Lehmer mean · Lablypedia

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L_{p} (x) = \frac{\sum_{k = 1}^{n} x_{k}^{p}}{\sum_{k = 1}^{n} x_{k}^{p - 1}} .

The weighted Lehmer mean with respect to a tuple $w$ of positive weights is defined as:

L_{p, w} (x) = \frac{\sum_{k = 1}^{n} w_{k} \cdot x_{k}^{p}}{\sum_{k = 1}^{n} w_{k} \cdot x_{k}^{p - 1}} .

The Lehmer mean is an alternative to power means for interpolating between minimum and maximum via arithmetic mean and harmonic mean.

Properties

The derivative of $p \mapsto L_{p} (x)$ is non-negative

\frac{\partial}{\partial p} L_{p} (x) = \frac{(\sum_{j = 1}^{n} \sum_{k = j + 1}^{n} [x_{j} - x_{k}] \cdot [\ln (x_{j}) - \ln (x_{k})] \cdot {[x_{j} \cdot x_{k}]}^{p - 1})}{{(\sum_{k = 1}^{n} x_{k}^{p - 1})}^{2}},

thus this function is monotonic and the inequality

p \leq q ⟹ L_{p} (x) \leq L_{q} (x)

holds.

The derivative of the weighted Lehmer mean is:

\frac{\partial L_{p, w} (x)}{\partial p} = \frac{(\sum w x^{p - 1}) (\sum w x^{p} \ln x) - (\sum w x^{p}) (\sum w x^{p - 1} \ln x)}{(\sum w x^{p - 1})^{2}}

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Special cases

$lim_{p \to - \infty} L_{p} (x)$ is the minimum of the elements of $x$ .
$L_{0} (x)$ is the harmonic mean.
$L_{\frac{1}{2}} ((x_{1}, x_{2}))$ is the geometric mean of the two values $x_{1}$ and $x_{2}$ .
$L_{1} (x)$ is the arithmetic mean.
$L_{2} (x)$ is the contraharmonic mean.
$lim_{p \to \infty} L_{p} (x)$ is the maximum of the elements of $x$ . Sketch of a proof: Without loss of generality let $x_{1}, \dots, x_{k}$ be the values which equal the maximum. Then $L_{p} (x) = x_{1} \cdot \frac{k + {(\frac{x_{k + 1}}{x_{1}})}^{p} + \dots + {(\frac{x_{n}}{x_{1}})}^{p}}{k + {(\frac{x_{k + 1}}{x_{1}})}^{p - 1} + \dots + {(\frac{x_{n}}{x_{1}})}^{p - 1}}$

Applications

Signal processing

Like a power mean, a Lehmer mean serves a non-linear moving average which is shifted towards small signal values for small $p$ and emphasizes big signal values for big $p$ . Given an efficient implementation of a moving arithmetic mean called smooth you can implement a moving Lehmer mean according to the following Haskell code.

<span id="mwVA"></span><span class="nf" id="mwVQ">lehmerSmooth</span><span class="w" id="mwVg"> </span><span class="ow" id="mwVw">::</span><span class="w" id="mwWA"> </span><span class="kt" id="mwWQ">Floating</span><span class="w" id="mwWg"> </span><span class="n" id="mwWw">a</span><span class="w" id="mwXA"> </span><span class="ow" id="mwXQ">=></span><span class="w" id="mwXg"> </span><span class="p" id="mwXw">([</span><span class="n" id="mwYA">a</span><span class="p" id="mwYQ">]</span><span class="w" id="mwYg"> </span><span class="ow" id="mwYw">-></span><span class="w" id="mwZA"> </span><span class="p" id="mwZQ">[</span><span class="n" id="mwZg">a</span><span class="p" id="mwZw">])</span><span class="w" id="mwaA"> </span><span class="ow" id="mwaQ">-></span><span class="w" id="mwag"> </span><span class="n" id="mwaw">a</span><span class="w" id="mwbA"> </span><span class="ow" id="mwbQ">-></span><span class="w" id="mwbg"> </span><span class="p" id="mwbw">[</span><span class="n" id="mwcA">a</span><span class="p" id="mwcQ">]</span><span class="w" id="mwcg"> </span><span class="ow" id="mwcw">-></span><span class="w" id="mwdA"> </span><span class="p" id="mwdQ">[</span><span class="n" id="mwdg">a</span><span class="p" id="mwdw">]</span> <span class="nf" id="mweA">lehmerSmooth</span><span class="w" id="mweQ"> </span><span class="n" id="mweg">smooth</span><span class="w" id="mwew"> </span><span class="n" id="mwfA">p</span><span class="w" id="mwfQ"> </span><span class="n" id="mwfg">xs</span><span class="w" id="mwfw"> </span><span class="ow" id="mwgA">=</span> <span class="w" id="mwgQ"> </span><span class="n" id="mwgg">zipWith</span><span class="w" id="mwgw"> </span><span class="p" id="mwhA">(</span><span class="o" id="mwhQ">/</span><span class="p" id="mwhg">)</span> <span class="w" id="mwhw"> </span><span class="p" id="mwiA">(</span><span class="n" id="mwiQ">smooth</span><span class="w" id="mwig"> </span><span class="p" id="mwiw">(</span><span class="n" id="mwjA">map</span><span class="w" id="mwjQ"> </span><span class="p" id="mwjg">(</span><span class="o" id="mwjw">**</span><span class="n" id="mwkA">p</span><span class="p" id="mwkQ">)</span><span class="w" id="mwkg"> </span><span class="n" id="mwkw">xs</span><span class="p" id="mwlA">))</span> <span class="w" id="mwlQ"> </span><span class="p" id="mwlg">(</span><span class="n" id="mwlw">smooth</span><span class="w" id="mwmA"> </span><span class="p" id="mwmQ">(</span><span class="n" id="mwmg">map</span><span class="w" id="mwmw"> </span><span class="p" id="mwnA">(</span><span class="o" id="mwnQ">**</span><span class="p" id="mwng">(</span><span class="n" id="mwnw">p</span><span class="o" id="mwoA">-</span><span class="mi" id="mwoQ">1</span><span class="p" id="mwog">))</span><span class="w" id="mwow"> </span><span class="n" id="mwpA">xs</span><span class="p" id="mwpQ">))</span>

For big $p$ it can serve an envelope detector on a rectified signal.
For small $p$ it can serve an baseline detector on a mass spectrum.

Gonzalez and Woods call this a "contraharmonic mean filter" described for varying values of p (however, as above, the contraharmonic mean can refer to the specific case $p = 2$ ). Their convention is to substitute p with the order of the filter Q:

f (x) = \frac{\sum_{k = 1}^{n} x_{k}^{Q + 1}}{\sum_{k = 1}^{n} x_{k}^{Q}} .

Q=0 is the arithmetic mean. Positive Q can reduce pepper noise and negative Q can reduce salt noise.

Properties

Special cases

Applications

Signal processing

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