sinewavePDF {ADMUR}R Documentation

PDF of a truncated sinusoidal curve


Probability density function for a truncated sinusoidal curve.


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



Numeric vector of years

min, max

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


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


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


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


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.


	# 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)

[Package ADMUR version 1.0.3 Index]