Mixture model detection functions

David Lawrence Miller (& Len Thomas)

CREEM, University of St Andrews

Distance sampling

Distance sampling

Detection functions

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

Detection functions

Criteria for detection functions

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

Key plus adjustment models

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

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

K+A models

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

Detection functions

 

Monotonicity

Example: Humpback whales

Constrained optimisation

Software plug: mrds:::check.mono

Covariate models

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

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

Monotonicity for covariate models

Example: Long-finned pilot whales (I)

Example: Long-finned pilot whales (II)

 

Mixture models

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?

Revisit: Humpback whales (mrds)

Revisit: Humpback whales (mmds)

Revisit: Long-finned pilot whales (mrds)

Revisit: Long-finned pilot whales (mrds)

Revisit: Long-finned pilot whales (mmds)

In practice

Conclusions

monotonicity can cause problems

you might not realise

that there are problems

Paper submitted to PLoS ONE

R package on CRAN – mmds

References

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.

Acknowledgements

Thanks!

Talk available at:

converged.yt/talks/ncsu-mixtures/talk.html

Constraining mixture proportions

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

and

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

Constraining mixture proportions