# Spatial Warped Gaussian Processes: Estimation and Efficient Field Reconstruction

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## Abstract

**:**

## 1. Introduction

#### Contributions and Outline

- (1)
- In Section 2, we derive mathematical properties of the proposed warped GP models warping function known as the Tukey g-and-h transformation function. This is actually a class of warping functions with sub-classes related to skewness and kurtosis transformations. Since this transformation is not widely known, it is valuable for practitioners to see the basic properties of this class of functions in order to better understand the resulting properties of the warped GP that arise when using this transform. In this regard we explore the transformation and its limiting behaviour (Proposition 2), the existence of the inverse transform (Proposition 1), and its derivatives (Lemma 1). This set of results is instrumental for the analysis of the properties of the Tukey g-and-h process and for the construction of spatial field approximations that form the key application in this paper.
- (2)
- In Section 3, we derive important statistical properties of the warped GP process which include properties of the finite dimensional distributions such as raw and central moments, the multivariate moments (Corollary 1), the population mean and variance (Corollaries 3–4), its higher order multivariate cross moments of any order (Proposition 3), and the index of regular variation (Lemma 2), to determine conditions in terms of the parameters of the g-and-h transform for the existence of all higher order mixed moments. In particular, the cross moments are directly relevant to the construction of efficient spatial field reconstructions for multiple locations in space that are performed in the application section. The tail behaviour of the finite dimensional distributions is captured by the tail index analysis and is relevant to understand models with excess kurtosis relative to a GP model, which is induced by our class of warping function.
- (3)
- In Section 4, we derive the spatial field reconstruction estimators of the Tukey g- and-h random field under five different metrics, most of which are based on the predictive posterior density (Lemma 4).
- (4)
- In Section 5, a detailed set of numerical experiments are provided. This includes both real data cases studies as well as synthetic case studies which were developed to demonstrate key properties of the spatial field reconstruction methods and their accuracy and performance relative to a baseline GP model. In regards to the real data case studies, a sequence of experiments was developed which was based on data obtained from the National Climatic Data Center (NCDC) which included spatial observations of precipitation obtained from ground monitoring stations in the regions of interest. In particular, we took one month of hourly precipitation measurements to fit our warped GP model to obtain a spatial field reconstruction which was compared to a baseline GP model with no warping in order to demonstrate the benefits of a spatial warping framework.

## 2. Constructing a Flexible Warping Function for Spatial Gaussian Processes

- (1)
- Parsimonious and finite number of well interpreted parameters;
- (2)
- Well-separated roles for skewness, kurtosis and the tail behaviour parameters;
- (3)
- Accurate and computationally efficient parameter estimation and tractability;
- (4)
- Derivable and interpretable stochastic properties;
- (5)
- Flexible structures in the marginal, conditional and joint density shapes;
- (6)
- Good inferential properties;
- (7)
- Exact and computationally efficient data generating mechanisms.

**Definition**

**1**

**.**Consider a Gaussian random variable $W\sim \mathcal{N}\left(\right)open="("\; close=")">\mu ,\sigma $ and transformation $Y:=\mathcal{T}\left(\right)open="("\; close=")">W;\mathrm{\u0131}$ with parameter vector $\mathrm{\u0131}:=\left(\right)open="["\; close="]">{m}_{0},{m}_{1},\mu ,{\sigma}^{2},\eta ,\theta $, where ${m}_{0}\in \mathbb{R}$, ${m}_{1}\in {\mathbb{R}}^{+}$, $\mathbf{\eta}\in {\mathbb{R}}^{d}$ and $\theta \in \mathbb{R}$. Then the resultant transformed random variable Y will be from a Tukey law if the corresponding transformation $\mathcal{T}\left(\right)open="("\; close=")">W;\mathbf{\zeta}$ is given by

**Definition**

**2**

**.**

- (1)
- Skew transform:$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathcal{G}\left(\right)open="("\; close=")">W;g\\ =\frac{exp\left(\right)open="("\; close=")">gW}{-}gW.\hfill \end{array}\end{array}$$
- (2)
- Kurtosis transform:$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathcal{H}\left(\right)open="("\; close=")">W;h\\ =exp\left(\right)open="("\; close=")">\frac{h{W}^{2}}{2},\hfill \end{array}\end{array}$$
- (3)
- Skew-Kurtosis product transform:$$\begin{array}{c}\hfill \mathcal{F}\left(\right)open="("\; close=")">W;\eta \\ =\mathcal{G}\left(\right)open="("\; close=")">W;g\mathcal{H}\left(\right)open="("\; close=")">W;h\hfill & ,\end{array}$$

**Proposition**

**1**

**.**

**Proof.**

**Remark**

**1.**

**Lemma**

**1**

**.**

- $$\begin{array}{c}\hfill {\mathcal{T}}_{gh}^{\prime}\left(\right)open="("\; close=")">W;{\mathbf{\zeta}}_{\mathit{gh}}\\ =hW{\mathcal{T}}_{gh}\left(\right)open="("\; close=")">W;{\mathbf{\zeta}}_{\mathit{gh}}+exp\left(\right)open="("\; close=")">gW+\frac{h{W}^{2}}{2}\hfill & ,\end{array}$$
- $$\begin{array}{c}\hfill {\mathcal{T}}_{gh}^{\prime \prime}\left(\right)open="("\; close=")">W;{\mathbf{\zeta}}_{\mathit{gh}}\\ =h{\mathcal{T}}_{gh}\left(\right)open="("\; close=")">W;{\mathbf{\zeta}}_{\mathit{gh}}+hW{\mathcal{T}}_{gh}^{\prime}\left(\right)open="("\; close=")">W;{\mathbf{\zeta}}_{\mathit{gh}}\hfill & +exp\left(\right)open="("\; close=")">gW+\frac{h{W}^{2}}{2}\end{array},$$
- $$\begin{array}{c}\hfill {\mathcal{T}}_{gh}^{\prime \prime \prime}\left(\right)open="("\; close=")">W;{\mathbf{\zeta}}_{\mathit{gh}}\\ =hW{\mathcal{T}}_{gh}^{\prime \prime}\left(\right)open="("\; close=")">W;{\mathbf{\zeta}}_{\mathit{gh}}+2h{\mathcal{T}}_{gh}^{\prime}\left(\right)open="("\; close=")">W;{\mathbf{\zeta}}_{\mathit{gh}}\hfill & +exp\left(\right)open="("\; close=")">gW+\frac{h{W}^{2}}{2}\end{array}+h$$

#### Tukey Warped Spatial Gaussian Process

**Definition**

**3.**

**Definition**

**4.**

## 3. Properties of the Spatial Tukey Warped Gaussian Process

#### 3.1. Spatial Tukey Warped Gaussian Characterisation at a Single Point in Space

**Proposition**

**2**

**.**

**Proof.**

**Remark**

**2.**

**Lemma**

**2.**

**Proof.**

#### 3.2. Spatial Tukey Warped Gaussian Characterisation at Multiple Points in Space

**Lemma**

**3**

**.**

**Proof.**

**Proposition**

**3**

**.**

**Proof.**

#### 3.3. Practical cross Moment Special Cases for Spatial Field Reconstruction of Tukey Warped GP Models

**Corollary**

**1**

**.**

**Proposition**

**4**

**.**

**Proof.**

**Corollary**

**2**

**.**

**Proof.**

## 4. Field Reconstruction for Spatial Tukey Warped Gaussian Processes

**Lemma**

**4**

**.**

**Proof.**

**Corollary**

**3**

**.**

**Corollary**

**4**

**.**

#### 4.1. Spatial Field Reconstruction Risk Functionals and Loss Functions

- (1)
- Minimum Mean Squared Estimator (MMSE) estimatorThe MMSE estimator minimises the MSE by utilising a squared loss function:$$\begin{array}{c}\hfill \begin{array}{c}\hfill L\left(\right)open="("\; close=")">{Y}_{*},{\widehat{Y}}_{*}:={\left(\right)}^{{Y}_{*}}2& .\end{array}\end{array}$$Then, the MMSE spatial field reconstruction estimator is given by$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{Y}}_{*}^{\mathrm{MMSE}}& =\underset{y}{arg\; min}\mathbb{E}\left(\right)open="["\; close="]">L\left(\right)open="("\; close=")">{Y}_{*},{\widehat{Y}}_{*}|{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},{\mathbf{x}}_{*}\hfill \end{array}& \hfill \phantom{\rule{1.em}{0ex}}& =\underset{y}{arg\; min}\mathbb{E}\left(\right)open="["\; close="]">{\left(\right)}^{{Y}_{*}}2|{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},{\mathbf{x}}_{*}\hfill \end{array}\hfill \phantom{\rule{1.em}{0ex}}& ={\int}_{-\infty}^{\infty}{Y}_{*}p\left(\right)open="("\; close=")">{Y}_{*}|{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},{\mathbf{x}}_{*}\mathrm{d}\phantom{\rule{0.277778em}{0ex}}{Y}_{*}.\hfill $$The conditional expectation is the estimate that minimises the conditional MMSE and the minimum value is given by the conditional variance shown below:$$\begin{array}{c}\hfill {\sigma}_{Y*|{\mathbf{Y}}_{1:N}}^{2}=\mathbb{E}\left(\right)open="["\; close="]">{\left(\right)}^{{Y}_{*}}2& |{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},{\mathbf{x}}_{*}\\ .\end{array}$$
- (2)
- Maximum A-Posteriori (MAP) estimator:The MAP estimator is the mode of the predictive posterior density and utilizes the $0/1$ loss function:$$\begin{array}{c}\hfill \begin{array}{c}\hfill L\left(\right)open="("\; close=")">{Y}_{*},{\widehat{Y}}_{*}:=\left(\right)open="\{"\; close>\begin{array}{cc}0\hfill & ,\mathrm{If}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{Y}_{*}={\widehat{Y}}_{*}\hfill \\ 1\hfill & ,\mathrm{Otherwise}\hfill \end{array}\end{array}\end{array}$$Then, the MAP spatial field reconstruction estimator is given by$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{Y}}_{*}^{\mathrm{MAP}}& =\underset{{y}^{*}}{arg\; max}p\left(\right)open="("\; close=")">{Y}_{*}|{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},{\mathbf{x}}_{*}.\hfill \end{array}\end{array}$$
- (3)
- Spatial-Best Linear Unbiased Estimator (S-BLUE):The S-BLUE utilises the same loss function as the MMSE estimator, but is restricted to the linear family of estimators, given by:$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{Y}}_{*}^{\mathrm{S}\text{-}\mathrm{BLUE}}& =\underset{{a}_{*},{\mathbf{B}}_{*}}{arg\; min}\mathbb{E}\left(\right)open="["\; close="]">{\left(\right)}^{{Y}_{*}}2|{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},{\mathbf{x}}_{*}\hfill & ,\end{array}\end{array}$$
- (4)
- Spatial Regional and Level Exceedance estimators:There are a few ways of characterizing the spatial exceedance of these processes, either through the region or the location of the spatial exceedance given a user defined threshold and the tolerance. The other characteristic is the level at which the random process exceeds the given threshold at a given location. We define these characterizations mathematically as follows:
- (a)
- Spatial-Regional Right Tail Exceedance (S-RTE):$$\begin{array}{c}\hfill {D}_{\mathbf{x}}\left(\right)open="("\; close=")">{\mathbf{x}}_{*};{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},T,\alpha =\left(\right)open="\{"\; close="\}">{\mathbf{x}}_{*}:\mathrm{P}\left(\right)open="("\; close=")">{Y}_{*}T|{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},{\mathbf{x}}_{*}& \ge 1-\alpha \\ ,\end{array}$$
- (b)
- Spatial-Regional Left Tail Exceedance (S-LTE):$$\begin{array}{c}\hfill {D}_{\mathbf{x}}\left(\right)open="("\; close=")">{\mathbf{x}}_{*};{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},T,\alpha =\left(\right)open="\{"\; close="\}">{\mathbf{x}}_{*}:\mathrm{P}\left(\right)open="("\; close=")">{Y}_{*}T|{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},{\mathbf{x}}_{*}& \ge 1-\alpha \\ ,\end{array}$$
- (c)
- Spatial-Level Exceedance (S-LE):$$\begin{array}{c}\hfill {D}_{\alpha}=inf\left(\right)open="\{"\; close="\}">\alpha :\mathrm{P}\left(\right)open="("\; close=")">{Y}_{*}T|{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},{\mathbf{x}}_{*}\ge 1-\alpha & .\end{array}$$Here we are interested to find the minimum quantile level $\alpha $ at which the process exceeds the given threshold T at a given location ${\mathbf{x}}_{*}$. The function ${D}_{\alpha}\left(\right)open="("\; close=")">{\mathbf{x}}_{*};{\mathbf{Y}}_{1:N},{\mathbf{x}}_{1:\mathrm{N}\phantom{\rule{4.pt}{0ex}}},T$, i.e.,

#### 4.2. Spatial Field Reconstruction Estimator Derivations

**Lemma**

**5**

**.**

**Remark**

**3.**

**Remark**

**4.**

**Proposition**

**5**

**.**

**Proof.**

**Remark**

**5.**

**Proposition**

**6**

**.**

**Proof.**

**Proposition**

**7**

**.**

**Lemma**

**6**

**.**

- (1)
- Marginal Quantile Function:The univariate marginal quantile function of the Tukey g-and-h process at any arbitrary location ${\mathbf{x}}_{k}$ is described as follows:$$\begin{array}{c}\hfill {Q}_{{Y}_{k}}\left(\alpha \right)={\mathcal{T}}_{gh}\left(\right)open="("\; close=")">{q}_{k}\left(\alpha \right);{\mathrm{\u0131}}_{gh}\end{array}$$where ${q}_{k}\left(\alpha \right)={a}_{k}+\sqrt{\mathcal{C}\left(\right)open="("\; close=")">{\mathbf{x}}_{k},{\mathbf{x}}_{k}}$, and ${\Phi}^{-1}\left(\alpha \right)$ denotes the inverse cdf of the standard normal distribution.
- (2)
- Conditional Quantile Function:The univariate conditional quantile function of the Tukey g-and-h process at any arbitrary location ${\mathbf{x}}_{*}$ is described as follows:$$\begin{array}{c}\hfill {Q}_{{Y}_{*}|{Y}_{1:N}}\left(\alpha \right)={\mathcal{T}}_{gh}\left(\right)open="("\; close=")">{q}_{*}\left(\alpha \right);{\mathrm{\u0131}}_{gh}\end{array}$$

**Proof.**

**Proposition**

**8**

**.**

**Remark**

**6.**

**Proposition**

**9**

**.**

## 5. Experimental Results: Warped Gaussian Process Spatial Field Reconstructions

#### 5.1. Simulation Setup

- (1)
- Scenario 1: We compare the performance of the different estimators such as, MMSE (Equation (39)), S-MAP (Equation (41)), S-BLUE (Equation (35)) to the Gaussian Process estimate in various scenarios such as, small g and h with low spatial correlation, high spatial correlation and other such combinations of g, h and l.
- (2)
- Scenario 2: We characterize the exceedance estimation ability of the Tukey g-and-h process as shown in Proposition 8 (g,h), through spatial ROC curves compared to a Gaussian process.
- (3)
- Scenario 3: We adopt real data case studies for the US and we apply our estimators on the noisy observed US precipitation data sets [40]. We consider a spatial field reconstruction that involves latitudes between 34 and 43 degrees and longitudes between −110 and −100 degrees and we compare the different spatial field reconstruction estimators and their performance on real data versus a classical Gaussian process model.

#### 5.2. Scenario 1: Spatial Field Reconstruction on Synthetic Data

**Figure 2.**Plots of comparison of the N-MSE as a function of number of sensors (N). Top Subplot: $g=0.001$, $h=0.001$, $l=0.5$. Bottom Subplot: $g=1.2$, $h=0.2$, $l=0.5$.

#### 5.3. Scenario 2: Spatial Exceedance on Synthetic Data

**Figure 4.**Plots of ROC curves comparing the Tukey g-and-h distribution to the Gaussian process for different values of g and h.

#### 5.4. Scenario 3: Spatial Field Reconstruction on Real Spatial Sensor Data for Hourly Precipitation

**Figure 5.**USA map with sensor locations measuring precipitation in 4 states showing the average precipitation for the month of November 1994 at 850 locations. The values (in mm) are colour coded according to the ranges shown.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

## Appendix B. Proof of Proposition 2

## Appendix C. Proof of Proposition 2

## Appendix D. Proof of Lemma 3

## Appendix E. Proof of Proposition 3

## Appendix F. Proof of Proposition 5

## Appendix G. Proof of Proposition 6

## Appendix H. Proof of Lemma 6

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**Figure 1.**Comparison of the standard Gaussian distribution to the Tukey g-and-h distributions with values from $\left(\right)$.

**Figure 3.**Plots of ROC curves comparing the Tukey g-and-h distribution to the Gaussian process for different values of g and h.

**Figure 6.**The N-MAD comparison between the various Tukey g-and-h estimators and the Gaussian process.

**Figure 7.**The ratio R of the N-MSE of the S-BLUE estimate and the GP estimate at each of the test locations when $N=500$, i.e., 500 locations were used as the training set.

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Peters, G.W.; Nevat, I.; Nagarajan, S.G.; Matsui, T.
Spatial Warped Gaussian Processes: Estimation and Efficient Field Reconstruction. *Entropy* **2021**, *23*, 1323.
https://doi.org/10.3390/e23101323

**AMA Style**

Peters GW, Nevat I, Nagarajan SG, Matsui T.
Spatial Warped Gaussian Processes: Estimation and Efficient Field Reconstruction. *Entropy*. 2021; 23(10):1323.
https://doi.org/10.3390/e23101323

**Chicago/Turabian Style**

Peters, Gareth W., Ido Nevat, Sai Ganesh Nagarajan, and Tomoko Matsui.
2021. "Spatial Warped Gaussian Processes: Estimation and Efficient Field Reconstruction" *Entropy* 23, no. 10: 1323.
https://doi.org/10.3390/e23101323