# Towards Generic Simulation for Demanding Stochastic Processes

^{*}

## Abstract

**:**

## 1. Introduction

- Complex dependence structures that extend way beyond the Markov dependence, and incorporate long-range dependence and short-scale fractal (smoothness/roughness) behavior. This is achieved by using a symmetric moving average scheme, which can involve a large number of white noise terms, with their weights determined in an explicit analytical manner.
- Marginal distributions that extend beyond Gaussian and incorporate heavy tails, boundedness, and intermittence. This is achieved by using an appropriate number of cumulants, analytically determined from the distribution function, thus resulting in genuine simulation of the process (without a transformation).
- Time asymmetry (irreversibility), achieved by using a non-Gaussian distribution function, combined with an asymmetric moving average scheme, with the weights again determined in an explicit analytical manner.

## 2. Methods

#### 2.1. Preliminaries

#### 2.2. Moments and Cumulants

- The meaning of the parameters is the following.
- (a) Dimensional parameters, with dimensions identical to those of the stochastic variable $\underset{\_}{x}$: μ: mean; $\sigma >0$: standard deviation; $\lambda >0$: scale parameter; a, b: lower and upper bound of $\underset{\_}{x}$.
- (b) Dimensionless parameters: $\xi >0$: upper-tail index; $\zeta >0$: lower-tail index; $\varsigma >0$: additional shape parameter, ${P}_{i}\in \left[0,1\right]$: probability.

- The meaning of constants and standard functions is this: γ: Euler constant; ${\mathrm{B}}_{p}$: Bernoulli number of order p; $\mathsf{\delta}\left(x\right)$: Dirac delta function of $x$; $\mathsf{\Gamma}\left(a\right)$: gamma function of $a$; $\mathsf{\psi}\left(a\right)$: digamma function of $a$; $\mathrm{B}\left(a,b\right)$: beta function of $a,b$.
- Distributions named “half” have their “full” version whose density $f\left(x\right)$ and tail function $\overline{F}\left(x\right)$ are obtained by dividing those given in the tables by 2. The “half” version given in the tables corresponds to $\underset{\_}{x}\ge 0$, while in the “full” version $\underset{\_}{x}\in \mathbb{R}$. The moments ${\mu}_{p}^{\prime}$ of the “full” version is: (a) for even p, 0; (b) for odd p, equal to those of half version.
- All other distributions, defined for $\underset{\_}{x}\ge 0$ but not named “half”, can also be extended to the whole real line by replacing $x$ with $\left|x\right|$ and dividing $f\left(x\right)$ by 2. Again, the moments ${\mu}_{p}^{\prime}$ of this extended version is: (a) for even p, 0; (b) for odd p, equal to those of original version.

#### 2.3. Second Order Properties

- The generalized Cauchy-type FHK (FHK-C) with climacogram:$$\gamma \left(k\right)={\lambda}_{0}{\left(1+{\left(k/\alpha \right)}^{2M}\right)}^{\frac{H-1}{M}}$$
- The mixed Cauchy–Dagum-type FHK (FHK-CD) climacogram:$$\gamma \left(k\right)={\lambda}_{1}{\left(1+\frac{k}{\alpha}\right)}^{2H-2}+{\lambda}_{2}\left(1-{\left(1+\frac{\alpha}{k}\right)}^{-2{\rm M}}\right)$$

#### 2.4. Stochastic Simulation

- Consists of real numbers, despite the expression in (31) involving complex numbers;
- Satisfies precisely Equation (30); and
- Is easy and fast to calculate using the fast Fourier transform (FFT).

- From the continuous-time stochastic model, expressed through its climacogram $\gamma \left(k\right)$, we calculate its autocovariance function in discrete time (assuming time step $D$) by Equation (26). (This step is obviously omitted if the model is already expressed in discrete time through its autocovariance function).
- We choose an appropriate number of coefficients J that is a power of 2 and perform inverse FFT (using common software) to calculate the discrete-time power spectrum and the frequency function ${A}^{\mathrm{R}}\left(\omega \right)$ for an array of ${\omega}_{j}=j{w}_{1},j=0,1,\dots ,J,{w}_{1}:=1/JD$:$${s}_{\mathrm{d}}\left({\omega}_{j}\right)=2{c}_{0}+4{\displaystyle \sum}_{\eta =1}^{J}{c}_{\eta}\mathrm{cos}\left(2\mathsf{\pi}\eta {\omega}_{j}\right),\text{}{A}^{\mathrm{R}}\left({\omega}_{j}\right)=\sqrt{2{s}_{\mathrm{d}}\left({\omega}_{j}\right)}$$
- We choose $\vartheta \left(\omega \right)$ (see below) and we form the arrays (vectors) ${A}^{\mathrm{R}}$ and ${A}^{\mathrm{I}}$, both of size 2J indexed as $0,\dots ,2J\u20131$, with the superscripts R and I standing for the real and imaginary part of a vector of complex numbers, respectively:$${\left[{A}^{\mathrm{R}}\right]}_{j}=\{\begin{array}{cc}\hfill {A}^{\mathrm{R}}\left({\omega}_{j}\right)\mathrm{cos}\left(2\mathsf{\pi}\vartheta \left({\omega}_{j}\right)\right)/2,\hfill & \hfill j=0,\dots ,J\hfill \\ \hfill {\left[{A}^{\mathrm{R}}\right]}_{2J-j},\hfill & \hfill j=J+1,\dots ,2J-1\hfill \end{array}$$$${\left[{A}^{\mathrm{I}}\right]}_{j}=\{\begin{array}{cc}\hfill -{A}^{\mathrm{R}}\left({\omega}_{j}\right)\mathrm{sin}\left(2\mathsf{\pi}\vartheta \left({\omega}_{j}\right)\right)/2,\hfill & \hfill j=0,\dots ,J-1\hfill \\ \hfill 0\hfill & \hfill j=J\hfill \\ \hfill -{\left[{A}^{\mathrm{I}}\right]}_{2J-j}\hfill & \hfill j=J+1,\dots ,2J-1\hfill \end{array}$$
- We perform FFT on the vector ${A}^{\mathrm{R}}+\mathrm{i}{A}^{\mathrm{I}}$ (using common software), and get the real part of the result, which is precisely the sequence of ${a}_{\eta}$.

#### 2.5. Distribution Function Approximation

## 3. Applications and Results

#### 3.1. Simulating a Persistent Process with Exponential Distribution

#### 3.2. Simulating a Persistent Process with Uniform Distribution

#### 3.3. Simulating an Antipersistent Process with Uniform Distribution

#### 3.4. Simulating the Precipitation Process at the Hourly Time Scale

- Pareto distribution with discontinuity at the origin for small time scales (Table 5, Equation (46), left). The tail index ξ is constant for all time scales k, while the probability wet, ${P}_{1}^{\left(k\right)}$, and the state scale parameter, $\lambda \left(k\right)$, are functions of the time scale k.
- Continuous PBF distribution, possibly with discontinuity at zero, for large time scales (Table 5, Equation (46), right). In this case, a new parameter $\zeta \left(k\right)$ is introduced, which is again a function of time scale. The Pareto distribution is a special case of the PBF for $\zeta \left(k\right)=1$. In contrast to the Pareto distribution, whose density is a consistently decreasing function of $x$, the PBF tends to be bell-shaped for increasing $\zeta \left(k\right)$, a property consistent with empirical observation and reason.
- Constant mean $\mu $ of the time-averaged process.
- Climacogram of type FHK-CD (Equation (24)), where to reduce the number of parameters it is assumed that $M=1-H$, thus getting Equation (48) in Table 5. By inspection of Equation (48), it is seen that, as $k\to \infty $, $\gamma \left(k\right)\to 0$, which makes the process ergodic; for $k=0$, $\gamma \left(0\right)={\gamma}_{0}={\lambda}_{1}+{\lambda}_{2}$, which is finite, as required for physical consistency.
- Probability wet and dry, ${P}_{1}^{\left(k\right)}=1-{P}_{0}^{\left(k\right)}$, varying with time scale according to Equation (49) in Table 5. It is clarified that two different expressions are used for the small and the large scales, where the transition time scale from the Pareto to the PBF distribution is denoted as ${k}^{*}$. In the Pareto case, ${P}_{1}^{\left(k\right)}$ can be determined directly from the climacogram and the mean (left column of Equation (49) in Table 5). For the PBF case, an additional equation is required, which has been derived based on maximum entropy considerations [50] and involves an additional parameter $\theta $ ($0\le \theta \le 1)$. Continuity of the transition demands that $\zeta \left({k}^{*}\right)=1$.

- Mean intensity, μ = 0.0823 mm/h;
- Intensity scale parameters, ${\lambda}_{1}=0.00110{\text{}\mathrm{mm}}^{2}/{\mathrm{h}}^{2},{\lambda}_{2}=1.43{\text{}\mathrm{mm}}^{2}/{\mathrm{h}}^{2}$;
- Time scale parameter, $\alpha $ = 8.74 h;
- Hurst parameter, H = 0.92; fractal (smoothness) parameter, $M=1-H=0.08$;
- Exponent of the expression of probability dry/wet, $\theta $ = 0.787;
- Upper tail index, ξ = 0.121.

#### 3.5. Simulating the Precipitation Process at the Annual Time Scale

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Comparison with a Conventional Approach

**Figure A1.**Graphical depiction of the results of the simulation application for a synthetic example of an ARMA(1,1) model as an approximation of the FHK process in the case study of Section 3.1, with an exponential distribution: (

**a**) climacogram; (

**b**) autocorrelogram; (

**c**) marginal distribution. The figure should be viewed in comparison to Figure 1 (panels b–d, respectively).

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**Figure 1.**Graphical depiction of the results of the simulation application for a synthetic example of a persistent FHK process with exponential distribution: (

**a**) cumulants; (

**b**) climacogram; (

**c**) autocorrelogram; (

**d**) marginal distribution.

**Figure 2.**Probability mass function of the discretized white noise used in the simulation application for a synthetic example of a persistent FHK process with uniform distribution.

**Figure 3.**Graphical depiction of the results of the simulation application for a synthetic example of a persistent FHK process with uniform distribution: (

**a**) cumulants; (

**b**) climacogram; (

**c**) autocorrelogram; (

**d**) marginal distribution.

**Figure 4.**Probability mass function of the discretized white noise used in the simulation application for a synthetic example of an antipersistent FHK process with uniform distribution.

**Figure 5.**Graphical depiction of the results of the simulation application for a synthetic example of an antipersistent FHK process with uniform distribution: (

**a**) cumulants; (

**b**) climacogram; (

**c**) autocorrelogram; (

**d**) marginal distribution. Notice in panel (

**a**) that the first cumulant of $\underset{\_}{v}$ is out of the graph area as it is very large ( ${\kappa}_{1}^{\left(v\right)}=15.49$).

**Figure 6.**Graphical depiction of the results of the simulation application for a real-world case study for the precipitation process in Bologna at the hourly time scale, modelled as a persistent FHK process with Pareto distribution with discontinuity at zero: (

**a**) cumulants; (

**b**) climacogram; (

**c**) autocorrelogram; (

**d**) marginal distribution.

**Figure 7.**Plots of generated time series of precipitation in Bologna at hourly time scale: (

**a**) for a period of 2000 h (83 d); (

**b**) focus on the first 200 h (~8 d).

**Figure 8.**Graphical depiction of the results of the simulation application for a real-world case study for the precipitation process in Bologna at the annual time scale, modelled as a persistent FHK process with PBF distribution: (

**a**) cumulants; (

**b**) climacogram; (

**c**) autocorrelogram; (

**d**) marginal distribution.

Operation | Mathematical Relationship | Eqn. no. |
---|---|---|

Shift of origin | ${\kappa}_{p}\left[\underset{\_}{x}+c\right]=\{\begin{array}{cc}\hfill {\kappa}_{1}\left[\underset{\_}{x}\right]+c\hfill & \hfill p=1\hfill \\ \hfill {\kappa}_{p}\left[\underset{\_}{x}\right]\hfill & \hfill p>1\hfill \end{array}$ | (11) |

Multiplication by a constant $\left(a\right)$ | ${\kappa}_{p}\left[a\underset{\_}{x}\right]={a}^{p}{\kappa}_{p}\left[\underset{\_}{x}\right]$ | (12) |

Linear combination of independent variables | ${\kappa}_{p}\left[{a}_{1}{\underset{\_}{x}}_{1}+\dots +{a}_{r}{\underset{\_}{x}}_{r}\right]={a}_{1}^{p}\text{}{\kappa}_{p}\left[{\underset{\_}{x}}_{1}\right]+\dots +{a}_{r}^{p}\text{}{\kappa}_{p}\left[{\underset{\_}{x}}_{r}\right]$ | (13) |

$\mathrm{Conditioning}\text{}\mathrm{on}\text{}\mathrm{an}\text{}\mathrm{event}\text{}{A}_{1}$$\text{}\mathrm{with}\text{}\mathrm{probability}\text{}{P}_{1}:=P\left({A}_{1}\right)$$,\text{}\mathrm{where}\text{}\mathrm{the}\text{}\mathrm{complementary}\text{}\mathrm{event}\text{}{A}_{2}$$\text{}\mathrm{has}\text{}\mathrm{probability}\text{}1-{P}_{1}=P\left({A}_{2}\right)$ | ${\mu}_{p}^{\prime}\left[\underset{\_}{x}\right]={P}_{1}{\mu}_{p}^{\prime}\left[\underset{\_}{x}|{A}_{1}\right]+\left(1-{P}_{1}\right){\mu}_{p}^{\prime}\left[\underset{\_}{x}|{A}_{2}\right]$ | (14) |

$\mathrm{Conditioning}\text{}\mathrm{on}\text{}\mathrm{an}\text{}\mathrm{event}\text{}{A}_{1}$$\text{}\mathrm{with}\text{}\mathrm{probability}\text{}{P}_{1}:=P\left({A}_{1}\right)$$,\text{}\mathrm{where}\text{}\underset{\_}{x}=c$$\text{}\left(\mathrm{constant}\right)\text{}\mathrm{upon}\text{}\mathrm{the}\text{}\mathrm{complementary}\text{}\mathrm{event}\text{}{A}_{2}$ | ${\mu}_{p}^{\prime}\left[\underset{\_}{x}\right]={P}_{1}{\mu}_{p}^{\prime}\left[\underset{\_}{x}|{A}_{1}\right]+\left(1-{P}_{1}\right){c}^{p}$ | (15) |

$\mathrm{Conditioning}\text{}\mathrm{on}\text{}\mathrm{an}\text{}\mathrm{event}\text{}{A}_{1}$$\text{}\mathrm{with}\text{}\mathrm{probability}\text{}{P}_{1}:=P\left({A}_{1}\right)$$,\text{}\mathrm{where}\text{}\underset{\_}{x}=0$$\text{}\mathrm{upon}\text{}\mathrm{the}\text{}\mathrm{complementary}\text{}\mathrm{event}\text{}{A}_{2}$ | ${\mu}_{p}^{\prime}\left[\underset{\_}{x}\right]={P}_{1}{\mu}_{p}^{\prime}\left[\underset{\_}{x}|{A}_{1}\right]$ | (16) |

**Table 2.**Non-central moments and cumulants of common distributions with finite domain (all moments and cumulants exist).

Name, Domain | Probability Density or Distribution Function | Moments, ${\mu}_{p}^{\prime}$ | Cumulants, ${\kappa}_{p}$ |
---|---|---|---|

Impulse, $\underset{\_}{x}=\mu $ | $f\left(x\right)=\text{}\mathsf{\delta}\left(x-\mu \right)$ | ${\mu}^{p}$ | $\{\begin{array}{cc}\hfill \mu \hfill & \hfill p=1\hfill \\ \hfill 0\hfill & \hfill p>1\hfill \end{array}$ |

Finite number of impulses, $\underset{\_}{x}\in \left\{{x}_{1},\dots ,{x}_{n}\right\}$ | $f\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}{P}_{i}\mathsf{\delta}\left(x-{x}_{i}\right)$ | ${\displaystyle \sum}_{i=1}^{n}}{P}_{i}{x}_{i}^{p$ | |

Uniform, $a\le \underset{\_}{x}\le b$ | $f\left(x\right)=\frac{1}{b-a}$ | $\frac{{b}^{p+1}-{a}^{p+1}}{\left(p+1\right)\left(b-a\right)}$ | $\{\begin{array}{cc}\hfill {\mu}_{1}^{\prime}=\frac{a+b}{2}\hfill & \hfill p=1\hfill \\ \hfill \frac{{\left(b-a\right)}^{p}{\mathrm{B}}_{p}}{p}\hfill & \hfill p\text{}\mathrm{odd}\hfill \\ \hfill 0\hfill & \hfill p\text{}\mathrm{even}\hfill \end{array}$ |

$\mathrm{Beta},\text{}0\le \underset{\_}{x}\le b$ | $f\left(x\right)=\frac{{\left(\frac{x}{b}\right)}^{\zeta -1}{\left(1-\frac{x}{b}\right)}^{\varsigma -1}}{\mathsf{{\rm B}}\left(\zeta ,\varsigma \right)}$ | $\frac{\mathsf{\Gamma}\left(\zeta +\varsigma \right)\mathsf{\Gamma}\left(p+\zeta \right)}{\mathsf{\Gamma}\left(\zeta \right)\mathsf{\Gamma}\left(p+\zeta +\varsigma \right)}{b}^{p}$ | |

Kumaraswamy, $0\le \underset{\_}{x}\le b$ | $F\left(x\right)=1-{\left(1-{\left(\frac{x}{b}\right)}^{\zeta}\right)}^{\varsigma}$ | $\varsigma \mathsf{{\rm B}}\left(\varsigma ,1+\frac{p}{\zeta}\right){b}^{p}$ |

**Table 3.**Non-central moments and cumulants of common distributions with zero upper-tail index (all moments and cumulants exist).

Name, Domain | Probability Density or Distribution Function | Moments, ${\mu}_{p}^{\prime}$ | Cumulants, ${\kappa}_{p}$ |
---|---|---|---|

Poisson $\underset{\_}{x}=j,j\in {\mathbb{N}}_{0}$ | $f\left(x\right)={e}^{-\varsigma}{\displaystyle \sum _{j=0}^{\infty}}\frac{{\varsigma}^{j}}{j!}\mathsf{\delta}\left(x-j\right)$ | $\varsigma $ | |

Exponential, $x\ge 0$ | $f\left(x\right)={\mathrm{e}}^{\u2013x/\mu}/\mu $ | $p!{\mu}^{p}$ | $\left(p-1\right)!{\mu}^{p}$ |

$\mathrm{Gamma},\text{}\underset{\_}{x}\ge 0$ | $f\left(x\right)=\frac{{\left(x/\lambda \right)}^{\zeta -1}{\mathrm{e}}^{\u2013x/\lambda}}{\lambda \mathsf{\Gamma}\left(\zeta \right)}$ | $\frac{\mathsf{\Gamma}\left(p+\zeta \right)}{\mathsf{\Gamma}\left(\zeta \right)}{\lambda}^{p}$ | $\zeta \left(p-1\right)!{\lambda}^{p}$ |

Generalized gamma, $\underset{\_}{x}\ge 0$ | $f\left(x\right)=\frac{1}{\lambda \mathsf{\Gamma}\left(\zeta /\varsigma \right)}{\left(\frac{x}{\lambda}\right)}^{\zeta -1}\mathrm{exp}\left(-{\left(\frac{x}{\lambda}\right)}^{\varsigma}\right)$ | $\frac{\mathsf{\Gamma}\left(p/\varsigma +\zeta /\varsigma \right)}{\mathsf{\Gamma}\left(\zeta /\varsigma \right)}{\lambda}^{p}$ | |

Weibull, $\underset{\_}{x}\ge 0$ | $F\left(x\right)=1-\mathrm{exp}\left(-{\left(\frac{x}{\lambda}\right)}^{\zeta}\right)$ | $\mathsf{\Gamma}\left(\frac{p}{\zeta}+1\right){\lambda}^{p}$ | |

Normal, $\underset{\_}{x}\in \mathbb{R}$ | $f\left(x\right)=\frac{\mathrm{exp}\left(-\frac{{\left(x-\mu \right)}^{2}}{2{\sigma}^{2}}\right)}{\sqrt{2\pi}\sigma}$ | $\{\begin{array}{cc}\hfill {\mu}_{1}^{\prime}=\mu ,\hfill & \hfill p=1\hfill \\ \hfill {\sigma}^{2}\hfill & \hfill p=2\hfill \\ \hfill 0\hfill & \hfill p>2\hfill \end{array}$ | |

Half-normal, $\underset{\_}{x}\ge 0$ | $f\left(x\right)=\frac{2}{\lambda \sqrt{2\mathsf{\pi}}}\mathrm{exp}\left(-\frac{{x}^{2}}{2{\lambda}^{2}}\right)$ | $\frac{{2}^{p/2}}{\sqrt{\mathsf{\pi}}}\mathsf{\Gamma}\left(\frac{p+1}{2}\right){\lambda}^{p}$ | |

Extended half-normal (Chi), $\underset{\_}{x}\ge 0$ | $f\left(x\right)=\frac{\sqrt{2}}{\lambda \mathsf{\Gamma}\left(\zeta /2\right)}{\left(\frac{{x}^{2}}{2{\lambda}^{2}}\right)}^{\frac{\zeta}{2}-\frac{1}{2}}\mathrm{exp}\left(-\frac{{x}^{2}}{2{\lambda}^{2}}\right)$ | ${2}^{p/2}\frac{\mathsf{\Gamma}\left(\frac{p+\zeta}{2}\right)}{\mathsf{\Gamma}\left(\frac{\zeta}{2}\right)}{\lambda}^{p}$ | |

$\mathrm{Lognormal}\text{}(\mathrm{ln}\underset{\_}{x}~\mathrm{N}\left(\mathrm{ln}\lambda ,\varsigma \right)$$),\text{}\underset{\_}{x}\ge 0$ | $f\left(x\right)=\frac{\mathrm{exp}\left(-\frac{1}{2{\varsigma}^{2}}{\left(\mathrm{ln}\left(\frac{x}{\lambda}\right)\right)}^{2}\right)}{\sqrt{2\mathsf{\pi}}\varsigma x}$ | ${\mathrm{e}}^{\frac{{p}^{2}{\varsigma}^{2}}{2}}{\lambda}^{p}$ | |

Extreme value type I (EV1), $\underset{\_}{x}\in \mathbb{R}$ | $F\left(x\right)=\mathrm{exp}\left(-{\mathrm{e}}^{-\frac{x}{\lambda}}\right)$ | $\frac{{\left(-1\right)}^{p}{\mathsf{\psi}}^{\left(p-1\right)}\left(1\right)}{p!}{\lambda}^{p}$ |

**Table 4.**Non-central moments of common distributions with upper-tail index ξ (moments and cumulants exist for $p<1/\xi $). Here, the cumulants do not have simple explicit expressions but can be readily calculated from Equation (9).

Name, Domain | Probability Density or Distribution Function | $\mathrm{Moments},\text{}{\mu}_{p}^{\prime}$ |
---|---|---|

Pareto $\underset{\_}{x}\ge 0$ | $F\left(x\right)=1-{\left(1+\xi \frac{x}{\lambda}\right)}^{-\frac{1}{\xi}}$ | $\mathrm{B}\left(\frac{1}{\xi}-p,p+1\right)\frac{{\lambda}^{p}}{{\xi}^{p+1}}$ |

Pareto-Burr-Feller (PBF) $\underset{\_}{x}\ge 0$ | $F\left(x\right)=1-{\left(1+\xi \zeta {\left(\frac{x}{\lambda}\right)}^{\zeta}\right)}^{-\frac{1}{\xi \zeta}}$ | $\mathsf{{\rm B}}\left(\frac{1}{\xi \zeta}-\frac{p}{\zeta},\frac{p}{\zeta}+1\right)\frac{{\lambda}^{p}}{{\left(\xi \zeta \right)}^{\frac{p}{\zeta}+1}}$ |

Dagum $\underset{\_}{x}\ge 0$ | $F\left(x\right)={\left(1+\frac{1}{\xi \zeta}{\left(\frac{x}{\lambda}\right)}^{-\frac{1}{\xi}}\right)}^{-\xi \zeta}$ | ${\left(\xi \zeta \right)}^{1-\xi p}\mathrm{B}\left(1-\xi p,\xi p+\xi \zeta \right){\lambda}^{p}$ |

Extreme value type II (EV2) $\underset{\_}{x}\ge 0$ | $F\left(x\right)=\mathrm{exp}\left(-\xi {\left(\frac{x}{\lambda}\right)}^{-\frac{1}{\xi}}\right)$ | $\mathsf{\Gamma}\left(1-p\xi \right){\left(\frac{\lambda}{\xi}\right)}^{p}$ |

Half Student $\underset{\_}{x}\ge 0$ | $f\left(x\right)=\frac{2{\left(1+{\left(\frac{x}{\lambda}\right)}^{2}\right)}^{-\frac{1}{2}-\frac{1}{2\xi}}}{\lambda \mathrm{B}\left(\frac{1}{2},\frac{1}{2\xi}\right)}$ | $\frac{\mathrm{B}\left(\frac{1}{2}+\frac{p}{2},\frac{1}{2\xi}-\frac{p}{2}\right)}{\mathrm{B}\left(\frac{1}{2},\frac{1}{2\xi}\right)}{\lambda}^{p}$ |

Half extended Student $\underset{\_}{x}\ge 0$ | $f\left(x\right)=\frac{2{\left({\left(\frac{x}{\lambda}\right)}^{2}\right)}^{\frac{\zeta}{2}-\frac{1}{2}}{\left(1+{\left(\frac{x}{\lambda}\right)}^{2}\right)}^{-\frac{\zeta}{2}-\frac{1}{2\xi}}}{\lambda \mathrm{B}\left(\frac{\zeta}{2},\frac{1}{2\xi}\right)}$ | $\frac{\mathrm{B}\left(\frac{1}{2\zeta}+\frac{p}{2},\frac{1}{2\xi}-\frac{p}{2}\right)}{\mathrm{B}\left(\frac{1}{2\zeta},\frac{1}{2\xi}\right)}{\lambda}^{p}$ |

Generalized beta prime (GBP) $\underset{\_}{x}\ge 0$ | $f\left(x\right)=\frac{\varsigma {\left(\frac{x}{\lambda}\right)}^{\zeta -1}{\left(1+{\left(\frac{x}{\lambda}\right)}^{\varsigma}\right)}^{-\frac{\zeta}{\varsigma}-\frac{1}{\xi \varsigma}}}{\lambda \mathrm{B}\left(\frac{\zeta}{\varsigma},\frac{1}{\xi \varsigma}\right)}$ | $\frac{\mathrm{B}\left(\frac{\zeta}{\varsigma}+\frac{p}{\varsigma},\frac{1}{\xi \varsigma}-\frac{p}{\varsigma}\right)}{\mathrm{B}\left(\frac{\zeta}{\varsigma},\frac{1}{\xi \varsigma}\right)}{\lambda}^{p}$ |

**Table 5.**Mathematical relationships of the ombrian model. The ombrian curves per se are given in the last row.

Quantity and Symbol | $\mathrm{Small}\text{}\mathrm{Scales},\text{}k\le {k}^{*}$ (Pareto) | $\mathrm{Large}\text{}\mathrm{Scales},\text{}k\ge {k}^{*}$ (PBF) | Eqn. no. |
---|---|---|---|

$\mathrm{Distribution}\text{}\mathrm{function},\text{}{F}^{\left(k\right)}\left(x\right)$ | $1-{P}_{1}^{\left(k\right)}{\left(1+\xi \frac{x}{\lambda \left(k\right)}\right)}^{-1/\xi}$ | $1-{P}_{1}^{\left(k\right)}{\left(1+{\xi}^{\prime}\zeta \left(k\right){\left(\frac{x}{\lambda \left(k\right)}\right)}^{\zeta \left(k\right)}\right)}^{-\frac{1}{{\xi}^{\prime}\zeta \left(k\right)}}$ | (46) |

$\mathrm{Mean},\text{}\mathrm{E}\left[{\underset{\_}{x}}^{\left(k\right)}\right]$ | $\mu $ | (47) | |

$\mathrm{Climacogram},\text{}\gamma \left(k\right)$ | ${\lambda}_{1}{\left(1+\frac{k}{\alpha}\right)}^{2H-2}+{\lambda}_{2}\left(1-{\left(1+\frac{\alpha}{k}\right)}^{2H-2}\right)$ | (48) | |

$\mathrm{Probability}\text{}\mathrm{wet},\text{}{P}_{1}^{\left(k\right)}$ | $\frac{1-\xi}{1/2-\xi}\frac{{\mu}^{2}}{\gamma \left(k\right)+{\mu}^{2}}$ | $1-{\left(1-{P}_{1}^{\left({k}^{*}\right)}\right)}^{{\left(k/{k}^{*}\right)}^{\theta}},\left(0\le \theta \le 1\right)$ | (49) |

Lower tail index (inverse), $\frac{1}{\zeta \left(k\right)}$ | $1$ | $\sqrt{\left(1-2\xi \right)\left({P}_{1}^{\left(k\right)}\left(\gamma \left(k\right)/{\mu}^{2}+1\right)-1\right)}$ | (50) |

Upper tail index, $\xi $ | $\xi $ | ${\xi}^{\prime}=\frac{\xi}{\zeta \left(k\right)}$ | (51) |

Scale parameter (inverse), $\frac{1}{\lambda \left(k\right)}$ | $\frac{{P}_{1}^{\left(k\right)}}{\mu \left(1-\xi \right)}$ | $\frac{{P}_{1}^{\left(k\right)}}{\mu}\left(1+\frac{1}{\left(1-\xi \right){\left(\zeta \left(k\right)\right)}^{2}}-\frac{1}{{\left(\zeta \left(k\right)\right)}^{\sqrt{2}}}\right)$ | (52) |

$\mathrm{Quantile},\text{}x$ | $\lambda \left(k\right)\frac{{\left(\text{}{P}_{1}^{\left(k\right)}T/k\right)}^{\xi}-1}{\xi}$ | $\lambda \left(k\right){\left(\frac{{\left({P}_{1}^{\left(k\right)}T/k\right)}^{\xi}-1}{\xi}\right)}^{\frac{1}{\zeta \left(k\right)}}$ | (53) |

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Koutsoyiannis, D.; Dimitriadis, P.
Towards Generic Simulation for Demanding Stochastic Processes. *Sci* **2021**, *3*, 34.
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Koutsoyiannis D, Dimitriadis P.
Towards Generic Simulation for Demanding Stochastic Processes. *Sci*. 2021; 3(3):34.
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Koutsoyiannis, Demetris, and Panayiotis Dimitriadis.
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