A function that can be used to calculate whether quantities are between fixed upper and lower bounds. A function that can be used to calculate whether quantities are between fixed upper and lower bounds.

If we have multiple instances of a variable we can estimate the probability distribution (pdf) for that variable using a process called kernel density estimation:

$P(s) = \sum_i K\left( \frac{s - s_i}{w} \right)$

In this equation $$K$$ is a symmetric funciton that must integrate to one that is often called a kernel function and $$w$$ is a smearing parameter. From a pdf calculated using kernel density estimation we can calculate the number/fraction of values between an upper and lower bound using:

$w(s) = \int_a^b \sum_i K\left( \frac{s - s_i}{w} \right)$

All the input to calculate a quantity like $$w(s)$$ is generally provided through a single keyword that will have the following form:

KEYWORD={TYPE UPPER= $$a$$ LOWER= $$b$$ SMEAR= $$\frac{w}{b-a}$$}

This will calculate the number of values between $$a$$ and $$b$$. To calculate the fraction of values you add the word NORM to the input specification. If the function keyword SMEAR is not present $$w$$ is set equal to $$0.5(b-a)$$. Finally, type should specify one of the kernel types that is present in plumed. These are listed in the table below:

 TYPE FUNCTION GAUSSIAN $$\frac{1}{\sqrt{2\pi}w} \exp\left( -\frac{(s-s_i)^2}{2w^2} \right)$$ TRIANGULAR $$\frac{1}{2w} \left( 1. - \left| \frac{s-s_i}{w} \right| \right) \quad \frac{s-s_i}{w}<1$$

Some keywords can also be used to calculate a descretized version of the histogram. That is to say the number of values between $$a$$ and $$b$$, the number of values between $$b$$ and $$c$$ and so on. A keyword that specifies this sort of calculation would look something like

KEYWORD={TYPE UPPER= $$a$$ LOWER= $$b$$ NBINS= $$n$$ SMEAR= $$\frac{w}{n(b-a)}$$}

This specification would calculate the following vector of quantities:

$w_j(s) = \int_{a + \frac{j-1}{n}(b-a)}^{a + \frac{j}{n}(b-a)} \sum_i K\left( \frac{s - s_i}{w} \right)$