Function Repository Resource:

PowerMean

Compute the mean of a list of numbers all taken to some power

Contributed by: Seth J. Chandler

ResourceFunction["PowerMean"][y,n,x]

takes x to the y^th power, then applies the Total function using level specification n, and divides the result by the Length of x.

ResourceFunction["PowerMean"][y,n]

represents an operator form of ResourceFunction["PowerMean"] that can be applied to x.

ResourceFunction["PowerMean"][y]

represents an operator form of ResourceFunction["PowerMean"] that, when applied to x, computes the mean at the top level.

ResourceFunction["PowerMean"][]

represents an operator form of ResourceFunction["PowerMean"] that, when applied to x, takes the mean of the squares.

Details and Options

Both x and y can be real, complex or symbolic.

The level specification n may take the form of:

an integer n, meaning the summation is performed at all levels down to level n;

an integer n wrapped with List, meaning the summation is performed at level n only; and

a List of two integers {n₁,n₂} meaning the summation is performed at all levels n₁ through n₂.

See the documentation for Total for a more complete explanation of how the level specification works there.

Examples

Basic Examples (4)

Compute the mean of the squares of a list:

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Compute the sum of the cubes of a list with symbolic parts:

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Compute the mean of the cubes of a symbolic array:

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Create an operator that when confronted with an expression computes the mean of its square roots:

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Scope (2)

The power may be complex, as may the list:

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The level specification can affect the results when the data has more than one dimension:

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The default is to apply at level 1:

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Apply the mean down to level 2:

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Apply the mean in the last two dimensions:

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Applications (3)

Use PowerMean to conduct ordinary least squares linear regression by finding the values of two parameters a and b that minimize the mean of the squared distances between the actual value of the independent variable and a value that depends on a and b:

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Use PowerMean to perform "Tikhonov" (ridge) regression:

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Use PowerMean to perform "LASSO" regression:

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Properties and Relations (1)

PowerMean[] is the same as the mean square of the results from the Norm function if the arguments it confronts are real-valued, but is not necessarily the same if the values it confronts are complex:

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$ResourceFunction["PowerMean"][][{3, 4, 5}] == Norm[{3, 4, 5}]^2/3$