Abstract
In this paper, a novel method is proposed to analyze multivariate longitudinal data that contains spatial location information. The method has the advantage of analyzing the relationship between curves at neighbor time points and observing the relationship between locations. We offer the spatial covariance function and use functional PCA to estimate unknown parameter functions. A detail solving process and theoretical properties are introduced. Based on the gradient descent method and leave-one-out cross-validation method, we estimate those unknown parameters and select the principal components respectively. Furthermore, compared with other four methods, the proposed method shows a better category effect on simulation studies and air quality data analysis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aguilera AM, Escabias M, Valderrama MJ (2006) Using principal components for estimating logistic regression with high-dimensional multicollinear data. Comput Stat Data Anal 50(8):1905–1924
Aguilera-Morillo MC, Durbán M, Aguilera AM (2017) Prediction of functional data with spatial dependence: a penalized approach. Stoch Environ Res Risk Assess 31(1):7–22
Arnone E, Azzimonti L, Nobile F, Sangalli L (2018) Modeling spatially dependent functional data via regression with diffierential regularization. J Multivar Anal 170:275–295
Arnone E, Sangalli LM, Vicini A (2021) Smoothing spatio-temporal data with complex missing data patterns. Stat Modell, 1471082X211057959
Chamroukhi F, Samé A, Govaert G, Aknin P (2010) A hidden process regression model for functional data description. Appl Curve Discrim Neurocomput 73(79):1210–1221
Delaigle A, Hall P, Bathia N (2012) Componentwise classification and clustering of functional data. Biometrika 99(2):299–313
Durban M, Lee D-J, Aguilera Morillo MdC, Aguilera AM (2022) From spatiotemporal smoothing to functional spatial regression: a penalized approach. Geostat Funct Data Anal, 403–423
Dvurechensky P, Gasnikov A, Kroshnin A (2018) Computational optimal transport: Complexity by accelerated gradient descent is better than by Sinkhorn’s algorithm. In: Proceedings of the 35th international conference on machine learning. PMLR, pp 1367–1376
Escabias M, Aguilera AM, Aguilera-Morillo MC (2014) Functional PCA and baseline logit models. J Classif 31(3):296–324
Ferraty F, Vieu P (2003) Curves discrimination: a nonparametric functional approach. Comput Stat Data Anal 44(1):161–173
Fisher RA (1936) The use of multiple measurements in taxonomic problems. Ann Eugen 7(2):179–188
Giraldo R, Delicado P, Mateu J (2010) Continuous time-varying kriging for spatial prediction of functional data: an environmental application. J Agric Biol Environ Stat 15(1):66–82
Goulard M, Voltz M (1993) Geostatistical interpolation of curves: a case study in soil science. In: Geostatistics Tróia’92. Springer, New York, pp 805–816
Hervás C, Silva M, Gutiéerrez PA, Serrano A (2008) Multilogistic regression by evolutionary neural network as a classification tool to discriminate highly overlapping signals: Qualitative investigation of volatile organic compounds in polluted waters by using headspace-mass spectrometric analysis. Chemom Intell Lab Syst 92(2):179–185
Huang H, Li Y, Guan Y (2014) Joint modeling and clustering paired generalized longitudinal trajectories with application to cocaine abuse treatment data. J Am Stat Assoc 109(508):1412–1424
Huang J, Pan X, Guo X, Li G (2018a) Health impact of China’s air pollution prevention and control action plan: an analysis of national air quality monitoring and mortality data. Lancet Planet Health 2(7):e313–e323
Huang T, Saporta G, Wang H, Wang S (2018b) Spatial functional linear model and its estimation method. arXiv:1811.00314
James GM (2002) Generalized linear models with functional predictors. J R Stat Soc Ser B (Stat Methodol) 64(3):411–432
James GM, Hastie TJ (2001) Functional linear discriminant analysis for irregularly sampled curves. J R Stat Soc Ser B (Stat Methodol) 63(3):533–550
Jiang H, Serban N (2012) Clustering random curves under spatial interdependence with application to service accessibility. Technometrics 54(2):108–119
Jiang R, Wang L, Bai Y (2022) Optimal model averaging estimator for multinomial logit models. Stat Theory Relat Fields, 1–14
Kerimray A, Baimatova N, Ibragimova OP, Bukenov B, Kenessov B, Plotitsyn P, Karaca F (2020) Assessing air quality changes in large cities during COVID-19 lockdowns: The impacts of traffic-free urban conditions in Almaty, Kazakhstan. Sci Total Environ 730:139179
Leng X, Müller H-G (2006) Classification using functional data analysis for temporal gene expression data. Bioinformatics 22(1):68–76
Li Y, Wang N, Carroll RJ (2013) Selecting the number of principal components in functional data. J Am Stat Assoc 108(504):1284–1294
Liang J, Bi G, Zhan C (2020) Multinomial and ordinal Logistic regression analyses with multi-categorical variables using R. Ann Transl Med 8(16):982
Liang D, Zhang H, Chang X, Huang H (2021) Modeling and regionalization of China’s PM2.5 using spatial-functional mixture models. J Am Stat Assoc 116(533):116–132
Luo X, Wang D, Zhou M, Yuan H (2019) Latent factor-based recommenders relying on extended stochastic gradient descent algorithms. IEEE Trans Syst Man Cybern Syst 51(2):916–926
Martínez-Hernández I, Genton MG (2020) Recent developments in complex and spatially correlated functional data. Braz J Prob Stat 34(2):204–229
Ndongo F B (2017) Spatio-temporal processes for functional data with application in climate monitoring. Ph.D. Thesis, Università degli studi di Bergamo
Poskitt DS, Sengarapillai A (2013) Description length and dimensionality reduction in functional data analysis. Comput Stat Data Anal 58:98–113
Ramsay JOSBW (2005) Functional data analysis, 2nd edn. Springer, New York
Rasmussen C E (2003) Gaussian processes in machine learning. In: Summer school on machine learning. Springer, New York, pp 63–71
Ruder S (2016) An overview of gradient descent optimization algorithms. arXiv:1609.04747
Russell BT, Porter WC (2021) Using spatial smoothing to model a functional regression estimator to points on a lattice with application to surface-level ozone in the Eastern United States. Environ Ecol Stat, 1–21
Shang HL (2014) A survey of functional principal component analysis. Adv Stat Anal 98(2):121–142
Vovan T (2018) Some results of classification problem by Bayesian method and application in credit operation. Stat Theory Relat Fields 2(2):150–157
Yao F, Müller H-G, Wang JL (2005a) Functional linear regression analysis for longitudinal data. Ann Stat 33(6):2873–2903
Yao F, Müller H-G, Wang JL (2005b) Functional data analysis for sparse longitudinal data. J Am Stat Assoc 100(470):577–590
Zhou L, Huang JZ, Carroll RJ (2008) Joint modelling of paired sparse functional data using principal components. Biometrika 95(3):601–619
Zhou L, Huang JZ, Martinez JG, Maity A, Baladandayuthapani V, Carroll RJ (2010) Reduced rank mixed effects models for spatially correlated hierarchical functional data. J Am Stat Assoc 105(489):390–400
Zhou ZR, Wang WW, Li Y, Jin KR, Wang XY, Wang ZW, Chen YS, Wang SJ, Hu J, Zhang HPZGZ, Hui Na LB, Chen XX, Zhang TS (2019) Indepth mining of clinical data: the construction of clinical prediction model with R. Ann Transl Med 7(23):796
Acknowledgements
Zhang’s research was partially sup ported by National Natural Science Foundation of China (11971171), the 111 Project (B14019) and Project of National Social Science Fund of China (15BTJ027) and the Open Research Fund of Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, ECNU.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A: Derivative of loss function (2.7)
Based on loss function (2.7), for each element of \(\varvec{\theta }_k\), we take the derivative and have
where \({\varvec{R}}_k=\varvec{R}_{kii^{\prime }}(\varvec{\Theta }_k)=\rho (\big \Vert \varvec{s}_i-{\varvec{s}}_{i^{\prime }}\big \Vert ;\varvec{\Theta }_k)\), according to the first derivative of \(\varvec{\theta }_k\), set the second derivative as the diagonal matrix \({\mathbf {H}}\), we calculate the second derivative with respect to \(\varvec{\theta }_k\), \({\mathbf {H}}\) and have the form of follows.
For \(k=k^{\prime }\), let \(\varvec{\Delta }_k=\alpha _k\varvec{1}+{\varvec{a}}_i^T\varvec{\Phi }{\varvec{b}}_k+f(\varvec{h}_i;\widetilde{\varvec{\Gamma }}_i)\varvec{d}_k^T\varvec{\Phi }{\varvec{c}}_k\), \(\varvec{\Delta }_{k^{\prime }}=\alpha _{k^{\prime }}{\varvec{1}}+\varvec{a}_i^T\varvec{\Phi }{\varvec{b}}_{k^{\prime }}+f(\varvec{h}_i;\widetilde{\varvec{\Gamma }}_i)\varvec{d}_{k^{\prime }}^T\varvec{\Phi }{\varvec{c}}_{k^{\prime }}\), and \(\varvec{\Delta }_l=\alpha _l{\varvec{1}}+\varvec{a}_i^T\varvec{\Phi }{\varvec{b}}_l+f(\varvec{h}_i;\widetilde{\varvec{\Gamma }}_i)\varvec{d}_l^T\varvec{\Phi }{\varvec{c}}_l\), for each element of \(\varvec{\theta }_k\), we derivate \(\varvec{\theta }_k\) with respect to the first derivative. For \(k\ne k^{\prime }\), the solving process is similar, for simplicity of presentation, we omit it here. It should be noticed that althrough the matrix \({\mathbf {H}}\) is difficult to calculate, we usually choose small k and small q in basic functions \(\psi _q(t)\) in actual situation.
Appendix B: Proof of theorem 1
As \(\varvec{\theta }_k=(\alpha _k{\varvec{1}},\varvec{a}_i,{\varvec{b}}_k,{\varvec{c}}_k,\varvec{d}_k,\varvec{\sigma }_{hk},\varvec{\Theta }_k)^T\), then
based on condition (1), (2), \(f_{\pi _{ik}}({\varvec{s}}_i,t)\) satisfy \(f_{\pi _{ik}}({\varvec{s}}_i,t)>0\) for \(t\in [0,T], {\varvec{s}}_i \in \mathcal {D} \subset \mathbb {R}^{2},i=1,\ldots ,n\), we derivate it with respect to \(\varvec{\theta }_k\),
and
for \(t\rightarrow \infty \), the integrate of the right hand of last part tends to 1. For \(\varvec{\theta }_k=(\alpha _k\varvec{1},{\varvec{a}}_i,{\varvec{b}}_k,{\varvec{c}}_k,\varvec{d}_k,\varvec{\sigma }_{hk},\varvec{\Theta }_k)^T\), taking \({\varvec{b}}_k\) for instance, we derivative \(f_{\pi _{ik}}({\varvec{s}}_i,t)\) with respect to \({\varvec{b}}_k\) and have
where \(\varvec{\Delta }_l=\alpha _l{\varvec{1}}+\varvec{a}_i^T\varvec{\Phi }{\varvec{b}}_l+f(\varvec{h}_i;\widetilde{\varvec{\Gamma }}_i)\varvec{d}_l^T\varvec{\Phi }{\varvec{c}}_l\), furthermore, \(\int _T\frac{d}{d\varvec{\theta }_k}\text {log} f_{\pi _{ik}}({\varvec{s}}_i,t)dt=\int _T\frac{1}{ f_{\pi _{ik}}({\varvec{s}}_i,t)}\frac{df_{\pi _{ik}}(\varvec{s}_i,t)}{d\varvec{\theta }_k}dt =\int _T(1- \frac{2}{\sum _{l=1}^K\text {exp}(\varvec{\Delta }_l)})dt\), the second derivative of \(f_{\pi _{ik}}({\varvec{s}}_i,t)\) with respect to \(\varvec{\theta }_k\) is
then
after a complicated calculation, as \(f_{\pi _{ik}}({\varvec{s}}_i,t)\) have second continuous derivative and \(\int _T \frac{d}{d \varvec{\theta }_k} f_{\pi _{ik}}({\varvec{s}}_i,t) dt=\frac{d}{d \varvec{\theta }_k}\int _T f_{\pi _{ik}}({\varvec{s}}_i,t)dt=0\). we have
where \(\varvec{\Delta }_k=\alpha _k{\varvec{1}}+\varvec{a}_i^T\varvec{\Phi }{\varvec{b}}_k+f(\varvec{h}_i;\widetilde{\varvec{\Gamma }}_i)\varvec{d}_k^T\varvec{\Phi }{\varvec{c}}_k\), let \(\omega =\text {exp}(\varvec{\Delta }_k)\) and \(H(\omega )=\frac{2\omega (1-\omega )}{(1+\omega )^3}\), for \(\omega \in (1,\infty )\), we get \(H(0)=0\) and \(\lim _{\omega \rightarrow \infty } H(\omega )=0\), then there exist M such that \(\int _T\frac{d^3}{d^3\varvec{\theta }_k}\text {log} f_{\pi _{ik}}({\varvec{s}}_i,t)dt\leqslant M\). According to A.1 and central limit theorem, there exist the Fisher information matrix
with
such that \(\sqrt{n}(\widehat{\varvec{\theta }}-\varvec{\theta }) {\mathop {\longrightarrow }\limits ^{L}} N(0, I(\varvec{\theta })^{-1})\).
Rights and permissions
About this article
Cite this article
Xu, T., Zhang, R. & Zhang, X. Estimation of spatial-functional based-line logit model for multivariate longitudinal data. Comput Stat 38, 79–99 (2023). https://doi.org/10.1007/s00180-022-01217-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-022-01217-4