Copulas

Function representing a joint distribution as a mapping from the CDFs of its marginals

Defines a general way to think about dependence between random variables

Starting to be used in ecology:

• Popovic, Hui, Warton, 2018. J. Multivar. Anal. 165, 86–100. doi: 10/dzx9
• Anderson et al 2019. Ecol. Evol. 44, 182. doi: 10/dzzb

Copulas

Model effects beyond the mean

Complex data often can't be modelled as conditional means + a mean-variance relationship

Distributional regression models have linear predictors for all parameters of the conditional distribution

y|ϑk

For a Gaussian response

\begin{align} \mu_i & = \beta^{\mu}_0 + \boldsymbol{x}_i^{\mathsf{T}}\boldsymbol{\beta}^{\mu}_j \\ \log(\sigma_i) & = \beta^{\sigma}_0 + \boldsymbol{x}_i^{\mathsf{T}}\boldsymbol{\beta}^{\sigma}_j \end{align}

Maximise penalised log-likelihood ⇨ β

Lake 227

Algal pigments well preserved in lake sediments

• Reflect phytoplankton standing crops in lakes
• Chlorophyll-a tracks planktonic sources
• β-carotene tracks planktonic & benthic sources

Lake 227

Gaussian copula with Gamma univariate marginal responses

\begin{align} \log(\mu_{\mathsf{Chl a}_{i}}) & = \beta^{\mu_{\mathsf{Chl a}}}_0 + f^{\mu_{\mathsf{Chl a}}}(\text{Year}_i) \\ \log(\mu_{\mathsf{\beta caro}_{i}}) & = \beta^{\mu_{\mathsf{\beta caro}}}_0 + f^{\mu_{\mathsf{\beta caro}}}(\text{Year}_i) \\ \log(\sigma_{\mathsf{Chl a}_{i}}) & = \beta^{\sigma_{\mathsf{Chl a}}}_0 \\ \log(\sigma_{\mathsf{\beta caro}_{i}}) & = \beta^{\sigma_{\mathsf{\beta caro}}}_0 \\ g(\theta_i) & = \beta^{\theta}_0 + f^{\theta}(\text{Year}_i) \end{align}

Lake 227 Model Fitting

Fitted with gjrm() from GJRM package

Marra, G., Radice, R., 2017. Comput. Stat. Data Anal. 112, 99–113. doi: 10/dzzc

Lake 227 — μ

Fitted mean functions for each response

Lake 227 — Kendall's τ

Estimate \theta but can transform to Kendall's τ

Acknowledgements

Funding

Data

Lake 227 data from Peter Leavitt (U Regina)

Slides

• HTML Slide deck bit.ly/visec-copula-gam © Simpson (2020)
• RMarkdown Source