powLaw: Power law with a constant offset
In eehh-stanford/yada: Yet Another Demographic Analysis package

Description Usage Arguments Details Author(s)

powLaw calculates the mean (h). powLawSigma calculates the noise (sigma, or sig for short). powLawDensity calculates the density. powLawNegLogLikVect calculates a vector of negative log-likelihood. powLawNegLogLik calculates the negative log-likelihood (sum of powLawNegLogLikVect). fitPowLaw returns the maximum likelihood fit. simPowLaw creates simulated data.

1	powLaw(x, th_w, transformVar = F)

`x`	Vector of independent variable observations
`th_w`	Vector of parameters with ordering [a,r,b,s,kappa]
`transformVar`	Whether a transformation of the parameterization is needed [Default FALSE]
`w`	Vector of dependent variable observations
`a`	Multiplicative coefficient
`r`	Scaling exponent
`b`	Offset
`s`	Baseline noise
`kappa`	Slope of noise [Optional]
`hetSpec`	Specification for the heteroskedasticity [Default 'none']

We assume that the response variable w is distributed as

w ~ N(h(x),sig(x)^2)

where x is the independent variable, h(x) the mean, sig(x) the noise, and N denotes a normal distribution. If sig is independent of x, the model is homoskedastic. Otherwise, it is heteroskedastic. For an observation (w,x), the likelihood is

l_w = 1/sqrt(2*pi)/sig*exp(-0.5*(w-h)^2/sig^2)

The negative log-likelihood is

eta_w = 0.5*log(2*pi) + log(sig) + 0.5*(w-h)^2/sig^2

For the mean and noise, we adopt the parametric forms

h = a*x^r + b

and

sig = s

[hetSepc = 'none']

sig = s*(1 + kappa*x)

[hetSpec = 'sd_x']

sig = s*(1 + kappa*a*x^r)

[hetSpec = 'sd_resp']