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157 Buckland et al.40 suggest that the modeling process for the analysis of line or point transect data can be visualized as hav- ing two steps. The ï¬rst involves selecting a key function as the starting point (Figure 51), starting with the uniform or half- normal. The uniform model has no parameters,40 while the half-normal has one unknown parameter that has to be esti- mated from the data. The second step is to adjust the key function with a series expansion. Buckland et al.40 suggest using (1) the cosine series, (2) simple polynomials, or (3) the Hermite polynomials. All three are linear in their parame- ters.40 Given in Figure 52 are the key function and the series expansion. A P P E N D I X F Distance Sampling
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. Figure 52. Series expansions for adjusting key functions. Key functions Series expansion wUniform, 1/ Cosine, cos Uniform, 1/ Simpl a j y w w j j m = â âââ ââ â1 Ï e polynomial, cos Half-nor a y w j j m j = â âââ ââ â1 2 mal, exp Cosine, cos( / )â âââ = ây a j y w j j m 2 2 2 2Ï Ï ââ â âHalf-normal, exp Hermite polyn( / )y2 22Ï omial, where Hazard-ra a H y y y j j j m s s 2 2= â ( ) = / Ï te, 1 exp Cosine, cosâ â â â = â( ( / ) )y a j y w b j j m Ï Ï 2 ââ ââ â â â âHazard-rate, 1 exp Simple p( ( / ) )y bÏ loynomial, a y w j j m j = â âââ ââ â2 2