sinewavePDF {ADMUR}

PDF of a truncated sinusoidal curve

Description

Probability density function for a truncated sinusoidal curve.

Usage

sinewavePDF(x, min, max, f, p, r)

Arguments

x

Numeric vector of years

min, max

Lower and upper \(x\) limits of the distribution

f

Numeric frequency (cycles per unit \(x\)).

p

Numeric between \(0\) and \(2\pi\), giving the cycle position (in radians) at \(x = 0\).

r

Numeric between 0 and 1, determining how flat the distribution is.

Details

The usual function to describe a sine wave is \(f(x) = A\sin(2\pi f x + p)\), where \(A\) is the amplitude, \(f\) is the frequency (cycles per year), and \(p\) is the cycle position (in radians) at \(x = 0\), and therefore oscillates above and below the x-axis.

However, a sinusoidal PDF must by definition always be non-negative, which can conceptually be considered as a sine wave stacked on top of a uniform distribution with a height \(A + k\), where \(k >= 0\). Since the PDF is \(f(x)\) divided by the area below the curve, A and k simplify to a single parameter \(r\) that determines the relative proportions of the uniform and sinusoidal components, such that:

when \(r = 0\) the amplitude of the sine wave component is zero, and the overall PDF is just a uniform distribution between min and max.

when \(r = 1\) the uniform component is zero, and the minima of the sine wave touches zero. This does not necessarily mean the PDF minimum equals zero, since a minimum point of the sine wave may not occur with PDF domain (truncated between min and max).

Therefore the formula for the PDF is:

\[\frac{1 + \sin(2\pi f x + p) - \ln(r)}{(x_{max} - x_{min})(1 - \ln(r)) + (\frac{1}{2\pi f})[\cos(2\pi f x_{min} - p) - \cos(2\pi f x_{max} - p)]}\]

where \(x =\) years, and \(x_{min}\) and \(x_{max}\) determine the truncated date range.

Examples

	# A sinewave with a period of 700 years
	x <- seq(1500,4500, length.out=1000)
	y <- sinewavePDF(x, min=2000, max=4000, f=1/700, p=0, r=0.2)
	plot(x,y,type='l')