David Lawrence Miller (& Len Thomas)

CREEM, University of St Andrews

Distance sampling

- Essentially want to compute “correction factor”
- Model \(\mathbb{P} \left[ \text{animal detected } \vert \text{ object at distance } y\right] = g(y;\boldsymbol{\theta})\)
- Calculate the average probability of detection:

\[ P_a = \frac{1}{w} \int_0^w g(y;\boldsymbol{\theta}) \text{d}y \]

- Horvitz-Thompson-type estimators \(\Rightarrow \hat{N}\)

- flexible (
*wide variety of shapes*) - efficient (
*few parameters)* - flat at 0 distance (
*detect what’s in front of you)* - monotonic non-increasing with increasing \(y\) (
*not easier to see far away things)*

\[ g(y) = k(y)\left( 1+ \alpha_K(y)\right) \]

key | \(k(y)\) | adjustment | \(\alpha_K(y)\) |
---|---|---|---|

uniform | \(1/w\) | cosine | \(\sum_{k=1}^K a_k \cos(k \pi y/w)\) |

Simple polynomial | \(\sum_{k=1}^K a_k (y/w)^{2k}\) | ||

half-normal | \(\exp\left(-\frac{y^2}{2 \sigma^2}\right)\) | cosine | \(\sum_{k=2}^K a_k \cos(k \pi y/w)\) |

Hermite polynomial | \(\sum_{k=2}^K a_k H_{2k}(y/\sigma)\) | ||

hazard-rate | \(1-\exp\left[-\left(\frac{y}{\sigma}\right)^{-b}\right]\) | cosine | \(\sum_{k=2}^K a_k \cos(k \pi y/w)\) |

Simple polynomial | \(\sum_{k=2}^K a_k (y/w)^{2k}\) |

For example, a half-normal w. cosine adjustments \[ g(y) = \exp\left(-\frac{y^2}{2 \sigma^2}\right) \left(1 + \sum_{k=2}^K a_k \cos(k \pi y/w)\right) \]

- Select number of adjustments (\(K\)) by forward-AIC
- Estimate \(\sigma\), \(\{a_k\}\) by maximum likelihood
- Can include covariates other than distance (see later)

Monotonicity

- K+A models are not necessarily monotonic
- check \(g(y_i)\geq g(y_{i+1})\) and that \(g(y_{i+1})\geq 0\) for \(i=1,\ldots,M-1\)
- perform optimisation s.t. constraints
- NLPQL (DISTANCE; Schittkowski, 1986) or SOLNP (
`mrds`

; Ye, 1987) - \(M\)++ \(\Rightarrow\) computing time ++

`mrds:::check.mono`

- Including covariates in the detection function makes this worse
- Observe covariates \(\mathbf{z}_i\) for observation \(i\)
- Affect detection via scale parameter

\[ \sigma_{i} = \exp\left( \beta_{0} + \sum_{k=1}^K \beta_k z_{ik}\right) \]

- Estimate \({\beta_k}\)
- Average detection per unique \(\mathbf{z}_i\):

\[ p(\mathbf{z}_i) = \frac{1}{w} \int_0^w g(y,\mathbf{z}_i;\boldsymbol{\theta}) \text{d}y \]

- constrained optimisation not possible
- too many combinations!
- continuous covariates – which values to check?
- “hard” optimisation problem

Mixture models

Sum of \(J\) monotonic detection functions:

\[ g(y,\mathbf{z}; \boldsymbol{\theta}, \boldsymbol{\phi}) = \sum_{j=1}^J \phi_j g_j(y,\mathbf{z}; \boldsymbol{\theta}_j), \]

\[ \text{where} \sum_j \phi_j = 1 \]

Letting \(g\) be half-normal:

\[ g(y,\mathbf{z}; \boldsymbol{\theta}, \boldsymbol{\phi}) = \sum_{j=1}^J \phi_j \exp \Big( - \frac{y^2}{2\sigma_j(\mathbf{z})^2} \Big). \]

Any sum of monotone functions is a monotone function

How do mixtures do?

`mrds`

)`mmds`

)`mrds`

)`mrds`

)`mmds`

)- Include mixtures as a candidate model
- Select number of components by AIC
- Compare with K+A via AIC as usual
- (Large simulation showed good results vs.
`mrds`

)

Conclusions

monotonicity can cause problems

you might not realise

that there are problems
Paper submitted to PLoS ONE

R package on CRAN –

`mmds`

Pike, D G, T Gunnlaugsson, A G Vikingsson, G Desportes, and B Mikkelson. An Estimate of the Abundance of Long-Finned Pilot Whales Globicephala Melas From the NASS-2001 Shipboard Survey, North Atlantic Marine Mammal Commission (NAMMCO) Scientific Committee Working Group on Abundance Estimates, 2003.

Schittkowski, K. NLPQL: a Fortran Subroutine for Solving Constrained Nonlinear Programming Problems. Annals of Operations Research 5 (1986): 485–500.

Williams, R, and L Thomas. Distribution and Abundance of Marine Mammals in the Coastal Waters of British Columbia, Canada. Journal of Cetacean Research and Management 9, no. 1 (2007): 15.

Ye, Y. Interior Algorithms for Linear, Quadratic, and Linearly Constrained Convex Programming. Stanford University, 1987.

- Humpback data from Rob Williams & Raincoast Conservation Foundation.
- Long-finned pilot whale data (NASS-2001) from Daniel Pike, Gísli Vikingsson and Bjarni Mikkelsen at the Marine Research Institute, Iceland.

Talk available at:

converged.yt/talks/ncsu-mixtures/talk.html- Due to David Borchers
- Estimate \(\alpha_p\) rather than \(\phi_j\)

\[ \phi_j = F(\sum_{p=1}^j e^{\alpha_p}) - F(\sum_{p=1}^{j-1} e^{\alpha_p}) \qquad \text{for } 1\leq j \leq J-1 \]

and

\[ \phi_J = 1-\sum_{j=1}^{J-1} \phi_j \]

- \(F\) is any continuous CDF on \((0,\infty]\) (\(\Gamma(3,2)\))
- \(\exp\) ensures \(e^{\alpha_p}\geq0\) \(\Rightarrow \alpha_p \in \mathbb{R}\)
- \(\sum \Rightarrow\) ordered the \(\phi_j\)
- CDF ensures that the \(\phi_j\)s sum to \(1\).