A Spatial Mixture Model for Spaceborne Lidar Observations Over Mixed Forest and Non-forest Land Types

Abstract

The Global Ecosystem Dynamics Investigation (GEDI) is a spaceborne lidar instrument that collects near-global measurements of forest structure. While expansive in scope, GEDI samples are spatially sparse and cover a small fraction of the land surface. Converting the sparse samples into spatially complete predictive maps is of practical importance for a number of ecological studies. A complicating factor is that GEDI collects measurements over forested and non-forested land alike, with no automatic labeling of the land type. Such classification is important, as it categorically influences the probability distribution of the spatial process and the ecological interpretation of the observations/predictions. We propose and implement a spatial mixture model, separating the observations and the greater spatial domain into two latent classes. The latent classes are governed by a Bernoulli spatial process, with spatial effects driven by a Gaussian process. Within each class, the process is governed by a separate spatial model, describing the unique probabilistic attributes. Model predictions take the form of scalar predictions of the GEDI observables as well as discrete labeling of the class membership. Inference is conducted through a Bayesian paradigm, yielding rich quantification of prediction and uncertainty through posterior predictive distributions. We demonstrate the method using GEDI data over Wollemi National Park, Australia, using optical data from Landsat 8 as model covariates. When compared to a single spatial model, the mixture model achieves much higher posterior predictive densities on the true value. When compared to a random forest model, a common algorithmic approach in the remote sensing community, the random forest achieves better absolute prediction accuracy for prediction locations far from observed training data locations, but at the expense of location-specific assessments of uncertainty. The unsupervised binary classifications of the mixture model appear broadly ecologically interpretable as forest and non-forest when compared to optical imagery, but further comparison to ground-truth data is required.

Gibbs Samplers

Here we give exact details on the Gibbs sampler for the typical and mixture model. First presenting our notation, let ${{\varvec{s}}}= [s_1 \cdots s_n]^T$ be the vector of n observation locations, ${{\varvec{y}}}({{\varvec{s}}})$ be the vector of observed log-RH98 and ${{\varvec{X}}}({{\varvec{s}}})$ be the $n\times p$ matrix of optical intensities.

1.1 Typical Spatial Model

The Gibbs sampler for the typical spatial model iterates as follows:

• 1. Sample spatial effects ${{\varvec{w}}}|{{\varvec{y}}}({{\varvec{s}}}), {{\varvec{X}}}({{\varvec{s}}}), \mu , {\varvec{\beta }},\theta , \tau ^2$, which is multivariate normal with mean and variance

  $$\begin{aligned} \textrm{E}[{{\varvec{w}}}| \ldots ]&=({{\varvec{Q}}}(\theta ) + \frac{1}{\tau ^2}{{\varvec{A}}}({{\varvec{s}}})^T{{\varvec{A}}}({{\varvec{s}}}))^{-1}{{\varvec{A}}}({{\varvec{s}}})^T\frac{1}{\tau ^2}({{\varvec{y}}}({{\varvec{s}}}) - \mu {\varvec{1}} - {{\varvec{X}}}({{\varvec{s}}}){\varvec{\beta }}), \end{aligned}$$

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  $$\begin{aligned} \textrm{Var}[{{\varvec{w}}}| \ldots ]&= ({{\varvec{Q}}}(\theta ) + \frac{1}{\tau ^2}{{\varvec{A}}}({{\varvec{s}}})^T{{\varvec{A}}}({{\varvec{s}}}))^{-1}. \end{aligned}$$

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• 2. Sample jointly mean parameters $\mu ,{\varvec{\beta }}| {{\varvec{y}}}({{\varvec{s}}}), {{\varvec{X}}}({{\varvec{s}}}), \theta , \tau ^2$ marginalizing over ${{\varvec{w}}}$, which is multivariate normal with mean and variance

  $$\begin{aligned} \textrm{E}\begin{bmatrix} \mu \\ {\varvec{\beta }}\end{bmatrix}&= \left( {\tilde{{{\varvec{X}}}}}({{\varvec{s}}})^T{\varvec{\Sigma }}({{\varvec{s}}};\theta )^{-1}{\tilde{{{\varvec{X}}}}}({{\varvec{s}}})\right) ^{-1}{\tilde{{{\varvec{X}}}}}({{\varvec{s}}})^T{\varvec{\Sigma }}({{\varvec{s}}};\theta )^{-1}{{\varvec{y}}}({{\varvec{s}}}), \end{aligned}$$

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  $$\begin{aligned} \textrm{Var}\begin{bmatrix} \mu \\ {\varvec{\beta }}\end{bmatrix}&= \left( {\tilde{{{\varvec{X}}}}}({{\varvec{s}}})^T{\varvec{\Sigma }}^{-1}{\tilde{{{\varvec{X}}}}}({{\varvec{s}}})\right) ^{-1}, \end{aligned}$$

  (25)

  where ${\varvec{\Sigma }}({{\varvec{s}}};\theta ) = {{\varvec{A}}}({{\varvec{s}}}){{\varvec{Q}}}(\theta )^{-1}{{\varvec{A}}}({{\varvec{s}}})^T + \tau ^2{{\varvec{I}}}$ and ${\tilde{{{\varvec{X}}}}}({{\varvec{s}}}) = [\textbf{1},~{{\varvec{X}}}({{\varvec{s}}})]$. Using the Woodbury matrix identity,

  $$\begin{aligned} {\varvec{\Sigma }}^{-1} = \frac{1}{\tau ^2}{{\varvec{I}}}- \frac{1}{\tau ^4}{{\varvec{A}}}({{\varvec{s}}})^T\left( {{\varvec{Q}}}(\theta ) + \frac{1}{\tau ^2}{{\varvec{A}}}({{\varvec{s}}})^T{{\varvec{A}}}({{\varvec{s}}})\right) ^{-1}{{\varvec{A}}}({{\varvec{s}}}). \end{aligned}$$

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• 3. Sample the noise variance $\tau ^2|{{\varvec{y}}}({{\varvec{s}}}), {{\varvec{X}}}({{\varvec{s}}}), \mu , {\varvec{\beta }},{{\varvec{w}}}$, which is inverse-gamma with shape n/2 and rate

  $$\begin{aligned} \frac{1}{2}\left( {{\varvec{y}}}({{\varvec{s}}}) - \mu {\varvec{1}} - {{\varvec{X}}}({{\varvec{s}}}){\varvec{\beta }}- {{\varvec{A}}}({{\varvec{s}}}){{\varvec{w}}}\right) ^T\left( {{\varvec{y}}}({{\varvec{s}}}) - \mu {\varvec{1}} - {{\varvec{X}}}({{\varvec{s}}}){\varvec{\beta }}- {{\varvec{A}}}({{\varvec{s}}}){{\varvec{w}}}\right) . \end{aligned}$$

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• 4. Sample jointly the Matérn parameters $\theta = \{\sigma ^2, \phi \}$ using a Metropolis–Hastings step on target density

  $$\begin{aligned} p(\theta |{{\varvec{w}}}) \propto |{{\varvec{Q}}}(\theta )|^{1/2}\exp \left( -\frac{1}{2}{{\varvec{w}}}^T{{\varvec{Q}}}(\theta ){{\varvec{w}}}\right) . \end{aligned}$$

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1.2 Spatial Mixture Model

Let ${{\varvec{s}}}_j = \{s_i~:~z(s_i) = j\}$ for $j\in \{0,1\}$ be the sub-vector of observations corresponding to either class. The Gibbs sampler for the spatial mixture model iterates as follows.

• 1. Sample classifications ${{\varvec{z}}}(s_i)|\ldots ;~i\in \{1,\ldots ,n\}$, which are Bernoulli distributed with conditional probabilities

  $$\begin{aligned} \pi ^*(s_i) = \frac{\pi (s_i)f\left( y(s_i)| z(s_i) =1\right) }{\pi (s_i)f\left( y(s_i)| z(s_i) =1\right) + (1-\pi (s_i))f\left( y(s_i)| z(s_i) =0\right) }, \end{aligned}$$

  (29)

  where $f\left( \cdot |z(s_i) = j\right)$ is a normal density with mean $\mu _j + {{\varvec{x}}}(s_i)^T{\varvec{\beta }}_j + {{\varvec{A}}}(s_i){{\varvec{w}}}_j$ and variance $\tau _j^2$ for $j\in \{0,1\}$, and $\pi (s_i)$ is Eq. (4) evaluated at current parameter and effect values.

• 2. Sample spatial effects ${{\varvec{w}}}_j|{{\varvec{y}}}({{\varvec{s}}}_j), {{\varvec{X}}}({{\varvec{s}}}_j), \mu _j, {\varvec{\beta }}_j,\theta _j, \tau ^2_j$ for $j\in \{0,1\}$, which are multivariate normal with mean and variance

  $$\begin{aligned} \textrm{E}[{{\varvec{w}}}_j| \ldots ]&=({{\varvec{Q}}}(\theta _j) + \frac{1}{\tau _j^2}{{\varvec{A}}}({{\varvec{s}}}_j)^T{{\varvec{A}}}({{\varvec{s}}}_j))^{-1}{{\varvec{A}}}({{\varvec{s}}}_j)^T\frac{1}{\tau _j^2}({{\varvec{y}}}({{\varvec{s}}}_j) - \mu _j{\varvec{1}} - {{\varvec{X}}}({{\varvec{s}}}_j){\varvec{\beta }}_j), \end{aligned}$$

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  $$\begin{aligned} \textrm{Var}[{{\varvec{w}}}_j| \ldots ]&= ({{\varvec{Q}}}(\theta _j) + \frac{1}{\tau _j^2}{{\varvec{A}}}({{\varvec{s}}}_j)^T{{\varvec{A}}}({{\varvec{s}}}_j))^{-1}. \end{aligned}$$

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• 3. Sample jointly mean parameters $\mu _j,{\varvec{\beta }}_j| {{\varvec{y}}}({{\varvec{s}}}_j), {{\varvec{X}}}({{\varvec{s}}}_j), \theta _j, \tau _j^2$ marginalizing over ${{\varvec{w}}}_j$ for $j\in \{0,1\}$, which are multivariate normal with mean and variance

  $$\begin{aligned} \textrm{E}\begin{bmatrix} \mu _j \\ {\varvec{\beta }}_j \end{bmatrix}&= \left( {\tilde{{{\varvec{X}}}}}({{\varvec{s}}}_j)^T{\varvec{\Sigma }}({{\varvec{s}}}_j;\theta _j)^{-1}{\tilde{{{\varvec{X}}}}}({{\varvec{s}}}_j)\right) ^{-1}{\tilde{{{\varvec{X}}}}}({{\varvec{s}}}_j)^T{\varvec{\Sigma }}({{\varvec{s}}}_j;\theta _j)^{-1}{{\varvec{y}}}({{\varvec{s}}}_j), \end{aligned}$$

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  $$\begin{aligned} \textrm{Var}\begin{bmatrix} \mu _j \\ {\varvec{\beta }}_j \end{bmatrix}&= \left( {\tilde{{{\varvec{X}}}}}({{\varvec{s}}}_j)^T{\varvec{\Sigma }}({{\varvec{s}}}_j;\theta _j)^{-1}{\tilde{{{\varvec{X}}}}}({{\varvec{s}}}_j)\right) ^{-1}, \end{aligned}$$

  (33)

  where ${\varvec{\Sigma }}({{\varvec{s}}}_j;\theta _j) = {{\varvec{A}}}({{\varvec{s}}}_j){{\varvec{Q}}}(\theta _j)^{-1}{{\varvec{A}}}({{\varvec{s}}}_j)^T + \tau _j^2{{\varvec{I}}}$ and ${\tilde{{{\varvec{X}}}}}({{\varvec{s}}}_j) = [\textbf{1},~{{\varvec{X}}}({{\varvec{s}}}_j)]$. Using the Woodbury matrix identity,

  $$\begin{aligned} {\varvec{\Sigma }}({{\varvec{s}}}_j;\theta _j)^{-1} = \frac{1}{\tau _j^2}{{\varvec{I}}}- \frac{1}{\tau _j^4}{{\varvec{A}}}({{\varvec{s}}}_j)^T\left( {{\varvec{Q}}}(\theta _j) + \frac{1}{\tau _j^2}{{\varvec{A}}}({{\varvec{s}}}_j)^T{{\varvec{A}}}({{\varvec{s}}}_j)\right) ^{-1}{{\varvec{A}}}({{\varvec{s}}}_j). \end{aligned}$$

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• 4. Sample the noise variance $\tau _j^2|{{\varvec{y}}}({{\varvec{s}}}_j), {{\varvec{X}}}({{\varvec{s}}}_j), \mu _j, {\varvec{\beta }}_j,{{\varvec{w}}}_j$ for $j\in \{0,1\}$, which are inverse-gamma with shape $n_j/2$ (where $n_j$ is the length of ${{\varvec{s}}}_j$) and rate

  $$\begin{aligned} \frac{1}{2}\left( {{\varvec{y}}}({{\varvec{s}}}_j) - \mu _j{\varvec{1}} - {{\varvec{X}}}({{\varvec{s}}}_j){\varvec{\beta }}_j - {{\varvec{A}}}({{\varvec{s}}}_j){{\varvec{w}}}_j\right) ^T\left( {{\varvec{y}}}({{\varvec{s}}}_j) - \mu _j{\varvec{1}} - {{\varvec{X}}}({{\varvec{s}}}_j){\varvec{\beta }}_j - {{\varvec{A}}}({{\varvec{s}}}_j){{\varvec{w}}}_j\right) .\nonumber \\ \end{aligned}$$

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• 5. Sample jointly $\mu _p,{\varvec{\beta }}_p, {{\varvec{w}}}_p| {{\varvec{z}}}(s), \theta _z$ using a Laplace approximation. Let ${{\varvec{b}}}= [\mu _p,{\varvec{\beta }}_p, {{\varvec{w}}}_p]^T$ and define design matrix ${\tilde{{{\varvec{A}}}}}=[\textbf{1}, {{\varvec{X}}}({{\varvec{s}}}),{{\varvec{A}}}({{\varvec{s}}})]$ and the prior precision ${\tilde{{{\varvec{Q}}}}}(\theta _p) = \textrm{blockdiag}\{\textbf{0}_{p+1}, {{\varvec{Q}}}(\theta _p)\}$, where $\textbf{0}_{p+1}$ is a $p+1\times p+1$ matrix of zeros, representing the prior precision of the scalar intercept $\mu _z$ and p regression coefficients ${\varvec{\beta }}_z$. A Newton–Raphson routine is used to find posterior mode ${\hat{{{\varvec{b}}}}}$. Letting ${{\varvec{b}}}_{(0)}$ be some initial value, we iterate

  $$\begin{aligned} {{\varvec{b}}}_{(j+1)}&= {{\varvec{b}}}_{(j)} - {{\varvec{H}}}\left( f({{\varvec{b}}}_{(j)}|{{\varvec{z}}}({{\varvec{s}}}),\theta )\right) ^{-1}\frac{\partial f}{\partial {{\varvec{b}}}}({{\varvec{b}}}_{(j)}|{{\varvec{z}}}({{\varvec{s}}}),\theta ) \end{aligned}$$

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  $$\begin{aligned}&= {{\varvec{b}}}_{(j)} + \left( {\tilde{{{\varvec{Q}}}}}(\theta _p) + {\tilde{{{\varvec{A}}}}}^T{{\varvec{D}}}({{\varvec{b}}}_{(j)}){\tilde{{{\varvec{A}}}}}\right) ^{-1}\left( {\tilde{{{\varvec{A}}}}}^T\left( {{\varvec{z}}}({{\varvec{s}}}) - {{\varvec{p}}}({{\varvec{s}}}|{{\varvec{b}}}_{(j)})\right) - {\tilde{{{\varvec{Q}}}}}(\theta _p){{\varvec{b}}}_{(j)}\right) , \end{aligned}$$

  (37)

  where ${{\varvec{D}}}({{\varvec{b}}}_{(j)}) = \textrm{diag}\{\pi (s_i|{{\varvec{b}}}_{(j)})(1 - \pi (s_i|{{\varvec{b}}}_{(j)})\}$, until a convergence criterion is met. Then the Laplace approximation is

  $$\begin{aligned} {{\varvec{b}}}|{{\varvec{z}}}(s),\theta _z \sim \textrm{MVN}\left( {\hat{{{\varvec{b}}}}},~~\left( {\tilde{{{\varvec{Q}}}}}(\theta _z) + {\tilde{{{\varvec{A}}}}}^T{{\varvec{D}}}({\hat{{{\varvec{b}}}}}){\tilde{{{\varvec{A}}}}}\right) ^{-1}\right) . \end{aligned}$$

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• 6. Sample jointly the Matérn parameters $\theta _j = \{\sigma _j^2, \phi _j\}$ for $j\in \{0,1,z\}$, using Metropolis-Hastings steps on the target densities

  $$\begin{aligned} p(\theta _j|{{\varvec{w}}}_j) \propto |{{\varvec{Q}}}(\theta _j)|^{1/2}\exp \left( -\frac{1}{2}{{\varvec{w}}}_j^T{{\varvec{Q}}}(\theta _j){{\varvec{w}}}_j\right) . \end{aligned}$$

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