A variation of quantile() that can be applied to weighted samples.
weighted_quantile(
x,
probs = seq(0, 1, 0.25),
weights = NULL,
n = NULL,
na.rm = FALSE,
type = 7
)
weighted_quantile_fun(x, weights = NULL, n = NULL, na.rm = FALSE, type = 7)numeric vector: sample values
numeric vector: probabilities in \([0, 1]\)
Weights for the sample. One of:
numeric vector of same length as x: weights for corresponding values in x,
which will be normalized to sum to 1.
NULL: indicates no weights are provided, so unweighted sample quantiles
(equivalent to quantile()) are returned.
Presumed effective sample size. If this is greater than 1 and
continuous quantiles (type >= 4) are requested, flat regions may be added
to the approximation to the inverse CDF in areas where the normalized
weight exceeds 1/n (i.e., regions of high density). This can be used to
ensure that if a sample of size n with duplicate x values is summarized
into a weighted sample without duplicates, the result of weighted_quantile(..., n = n)
on the weighted sample is equal to the result of quantile() on the original
sample. One of:
NULL: do not make a sample size adjustment.
numeric: presumed effective sample size.
function or name of function (as a string): A function applied to
weights (prior to normalization) to determine the sample size. Some
useful values may be:
"length": i.e. use the number of elements in weights (equivalently
in x) as the effective sample size.
"sum": i.e. use the sum of the unnormalized weights as the sample
size. Useful if the provided weights is unnormalized so that its
sum represents the true sample size.
logical: if TRUE, corresponding entries in x and weights
are removed if either is NA.
integer between 1 and 9: determines the type of quantile estimator to be used. Types 1 to 3 are for discontinuous quantiles, types 4 to 9 are for continuous quantiles. See Details.
weighted_quantile() returns a numeric vector of length(probs) with the
estimate of the corresponding quantile from probs.
weighted_quantile_fun() returns a function that takes a single argument,
a vector of probabilities, which itself returns the corresponding quantile
estimates. It may be useful when weighted_quantile() needs to be called
repeatedly for the same sample, re-using some pre-computation.
Calculates weighted quantiles using a variation of the quantile types based
on a generalization of quantile().
Type 1--3 (discontinuous) quantiles are directly a function of the inverse CDF as a step function, and so can be directly translated to the weighted case using the natural definition of the weighted ECDF as the cumulative sum of the normalized weights.
Type 4--9 (continuous) quantiles require some translation from the definitions
in quantile(). quantile() defines continuous estimators in terms of
\(x_k\), which is the \(k\)th order statistic, and \(p_k\), which is a function of \(k\)
and \(n\) (the sample size). In the weighted case, we instead take \(x_k\) as the \(k\)th
smallest value of \(x\) in the weighted sample (not necessarily an order statistic,
because of the weights). Then we can re-write the formulas for \(p_k\) in terms of
\(F(x_k)\) (the empirical CDF at \(x_k\), i.e. the cumulative sum of normalized
weights) and \(f(x_k)\) (the normalized weight at \(x_k\)), by using the
fact that, in the unweighted case, \(k = F(x_k) \cdot n\) and \(1/n = f(x_k)\):
\(p_k = \frac{k}{n} = F(x_k)\)
\(p_k = \frac{k - 0.5}{n} = F(x_k) - \frac{f(x_k)}{2}\)
\(p_k = \frac{k}{n + 1} = \frac{F(x_k)}{1 + f(x_k)}\)
\(p_k = \frac{k - 1}{n - 1} = \frac{F(x_k) - f(x_k)}{1 - f(x_k)}\)
\(p_k = \frac{k - 1/3}{n + 1/3} = \frac{F(x_k) - f(x_k)/3}{1 + f(x_k)/3}\)
\(p_k = \frac{k - 3/8}{n + 1/4} = \frac{F(x_k) - f(x_k) \cdot 3/8}{1 + f(x_k)/4}\)
Then the quantile function (inverse CDF) is the piece-wise linear function defined by the points \((p_k, x_k)\).