# Feasibility of Multi-Year Forecast for the Colorado River Water Supply: Time Series Modeling

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data and Methodology

#### 2.2. Univariate Time Series Models

#### 2.2.1. ARMA (1, 1)

#### 2.2.2. Sparse AR (19)

#### 2.3. ARMA Models with Exogenous Variables

#### 2.3.1. SARX (19, 1, 0)

#### 2.3.2. ARMAX (19, 1, 2, 0)

#### 2.4. Cross-Validation

#### 2.5. Benchmark Modeling Using the 10-Year Moving-Average Data

- Ten-Year Moving-Average Univariate Model AR (19)The best univariate model chosen for the 10-year moving-average WS is an AR (19) model; this is consistent with the raw-data model in Section 2.2. The fitted model is$$\begin{array}{c}{\tilde{X}}_{t}=-0.1992{\tilde{X}}_{t-1}-0.0514{\tilde{X}}_{t-2}+0.1731{\tilde{X}}_{t-3}+0.0887{\tilde{X}}_{t-4}-0.0282{\tilde{X}}_{t-5}+\hfill \\ 0.0334{\tilde{X}}_{t-6}-0.1966{\tilde{X}}_{t-7}+0.1174{\tilde{X}}_{t-8}-0.0532{\tilde{X}}_{t-9}-0.5161{\tilde{X}}_{t-10}+\hfill \\ 0.1626{\tilde{X}}_{t-11}-0.1017{\tilde{X}}_{t-12}+0.3447{\tilde{X}}_{t-13}+0.0144{\tilde{X}}_{t-14}+0.115{\tilde{X}}_{t-15}-\hfill \\ 0.1354{\tilde{X}}_{t-16}-0.281{\tilde{X}}_{t-17}-0.0375{\tilde{X}}_{t-18}-0.2447{\tilde{X}}_{t-19}+{Z}_{t},\hfill \end{array}$$
- Ten-Year Moving-Average ARMAX (13, 10, 7, 0)One of the ARMAX models chosen had an ARMA (13, 10) base, and it includes 7 GSL elevation lags. The model equation is$$\begin{array}{c}{\tilde{X}}_{t}=-0.0493{\tilde{S}}_{t}+1.772{\tilde{S}}_{t-1}-2.4889{\tilde{S}}_{t-2}+1.1278{\tilde{S}}_{t-3}-0.6349{\tilde{S}}_{t-4}-0.4446{\tilde{S}}_{t-5}\hfill \\ \phantom{\rule{8em}{0ex}}+1.4953{\tilde{S}}_{t-6}-0.7594{\tilde{S}}_{t-7}+0.184{\tilde{X}}_{t-3}+0.2239{\tilde{X}}_{t-13}\hfill \\ \phantom{\rule{8em}{0ex}}-0.6748{Z}_{t-10}+{Z}_{t}\hfill \end{array}$$
- Ten-Year Moving-Average ARMAX (19, 10, 7, 0)The other competitive ARMAX model for the ten-year moving-average WS also included 7 GSL lags and had an ARMA (19, 10) base. The equation is estimated as$$\begin{array}{c}{\tilde{X}}_{t}=-0.0493{\tilde{S}}_{t}+1.7720{\tilde{S}}_{t-1}-2.4889{\tilde{S}}_{t-2}+1.1278{\tilde{S}}_{t-3}-0.6349{\tilde{S}}_{t-4}\hfill \\ \phantom{\rule{8em}{0ex}}-0.4446{\tilde{S}}_{t-5}+1.4953{\tilde{S}}_{t-6}-0.7594{\tilde{S}}_{t-7}+0.1837{\tilde{X}}_{t-3}\hfill \\ \phantom{\rule{8em}{0ex}}+0.2223{\tilde{X}}_{t-13}+0.0153{\tilde{X}}_{t-18}-0.1264{\tilde{X}}_{t-19}-0.6690{Z}_{t-10}+{Z}_{t}\hfill \end{array}$$

## 3. Prediction Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) The autocorrelation function (ACF) and (

**b**) partial autocorrelation function (PACF) of the first-order difference of the Colorado River WS time series. The two blue dashed lines on each plot denote the corresponding 95% confidence interval.

**Figure 3.**The cross-autocorrelation function (CACF) of the differenced Colorado River and Great Salt Lake (GSL) series. Note the significant cross-autocorrelation at lag −1. The two blue dashed lines denote the 95% confidence interval.

**Figure 4.**Autocorrelation function (ACF) plots of model residuals for (

**a**) MA (1) and (

**b**) AR (2). Blue dashed lines denote the 95% confidence intervals.

**Figure 5.**Diagnosis plots of the ARMA (1,1) model: (

**top**) overlaid plot of the observed annual water supply anomaly ${X}_{t}$ (black), model fitted value (blue) and its 95% confidence interval (grey); (

**left**) residual plot; (

**right**) ACF of the residual.

**Figure 6.**Diagnosis plots of the SAR (19) model: (

**top**) overlaid plot of the observed annual water supply anomaly ${X}_{t}$ (black), model fitted value (blue) and its 95% confidence interval (grey); (

**left**) residual plot; (

**right**) ACF of the residual.

**Figure 7.**Diagnosis plots of the SARX (19,1,0) model: (

**top**) overlaid plot of the observed annual water supply anomaly ${X}_{t}$ (black), model fitted value (blue) and its 95% confidence interval (grey); (

**left**) residual plot; (

**right**) ACF of the residual.

**Figure 8.**Diagnosis plots of the ARMAX (19,1,2,0) model: (

**top**) overlaid plot of the observed annual water supply anomaly ${X}_{t}$ (black), model fitted value (blue) and its 95% confidence interval (grey); (

**left**) residual plot; (

**right**) ACF of the residual.

**Figure 9.**Plot of the 10-year moving-average water supply in million acre feet (maf) of the Colorado River (

**a**) and plot of the ten-year moving average of Great Salt Lake water level (

**b**).

**Figure 10.**Plot of the observed values (black) with the predictions (red) ten years out for (

**A**) SAR (19) model, (

**B**) SARX (19, 1, 0) model, and (

**C**) ARMAX (19, 1, 2, 0) model in million acre feet (maf). The gray shaded area denotes the 95% predicting confidence interval.

**Figure 11.**The observed values for the 10-year moving average (black) shown with the predictions (red) 10 years out for the best annual model and the predictions (blue) 10 years out for the best moving-average model.

**Table 1.**Cross-validation results for the four models developed for the annual Colorado River Water Supply data. Root mean squared error (RMSE) is shown on top while the skill score (SS) is shown on the bottom. The columns represent the ten forecasting horizons (h = 1, 2, … 10). Highlighted values are the best statistic for each forecasting horizon.

RMSE | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

ARMA | 5.0708 | 6.9829 | 6.9649 | 6.9612 | 6.945 | 6.9467 | 6.9728 | 6.9438 | 6.9472 | 6.9208 |

SAR | 2.8673 | 4.4851 | 4.5344 | 4.319 | 4.1826 | 4.1037 | 4.318 | 5.1207 | 5.2668 | 5.2246 |

SARX | 3.1274 | 4.3809 | 4.6646 | 4.157 | 3.7697 | 3.7116 | 4.1295 | 5.0421 | 5.1573 | 5.1757 |

ARMAX | 2.877 | 4.3947 | 4.9787 | 5.1741 | 4.0544 | 3.9955 | 3.7328 | 4.4858 | 4.9159 | 6.2473 |

SS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

ARMA | 0.4719 | −0.0107 | −0.0080 | 0.0029 | 0 | 0 | 0 | 0 | 0 | 0 |

SAR | 0.8311 | 0.583 | 0.5727 | 0.6162 | 0.6373 | 0.651 | 0.6165 | 0.4562 | 0.4252 | 0.4301 |

SARX | 0.7991 | 0.6022 | 0.5479 | 0.6444 | 0.7054 | 0.7145 | 0.6493 | 0.4727 | 0.4489 | 0.4407 |

ARMAX | 0.83 | 0.5997 | 0.4849 | 0.4491 | 0.6592 | 0.6692 | 0.7134 | 0.5827 | 0.4993 | 0.1852 |

**Table 2.**Cross-validation results for the three models developed for the 10-year moving average data. Root mean squared error is shown on top while the skill score is shown on the bottom. The columns represent the ten forecasting horizons (h = 1, 2, … 10). Highlighted values are the best statistic for each forecasting horizon.

RMSE | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

AR(19) | 0.5522 | 0.5374 | 0.547 | 0.4876 | 0.5062 | 0.4872 | 0.4805 | 0.4951 | 0.5087 | 0.4506 |

ARMAX1 | 0.4152 | 0.4123 | 0.4079 | 0.4268 | 0.4297 | 0.4266 | 0.4316 | 0.4369 | 0.4315 | 0.4312 |

ARMAX2 | 0.4153 | 0.419 | 0.4125 | 0.4197 | 0.4208 | 0.4181 | 0.4185 | 0.4231 | 0.4243 | 0.4265 |

SS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

AR(19) | 0.0181 | 0.0802 | 0.0541 | 0.2458 | 0.1811 | 0.235 | 0.2537 | 0.2091 | 0.1605 | 0.3391 |

ARMAX1 | 0.445 | 0.4585 | 0.4739 | 0.4221 | 0.4099 | 0.4135 | 0.3977 | 0.3841 | 0.3959 | 0.3949 |

ARMAX2 | 0.4448 | 0.4409 | 0.4621 | 0.4413 | 0.434 | 0.4367 | 0.4337 | 0.4223 | 0.416 | 0.4081 |

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**MDPI and ACS Style**

Plucinski, B.; Sun, Y.; Wang, S.-Y.S.; Gillies, R.R.; Eklund, J.; Wang, C.-C.
Feasibility of Multi-Year Forecast for the Colorado River Water Supply: Time Series Modeling. *Water* **2019**, *11*, 2433.
https://doi.org/10.3390/w11122433

**AMA Style**

Plucinski B, Sun Y, Wang S-YS, Gillies RR, Eklund J, Wang C-C.
Feasibility of Multi-Year Forecast for the Colorado River Water Supply: Time Series Modeling. *Water*. 2019; 11(12):2433.
https://doi.org/10.3390/w11122433

**Chicago/Turabian Style**

Plucinski, Brian, Yan Sun, S.-Y. Simon Wang, Robert R. Gillies, James Eklund, and Chih-Chia Wang.
2019. "Feasibility of Multi-Year Forecast for the Colorado River Water Supply: Time Series Modeling" *Water* 11, no. 12: 2433.
https://doi.org/10.3390/w11122433