Solutions To Variational Problems
Several measures of central tendency can be characterized as solving a variational problem, in the sense of the calculus of variations, namely minimizing variation from the center. That is, given a measure of statistical dispersion, one asks for a measure of central tendency that minimizes variation: such that variation from the center is minimal among all choices of center. In a quip, "dispersion precedes location". In the sense of Lp spaces, the correspondence is:
| Lp | dispersion | central tendency |
|---|---|---|
| L1 | average absolute deviation | median |
| L2 | standard deviation | mean |
| L∞ | maximum deviation | midrange |
Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any other point. The uniqueness of this characterization of mean follows from convex optimization. Indeed, for a given (fixed) data set x, the function
represents the dispersion about a constant value c relative to the L2 norm. Because the function ƒ2 is a strictly convex coercive function, the minimizer exists and is unique.
Note that the median in this sense is not in general unique, and in fact any point between the two central points of a discrete distribution minimizes average absolute deviation. The dispersion in the L1 norm, given by
is not strictly convex, whereas strict convexity is needed to ensure uniqueness of the minimizer. In spite of this, the minimizer is unique for the L∞ norm.
Read more about this topic: Average
Famous quotes containing the words solutions and/or problems:
“The anorexic prefigures this culture in rather a poetic fashion by trying to keep it at bay. He refuses lack. He says: I lack nothing, therefore I shall not eat. With the overweight person, it is the opposite: he refuses fullness, repletion. He says, I lack everything, so I will eat anything at all. The anorexic staves off lack by emptiness, the overweight person staves off fullness by excess. Both are homeopathic final solutions, solutions by extermination.”
—Jean Baudrillard (b. 1929)
“I believe that if we are to survive as a planet, we must teach this next generation to handle their own conflicts assertively and nonviolently. If in their early years our children learn to listen to all sides of the story, use their heads and then their mouths, and come up with a plan and share, then, when they become our leaders, and some of them will, they will have the tools to handle global problems and conflict.”
—Barbara Coloroso (20th century)