Motivation

• Understanding the patterns and drivers for Coral cover of benthic community in Abrolhos from 2006 to 2018.

• To investigate the heterogeneity of the response variable under different sites and years.

• To combine temporal dependency and hierarchical structure at the same time.

Static scenary

Cover Coral vs Turbity value

Static vs dynamic scenary

Cover Coral vs Turbity value Hiearchy \(\Leftrightarrow\) Longitudinal :	Hiearchy and Dynamic

Trajectories: Data over year

Dynamic Cover Coral	Turbity value by sites and years

Objectives

• To introduce a dynamic hierarchy based on a quantile regression approach;

• To achieve flexibility with the proposed model, so that it can be useful in many different settings;

• Quantile prediction of Coral cover based on drivers values by sites and years;

Proposal

• Combine approach of (Kotz, Kozubowski, and Podgórski 2012), (Gelfand, Sahu, and Carlin 1996) and (Gonçalves, Migon, and Bastos 2020);

• The filtered information throught the \(\tau^{th}\) quantile from random variable response \(y\) into two components: Hierarchical and dynamic;

• This decomposition allows us to obtain a flexible/comparative proposal;

• A robust model for heteroscedasticity and outliers.

Model

\[\begin{eqnarray}\label{eq:yhierar} \begin{aligned} y_{ij} & = X^{\prime}_{ij}\gamma_{i} + F^{\prime}_{ij}\theta_{j} + \mathcal{Q}, \quad i = 1,\ldots, I,j=1,\ldots,J;\\ \gamma_{i} &= \gamma+ v_{i}, \quad v_{i} \sim \mathcal{N}({0},V);\\ \theta_{j} &= G_{j} \theta_{j-1}+ w_{j}, \quad w_{j} \sim \mathcal{N}({0},W);\\ \theta_{0}\mid \mathcal{D}_{0} &\sim \mathcal{N}(m_{0},C_{0}). \end{aligned} \end{eqnarray}\]

• \(\mathcal{Q}\) term is what enables the quantile part.

• \(I-\)dimensional vector \(\gamma = (\alpha^{\prime}, \beta^{\prime})\), \((p+q_2)-\)dimensional vector \(\theta_{j}= (\mu^{\prime}_{j},\beta^{\prime}_{2j});\)

• \(\mu_{1j}\), the first term of \(\theta_j\), represents the mean level of the trajectories of the \(\tau^{th}\) quantile;

Aplication

Fitting the proposal using Stan in R: Summary of the estimated parameters

\(\alpha\)
\(\beta\) \(\mu_{1j}\)

Predicition of Cover Coral by year: 2018

Predicition of Cover Coral by Quantile 50%

Predicition of Coral cover in one Site over the years

Comparison of models: selection methods

Conclusion

• The methods is flexible

• The quantile to be chosen depends on the specific goals of the analyst;

• Under quantile approach the proposed model is robust to heteroscedasticity and outliers;

• The proposed model is advantageous when the hierarchy and dynamic components are present;

• The proposal allows the predictions to vary across the quantiles, years and sites.

Thanks

• Foundation of Espírito Santo (Financial help)

• Guido Alberti Moreira (revision)

• Juliana Gonçalves (Abrolhos shape file)

• virtual International Statistical Ecology Conference

• Organizing committee

Contact information

pamela@dme.ufrj.br