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')