Internals: prediction

The aim of the document is to explain how predictions are calculated in mrds.

It’s assumed you have some familiarity with mrds and a lot of familiarity with R. This is not supposed to be a replacement for the mrds documentation, but rather further explanation for those hacking on mrds.

Terms of reference

• $y$ refers to distance: this could be perpendicular or radial.
• $w$ is the right truncation distance.
• $g(y, \mathbf{z};\mathbf{\theta})$ a general detection function (missing the $\mathbf{z}$ if there are no covariates), we assume that the detection function has some parameters ($\mathbf{\theta}$).

What are “predictions” in mrds?

Before explaining what the code does, it’s worth thinking about what a prediction means in the context of distance sampling and mrds. These mean different things depending on the model that we are thinking about. In general we are thinking about who quantities: probabilities of detection and effective strip width/area.

• Probabilities:
• Conventional distance sampling (CDS) detection functions: for CDS models, we are thinking about the average probability of detection. That is, we have a model for $\mathbb{P}(\text{animal detected } \vert \text{ animal at distance } x)$ (the detection function) and we want to integrate out distance from the model to obtain $\mathbb{P}(\text{animal detected})$.
• Multiple covariate distance sampling (MCDS) detection functions: as with CDS models, we wish to integrate distance out of the model, however the resulting probabilities are conditional on the observed values of the non-distance covariates, since we don’t know their distributions.
• MRDS full independence detection functions:
• MRDS point independence detection functions:

Predictions for mrds model types

Predictions from predict.ds

Predictions from predict.ds can be one of two things, either:

1) the predicted average probability of detection for a given set of covariates. That is: $\hat{p}(\mathbf{z}; \boldsymbol{\theta}) = \int_0^w g(y, \mathbf{z}; \boldsymbol{\theta}) \pi(y) \text{d}y,$ where for line transects $\pi(y) = \frac{1}{w}$ and for point transects $\pi(y)=\frac{2y}{w^2}$. 2) the predicted effective strip width for a given set of covariates. That is: $\hat{\mu}(\mathbf{z}; \mathbf{\theta}) = \int_0^w g(y, \mathbf{z}; \mathbf{\theta}) \text{d}y,$

Giving either of these quantities a subscript $i$ will denote that they are fitted values (i.e. predictions at the observed data): $\hat{p}_i(\mathbf{z}; \mathbf{\theta})$ or $\hat{\mu}_i(\mathbf{z}; \mathbf{\theta})$.

Predictions from predict.io.fi

For full independence models, we fit a GLM to the detections (see Fitting MRDS), but the probabilities in the model are in fact $\pi_{ij} = \mathbb{P}(\text{animal } i \text{ detected by observer } j \quad\vert \text{ detected by at least one observer})$, so in order to obtain the probabilities we want, we must calculate: $\hat{p}_{MR}(0,\mathbf{z};\hat{\boldsymbol{\theta}}) = \hat{p}_{GLM}(0,\mathbf{z};\hat{\boldsymbol{\theta}} \vert \texttt{observer==1}) + \hat{p}_{GLM}(0,\mathbf{z};\hat{\boldsymbol{\theta}} \vert \texttt{observer==2}) -\\ \hat{p}_{GLM}(0,\mathbf{z};\hat{\boldsymbol{\theta}} \vert \texttt{observer==1}) \hat{p}_{GLM}(0,\mathbf{z}; \hat{\boldsymbol{\theta}} \vert \texttt{observer==2}),$ the conditional probability of an animal being seen by either observer. See Section 6.3.2, Laake and Borchers (2004) (in particular equation 6.22). For brevity we write: $\hat{p}_{MR}(0,\mathbf{z};\hat{\boldsymbol{\theta}}) = \hat{p}_1(0;\mathbf{z}) + \hat{p}_2(0;\mathbf{z}) - \hat{p}_1(0;\mathbf{z}) \hat{p}_2(0;\mathbf{z}).$

When the option integrate=FALSE is passed a three element list is returned with elements:

• $\hat{p}_{MR}(0,\mathbf{z};\hat{\boldsymbol{\theta}})$
• $\hat{p}_{GLM}(0,\mathbf{z};\hat{\boldsymbol{\theta}} \vert \texttt{observer==1})$
• $\hat{p}_{GLM}(0,\mathbf{z};\hat{\boldsymbol{\theta}} \vert \texttt{observer==2})$

for each animal.

When option integrate=TRUE, a vector of integrated average detection probabilities for each observation are returned. In this case the logistic function must be integrated over the range of the distances (this is calculated by pdot.dsr.integrate.logistic). We therefore calculate: $\int_0^w \hat{p}_1(x;\mathbf{z}) + \hat{p}_2(x;\mathbf{z}) - \hat{p}_1(x;\mathbf{z}) \hat{p}_2(x;\mathbf{z}) dx,$ where $\frac{\exp(X\beta)}{1+\exp(X\beta)}$ where some column of the design matrix $X$ are the distances. So, during the integration we hold everything else fixed and vary distance to perform the integration.

• what if distance is not in the model?

Predictions from predict.io

For independent observer methods with point independence, we need to calculate $\hat{p}\text{MR}(0,\mathbf{z};\boldsymbol{\theta})$ (the intercept or apex) from the mark-recapture part of the model and then $\hat{p}\cdot(\mathbf{z};\boldsymbol{\theta})$ , the average detection probability at covariates $\mathbf{z}$, from the detection function part of the model. We then multiply these two quantities: $\hat{p}(\mathbf{z};\boldsymbol{\theta}) = \hat{p}_{MR}(0,\mathbf{z};\boldsymbol{\theta}) \hat{p}_\cdot(\mathbf{z};\boldsymbol{\theta})$ to get the predictions. That is: the predicted probability of detection given covariates and that the object was seen by observer 1 (NB predictions will only be made for newdata$observer==1 in predict.io). So,$\hat{p}\cdot(\mathbf{z};\boldsymbol{\theta})$is as$\hat{p}(\mathbf{z}; \boldsymbol{\theta})$in 1, above. The other part of the prediction,$\hat{p}{MR}(0,\mathbf{z};\boldsymbol{\theta})\$ is calculated from the GLM part of the model and is given above.

References

• Borchers, DL, JL Laake, C Southwell, and CGM Paxton. Accommodating Unmodeled Heterogeneity in Double‐Observer Distance Sampling Surveys. Biometrics 62, no. 2 (2006): 372–378. doi:10.1111/j.1541-0420.2005.00493.x
• Laake, JL, and DL Borchers. Methods for Incomplete Detection at Zero Distance. In Advanced Distance Sampling, edited by ST Buckland, DR Anderson, KP Burnham, JL Laake, DL Borchers, and L Thomas, 48–70, Oxford University Press, 2004.