bayesian mixture models: Theory and methods. In Handbook of mixture analysis , pages 53--72. Chapman and Hall/CRC

    2019

  89. [97]

    and Mengersen, K

    Rousseau, J. and Mengersen, K. (2011). Asymptotic behaviour of the posterior distribution in overfitted mixture models. Journal of the Royal Statistical Society Series B: Statistical Methodology , 73(5):689--710

    2011

  90. [98]

    Rushworth, A., Lee, D., and Sarran, C. (2017). An adaptive spatiotemporal smoothing model for estimating trends and step changes in disease risk. Journal of the Royal Statistical Society Series C: Applied Statistics , 66(1):141--157

    2017

  91. [99]

    K., Gelfand, A

    Sahu, S. K., Gelfand, A. E., and M, D. (2006). Spatio-temporal modeling of fine particulate matter. Journal of Agricultural, Biological, and Environmental Statistics , 11:61--86

    2006

  92. [100]

    Sarang, P. (2023). Centroid-Based Clustering , pages 171--183. Springer International Publishing, Cham

    2023

  93. [101]

    Sethuraman, J. (1994). A constructive definition of D irichlet priors. Statistica sinica , pages 639--650

    1994

  94. [102]

    Stephens, M. (2000a). Bayesian analysis of mixture models with an unknown number of components--an alternative to reversible jump methods. The Annals of Statistics , 28(1):40--74

    2000

  95. [103]

    Stephens, M. (2000b). Dealing with label switching in mixture models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 62(4):795--809

    2000

  96. [104]

    R., M \"u ller, P., and Sans \'o , B

    Stroud, J. R., M \"u ller, P., and Sans \'o , B. (2001). Dynamic models for spatiotemporal data. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 63(4):673--689

    2001

  97. [105]

    Torabi, M. (2014). Spatiotemporal modeling of odds of disease. Environmetrics , 25(5):341--350

    2014

  98. [106]

    D., and Hosack, G

    Vanhatalo, J., Foster, S. D., and Hosack, G. R. (2021). Spatiotemporal clustering using G aussian processes embedded in a mixture model. Environmetrics , 32(7):e2681

    2021

  99. [107]

    Wade, S. (2023). Bayesian cluster analysis. Philosophical Transactions of the Royal Society A , 381(2247):20220149

    2023

  100. [108]

    and Ghahramani, Z

    Wade, S. and Ghahramani, Z. (2018). Bayesian cluster analysis: Point estimation and credible balls (with discussion). Bayesian Analysis , 13(2):559--626

    2018

  101. [109]

    A., Carlin, B

    Waller, L. A., Carlin, B. P., Xia, H., and Gelfand, A. E. (1997). Hierarchical spatio-temporal mapping of disease rates. Journal of the American Statistical association , 92(438):607--617

    1997

  102. [110]

    Wan, Y., Xu, M., Huang, H., and Xi Chen, S. (2021). A spatio-temporal model for the analysis and prediction of fine particulate matter concentration in B eijing. Environmetrics , 32(1):e2648

    2021

  103. [111]

    Wang, F., Duan, C., Li, Y., Huang, H., and Shia, B.-C. (2024). Spatiotemporal varying coefficient model for respiratory disease mapping in T aiwan. Biostatistics , 25(1):40--56

    2024

  104. [112]

    and Harrison, J

    West, M. and Harrison, J. (2006). Bayesian forecasting and dynamic models . Springer Science & Business Media

    2006

  105. [113]

    World health statistics 2022: monitoring health for the SDGs, sustainable development goals

    World Health Organization (2022). World health statistics 2022: monitoring health for the SDGs, sustainable development goals . World Health Organization

    2022

  106. [114]

    and Luo, X

    Wu, Q. and Luo, X. (2022). Nonparametric B ayesian two-level clustering for subject-level single-cell expression data. Statistica Sinica , 32(4):1835--1856

    2022

  107. [115]

    and Xu, Y

    Xie, F. and Xu, Y. (2020). Bayesian repulsive G aussian mixture model. Journal of the American Statistical Association , 115(529):187--203

    2020

  108. [116]

    Xu, Y., M \"u ller, P., and Telesca, D. (2016). Bayesian inference for latent biologic structure with determinantal point processes ( DPP ). Biometrics , 72(3):955--964

    2016

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.