extended_mean: Extended mean

Description

Calculate a generalized logarithmic mean / extended mean.

Usage

extended_mean(r, s)
generalized_logmean(r)
logmean(a, b, tol = .Machine$double.eps^0.5)

Value

generalized_logmean() and extended_mean() return a function:

function(a, b, tol = .Machine$double.eps^0.5){...}

This computes the component-wise generalized logarithmic mean of order r, or the extended mean of orders r and s, of a and b.

logmean() returns a numeric vector, the same length as max(length(a), length(b)), giving the component-wise logarithmic mean of a and b.

Arguments

r, s: A finite number giving the order of the generalized logarithmic mean / extended mean.
a, b: A strictly positive numeric vector.
tol: The tolerance used to determine if a == b.

Details

The function extended_mean() returns a function to compute the component-wise extended mean of a and b of orders r and s. See Bullen (2003, p. 393) for a definition. This is also called the difference mean, Stolarsky mean, or extended mean-value mean.

The function generalized_logmean() returns a function to compute the component-wise generalized logarithmic mean of a and b of order r. See Bullen (2003, p. 385) for a definition, or https://en.wikipedia.org/wiki/Stolarsky_mean. The generalized logarithmic mean is a special case of the extended mean, corresponding to extended_mean(r, 1)(), but is more commonly used for price indexes.

The function logmean() returns the ordinary component-wise logarithmic mean of a and b, and corresponds to generalized_logmean(1)().

Both a and b should be strictly positive. This is not enforced, but the results may not make sense when the generalized logarithmic mean / extended mean is not defined. The usual recycling rules apply when a and b are not the same length.

By definition, the generalized logarithmic mean / extended mean of a and b is a when a == b. The tol argument is used to test equality by checking if abs(a - b) <= tol. The default value is the same as all.equal(). Setting tol = 0 tests for exact equality, but can give misleading results when a and b are computed values. In some cases it's useful to multiply tol by a scale factor, such as max(abs(a), abs(b)). This often doesn't matter when making price indexes, however, as a and b are usually around 1.

References

Balk, B. M. (2008). Price and Quantity Index Numbers. Cambridge University Press.

Bullen, P. S. (2003). Handbook of Means and Their Inequalities. Springer Science+Business Media.

Stolarsky, K. B. (1975). Generalizations of the Logarithmic Mean. Mathematics Magazine, 48(2): 87-92.

Examples

Run this code

x <- 8:5
y <- 1:4

#---- Comparing logarithmic means and generalized means ----

# The arithmetic and geometric means are special cases of the
# generalized logarithmic mean

all.equal(generalized_logmean(2)(x, y), (x + y) / 2)
all.equal(generalized_logmean(-1)(x, y), sqrt(x * y))

# The logarithmic mean lies between the arithmetic and geometric means
# because the generalized logarithmic mean is increasing in r

all(logmean(x, y) < (x + y) / 2) &
  all(logmean(x, y) > sqrt(x * y))

# The harmonic mean cannot be expressed as a logarithmic mean, but can
# be expressed as an extended mean

all.equal(extended_mean(-2, -1)(x, y), 2 / (1 / x + 1 / y))

# The quadratic mean is also a type of extended mean

all.equal(extended_mean(2, 4)(x, y), sqrt(x^2 / 2 + y^2 / 2))

# As are heronian and centroidal means

all.equal(
  extended_mean(0.5, 1.5)(x, y),
  (x + sqrt(x * y) + y) / 3
)
all.equal(
  extended_mean(2, 3)(x, y),
  2 / 3 * (x^2 + x * y + y^2) / (x + y)
)

#---- Approximating the logarithmic mean ----

# The logarithmic mean can be approximated as a convex combination of
# the arithmetic and geometric means that gives more weight to the
# geometric mean

approx1 <- 1 / 3 * (x + y) / 2 + 2 / 3 * sqrt(x * y)
approx2 <- ((x + y) / 2)^(1 / 3) * (sqrt(x * y))^(2 / 3)

approx1 - logmean(x, y) # always a positive approximation error
approx2 - logmean(x, y) # a negative approximation error

# A better approximation

correction <- (log(x / y) / pi)^4 / 32
approx1 / (1 + correction) - logmean(x, y)

#---- Some identities ----

# A useful identity for turning an additive change into a proportionate
# change

all.equal(logmean(x, y) * log(x / y), x - y)

# Works for other orders, too

r <- 2

all.equal(
  generalized_logmean(r)(x, y)^(r - 1) * (r * (x - y)),
  (x^r - y^r)
)

# Some other identities

all.equal(
  generalized_logmean(-2)(1, 2),
  (harmonic_mean(1:2) * geometric_mean(1:2)^2)^(1 / 3)
)

all.equal(
  generalized_logmean(0.5)(1, 2),
  (arithmetic_mean(1:2) + geometric_mean(1:2)) / 2
)

all.equal(
  logmean(1, 2),
  geometric_mean(1:2)^2 * logmean(1, 1 / 2)
)

#---- Integral representations of the logarithmic mean ----

logmean(2, 3)

integrate(function(t) 2^(1 - t) * 3^t, 0, 1)$value
1 / integrate(function(t) 1 / (2 * (1 - t) + 3 * t), 0, 1)$value

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