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Moment and Bayesian wavelet regression from spatially correlated functional data

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Abstract

A new wavelet-based estimation methodology, in the context of spatial functional regression, is proposed to discriminate between small-scale and large scale variability of spatially correlated functional data, defined by depth-dependent curves. Specifically, the discrete wavelet transform of the data is computed in space and depth to reduce dimensionality. Moment-based regression estimation is applied for the approximation of the scaling coefficients of the functional response. While its wavelet coefficients are estimated in a Bayesian regression framework. Both regression approaches are implemented from the empirical versions of the scaling and wavelet auto-covariance and cross-covariance operators, characterizing the correlation structure of the spatial functional response. Weather stations in ocean islands display high spatial concentration. The proposed estimation methodology overcomes the difficulties arising in the estimation of ocean temperature field at different depths, from long records of ocean temperature measurements in these stations. Data are collected from The World-Wide Ocean Optics Database. The performance of the presented approach is tested in terms of 10-fold cross-validation, and residual spatial and depth correlation analysis. Additionally, an application to soil sciences, for prediction of electrical conductivity profiles is also considered to compare this approach with previous related ones, in the statistical analysis of spatially correlated curves in depth.

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Acknowledgments

This work has been supported in part by project MTM2012-32674 (co-funded with FEDER) of the DGI, MEC, Spain. We would like to thank Professors Yang, Wikley, Holanz, Myersx, and Sudduth for sending and allowing us to use their dataset to illustrate the estimation methodology proposed in this paper, which has been inspired and motivated by their proposal.

Author information

Authors and Affiliations

  1. Department of Statistics and Operations Research,Faculty of Sciences, University of Granada, Campus Fuente Nueva s/n, 18071, Granada, Spain

    R. Espejo & M. D. Ruiz-Medina

  2. Quantitative Methods for Economics and Business, Faculty of Economy, University of Granada, Campus of Cartuja s/n, Granada, Spain

    R. M. Fernández-Pascual

Authors
  1. R. M. Fernández-Pascual
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  2. R. Espejo
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  3. M. D. Ruiz-Medina
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Corresponding author

Correspondence to M. D. Ruiz-Medina.

Appendix

Appendix

In this appendix, for the ocean temperature, and the two covariates studied, salinity and relative fluorescence, we show the sample auto-correlation and cross-correlation functions, over the ten years analyzed, at nodes 1, 6 and 16, for certain depth intervals (see Figs. 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70 below), for the first region analyzed with longitude–latitude interval \([-85, -45] x [-30, -60],\) as well as for the second region with longitude–latitude interval \([-60, -52]\times [-35, -55]).\) In both regions, it can be appreciated the absence of downward sloping, or gradually decay for increasing lags, that supports the stationarity assumption in time for the above three magnitudes.

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Region 1. Ocean temperature. Sample ACF for 16 depth intervals at node 1

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Fig. 36
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Region 1. Salinity. Sample ACF for 16 depth intervals at node 1

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Region 1. Relative fluorescence. Sample ACF for 16 depth intervals at node 1

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Fig. 38
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Region 1. Ocean temperature. Sample ACF for 16 depth intervals at node 6

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Region 1. Salinity. Sample ACF for 16 depth intervals at node 6

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Region 1. Relative fluorescence. Sample ACF for 16 depth intervals at node 6

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Region 1. Ocean temperature. Sample ACF for 16 depth intervals at node 16

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Region 1. Salinity. Sample ACF for 16 depth intervals at node 16

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Fig. 43
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Region 1. Relative fluorescence. Sample ACF for 16 depth intervals at node 16

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Region 1. Ocean temperature. Sample cross correlation function for certain depth intervals at node 1

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Fig. 45
figure 45

Region 1. Salinity. Sample cross correlation function for certain depth intervals at node 1

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Fig. 46
figure 46

Region 1. Relative fluorescence. Sample cross correlation function for certain depth intervals at node 1

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Fig. 47
figure 47

Region 1. Ocean temperature. Sample cross correlation function for certain depth intervals at node 6

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Fig. 48
figure 48

Region 1. Salinity. Sample cross correlation function for certain depth intervals at node 6

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Fig. 49
figure 49

Region 1. Relative fluorescence. Sample cross correlation function for certain depth intervals at node 6

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Fig. 50
figure 50

Region 1. Ocean temperature. Sample cross correlation function for certain depth intervals at node 16

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Fig. 51
figure 51

Region 1. Salinity. Sample cross correlation function for certain depth intervals at node 16

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Fig. 52
figure 52

Region 1. Relative fluorescence. Sample cross correlation function for certain depth intervals at node 16

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Fig. 53
figure 53

Region 2. Ocean temperature. Sample ACF for 16 depth intervals at node 1

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Fig. 54
figure 54

Region 2. Salinity. Sample ACF for 16 depth intervals at node 1

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Fig. 55
figure 55

Region 2. Relative fluorescence. Sample ACF for 16 depth intervals at node 1

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Fig. 56
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Region 2. Ocean temperature. Sample ACF for 16 depth intervals at node 6

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Fig. 57
figure 57

Region 2. Salinity. Sample ACF for 16 depth intervals at node 6

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Fig. 58
figure 58

Region 2. Relative fluorescence. Sample ACF for 16 depth intervals at node 6

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Fig. 59
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Region 2. Ocean temperature. Sample ACF for 16 depth intervals at node 16

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Fig. 60
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Region 2. Salinity. Sample ACF for 16 depth intervals at node 16

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Fig. 61
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Region 2. Relative fluorescence. Sample ACF for 16 depth intervals at node 16

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Fig. 62
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Region 2. Ocean temperature. Sample cross correlation function for certain depth intervals at node 1

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Fig. 63
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Region 2. Salinity. Sample cross correlation function for certain depth intervals at node 1

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Fig. 64
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Region 2. Relative fluorescence. Sample cross correlation function for certain depth intervals at node 1

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Fig. 65
figure 65

Region 2. Ocean temperature. Sample cross correlation function for certain depth intervals at node 6

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Fig. 66
figure 66

Region 2. Salinity. Sample cross correlation function for certain depth intervals at node 6

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Fig. 67
figure 67

Region 2. Relative fluorescence. Sample cross correlation function for certain depth intervals at node 6

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Fig. 68
figure 68

Region 2. Ocean temperature. Sample cross correlation function for certain depth intervals at node 16

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Fig. 69
figure 69

Region 2. Salinity. Sample cross correlation function for certain depth intervals at node 16

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Fig. 70
figure 70

Region 2. Relative fluorescence. Sample cross correlation function for certain depth intervals at node 16

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Fernández-Pascual, R.M., Espejo, R. & Ruiz-Medina, M.D. Moment and Bayesian wavelet regression from spatially correlated functional data. Stoch Environ Res Risk Assess 30, 523–557 (2016). https://doi.org/10.1007/s00477-015-1130-5

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