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 density function 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 function that must integrate to one that is often called a kernel function and $w$ is a smearing parameter. From a probability density function 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 discrete 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:

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