Mean, variance, skewness and kurtosis in one pass (C backend)

A single streaming pass (Welford's recurrence, extended to the third and fourth central moments) over x. One pass matters for capsule members large enough that reading the column twice is the expensive part.

Usage

core_moments(x)

Arguments

x

Numeric vector (coerced with as.numeric()) .

Value

A named length-4 numeric: mean, variance, skewness, kurtosis. skewness needs at least 3 observations and kurtosis at least 4; both are NaN below that, as is everything if x contains NA/NaN.

Details

The variance uses the n - 1 denominator, matching stats::var(). The shape statistics use the sample moment definitions \(m_3 / m_2^{3/2}\) and \(m_4 / m_2^2 - 3\), with \(m_k\) the k-th central moment divided by n – so kurtosis is reported as EXCESS kurtosis and a normal sample sits near zero, not near three.

References

Welford BP (1962). Note on a method for calculating corrected sums of squares and products. Technometrics 4(3), 419–420. doi:10.1080/00401706.1962.10490022

Examples

core_moments(c(2, 4, 4, 4, 5, 5, 7, 9))
#>      mean  variance  skewness  kurtosis 
#>  5.000000  4.571429  0.656250 -0.218750 

# The first two entries agree with base R.
m <- core_moments(1:10)
all.equal(m[["mean"]], mean(1:10))
#> [1] TRUE
all.equal(m[["variance"]], stats::var(1:10))
#> [1] TRUE

# A symmetric sample has no skew; excess kurtosis is near 0 for normal
# data and positive for a heavy-tailed sample.
core_moments(c(-2, -1, 0, 1, 2))[["skewness"]]
#> [1] 0
core_moments(c(rep(0, 20), -8, 8))[["kurtosis"]] > 0
#> [1] TRUE

# Too short to define a shape statistic: NaN rather than a guess.
core_moments(c(1, 2))
#>     mean variance skewness kurtosis 
#>      1.5      0.5      NaN      NaN