integration.constant: Compute the integration constant for distance density functions

Description

Using numerical integration, this function computes the area under a distance function between two limits (w.lo and w.hi).

Usage

integration.constant(
  dist,
  density,
  a,
  covars,
  w.lo,
  w.hi,
  series,
  expansions,
  pointSurvey
)

Value

A scalar (or vector of scalars if covariates are present) that is the area under density between w.lo and w.hi. This scalar can be used as a divisor to scale density such that it integrates to 1.0. If x = density(...), then x / integration.constant(density, ...) will integrate to 1.0.

Arguments

dist

Vector of detection distance values.

density

A likelihood function for which the integration constant is sought. This function must be capable of evaluating values between w.lo and w.hi and have the following parameters:

a = Parameter vector.
dist = Vector of distances.
covars = If the density allows covariates, the covariate matrix.
w.lo = Lower limit or left truncation value.
w.hi = Upper limit or right truncation value.
series = Form of the series expansions, if any.
expansions = Number of expansion terms.
scale = Whether to scale function to integrate to 1.

a

Vector of parameters to pass to density.

covars

Matrix of covariate values.

w.lo

The lower limit of integration, or the left truncation value for perpendicular distances.

w.hi

The upper limit of integration, or the right truncation value for perpendicular distances.

series

The series to use for expansions. If expansions > 0, this string specifies the type of expansion. Valid values at present are 'simple', 'hermite', and 'cosine'.

expansions

Number of expansions in density.

pointSurvey

Boolean. TRUE if point transect data, FALSE if line transect data.

Details

The trapezoid rule is used to numerically integrate density from w.lo to w.hi. Two-hundred (200) equal-sized trapezoids are used in the integration. The number of trapezoids to use is fixed and cannot be changed without re-writing this routine.

Examples

Run this code

# Can put any number for first argument (1 used here)
scl <- integration.constant(dist=units::set_units(1,"m")
                          , density=logistic.like
                          , covars = NULL
                          , pointSurvey = FALSE
                          , w.lo = units::set_units(0,"m")
                          , w.hi = units::set_units(100,"m")
                          , expansions = 0
                          , a=c(75,25))
print(scl) # Should be 75.1

x <- units::set_units(seq(0,100,length=200), "m")
y <- logistic.like( c(75,25), x, scale=FALSE ) / scl
int.y <- (x[2]-x[1]) * sum(y[-length(y)]+y[-1]) / 2  # the trapezoid rule, should be 1.0
print(int.y) # Should be 1