#
Expectation Values and Variance Based on L^{p}-Norms

## Abstract

**:**

## 1. Introduction

## 2. The Generalized Formal Scheme of Means Characterization

#### 2.1. The Means Characterization Based on Optimization Methods

#### 2.2. Formal Scheme of Means Characterization

## 3. The Concept of ${\mathcal{L}}^{p}$-Expectation Values

#### 3.1. The Non-Euclidean Norm Operator ${\widehat{\mathcal{L}}}_{p}$

- (
**i**) - The non-Euclidean mean of ${\left\{{y}_{k}\right\}}_{k=1}^{W}$ is the Euclidean mean of ${\{{\widehat{\mathcal{L}}}_{p}\left({y}_{k}\right)\}}_{k=1}^{W}$$${\langle y\rangle}_{p}={\langle {\widehat{\mathcal{L}}}_{p}\left(y\right)\rangle}_{2}=\sum _{k=1}^{W}{\mathsf{p}}_{k}{\widehat{\mathcal{L}}}_{p}\left({y}_{k}\right)\phantom{\rule{3.33333pt}{0ex}}$$
- (
**ii**) - Zero-mean of ${\{\frac{\partial}{\partial \beta}{\widehat{\mathcal{L}}}_{p}\left({y}_{k}\right)\}}_{k=1}^{W}$,$$0=\sum _{k=1}^{W}{\mathsf{p}}_{k}\frac{\partial}{\partial \beta}{\widehat{\mathcal{L}}}_{p}\left({y}_{k}\right)\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\mathrm{or}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\frac{\partial}{\partial \beta}{\langle y\rangle}_{p}=\sum _{k=1}^{W}\frac{\partial {\mathsf{p}}_{k}}{\partial \beta}{\widehat{\mathcal{L}}}_{p}\left({y}_{k}\right)\phantom{\rule{3.33333pt}{0ex}}$$
- (
**iii**) - Norm-derivative (Equation (7)):$$\begin{array}{c}\hfill \frac{\partial {\langle y\rangle}_{p}}{\partial p}=\sum _{k=1}^{W}\frac{\partial {\mathsf{p}}_{k}}{\partial p}{\widehat{\mathcal{L}}}_{p}\left({y}_{k}\right)+\sum _{k=1}^{W}{\mathsf{p}}_{k}{\widehat{\mathcal{L}}}_{p}({y}_{k}-{\langle y\rangle}_{p})ln|{y}_{k}-{\langle y\rangle}_{p}|\phantom{\rule{3.33333pt}{0ex}}\end{array}$$$$\begin{array}{c}\hfill \Rightarrow \frac{\partial {\langle y\rangle}_{p}}{\partial p}=\sum _{k=1}^{W}{\mathsf{p}}_{k}{\widehat{\mathcal{L}}}_{p}({y}_{k}-{\langle y\rangle}_{p})ln|{y}_{k}-{\langle y\rangle}_{p}|\phantom{\rule{3.33333pt}{0ex}},\mathrm{if}\phantom{\rule{3.33333pt}{0ex}}\frac{\partial {\mathsf{p}}_{k}}{\partial p}=0\phantom{\rule{3.33333pt}{0ex}}\end{array}$$
- (
**iv**) - In the Euclidean case, ${\widehat{\mathcal{L}}}_{p}$ degenerates to the identity operator ${\widehat{\mathcal{L}}}_{p=2}=\widehat{1}$ .
- (
**v**) - Linear operations: ${\widehat{\mathcal{L}}}_{p}(\lambda {y}_{k}+c)=\lambda {\widehat{\mathcal{L}}}_{p}\left({y}_{k}\right)+c,\forall \phantom{\rule{0.166667em}{0ex}}\lambda ,c\in \Re $.Hence, ${\langle y-{\langle y\rangle}_{p}\rangle}_{p}={\langle {\widehat{\mathcal{L}}}_{p}(y-{\langle y\rangle}_{p})\rangle}_{2}=0$, which reads Equation (8).
- (
**vi**) - Non-additivity of ${\widehat{\mathcal{L}}}_{p}$: ${\widehat{\mathcal{L}}}_{p}({y}_{k}+{z}_{k})\ne {\widehat{\mathcal{L}}}_{p}\left({y}_{k}\right)+{\widehat{\mathcal{L}}}_{p}\left({z}_{k}\right)$.
- (
**vii**) - Inverse of non-Euclidean norm operator, ${\widehat{\mathcal{L}}}_{p}^{-1}$, ${\widehat{\mathcal{L}}}_{p}^{-1}{\widehat{\mathcal{L}}}_{p}={\widehat{\mathcal{L}}}_{p}{\widehat{\mathcal{L}}}_{p}^{-1}=\widehat{1}$,$${\widehat{\mathcal{L}}}_{p}^{-1}\left({y}_{k}\right)=\frac{|{y}_{k}-{\langle y\rangle}_{p}{|}^{\frac{1}{p-1}}sign({y}_{k}-{\langle y\rangle}_{p})}{{\left[(p-1){\varphi}_{p}\right]}^{\frac{1}{1-p}}}+{\langle y\rangle}_{p}\phantom{\rule{3.33333pt}{0ex}}$$

#### 3.2. The Non-Euclidean ${\mathcal{L}}^{p}$-Mean Estimator and Its Expectation Value

#### 3.3. Examples

#### 3.3.1. Gas at Thermal Equilibrium

#### 3.3.2. Plasma Out of Thermal Equilibrium

**Figure 1.**The kappa distribution of energy, $P(\epsilon ;T;{\kappa}_{0};f)g\left(\epsilon \right)\times \left({k}_{B}T\right)$, depicted in a log-log scale for ${\kappa}_{0}=0.01$ (red), $0.1$ (blue), 1 (green), 10 (magenta), and for $f=3$.

#### 3.3.3. D-Dimensional Quantum Harmonic Oscillator in Thermal Equilibrium

**Figure 2.**The dependence of the internal energy on the degrees of freedom f and the kappa index ${\kappa}_{0}$. (

**a**) Plot of ${\langle \epsilon \rangle}_{p}/(1/2\phantom{\rule{3.33333pt}{0ex}}{k}_{B}T)$ vs. the degrees of freedom f, for $p=1.5$ (red), 2 (blue), $2.5$ (green), and ${\kappa}_{0}=1.5$ (solid), and 100 (dash). (

**b**) Plot of ${\langle \epsilon \rangle}_{p}/(f/2\phantom{\rule{3.33333pt}{0ex}}{k}_{B}T)$, vs. f, for the same p and ${\kappa}_{0}$. (

**c**) The same plot as (b), but vs. ${\kappa}_{0}$ and for $f=3$. We observe larger deviation from the Euclidean norm for smaller kappa indices. Note that the graph is restricted to ${\kappa}_{0}>p-2$ (see text).

**Figure 3.**The internal energy and heat capacity of the D-dimensional quantum harmonic oscillator for non-Euclidean norms ($\forall \phantom{\rule{0.166667em}{0ex}}p$) at thermal equilibrium (${\kappa}_{0}\to \infty $). (

**a**), (

**b**) The internal energy ${\langle \epsilon \rangle}_{p}/\left(\hslash \omega \right)$ and heat capacity ${c}_{V{,}_{p}}$, are respectively plotted vs. the temperature ${k}_{B}T/\left(\hslash \omega \right)$, for $p=1.8$ (red), 2 (blue), $2.5$ (green), and $D=2$. (

**c**), (

**d**) Similar to (a) and (b), but plotted as functions of the dimensionality D, for ${k}_{B}T/\left(\hslash \omega \right)=1$.

## 4. The Non-Euclidean ${\mathcal{L}}^{p}$-Variance of the ${\mathcal{L}}^{p}$-Expectation Value

#### 4.1. Preliminaries: Formulations

#### 4.2. Examples

#### 4.2.1. Example 1: Gaussian distribution

#### 4.2.2. Example 2: Generalized Gaussian distribution

#### 4.3. Justification of the ${\mathcal{L}}^{p}$-Variance Expression

## 5. Further Analytical and Numerical Examples

#### 5.1. Analytical Example: The Spectrum of the ${\mathcal{L}}^{p}$ Means and Its Degeneration

#### 5.2. Numerical Example: Earth’s Magnetic Field

**Figure 4.**The magnitude of the Earth’s total magnetic field. (

**a**) The time series recorded between 1/1/2008 and 1/2/2008. (

**b**) The relevant distribution $\mathsf{p}\left(B\right)$ is roughly symmetric. As a result, the numerically calculated ${\mathcal{L}}^{p}$-expectation values, ${\langle B\rangle}_{p}$, configure a narrow spectrum within the interval between the two horizontal dotted lines, where the dependence of ${\langle B\rangle}_{p}$-values on the p-norm is shown within the magnified inset (

**c**).

**Figure 5.**The ${\mathcal{L}}^{p}$-expectation value of the magnitude of the Earth’s total magnetic field (shown in Figure 4a), ${\langle B\rangle}_{p}$, together with its error $\delta {\langle B\rangle}_{p}$, are depicted as functions of the p-norm (panels (

**a**) and (

**b**), respectively). A local minimum of the error is found close to the Euclidean norm, i.e., for $p\approx 2.05$, as it is shown within the magnified inset (

**c**).

**Figure 6.**The error $\delta {\langle B\rangle}_{p}$ is depicted with additive equidistributed noise inserted into the ${\left\{{B}_{i}\right\}}_{i=1}^{N}$ values ($N=46080$). The amplitude of the noise is equal to the resolution of the values ${\left\{{B}_{i}\right\}}_{i=1}^{N}$. Namely, we set $\u03f5=-0.01$ nT (

**a**), and $\u03f5=+0.01$ nT (

**c**). In panel (

**b**) we depict the unperturbed error for convenience. The magnified panels (

**d**), (

**e**) and (

**f**) of the respective panels (a), (b) and (c) demonstrate the minimum error at $p\approx 2.05$ that remains unaffected, for amplitudes of additive noise less or equal to the reading error, i.e., $|\u03f5\le 0.01|$ nT, in contrast to the fluctuations, appearing for $p-1\to 0$, which are affected by the additive noise.

## 6. Conclusions

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Livadiotis, G.
Expectation Values and Variance Based on *L ^{p}*-Norms.

*Entropy*

**2012**,

*14*, 2375-2396. https://doi.org/10.3390/e14122375

**AMA Style**

Livadiotis G.
Expectation Values and Variance Based on *L ^{p}*-Norms.

*Entropy*. 2012; 14(12):2375-2396. https://doi.org/10.3390/e14122375

**Chicago/Turabian Style**

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2012. "Expectation Values and Variance Based on *L ^{p}*-Norms"

*Entropy*14, no. 12: 2375-2396. https://doi.org/10.3390/e14122375