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

For a continuous description of data

$y\left(t\right)\in {D}_{y}\subseteq \mathbb{R}$, where

$t\in {D}_{t}\subseteq \mathbb{R}$ is a continuous index, the

${\mathcal{L}}^{p}$-expectation value

${\langle y\rangle}_{p}$ is given by

Given the probability distribution of

y-values,

$\mathsf{p}\left(y\right)$, the

${\mathcal{L}}^{p}$-expectation value is given by Equation (

32). Consider now the equidistribution of data in the interval

$[0,1]$. From Equation (

32) we have

The fact that

${\langle y\rangle}_{p}$ is independent of

p, is a general result of symmetric probability distributions. Indeed, consider a distribution

$\mathsf{p}\left(y\right)$,

$y\in [-c,c]$, symmetric at

$y=0$,

i.e.,

$\mathsf{p}(-y)=\mathsf{p}\left(y\right)\forall \phantom{\rule{0.166667em}{0ex}}y\in [-c,c]$. Then,

(the

sign is odd function, while

${\mathsf{p}\left(y\right)\left|y\right|}^{p-1}$ is even). Hence, given the uniqueness of

${\langle y\rangle}_{p}$ for a given

p, we conclude in

${\langle y\rangle}_{p}=0,\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}p\ge 1$.

Therefore, in the case where the distribution

$\mathsf{p}\left(y\right)$ is symmetric, the whole set of

${\langle y\rangle}_{p}$-values degenerate to one single value, which can be found thus, by the usual Euclidean norm, namely

${\langle y\rangle}_{p}={\langle y\rangle}_{2}$. (The opposite statement is also true.) However, when

$\mathsf{p}\left(y\right)$ is asymmetric, a spectrum-like range of different

${\langle y\rangle}_{p}$-values is generated [

7]. For example, the distribution

$\mathsf{p}\left(y\right)\simeq 1+\delta (1-3{y}^{2})$ in the interval

$y\in [0,1]$ is symmetric for

$\delta =0$, but becomes asymmetric for

$0<\delta <<1$. Then, we find

${\langle y\rangle}_{p}\simeq \frac{1}{2}-\frac{3}{4}\frac{1}{p+1}\delta +\mathit{O}\left({\delta}^{2}\right)$.

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

We consider the time series of the Earth’s magnetic field magnitude (in nT). In particular, we focus on a stationary segment recorded by the GOES-12 satellite between the month 1/1/2008 and 1/2/2008, that is a sampling of one measurement per minute, constituting a segment of

$N=46,080$ data points, depicted in

Figure 4a. This segment is characterized by a roughly symmetric distribution

$\mathsf{p}\left(B\right)$ (in nT

${}^{-1}$), depicted in

Figure 4b, resulting to a narrow spectrum of

${\langle B\rangle}_{p}$-values, depicted in

Figure 4c. Notice that the extreme values of

${\langle B\rangle}_{p}$ do not coincide with the respective for

$p=1$ and

$p\to \infty $. Indeed, a minimum of the

${\mathcal{L}}^{p}$-expectation values, that is

${\langle B\rangle}_{p,}{}_{min}\approx 96.614$ nT, can be found for the non-Euclidean norm

$p\approx 5.63$, while a maximum value of about

${\langle B\rangle}_{p,}{}_{max}\approx 98.819$ nT, is located at

$p\approx 25.83$. As

$p\to \infty $,

${\langle B\rangle}_{p}$ tends to

$({B}_{min}+{B}_{max})/2\approx 98.535$ nT. Hence, it is evident that

${\mathcal{L}}^{p}$ means

${\mu}_{p}$ are not indispensably monotonic functions of

p.

**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 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**).

The expectation value

${\langle B\rangle}_{p}$ is given by the estimator

${\widehat{\mu}}_{{p,}_{N}}({\left\{{B}_{i}\right\}}_{i=1}^{N};p)$. On the other hand, the error

$\delta {\langle B\rangle}_{p}$ is given by the square root of the variance

${\widehat{{S}^{\phantom{\rule{0.166667em}{0ex}}2}}}_{{p,}_{N}}$ of the estimator

${\widehat{\mu}}_{{p,}_{N}}({\left\{{B}_{i}\right\}}_{i=1}^{N};p)$, that is

In

Figure 5a,b, the

${\mathcal{L}}^{p}$-expectation value of the Earth’s magnetic field magnitude (shown in

Figure 4a),

${\langle B\rangle}_{p}$, together with its error

$\delta {\langle B\rangle}_{p}$, are respectively depicted as functions of the

p-norm. A local minimum of the error

$\delta {\langle B\rangle}_{p}$ can be detected for

$p\approx 2.05$, for which

${\langle B\rangle}_{p}\approx 97.88$ nT and

$\delta {\langle B\rangle}_{p,}{}_{min}\approx 0.071$ nT, shown in the magnified inset of

Figure 5c.

**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 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**).

For $p\gtrsim 2.05$ the error increases as p increases, until it reaches a local maximum at $p\approx 7.88$, for which ${\langle B\rangle}_{p}\approx 97.07$ nT and $\delta {\langle B\rangle}_{p,}{}_{max}\approx 0.091$ nT. Then, for $p\gtrsim 7.88$ the error decreases monotonically as p increases. We can readily derive that $\delta {\langle B\rangle}_{p}\approx \frac{1}{2}({B}_{max}-{B}_{min})\frac{1}{\sqrt{N}}\frac{1}{\sqrt{p-1}}\to 0$, as $p\to \infty $.

For

$p\lesssim 2.05$ the error increases as

p decreases, but numerous fluctuations appear that become more dense as

$p\to 1$. This “instability cloud” is due to the reading errors of the data values, with their effect being magnified as

$p-1$ tends to zero. This effect can be demonstrated in

Figure 6, where the error

$\delta {\langle B\rangle}_{p}$ is depicted when an additive noise is inserted into the

${\left\{{B}_{i}\right\}}_{i=1}^{N}$ values. In particular, we consider the perturbed values

${\{{\tilde{B}}_{i}\}}_{i=1}^{N}$, where

${\tilde{B}}_{i}\equiv {B}_{i}+\u03f5\phantom{\rule{0.166667em}{0ex}}{r}_{i}$,

$\forall \phantom{\rule{0.166667em}{0ex}}i=1,\dots ,N$, with

${\left\{{r}_{i}\right\}}_{i=1}^{N}$, being equidistributed in

$[0,1]$ and

$\left|\u03f5\right|$ is the amplitude of the perturbations. We set

$\left|\u03f5\right|=0.01$ nT, which is equal to the resolution (reading error) of the values

${\left\{{B}_{i}\right\}}_{i=1}^{N}$. In

Figure 6a we set

$\u03f5=-0.01$ nT, while in

Figure 6c we set

$\u03f5=+0.01$ nT. In

Figure 6b we depict the unperturbed error for convenience. We observe that the “instability cloud”, occurred for

$p\to 1$, is different for the three cases

$\u03f5=-0.01$, 0, and

$+0.01$ (nT). However, the minimum at

$p\approx 2.05$ remains unaffected.

The existence of a local minimum of the error, such as the minimum at

$p\approx 2.05$, is of great importance. It suggests that for this specific norm, the expectation value comes with the minimum error. Therefore, after the total deviations minimization that leads to the normal Equation (

1) from which the optimal parameter

${\alpha}^{*}\left(p\right)$ is derived, the optimization is completed by determining the specific norm

${p}^{*}$ for which the variance

${\widehat{{\sigma}^{2}}}_{p}\left(p\right)$ has a local minimum (if that exists). The importance of the local minimum of

${\sigma}^{2}{}_{p}\left(p\right)$ is that for any deviation of the norm at

$p={p}^{*}$, either

$p<{p}^{*}$, or

$p>{p}^{*}$, the error increases, introducing thus, a type of norm-stability. Hence, the

${\mathcal{L}}^{p}$-norm that corresponds to the minimized error,

$p\approx 2.05$, is distinguished. It is interesting that the norm

$p\approx 2.05$ is very close to the Euclidean one (

$p=2$). However, it has to be stressed out that there is not any universally preferred norm, since this is dependent on the specific data values.

Both the diagrams of the ${\mathcal{L}}^{p}$ mean ${\mu}_{p}\left(p\right)$ and its relevant error $\delta {\mu}_{p}\left(p\right)$, depicted in terms of the norm p, constitute a “metricogram". In a metricogram, we are able to observe the whole spectrum of the ${\mathcal{L}}^{p}$ mean and its error, and to recognize the preferable norms.

**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.

**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.