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arxiv: 2605.22595 · v1 · pith:UP3YXSWXnew · submitted 2026-05-21 · 📊 stat.ME

A new class of functional conditional autoregressive models

Sooran Kim This is my paper

Pith reviewed 2026-05-22 03:48 UTC · model grok-4.3

classification 📊 stat.ME
keywords functional dataspatial dependenceconditional autoregressive modelscovariance operatorsuperconsistencyasymptotic normalityexpanding lattice
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The pith

A new conditional autoregressive model for functional data yields consistent covariance estimates and superconsistent inference on spatial dependence.

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

The paper introduces a class of models for functional data that depend on neighbors through conditional means, fully specified by a covariance operator and one spatial dependence parameter. Estimation first recovers the covariance operator from conditionally centered observations, then maximizes a likelihood on projected data to recover the dependence parameter, and finally applies a profile approach for the joint estimators. Under an expanding lattice regime the covariance estimator is consistent where naive marginal centering fails, and the dependence parameter estimator is superconsistent and asymptotically normal. The asymptotic normality supplies the first route to formal statistical inference on the strength of spatial dependence for functional observations. Simulations confirm the rates, and the method is applied to weekly pollution trajectories across Midwestern counties.

Core claim

The proposed functional conditional autoregressive model, defined by conditional means given neighboring functional observations and parameterized by a covariance operator together with a spatial dependence parameter, admits an estimator of the covariance operator that is consistent when formed from conditionally centered data and an estimator of the spatial dependence parameter that is superconsistent and asymptotically normal under expanding lattice asymptotics, thereby enabling statistical inference for spatial dependence in functional data.

What carries the argument

The conditional mean given neighboring functional observations, together with the covariance operator and scalar spatial dependence parameter that characterize the entire dependence structure.

If this is right

  • The covariance estimator is consistent only when formed from conditionally centered data rather than from marginally centered data.
  • The spatial dependence parameter estimator converges at a superconsistent rate and is asymptotically normal.
  • Formal statistical inference on the magnitude of spatial dependence becomes available for functional data.
  • The procedure remains computationally feasible for moderate numbers of spatial locations and functional observations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same conditional centering step could be tested on non-lattice spatial arrangements such as irregular monitoring networks.
  • Asymptotic normality of the dependence estimator would support hypothesis tests for the presence or absence of spatial dependence in new functional datasets.
  • The model structure suggests a natural route to adding covariate effects while retaining the superconsistency property for the dependence parameter.

Load-bearing premise

The observed functional data are generated exactly from the proposed conditional autoregressive model under an expanding lattice regime with suitable regularity on the covariance operator and the spatial dependence structure.

What would settle it

A large-scale simulation on expanding lattices in which the spatial dependence parameter estimator fails to converge faster than the usual root-n rate or in which its studentized distribution deviates systematically from normality would contradict the central claims.

Figures

Figures reproduced from arXiv: 2605.22595 by Sooran Kim.

Figure 1
Figure 1. Figure 1: Rejection rates of the proposed test statistic ( [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: PM2.5 concentrations in 1054 counties in the Midwestern United States. (a) Weekly curves over 53 weeks; (b) Average concentration for 53 weeks. from (7). It reveals that the estimated covariance operator Γˆ FCAR exhibits greater structural complexity, with stronger values concentrated near the diago￾nal, which indicate pronounced short-range temporal correlations. On the other hand, the naive covariance es… view at source ↗
Figure 3
Figure 3. Figure 3: Heatmaps of estimated covariance operators from PM [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Heatmap of the element-wise differences between the estimated co [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Estimated parameter function ˆα from PM2.5 data in the Midwestern United States. 7 Conclusion This article introduces a new class of FCAR models, specifically designed for spatially dependent functional data. We develop a profile-based estimation pro￾cedure that alternates between estimating the covariance operator using condi￾tionally centered data and estimating the spatial dependence parameter through m… view at source ↗
read the original abstract

We introduce a new class of conditional autoregressive models for spatially dependent functional data, formulated through conditional means given neighboring functional observations and characterized by a covariance operator and a spatial dependence parameter. Our estimation strategy consists of three components: (i) estimating the covariance operator using conditionally centered data, (ii) estimating the spatial dependence parameter by maximizing the likelihood of projected observations, and (iii) applying a novel profile-based approach to obtain the final estimators. Under an expanding lattice framework, we establish two key theoretical results. First, we establish the consistency of the proposed covariance estimator, which is not attainable using naive methods based on marginally centered data. Second, we prove that the spatial dependence parameter estimator is superconsistent and asymptotically normal, where the latter property enables statistical inference for spatial dependence in functional data -- a contribution that is novel in the existing literature. Numerical studies support the theoretical results and demonstrate the computational efficiency of our method. Finally, we illustrate its practical utility by analyzing weekly PM$_{2.5}$ concentration trajectories in 2019 across counties in the Midwestern United States.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces a new class of conditional autoregressive models for spatially dependent functional data, defined through conditional means given neighboring observations and involving a covariance operator together with a spatial dependence parameter. Estimation proceeds in three steps: covariance estimation via conditional centering, maximization of a projected-data likelihood for the spatial parameter, and a profile approach for the final estimators. Under an expanding-lattice asymptotic regime with regularity conditions, the authors establish consistency of the conditionally centered covariance estimator and superconsistency plus asymptotic normality of the spatial-dependence-parameter estimator. The claims are illustrated by simulations and an application to weekly PM2.5 trajectories across Midwestern counties.

Significance. If the stated results hold, the work provides a useful addition to spatial functional data analysis by supplying a conditional autoregressive framework that yields consistent covariance estimation and, crucially, permits inference on spatial dependence through asymptotic normality—an aspect noted as novel. The explicit contrast with naive marginal centering and the choice of an expanding-lattice regime are technically appropriate. The computational efficiency reported in simulations and the environmental application further support practical relevance.

minor comments (3)
  1. [Section 2] Section 2: The precise definition of the neighborhood structure on the lattice and the manner in which the covariance operator enters the conditional mean should be accompanied by a short illustrative diagram or explicit low-dimensional example to aid readability.
  2. [Table 2] Table 2: The simulation results for the spatial-parameter estimator report bias and MSE but omit the empirical coverage of the asymptotic confidence intervals; adding this column would directly illustrate the utility of the normality result.
  3. [Section 6] Section 6: In the PM2.5 application the authors do not report sensitivity checks with respect to the number of retained principal components or the choice of projection dimension; a brief robustness table would strengthen the empirical section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment that led to the recommendation of minor revision. The referee's summary accurately captures the key elements of our work on conditional autoregressive models for spatially dependent functional data, the three-step estimation procedure, and the theoretical results under the expanding-lattice regime.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines a conditional autoregressive model for functional data, proposes a three-step estimator (conditional centering for covariance, projected likelihood for spatial parameter, profile refinement), and derives consistency plus superconsistency/asymptotic normality under an expanding-lattice regime with stated regularity conditions on the covariance operator. None of these steps reduces by construction to fitted quantities or prior self-citations; the consistency claim is explicitly contrasted with naive marginal centering, and the superconsistency result is presented as a novel theoretical contribution enabled by the model structure rather than assumed or fitted into existence. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; the model rests on standard functional data and spatial process assumptions that are not enumerated here.

free parameters (1)
  • spatial dependence parameter
    Core parameter of the conditional autoregressive structure, estimated via likelihood maximization on projected observations.
axioms (2)
  • domain assumption Functional observations follow the conditional autoregressive model with a well-defined covariance operator and spatial dependence structure.
    Foundational modeling assumption invoked to define the conditional means and enable the estimation and theory.
  • domain assumption Data are observed on an expanding lattice with appropriate regularity conditions.
    Asymptotic framework required for the consistency and asymptotic normality results.

pith-pipeline@v0.9.0 · 5705 in / 1384 out tokens · 51340 ms · 2026-05-22T03:48:48.856992+00:00 · methodology

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Reference graph

Works this paper leans on

140 extracted references · 140 canonical work pages · 1 internal anchor

  1. [1]

    , Broze, L

    Ahmed, M.-S. , Broze, L. , Dabo-Niang, S. & Gharbi, Z. (2022). Quasi-maximum likelihood estimators for functional linear spatial autoregressive models. Geostatistical Functional Data Analysis , 286--328

    work page 2022

  2. [2]

    , Carlin, B

    Banerjee, S. , Carlin, B. P. & Gelfand, A. E. (2003). Hierarchical Modeling and Analysis for Spatial Data. Chapman and Hall/CRC

    work page 2003

  3. [3]

    , Feldman, D

    Beenstock, M. , Feldman, D. & Felsenstein, D. (2012). Testing for unit roots and cointegration in spatial cross-section data. Spatial Economic Analysis 7, 203--222

    work page 2012

  4. [4]

    Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society: Series B (Methodological) 36, 192--225

    work page 1974

  5. [5]

    , Shang, H

    Beyaztas, U. , Shang, H. L. , Sezer, G. B. , Mandal, A. , Zoh, R. S. & Tekwe, C. D. (2025). Spatial function-on-function regression. Journal of Agricultural, Biological and Environmental Statistics , 1--19

    work page 2025

  6. [6]

    , Ferraty, F

    Cardot, H. , Ferraty, F. , Mas, A. & Sarda, P. (2003). Testing hypotheses in the functional linear model. Scandinavian Journal of Statistics 30, 241--255

    work page 2003

  7. [7]

    Carlin, B. P. & Banerjee, S. (2003). Hierarchical multivariate CAR models for spatio-temporally correlated survival data. Bayesian Statistics 7, 45--63

    work page 2003

  8. [8]

    & Bernardinelli, L

    Clayton, D. & Bernardinelli, L. (1996). Bayesian methods for mapping disease risk. In Geographical and Environmental Epidemiology: Methods for Small Area Studies. Oxford University Press

    work page 1996

  9. [9]

    Cressie, N. (2015). Statistics for Spatial Data. John Wiley & Sons

    work page 2015

  10. [10]

    & Hall, P

    Delaigle, A. & Hall, P. (2012). Achieving near perfect classification for functional data. Journal of the Royal Statistical Society Series B: Statistical Methodology 74, 267--286

    work page 2012

  11. [11]

    , Giraldo, R

    Delicado, P. , Giraldo, R. , Comas, C. & Mateu, J. (2010). Statistics for spatial functional data: some recent contributions. Environmetrics: The official journal of the International Environmetrics Society 21, 224--239

    work page 2010

  12. [12]

    Dou, W. W. , Pollard, D. & Zhou, H. H. (2012). Estimation in functional regression for general exponential families. Annals of Statistics , 2421--2451

    work page 2012

  13. [13]

    , Horv \'a th, L

    Fremdt, S. , Horv \'a th, L. , Kokoszka, P. & Steinebach, J. G. (2014). Functional data analysis with increasing number of projections. Journal of Multivariate Analysis 124, 313--332

    work page 2014

  14. [14]

    , Guyon, X

    Gaetan, C. , Guyon, X. et al. (2010). Spatial Statistics and Modeling, vol. 90. Springer

    work page 2010

  15. [15]

    Gelfand, A. E. & Vounatsou, P. (2003). Proper multivariate conditional autoregressive models for spatial data analysis. Biostatistics 4, 11--15

    work page 2003

  16. [16]

    & Horowitz, J

    Hall, P. & Horowitz, J. L. (2007). Methodology and convergence rates for functional linear regression. Annals of statistics 35, 70--91

    work page 2007

  17. [17]

    Hamilton, J. D. (1994). Time Series Analysis. Princeton, NJ: Princeton University Press

    work page 1994

  18. [18]

    & Kokoszka, P

    Horv \'a th, L. & Kokoszka, P. (2012). Inference for Functional Data with Applications, vol. 200. Springer Science & Business Media

    work page 2012

  19. [19]

    , Kokoszka, P

    Horv \'a th, L. , Kokoszka, P. & Reimherr, M. (2009). Two sample inference in functional linear models. Canadian Journal of Statistics 37, 571--591

    work page 2009

  20. [21]

    & Eubank, R

    Hsing, T. & Eubank, R. (2015). Theoretical Foundations of Functional Data Analysis, with An Introduction to Linear Pperators, vol. 997. John Wiley & Sons

    work page 2015

  21. [22]

    Khoo, T. H. , Pathmanathan, D. & Dabo-Niang, S. (2023). Spatial autocorrelation of global stock exchanges using functional areal spatial principal component analysis. Mathematics 11, 674

    work page 2023

  22. [23]

    & Reimherr, M

    Kokoszka, P. & Reimherr, M. (2017). Introduction to Functional Data Analysis. Chapman and Hall/CRC

    work page 2017

  23. [24]

    , Staicu, A.-M

    Kong, D. , Staicu, A.-M. & Maity, A. (2016). Classical testing in functional linear models. Journal of Nonparametric Statistics 28, 813--838

    work page 2016

  24. [25]

    Lee, D. (2011). A comparison of conditional autoregressive models used in bayesian disease mapping. Spatial and spatio-temporal epidemiology 2, 79--89

    work page 2011

  25. [26]

    Lee, L.-F. (2004). Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models. Econometrica 72, 1899--1925

    work page 2004

  26. [27]

    , Zhang, H

    Liang, D. , Zhang, H. , Chang, X. & Huang, H. (2021). Modeling and regionalization of China’s PM2. 5 using spatial-functional mixture models . Journal of the American Statistical Association 116, 116--132

    work page 2021

  27. [28]

    & Lin, Z

    Lin, Y. & Lin, Z. (2025). Hypothesis testing for functional linear models via bootstrapping. Bernoulli 31, 3309--3330

    work page 2025

  28. [29]

    Mardia, K. (1988). Multi-dimensional multivariate G aussian M arkov random fields with application to image processing. Journal of Multivariate Analysis 24, 265--284

    work page 1988

  29. [30]

    & Genton, M

    Mart \' nez-Hern \'a ndez, I. & Genton, M. G. (2020). Recent developments in complex and spatially correlated functional data. Brazilian Journal of Probability and Statistics 34, 204--229

    work page 2020

  30. [31]

    Moran, P. A. (1950). Notes on continuous stochastic phenomena. Biometrika 37, 17--23

    work page 1950

  31. [32]

    , Giraldo, R

    Pineda-R \' os, W. , Giraldo, R. & Porcu, E. (2019). Functional sar models: With application to spatial econometrics. Spatial statistics 29, 145--159

    work page 2019

  32. [33]

    , Liebl, D

    Po , D. , Liebl, D. , Kneip, A. , Eisenbarth, H. , Wager, T. D. & Barrett, L. F. (2020). Superconsistent estimation of points of impact in non-parametric regression with functional predictors. Journal of the Royal Statistical Society Series B: Statistical Methodology 82, 1115--1140

    work page 2020

  33. [34]

    , Irpino, A

    Romano, E. , Irpino, A. & Mateu, J. (2022). Spatial functional data analysis for probability density functions: Compositional functional data vs. distributional data approach. Geostatistical Functional Data Analysis , 128--153

    work page 2022

  34. [35]

    Sain, S. R. , Furrer, R. & Cressie, N. (2011). A spatial analysis of multivariate output from regional climate models. The Annals of Applied Statistics , 150--175

    work page 2011

  35. [36]

    , Wen, Y

    Shen, X. , Wen, Y. , Cui, Y. & Lu, Q. (2022). A conditional autoregressive model for genetic association analysis accounting for genetic heterogeneity. Statistics in medicine 41, 517--542

    work page 2022

  36. [37]

    Stein, M. L. (1999). Interpolation of spatial data: some theory for kriging. Springer Science & Business Media

    work page 1999

  37. [38]

    , Wang, T

    Tang, C. , Wang, T. & Zhang, P. (2022). Functional data analysis: An application to COVID-19 data in the United States in 2020 . Quantitative Biology 10, 172--187

    work page 2022

  38. [39]

    , Reid, N

    Varin, C. , Reid, N. & Firth, D. (2011). An overview of composite likelihood methods. Statistica Sinica , 5--42

    work page 2011

  39. [40]

    Ver Hoef, J. M. , Peterson, E. E. , Hooten, M. B. , Hanks, E. M. & Fortin, M.-J. (2018). Spatial autoregressive models for statistical inference from ecological data. Ecological Monographs 88, 36--59

    work page 2018

  40. [41]

    Wall, M. M. (2004). A close look at the spatial structure implied by the car and sar models. Journal of statistical planning and inference 121, 311--324

    work page 2004

  41. [42]

    , Dai, X

    Yeon, H. , Dai, X. & Nordman, D. J. (2023). Bootstrap inference in functional linear regression models with scalar response. Bernoulli 29, 2599--2626

    work page 2023

  42. [43]

    , de Jong, R

    Yu, J. , de Jong, R. & Lee, L.-f. (2012). Estimation for spatial dynamic panel data with fixed effects: The case of spatial cointegration. Journal of Econometrics 167, 16--37

    work page 2012

  43. [44]

    Zhang, J.-T. (2011). Statistical inferences for linear models with functional responses. Statistica Sinica , 1431--1451

    work page 2011

  44. [45]

    & Chen, J

    Zhang, J.-T. & Chen, J. (2007). Statistical inferences for functional data

    work page 2007

  45. [46]

    , Baladandayuthapani, V

    Zhang, L. , Baladandayuthapani, V. , Zhu, H. , Baggerly, K. A. , Majewski, T. , Czerniak, B. A. & Morris, J. S. (2016). Functional CAR models for large spatially correlated functional datasets. Journal of the American statistical association 111, 772--786

    work page 2016

  46. [47]

    Berk, R., Brown, L., Buja, A., Zhang, K., and Zhao, L. (2013). Valid post-selection inference. The Annals of Statistics , pages 802--837

    work page 2013

  47. [48]

    Cressie, N. (2015). Statistics for Spatial Data . John Wiley & Sons

    work page 2015

  48. [49]

    and Kokoszka, P

    Horv \'a th, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications , volume 200. Springer Science & Business Media

    work page 2012

  49. [50]

    and Eubank, R

    Hsing, T. and Eubank, R. (2015). Theoretical Foundations of Functional Data Analysis, with An Introduction to Linear Pperators , volume 997. John Wiley & Sons

    work page 2015

  50. [51]

    and Reimherr, M

    Kokoszka, P. and Reimherr, M. (2017). Introduction to Functional Data Analysis . Chapman and Hall/CRC

    work page 2017

  51. [52]

    Magnus, J. R. et al. (1978). The Moments of Products of Quadratic Forms in Normal Variables . Univ., Instituut voor Actuariaat en Econometrie

    work page 1978

  52. [53]

    Yeon, H., Dai, X., and Nordman, D. J. (2023). Bootstrap inference in functional linear regression models with scalar response. Bernoulli , 29(4):2599--2626

    work page 2023

  53. [54]

    and Chen, J

    Zhang, J.-T. and Chen, J. (2007). Statistical inferences for functional data

    work page 2007

  54. [55]

    A., Majewski, T., Czerniak, B

    Zhang, L., Baladandayuthapani, V., Zhu, H., Baggerly, K. A., Majewski, T., Czerniak, B. A., and Morris, J. S. (2016). Functional CAR models for large spatially correlated functional datasets. Journal of the American statistical association , 111(514):772--786

    work page 2016

  55. [56]

    Berrendero, J. R. , Cuevas, A. & Torrecilla, J. L. (2016). On the use of reproducing kernel H ilbert spaces in functional classification. arXiv : 1507.04398v3

    work page arXiv 2016

  56. [57]

    Cox, D. R. (1972). Regression models and life tables (with Discussion) . J. R. Statist. Soc. B 34, 187--220

    work page 1972

  57. [58]

    Heard, N. A. , Holmes, C. C. & Stephens, D. A. (2006). A quantitative study of gene regulation involved in the immune response of A nopheline mosquitoes: A n application of B ayesian hierarchical clustering of curves. J. Am. Statist. Assoc. 101, 18--29

    work page 2006

  58. [59]

    R: A Language and Environment for Statistical Computing

    R Development Core Team (2025). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. ISBN 3-900051-07-0, http://www.R-project.org

    work page 2025

  59. [60]

    Journal of the Royal Statistical Society: Series B (Methodological) , volume=

    Spatial interaction and the statistical analysis of lattice systems , author=. Journal of the Royal Statistical Society: Series B (Methodological) , volume=. 1974 , publisher=

    work page 1974

  60. [61]

    Spatial and spatio-temporal epidemiology , volume=

    A comparison of conditional autoregressive models used in Bayesian disease mapping , author=. Spatial and spatio-temporal epidemiology , volume=. 2011 , publisher=

    work page 2011

  61. [62]

    Biostatistics , volume=

    Proper multivariate conditional autoregressive models for spatial data analysis , author=. Biostatistics , volume=. 2003 , publisher=

    work page 2003

  62. [63]

    Functional

    Zhang, Lin and Baladandayuthapani, Veerabhadran and Zhu, Hongxiao and Baggerly, Keith A and Majewski, Tadeusz and Czerniak, Bogdan A and Morris, Jeffrey S , journal=. Functional. 2016 , publisher=

    work page 2016

  63. [64]

    Biometrika , volume=

    On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables , author=. Biometrika , volume=. 1918 , publisher=

    work page 1918

  64. [65]

    2017 , publisher=

    Introduction to Functional Data Analysis , author=. 2017 , publisher=

    work page 2017

  65. [66]

    2015 , publisher=

    Theoretical Foundations of Functional Data Analysis, with An Introduction to Linear Pperators , author=. 2015 , publisher=

    work page 2015

  66. [67]

    2015 , publisher=

    Statistics for Spatial Data , author=. 2015 , publisher=

    work page 2015

  67. [68]

    1978 , publisher=

    The Moments of Products of Quadratic Forms in Normal Variables , author=. 1978 , publisher=

    work page 1978

  68. [69]

    Bernoulli , volume=

    Bootstrap inference in functional linear regression models with scalar response , author=. Bernoulli , volume=. 2023 , publisher=

    work page 2023

  69. [70]

    Statistica Sinica , pages=

    An overview of composite likelihood methods , author=. Statistica Sinica , pages=. 2011 , publisher=

    work page 2011

  70. [72]

    2007 , journal=

    Methodology and convergence rates for functional linear regression , author=. 2007 , journal=

    work page 2007

  71. [73]

    Annals of Statistics , pages=

    Estimation in functional regression for general exponential families , author=. Annals of Statistics , pages=. 2012 , publisher=

    work page 2012

  72. [74]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Achieving near perfect classification for functional data , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2012 , publisher=

    work page 2012

  73. [75]

    Journal of Multivariate Analysis , volume=

    Functional data analysis with increasing number of projections , author=. Journal of Multivariate Analysis , volume=. 2014 , publisher=

    work page 2014

  74. [76]

    Journal of Nonparametric Statistics , volume=

    Classical testing in functional linear models , author=. Journal of Nonparametric Statistics , volume=. 2016 , publisher=

    work page 2016

  75. [77]

    Environmetrics: The official journal of the International Environmetrics Society , volume=

    Statistics for spatial functional data: some recent contributions , author=. Environmetrics: The official journal of the International Environmetrics Society , volume=. 2010 , publisher=

    work page 2010

  76. [78]

    2020 , journal=

    Recent developments in complex and spatially correlated functional data , author=. 2020 , journal=

    work page 2020

  77. [79]

    2022 , publisher=

    Tang, Chen and Wang, Tiandong and Zhang, Panpan , journal=. 2022 , publisher=

    work page 2022

  78. [80]

    2021 , publisher=

    Liang, Decai and Zhang, Haozhe and Chang, Xiaohui and Huang, Hui , journal=. 2021 , publisher=

    work page 2021

  79. [81]

    Multi-dimensional multivariate

    Mardia, KV , journal=. Multi-dimensional multivariate. 1988 , publisher=

    work page 1988

  80. [82]

    Hierarchical multivariate

    Carlin, Bradley P and Banerjee, Sudipto , journal=. Hierarchical multivariate. 2003 , publisher=

    work page 2003

Showing first 80 references.