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157 APPENDIX F Distance Sampling Buckland et al.40 suggest that the modeling process for the mated from the data. The second step is to adjust the key analysis of line or point transect data can be visualized as hav- function with a series expansion. Buckland et al.40 suggest ing two steps. The first involves selecting a key function as the using (1) the cosine series, (2) simple polynomials, or (3) the starting point (Figure 51), starting with the uniform or half- Hermite polynomials. All three are linear in their parame- normal. The uniform model has no parameters,40 while the ters.40 Given in Figure 52 are the key function and the series half-normal has one unknown parameter that has to be esti- expansion.

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Figure 51. Functions useful in modeling distance data: (1) uniform, half-normal, and negative exponential, and (2) hazard-rate model for four different values of the shape parameter b. Key functions Series expansion m jy sine, a j cos w Uniform, 1/w Cos j =1 2j m y Uniform, 1/w Simple polynomial, a j cos j =1 w m jy Half-normal, exp (- y 2 / 2 2 ) Cosine, a j cos j =2 w m Half-normal, exp (- y 2 / 2 2 ) Hermite polynomial, a j H 2 j ( ys ) where y s = y / j =2 m jy Hazard-rate, 1 - exp (-( y / )- b ) Cosine, a j cos j =2 w 2j m y Hazard-rate, 1 - exp (-( y / )- b ) Simple ploynomial, a j j =2 w Figure 52. Series expansions for adjusting key functions.