# A Cautionary Note on the Reproduction of Dependencies through Linear Stochastic Models with Non-Gaussian White Noise

^{1}

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## Abstract

**:**

## 1. Introduction: A Glimpse of History

“Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.”—George Box and Norman Draper [1]

## 2. The Envelope Behavior of Linear Stochastic Models with Non-Gaussian White Noise

#### 2.1. The Thomas-Fiering Approach

#### 2.2. The Envelope Behavior in the Classical Univariate AR(1) Model

#### 2.3. From the Univariate to the Multivariate AR(1) Model

**A**

_{1}ensures that each individual process is a Markov process and significantly simplifies the parameter estimation procedure, since the lag-1 cross-correlations are not explicitly modeled. Its use is often advocated by the literature, since several authors suggest that lag-1 cross-correlations can be neglected [14,22,26,33,40,41]. Yet it is noted that while this simplification could be valid for processes considered at a coarse time scale (e.g., monthly or annual), it should be used with caution in cases of fine time scale processes (e.g., hourly) or for modeling phenomena characterized by cause-effect relationships (e.g., rainfall-runoff). Nevertheless, here we focus on the so-called multivariate Markov model (i.e., CMAR(1)). Regarding its parameter estimation and assuming that ${\mathit{A}}_{1}$ is a diagonal matrix of the form:

**B**is an $m\times m$, typically lower triangular, matrix (also known as the square root of ${\mathit{\Sigma}}_{\underset{\_}{\mathit{\epsilon}}}$) obtained by standard decomposition techniques (e.g., the Cholesky technique) or optimization approaches [23,42]. The latter methods are usually employed when

**B**is non-positive definite. Typically, such problems arise when the sample estimates of the required statistics are extracted from historical time series of different and/or limited lengths [11]. Nonetheless, given that ${\mathit{A}}_{1}$ is diagonal and assuming that ${\underset{\_}{\mathit{\epsilon}}}_{t}=\mathit{B}{\underset{\_}{\mathit{\xi}}}_{t}$, where ${\underset{\_}{\mathit{\xi}}}_{t}$ is an m-dimensional column-vector of i.i.d. RVs, the decomposition of Equation (11) can be given as follows:

**,**respectively; we remark that $\underset{\_}{\mathit{\xi}}$ coincides with the skewness coefficient, since the model assumes unit variance for $\underset{\_}{\mathit{\xi}}$. Similar to the univariate case, the white noise term is typically generated using the $\mathcal{P}$III distribution (Equation (5)). To illustrate the envelope behavior of the latter model, we rewrite the Equation (11) similarly to Equation (7), i.e.:

#### 2.4. The Envelope Behavior beyond AR Models

## 3. Real-World Case Study

^{3}/s) of river Achelous at Kremasta dam in Western Greece. It is assumed that the autocorrelation structure of the daily streamflow of each month can be described by a stationary AR(1) model. The historical monthly and daily time series are clearly characterized by non-Gaussian distributions and skewness coefficients ranging from 1.6 (June) up to 6.7 (October). Specifically, we generate daily synthetic time series with a length of 1000 years, using for each month a different AR(1) model with $\mathcal{P}$III white noise (i.e., AR(1)-$\mathcal{P}$III). The model very satisfactorily reproduced the target historical marginal statistics of each month (Table A4), as well as the theoretical Markovian autocorrelation structure (see Figure A2 for a comparison among the empirical, synthetic, and theoretical ACFs), which however deviates from the empirical ACF for some months, showing a more persistent behavior. Yet a comparison of the lag-1 dependence patterns between the synthetic and the historical data, using scatter plots for each month (Figure 7), reveals the omnipresence of the envelope behavior. Evidently the model generates unrealistic dependence patterns that are far from the historical ones. The synthetic pairs of values are bounded by the theoretical envelope function (red line; embedded in each plot), while the historical pairs clearly extend beyond that line. In an effort to provide a quantitative metric, we calculate the empirical probability of a historical pair to lie below the envelope function. The overall mean value of this metric estimated from all months is 27%, while it ranges from 14% (in November) to 42% (in April).

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

**Table A1.**Scenario-based summary of theoretical (see Table 1 of the main manuscript; Section 2.2—“The envelope behavior in the classical univariate AR(1) model”) and simulated (synthetically generated; using an AR(1) with $\mathcal{P}$III white noise) statistics.

Scenario | Type | Mean (μ) | Variance (σ^{2}) | Skewness (C_{s}) | Autocorrelation (ρ_{1}) |
---|---|---|---|---|---|

Scenario A | Theoretical | 0.50 | 1.00 | 1.00 | 0.20 |

Simulated | 0.46 | 0.93 | 1.05 | 0.20 | |

Scenario B | Theoretical | 0.50 | 1.00 | 2.00 | 0.20 |

Simulated | 0.54 | 1.06 | 2.07 | 0.18 | |

Scenario C | Theoretical | 0.50 | 1.00 | 4.00 | 0.20 |

Simulated | 0.50 | 0.91 | 3.48 | 0.21 | |

Scenario D | Theoretical | 0.50 | 1.00 | 1.00 | 0.40 |

Simulated | 0.46 | 0.97 | 0.91 | 0.34 | |

Scenario E | Theoretical | 0.50 | 1.00 | 2.00 | 0.40 |

Simulated | 0.49 | 1.11 | 2.09 | 0.45 | |

Scenario F | Theoretical | 0.50 | 1.00 | 4.00 | 0.40 |

Simulated | 0.46 | 1.01 | 4.89 | 0.45 | |

Scenario G | Theoretical | 0.50 | 1.00 | 1.00 | 0.60 |

Simulated | 0.42 | 0.97 | 0.88 | 0.64 | |

Scenario H | Theoretical | 0.50 | 1.00 | 2.00 | 0.60 |

Simulated | 0.48 | 1.04 | 2.20 | 0.62 | |

Scenario I | Theoretical | 0.50 | 1.00 | 4.00 | 0.60 |

Simulated | 0.48 | 0.93 | 4.22 | 0.57 | |

Scenario J | Theoretical | 0.50 | 1.00 | 1.00 | 0.80 |

Simulated | 0.50 | 1.09 | 0.75 | 0.82 | |

Scenario K | Theoretical | 0.50 | 1.00 | 2.00 | 0.80 |

Simulated | 0.45 | 0.97 | 2.11 | 0.81 | |

Scenario L | Theoretical | 0.50 | 1.00 | 4.00 | 0.80 |

Simulated | 0.55 | 1.08 | 4.24 | 0.81 |

**Figure A1.**Scenario-based (see Table 1 of the main manuscript; Section 2.2—“The envelope behavior in the classical univariate AR(1) model”) comparison of synthetic (using the an AR(1) with $\mathcal{P}$III white noise) and theoretical autocorrelation function (ACF). The labels of each plot resemble the corresponding scenarios of the aforementioned table (see also Table A1).

**Table A2.**Summary of theoretical and simulated statistics for the first, zero-autocorrelated, bivariate AR(1) process with $\mathcal{P}$III white noise, employed in Section 2.3—“From the univariate to the multivariate AR(1) model” of the main manuscript.

Process | Type | Mean (μ) | Variance (σ^{2}) | Skewness (C_{s}) | Autocorrelation (ρ_{1}) |
---|---|---|---|---|---|

${\underset{\_}{x}}_{t}^{1}$ | Theoretical | 0.50 | 1.00 | 2.00 | 0.00 |

Simulated | 0.50 | 1.06 | 2.39 | 0.00 | |

${\underset{\_}{x}}_{t}^{2}$ | Theoretical | 0.50 | 1.00 | 2.50 | 0.00 |

Simulated | 0.51 | 1.14 | 2.95 | 0.00 | |

Theoretical cross-correlation (ρ_{0}) = 0.80|Simulated cross-correlation (ρ_{0}) = 0.79 |

**Table A3.**Summary of theoretical and simulated statistics for the second, autocorrelated, bivariate AR(1) process with $\mathcal{P}$III white noise, employed in Section 2.3—“From the univariate to the multivariate AR(1) model” of the main manuscript.

Process | Type | Mean (μ) | Variance (σ^{2}) | Skewness (C_{s}) | Autocorrelation (ρ_{1}) |
---|---|---|---|---|---|

${\underset{\_}{x}}_{t}^{1}$ | Theoretical | 0.50 | 1.00 | 2.00 | 0.70 |

Simulated | 0.52 | 1.08 | 2.00 | 0.70 | |

${\underset{\_}{x}}_{t}^{2}$ | Theoretical | 0.50 | 1.00 | 2.50 | 0.50 |

Simulated | 0.52 | 1.11 | 2.51 | 0.51 | |

Theoretical cross-correlation (ρ_{0}) = 0.80|Simulated cross-correlation (ρ_{0}) = 0.80 |

**Table A4.**Monthly-based summary of historical and simulated (synthetically generated using an AR(1) with $\mathcal{P}$III white noise) statistics of the real-world case study employed in Section 3—“Real world case study” of the main manuscript.

Month | Type | Mean (μ) | Variance (σ^{2}) | Skewness (C_{s}) | Autocorrelation (ρ_{1}) |
---|---|---|---|---|---|

January | Historical | 167.89 | 33,973.86 | 3.89 | 0.69 |

Simulated | 166.12 | 35,044.58 | 3.92 | 0.70 | |

February | Historical | 179.50 | 32,317.25 | 3.95 | 0.66 |

Simulated | 177.10 | 32,538.62 | 4.28 | 0.66 | |

March | Historical | 172.07 | 13,773.37 | 2.69 | 0.75 |

Simulated | 173.37 | 13,608.23 | 2.68 | 0.75 | |

April | Historical | 172.47 | 10,253.59 | 4.04 | 0.74 |

Simulated | 171.62 | 10,502.08 | 4.28 | 0.74 | |

May | Historical | 107.83 | 4055.14 | 2.29 | 0.77 |

Simulated | 110.20 | 4368.32 | 2.31 | 0.77 | |

June | Historical | 50.86 | 591.95 | 1.59 | 0.64 |

Simulated | 51.26 | 604.55 | 1.58 | 0.63 | |

July | Historical | 31.13 | 177.42 | 2.19 | 0.45 |

Simulated | 31.06 | 176.04 | 2.17 | 0.45 | |

August | Historical | 24.00 | 96.04 | 2.41 | 0.47 |

Simulated | 23.96 | 94.83 | 2.35 | 0.47 | |

September | Historical | 24.86 | 492.39 | 5.99 | 0.63 |

Simulated | 24.42 | 432.84 | 5.57 | 0.63 | |

October | Historical | 51.77 | 8883.06 | 6.70 | 0.60 |

Simulated | 50.71 | 7905.46 | 6.26 | 0.60 | |

November | Historical | 114.63 | 24,332.88 | 3.49 | 0.61 |

Simulated | 111.69 | 23,039.17 | 3.63 | 0.61 | |

December | Historical | 197.14 | 68,785.55 | 4.87 | 0.62 |

Simulated | 193.85 | 63,948.33 | 4.53 | 0.61 |

**Figure A2.**Monthly-based comparison of empirical (historical), synthetic (using AR(1) with $\mathcal{P}$III white noise), and theoretical autocorrelation functions (ACFs) of the real-world case study employed in Section 3—“Real-world case study” of the main manuscript.

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**Figure 1.**Comparison of the (

**A**) January–February, (

**B**) March–April, and (

**C**) September–October dependence patterns between historical and synthetic monthly runoff data (10

^{9}m

^{3}) of the Nile, at Aswan dam. Synthetic time series were generated by the cyclostationary Thomas-Fiering (TF) approach (adapted by Tsoukalas et al. [33]; the simulated negative values were not truncated to zero in order to avoid distortion of the dependence pattern). The red line (—) depicts the envelope equation of the TF model (when combined with $\mathcal{P}$III white noise. See also Appendix A).

**Figure 2.**Relationship between (

**A**) the target skewness coefficient of process ${\underset{\_}{x}}_{t}$ and the required skewness for white noise term ${\underset{\_}{\epsilon}}_{t}$ for a given lag-1 autocorrelation coefficient ${\rho}_{1}$; and (

**B**) the lag-1 autocorrelation coefficient ${\rho}_{1}$ and the required skewness coefficient of white noise term ${\underset{\_}{\epsilon}}_{t}$ to attain the target skewness coefficient of process ${\underset{\_}{x}}_{t}$.

**Figure 3.**Scatter plots depicting the simulated (using the TF model, i.e., the autoregressive model of order 1 (AR(1))-$\mathcal{P}$III) lag-1 dependence pattern among consecutive time steps (i.e., pair values (•) of the previous and current time steps). The labels of each plot resemble the corresponding scenarios of Table 1. The red line (—) depicts the envelope equation shown in the title of each plot.

**Figure 4.**Scatter plots depicting the simulated (using the contemporaneous multivariate autoregressive model of order 1 (CMAR(1) model) with $\mathcal{P}$III white noise) for (

**A**) and (

**B**) lag-1 dependence patterns of the zero-autocorrelated processes ${\underset{\_}{x}}_{t}^{1}$ and ${\underset{\_}{x}}_{t}^{2}$, respectively, for consecutive time steps (i.e., pair values (•) of the previous and current time steps). Panel (

**C**) depicts the contemporaneous dependence (lag-0) of ${\underset{\_}{x}}_{t}^{1}$ and ${\underset{\_}{x}}_{t}^{2}$. The red line (—) depicts the envelope equation shown in the title of each plot. Panel (

**D**) compares the simulated and theoretical autocorrelation function (ACF) of ${\underset{\_}{x}}_{t}^{1}$ while panel (

**E**) compares that of ${\underset{\_}{x}}_{t}^{2}$. Finally, panel (

**F**) compares the simulated and theoretical cross-correlation function (CCF) of ${\underset{\_}{x}}_{t}^{1}$ and ${\underset{\_}{x}}_{t}^{2}$.

**Figure 5.**Scatter plots depicting the simulated (using the CMAR(1) model with $\mathcal{P}$III white noise) for (

**A**) and (

**B**) lag-1 dependence pattern of the autocorrelated processes ${\underset{\_}{x}}_{t}^{1}$ and ${\underset{\_}{x}}_{t}^{2}$, respectively, for consecutive time steps (i.e., pair values (•) of the previous and current time steps), while panel (

**C**) depicts the contemporaneous dependence (lag-0) of ${\underset{\_}{x}}_{t}^{1}$ and ${\underset{\_}{x}}_{t}^{2}$. The red line (—) depicts the envelope equation shown in the title of each plot. Panel (

**D**) compares the simulated and theoretical ACF of ${\underset{\_}{x}}_{t}^{1}$ while panel (

**E**) compares that of ${\underset{\_}{x}}_{t}^{2}$. Lastly, panel (

**F**) compares the simulated and theoretical CCF of ${\underset{\_}{x}}_{t}^{1}$ and ${\underset{\_}{x}}_{t}^{2}$.

**Figure 6.**Scatter plots depicting the simulated lag-1 dependence pattern among consecutive time steps (i.e., pair values (•) of the previous and current time steps) obtained by: (

**A**) ARMA(1,1)-$\mathcal{P}$III; (

**B**) MA(32)-$\mathcal{P}$III; and (

**C**) SMA(32)-$\mathcal{P}$III models. Comparison of synthetic and theoretical autocorrelation function (ACF) obtained by: (

**D**) ARMA(1,1)-$\mathcal{P}$III; (

**E**) MA(32)-$\mathcal{P}$III; and (

**F**) SMA(32)-$\mathcal{P}$III models.

**Figure 7.**Scatter plots showing the lag-1 dependence pattern of the daily streamflow of the Achelous river at the Kremasta dam, Greece (orange dots; •) and of a synthetic time series generated using an AR(1)-$\mathcal{P}$III model (black dots; •). The red line (—) depicts the envelope equation embedded each plot.

**Table 1.**Summary of target statistics for all scenarios (in all cases, ${\mu}_{\underset{\_}{x}}=0.5$ and ${\sigma}_{\underset{\_}{x}}^{2}=1$).

Scenario | A | B | C | D | E | F | G | H | I | J | K | L | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${C}_{{s}_{\underset{\_}{x}}}$ | 1 | 2 | 4 | 1 | 2 | 4 | 1 | 2 | 4 | 1 | 2 | 4 | |

${\rho}_{1}={a}_{1}$ | 0.2 | 0.4 | 0.6 | 0.8 | |||||||||

${\mu}_{\underset{\_}{\epsilon}}$ | 0.4 | 0.3 | 0.2 | 0.1 | |||||||||

${\sigma}_{\underset{\_}{\epsilon}}^{2}$ | 0.96 | 0.84 | 0.64 | 0.36 | |||||||||

${C}_{{s}_{\underset{\_}{\epsilon}}}$ | 1.05 | 2.11 | 4.22 | 1.22 | 2.43 | 4.86 | 1.53 | 3.06 | 6.13 | 2.26 | 4.52 | 9.04 | |

$\mathcal{P}$III distribution | $\gamma $ | 3.596 | 0.899 | 0.225 | 2.706 | 0.677 | 0.169 | 1.706 | 0.426 | 0.107 | 0.784 | 0.196 | 0.049 |

$b$ | 0.517 | 1.033 | 2.067 | 0.557 | 1.114 | 2.229 | 0.613 | 1.225 | 2.450 | 0.678 | 1.356 | 2.711 | |

$c$ | −1.458 | −0.529 | −0.065 | −1.208 | −0.454 | −0.077 | −0.845 | −0.322 | −0.061 | −0.431 | −0.166 | −0.033 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Tsoukalas, I.; Papalexiou, S.M.; Efstratiadis, A.; Makropoulos, C.
A Cautionary Note on the Reproduction of Dependencies through Linear Stochastic Models with Non-Gaussian White Noise. *Water* **2018**, *10*, 771.
https://doi.org/10.3390/w10060771

**AMA Style**

Tsoukalas I, Papalexiou SM, Efstratiadis A, Makropoulos C.
A Cautionary Note on the Reproduction of Dependencies through Linear Stochastic Models with Non-Gaussian White Noise. *Water*. 2018; 10(6):771.
https://doi.org/10.3390/w10060771

**Chicago/Turabian Style**

Tsoukalas, Ioannis, Simon Michael Papalexiou, Andreas Efstratiadis, and Christos Makropoulos.
2018. "A Cautionary Note on the Reproduction of Dependencies through Linear Stochastic Models with Non-Gaussian White Noise" *Water* 10, no. 6: 771.
https://doi.org/10.3390/w10060771