Generalized Mean - Definition

Definition

If is a non-zero real number, we can define the generalized mean or power mean with exponent of the positive real numbers as:

M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p}

While for we assume that it's equal to the geometric mean (which is, in fact, the limit of means with exponents approaching zero, as proved below for the general case):

M_0(x_1, \dots, x_n) = \sqrt{\prod_{i=1}^n x_i}

Furthermore, for a sequence of positive weights with sum we define the weighted power mean as:

M_p(x_1,\dots,x_n) = \left(\sum_{i=1}^n w_i x_i^p \right)^{1/p}
M_0(x_1,\dots,x_n) = \prod_{i=1}^n x_i^{w_i}

The unweighted means correspond to setting all . For exponents equal to positive or negative infinity the means are maximum and minimum, respectively, regardless of weights (and they are actually the limit points for exponents approaching the respective extremes, as proved below):

M_\infty (x_1,\dots,x_n)=\max(x_1,\dots,x_n)
M_{-\infty}(x_1,\dots,x_n)=\min(x_1,\dots,x_n)
Proof that (geometric mean)

We can rewrite the definition of using the exponential function

M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)^{1/p}} \right) } = \exp{\left( \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} \right) }

In the limit, we can apply L'Hôpital's rule to the exponential component,

\lim_{p \to 0} \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} = \lim_{p \to 0} \frac{\sum_{i=1}^n w_i x_i^p \ln{x_i}}{\sum_{i=1}^n w_i x_i^p} = \sum_{i=1}^n w_i \ln{x_i} = \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)}

since . By the continuity of the exponential function, we can substitute back into the above relation to obtain

\lim_{p \to 0} M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)} \right)} = \prod_{i=1}^n x_i^{w_i} = M_0(x_1,\dots,x_n)

as desired.
Proof that

Assume (possibly after relabeling and combining terms together) that . Then

\lim_{p \to \infty} \sum_{i=1}^n w_i x_i^p = \lim_{p \to \infty} x_1^p \sum_{i=1}^n w_i \left( \frac{x_i}{x_1} \right)^p = \lim_{p \to \infty} w_1 x_1^p

because for . (Note that although for goes to infinity, it does so much slower than so that in the limit it can be ignored.) Thus

\lim_{p \to \infty} M_p(x_1,\dots,x_n) = \lim_{p \to \infty} \left(w_1 x_1^p \right)^{1/p} = x_1 = \max(x_1,\dots,x_n) = M_\infty (x_1,\dots,x_n)

since . The formula for follows from .

Read more about this topic:  Generalized Mean

Famous quotes containing the word definition:

    “I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)

    “Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
    —Nadine Gordimer (b. 1923)