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A Bayesian Nonparametric Approach to Unmixing Detrital Geochronologic Data

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Abstract

Sedimentary deposits constitute the primary record of changing environmental conditions that have acted on Earth’s surface over geologic time. Clastic material is eroded from source locations (parents) in sediment routing systems and deposited at sink locations (children). Both parents and children have characteristics that vary across many different dimensions, including grain size, chemical composition, and the geochronologic age of constituent detrital minerals. During transport, sediment from different parents is mixed together to form a child, which in turn may serve as the parent for other sediment farther down-system or later in time when buried sediment is exhumed. The distribution of detrital mineral ages observed in parent and child sediments allows for investigation of the proportion of each parent in the child sediment, which reflects the properties of the sediment routing system. To model the proportion of dates in a child sample that comes from each of the parent distributions, we use a Bayesian mixture of Dirichlet processes. This model enables us to estimate the mixing proportions with associated uncertainty while making minimal assumptions. We also present an extension to the model whereby we reconstruct unobserved parent distributions from multiple observed child distributions using mixtures of Dirichlet processes. The model accounts for uncertainty in both the number of mineral formation events that constitute each parent distribution and the mixing proportions of each parent distribution that constitutes a child distribution. To demonstrate the model, we perform analyses using simulated data where the true age distribution is known as well as using a real-world case study from the coast of central California, USA.

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Appendix

Appendix

In the body of the text, we explain in detail how the scientific understanding of the geologic processes that generated the data led to the development of the statistical model. Here in the appendix, we define the statistical models directly without the scientific motivation for clarity of model definition.

1.1 Top-Down Mixing Model

In the top-down mixing model, observations are made on both the child sediments and parent sediments. The \(n_y\) observed age measurements for the child sediment are given by the vector \(\mathbf {y} = (y_1, \ldots , y_{n_y})'\) and are reported with an observed analytic measurement standard deviation \(\varvec{\sigma }_y = (\sigma _{y1}, \ldots , \sigma _{yn_y})'\) associated with each observation. Assuming a normal distribution for the measurement process, the latent true ages are defined as \(\tilde{\mathbf {y}} = (\tilde{y}_1, \ldots , \tilde{y}_{n_y})'\) and are modeled by

$$\begin{aligned} \mathbf {y} | \tilde{\mathbf {y}}, \varvec{\sigma }_y^2&\sim {\text {N}} (\mathbf {y} | \tilde{\mathbf {y}}, {\text {diag}} ( \varvec{\sigma }_y^2 ) ), \end{aligned}$$

where \(N(\mathbf {x} | \varvec{\mu }, \varvec{\varSigma })\) is a multivariate normal distribution with data vector \(\mathbf {x}\), mean vector \(\varvec{\mu }\), and covariance matrix \(\varvec{\varSigma }\). The notation \({\text {diag}} (\varvec{\sigma ^2})\) represents a diagonal covariance matrix with iith element \(\sigma ^2_i\) and off-diagonal elements all equal to 0.

Likewise, the \(b = 1, \ldots , B\) parent observations each comprise \(n_b\) observations and are given by the vector \(\mathbf {z}_b = (z_{b1}, \ldots z_{bn_b})'\) and are reported with an observed analytic measurement standard deviation \(\varvec{\sigma }_b = (\sigma _{b1}, \ldots , \sigma _{bn_b})'\) associated with each observation. Assuming a normal distribution for the measurement process, the latent true ages are defined as \(\tilde{\mathbf {z}}_b = (\tilde{z}_{b1}, \ldots , \tilde{z}_{bn_b})'\) and are modeled by

$$\begin{aligned} \mathbf {z}_b | \tilde{\mathbf {z}}_b, \varvec{\sigma }_{b}^2&\sim {\text {N}} (\mathbf {z}_b | \tilde{\mathbf {z}}_b, {\text {diag}} ( \varvec{\sigma }_{b}^2 ) ). \end{aligned}$$

The latent parent age distributions for the \(b = 1, \ldots , B\) parents are modeled using a finite mixture of K Gaussian distributions

$$\begin{aligned} \tilde{\mathbf {z}}_b | \varvec{\mu }, \varvec{\sigma }^2, \varvec{p}_{b}&\sim \prod _{i=1}^{n_b} \sum _{k=1}^K p_{bk} {\text {N}} \left( \tilde{z}_{ib} \big | \mu _{k}, \sigma ^2_{k} \right) , \end{aligned}$$

where \(\varvec{\mu } = (\mu _1, \ldots , \mu _K)'\) and \(\varvec{\sigma }^2 = (\sigma ^2_1, \ldots , \sigma ^2_K)'\) are the mean and variance of the mixture distributions (which are shared across each of the parents) and \(\mathbf {p}_{b} = (p_{b1}, \ldots , p_{bK})'\) are mixture weights where for \(k = 1, \ldots , K,\) \(p_{bk} > 0\) and \(\sum _{k=1}^K p_{bk} = 1\). For \(k = 1, \ldots , K\), the mixture distribution means are assigned independent, vague priors \(N(\mu _\mu = 150 \text{ Myr }, \sigma ^2_\mu = 150^2 \text{ Myr}^2)\), where Myr represents a million years. To ensure the mixing distributions are relatively concentrated with respect to geologic time, the mixing kernel standard deviations are assigned independent truncated half-Cauchy priors \(\sigma _k \sim {\text {Cauchy}}^+(0, 25 \text{ Myr})I\{0< \sigma _k < 50 \text{ Myr }\}\), which enforces the mixing distribution scales (which represent geologic mineral formation events) to be small relative to the range of dates from about 0 to 300 Myr.

For each parent \(b = 1, \ldots , B\), the mixing probabilities \(\mathbf {p}_b\) are modeled by introducing \(k = 1, \ldots , K-1\) independent and identically distributed random variables \(\tilde{p}_{bk} \sim Beta(1, \alpha _b)\) and transforming the \(\tilde{p}_{bk}\)s by

$$\begin{aligned} p_{b k}&= {\left\{ \begin{array}{ll} \tilde{p}_{b1} &{} \text{ for } k = 1,\\ \tilde{p}_{bk} \prod _{s=1}^{k-1} (1 - \tilde{p}_{bs}) &{} \text{ for } k=2, \ldots , K-1, \\ \prod _{s=1}^{k-1} (1 - p_{bs}) &{} \text{ for } k = K. \end{array}\right. } \end{aligned}$$

which induces a finite approximation to the stick-breaking representation of a Dirichlet process so long as K is chose large enough (Section 3 Ishwaran and James 2001 and Ishwaran and Zarepour (2002)). For \(b = 1, \ldots , B\), \(\alpha _b\) is assigned a gamma(1, 1) prior with \( \varvec{\alpha } = (\alpha _1, \ldots , \alpha _B)'\) defined as the vector of concentration parameters.

Combining the parent distributions, the unobserved, latent ages are modeled using the finite mixture of mixtures

$$\begin{aligned} \tilde{\mathbf {y}} | \varvec{\mu }, \varvec{\sigma }^2, \{\varvec{p}_b \}_{b=1}^B, \varvec{\phi }&\sim \prod _{i=1}^{n_y} \sum _{b=1}^B \phi _b \sum _{k=1}^K p_{bk} {\text {N}} \left( \tilde{y}_i \big | \mu _{b}, \sigma _{b}^2 \right) , \end{aligned}$$

where the notation \(\{\mathbf {p}_b\}_{b=1}^{B}\) denotes the set of parameters \(\{ \mathbf {p}_1, \ldots , \mathbf {p}_B\}\). The parameter \(\varvec{\phi } = (\phi _1, \ldots , \phi _B)'\) models the proportion of the child sediment \(\phi _b\) that comes from parent b where \(\phi _b > 0\) and \(\sum _{b=1}^B \phi _b = 1\). The mixing proportion \(\varvec{\phi }\) is assigned a \(Dirichlet(\alpha _\phi \mathbf {1})\) prior where \(\mathbf {1}\) is a vector of ones of length B and \(\alpha _\phi \) is assigned a gamma(1, 1) prior.

All combined, the top-down mixing model posterior is

$$\begin{aligned}&\left[ \tilde{\mathbf {y}}, \{ \tilde{\mathbf {z}}_b \}_{b=1}^B, \varvec{\mu }, \varvec{\sigma }^2, \{ \varvec{p}_b \}_{b=1}^B, \varvec{\phi }, \alpha _\phi , \varvec{\alpha } \big | \mathbf {y}, \varvec{\sigma }_y^2, \{ \mathbf {z}_b\}_{b=1}^B, \{\varvec{\sigma }^2_b \}_{b=1}^{B} \right] \\&\quad \propto \left[ \mathbf {y} \big | \tilde{\mathbf {y}}, \varvec{\sigma }_y^2 \right] \prod _{b=1}^B \left[ \mathbf {z}_b \big | \tilde{\mathbf {z}}_b, \varvec{\sigma }_b^2 \right] \\&\qquad \times \left[ \tilde{\mathbf {y}} \big | \varvec{\mu }, \varvec{\sigma }^2, \{ \varvec{p}_b \}_{b=1}^B, \varvec{\phi } \right] \prod _{b=1}^B \left[ \tilde{\mathbf {z}}_b \big | \varvec{\mu }, \varvec{\sigma }^2, \varvec{p}_b \right] \\&\qquad \times \left[ \varvec{\mu } \right] \left[ \varvec{\sigma }^2 \right] \left[ \varvec{\phi } | \alpha _{\phi } \right] \left[ \alpha _{\phi } \right] \left( \prod _{b=1}^B \left[ \varvec{p}_b \big | \alpha _b \right] \left[ \alpha _b \right] \right) , \end{aligned}$$

where each line on the right-hand side of the proportional symbol is the data, process, and prior model, respectively.

1.2 Bottom-Up Unmixing Model

In the bottom-up unmixing model, observations are made on \(d = 1, \ldots , D\) child sediments whereas the parent sediments are unobserved. For each of the \(d = 1, \ldots , D\) children, the \(n_d\) observed age measurements are given by the vector \(\mathbf {y}_d = (y_{d1}, \ldots , y_{dn_d})'\) and are reported with an observed analytic measurement standard deviation \(\varvec{\sigma }_d = (\sigma _{d1}, \ldots , \sigma _{dn_d})'\) associated with each observation. Assuming a normal distribution for the measurement process, the latent true ages are defined as \(\tilde{\mathbf {y}}_d = (\tilde{y}_{d1}, \ldots , \tilde{y}_{dn_d})'\) and are modeled by

$$\begin{aligned} \mathbf {y}_d | \tilde{\mathbf {y}}_d, \varvec{\sigma }_d^2&\sim {\text {N}} (\mathbf {y}_d | \tilde{\mathbf {y}}_d, {\text {diag}} ( \varvec{\sigma }_d^2 ) ), \end{aligned}$$

As none of the parent ages are observed, the latent parent age distributions for the \(b = 1, \ldots , B\) parents are represented as a finite mixture of K Gaussian distributions

$$\begin{aligned} \sum _{k=1}^K p_{bk} {\text {N}} \left( \mu _{k}, \sigma ^2_{k} \right) , \end{aligned}$$

where \(\varvec{\mu } = (\mu _1, \ldots , \mu _K)'\) and \(\varvec{\sigma }^2 = (\sigma ^2_1, \ldots , \sigma ^2_K)'\) are the mean and variance of the mixture distributions (which are shared across each of the parents) and \(\mathbf {p}_{b} = (p_{b1}, \ldots , p_{bK})'\) are mixture weights where for \(k = 1, \ldots , K,\) \(p_{bk} > 0\) and \(\sum _{k=1}^K p_{bk} = 1\). For \(k = 1, \ldots , K\), the mixture distribution means are assigned independent, vague priors \(N(\mu _\mu = 150 \text{ Myr }, \sigma ^2_\mu = 150^2 \text{ Myr}^2)\) where Myr represents a million years. To ensure the mixing distributions are relatively concentrated with respect to geologic time, the mixing kernel standard deviations are assigned independent truncated half-Cauchy priors \(\sigma _k \sim {\text {Cauchy}}^+(0, 25 \text{ Myr})I\{0< \sigma _k < 50 \text{ Myr }\}\), which enforces the mixing distribution scales (which represent geologic mineral formation events) to be small relative to the range of dates from about 0 to 300 Myr.

Because none of the parent ages are observed, the parent distributions are estimated entirely using child sediment observations. Assuming a fixed and known number of parents B, the bottom-up process model for the dth child is

$$\begin{aligned} \tilde{\mathbf {y}}_{d} | \varvec{\mu }, \varvec{\sigma ^2}, \{ \varvec{p}_b \}_{b=1}^B, \varvec{\phi }_d&\sim \prod _{i1=}^{n_d} \sum _{b=1}^B \phi _{db} \sum _{k=1}^K p_{bk} {\text {N}}(\tilde{y}_{id} | \mu _{k}, \sigma ^2_{k}), \end{aligned}$$

where the B-dimensional vector of mixture proportions \(\varvec{\phi }_d = \left( \phi _{d1}, \ldots , \phi _{dB} \right) '\) models the proportion of the dth child sediment that can be attributed to each of the B parents where for \(b = 1, \ldots , B\), \(\phi _{db}>0\) and \(\sum _{b=1}^B \phi _{db} = 1\). For each of the \(d = 1, \ldots , D\) children, the mixing proportions \(\varvec{\phi }_d\) are assigned an independent \(Dirichlet(\alpha _{d} \mathbf {1})\) prior, where \(\mathbf {1}\) is a vector of ones of length B and each \(\alpha _{d}\) is assigned a gamma(1, 1) prior. The priors for \(\varvec{\mu }\), \(\varvec{\sigma ^2}\), and \(\{\mathbf {p}_{b}\}_{b=1}^B\) (and their respective hyperparameters \(\varvec{\alpha } = (\alpha _1, \ldots , \alpha _B)'\)) are the same as in the top-down mixing model.

Thus, the bottom-up unmixing model posterior distribution is

$$\begin{aligned}&\left[ \{ \tilde{\mathbf {y}}_d \}_{d=1}^D, \varvec{\mu }, \varvec{\sigma }^2, \{ \varvec{p}_b \}_{b=1}^B, \{ \varvec{\phi }_d \}_{d=1}^D, \varvec{\alpha }_{\phi }, \varvec{\alpha } \big | \{ \mathbf {y}_d\}_{d=1}^D, \{\varvec{\sigma }^2_{d} \}_{d=1}^D \right] \\&\quad \propto \prod _{d=1}^D \left[ \mathbf {y}_d \big | \tilde{\mathbf {y}}_d, \varvec{\sigma }_{d}^2 \right] \\&\qquad \times \prod _{d=1}^D \left[ \tilde{\mathbf {y}}_d \big | \varvec{\mu }, \varvec{\sigma }^2, \varvec{\phi }_d, \{ \varvec{p}_b \}_{b=1}^B \right] \\&\qquad \times \left[ \varvec{\mu }\right] \left[ \varvec{\sigma }^2 \right] \left( \prod _{b=1}^B \left[ \varvec{p}_b \big | \alpha _b \right] \left[ \alpha _b \right] \right) \left( \prod _{d=1}^D \left[ \varvec{\phi }_d | \alpha _{d} \right] \left[ \alpha _{d} \right] \right) , \end{aligned}$$

where each line on the right-hand side of the proportional symbol is the data, process, and prior model, respectively.

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Tipton, J.R., Sharman, G.R. & Johnstone, S.A. A Bayesian Nonparametric Approach to Unmixing Detrital Geochronologic Data. Math Geosci 54, 151–176 (2022). https://doi.org/10.1007/s11004-021-09961-x

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