# A Robust Regime Shift Change Detection Algorithm for Water-Flow Dynamics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Region

^{2}, which is just under a quarter of Alberta’s province [27] and comprises numerous named/unnamed rivers and lakes. Both cold and warm seasons are available in this region. During cold months, a substantial portion of precipitation is in the form of snow, while during the warm months, meltwater and rainfall merge and contribute to river streamflow. Water from sub-basins likewise amalgamates with the main river as the river flows in the direction of Lake Athabasca [26,28]. The ARB region is shown in Figure 1, whose background gradient color was generated utilizing Shuttle Radar Topography Mission (SRTM) data sampled at 30 m. The ARB region includes three main subregions, namely, lower, middle, and upper [29].

#### 2.2. Streamflow Data

#### 2.3. Methods

#### 2.3.1. Seasonal Modeling and Gap Filling of Hydrometric Data Time Series

_{t}is assumed to be a Gaussian white-noise process with zero average and variance σ

^{2}[34,35,36]. For time series y

_{t}, the mathematical formulation of SARIMA can be expressed as follows [34,35,36]:

- $\Phi $: Seasonal Autoregressive Parameter,
- $\varphi $: Autoregressive Parameter,
- ${\mathsf{\Delta}}^{d}$: the difference operator where d specifies the order of differencing,
- ${\mathsf{\Delta}}_{s}{}^{D}$: the seasonal difference operator where D is the order of seasonal differencing,
- $\theta $: Moving Average Parameter,
- $\Theta $: Seasonal Moving Average Parameter.

#### 2.3.2. Regime Shift Change Detection (RSCD)

- ${\mu}_{j}$ and ${\mu}_{k}$ are average values of a subset of $X$,
- n = n(X) is the number of elements of set X,
- $m=n\left(Y\right)$ is the number of elements of set $Y$,
- $0\le \mathrm{GR}\left({x}_{j},{x}_{k}\right)\le 1$ for $0\le j,k\le n$,
- $0\le \mathrm{RD}\left({\mu}_{j},{\mu}_{k}\right)\le 1$ for $0\le j,k\le m+1$.

#### 2.3.3. RSCD Thresholds

Algorithm 1 Regime Shift Change Detection (RSCD) algorithm, * Refine Method (RM). This parameter determines the two specializations of this method. ** x_{r} is a point right after point c_{(j+1)} |

Input: $X={x}_{0},{x}_{1},\dots ,{x}_{n}$, μ, σ, ε, and RM *Output: Periods P$Y=\left\{{y}_{0},{y}_{1},\dots ,{y}_{m}\right\}$ ← potential change point candidates through $\left|{x}_{i}-\mu \right|>1.5\sigma $ with 1 ≤ j ≤ n − 1; if n(Y) ≥ 2 then$C=\left\{{c}_{0},{c}_{1},\dots ,{c}_{k}\right\}$ ← consecutive potential change point candidates; while n(C) ≥ 2 dofor (${c}_{j},{c}_{j+1}$) doif there is a period after ${c}_{j+1}$ thenif $\mathrm{GR}\left({\mu}_{l},{c}_{j}\right)$ > $\mathrm{GR}\left({c}_{j+1},{\mu}_{r}\right)$ thenRemove ${c}_{j+1}$ from CelseRemove ${c}_{j}$ from Cendelseif GR(${\mu}_{l}$, ${c}_{j}$) > GR(${c}_{j+1}$, ${x}_{r}$) ** thenRemove ${c}_{j+1}$ from CelseRemove ${c}_{j}$ from CendendendUpdate C endUpdate Y endif n(Y) ≥ 1 then$P=\left\{{p}_{0},{p}_{1},\dots ,{p}_{m},{p}_{m+1}\right\}$ ← periods defined using m change points; while n(P) ≥ 2 dofor $\left({p}_{j},{p}_{j+1}\right)$ doif RM = Relative-Difference thenif RD(${\mu}_{j},{\mu}_{j+1}$) < ε thenMerge periods ${p}_{j}$ and ${p}_{j+1}$ endendif RM = Growth Rate thenif GR(${\mu}_{j},{\mu}_{j+1}$) < ε thenMerge periods ${p}_{j}$ and ${p}_{j+1}$ endendendUpdate P endelseThere is no regime shift. End |

#### 2.3.4. RSCD for Newly Observed Data

_{0≤i≤n}{x

_{i}} or min

_{0≤i≤n}{x

_{i}} is smaller than some value from set X*, Algorithm 1 needs to be applied again on $X\cup {X}^{*}=\left\{{x}_{0},{x}_{1},\dots ,{x}_{n},{x}_{n+1},{x}_{n+2},\dots \right\}$. However, using the RSCD with Growth Rate (RSCD-GR), only the newly observed data needs to be investigated.

## 3. Results

#### 3.1. Seasonal Modeling and Gap Filling of Hydrometric Data Time Series

#### 3.2. Regime Shift Change Detection (RSCD)

#### 3.3. RSCD Algorithm for the Test Set

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ARB | Athabasca River Basin |

ARIMA | Autoregressive Integrated Moving Average |

GR | Growth Rate |

MA | Moving-Average |

RFR | Random Forest Regressor |

RSCD | Regime Shift Change Detection |

RD | Relative Difference |

RSCD-GR | RSCD with Growth Rate |

RSCD-RD | RSCD with Relative Difference |

SARIMA | Seasonal Autoregressive Integrated Moving Average |

SRTM | Shuttle Radar Topography Mission |

RAMP | The Regional Aquatics Monitoring Program |

WSC | Water Survey of Canada |

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**Figure 1.**A map of the Athabasca River Basin (ARB). The gradient background color was produced using Shuttle Radar Topography Mission (SRTM) data sampled at 30 m. The right corner window demonstrates the location of the ARB within Canada. Stations selected for training the RSCD algorithm are exhibited with circles, whereas stations selected for testing the algorithm are illustrated with stars.

**Figure 2.**Averaged normalized monthly streamflow data distribution by train (

**a**) and test (

**b**) sets. Cold months are highlighted with blue color, and warm months are shaded by pink color. In part (

**a**), for the month of April, blue and pink colors are overlapped as this month was considered a warm month for 07BE001 and 07DA001 but a cold month for 07AA002, 07AD002, and 07AE001. For part (

**b**), January, February, March, November, and December are regarded as cold months, while April, May, June, July, August, September, and October are considered as warm months.

**Figure 3.**Set T distribution plot for each station separated by warm and cold months: (

**a**,

**c**) density data distribution for set T. (

**b**,

**d**) box plot data distribution for set T.

**Figure 4.**The observed and extended streamflow (m

^{3}/s) monthly time series through SARIMA (solid blue line) and RFR (solid red line) modeling for 07AA002 (Athabasca River near Jasper), 07AD002 (Athabasca River at Hinton), 07AE001 (Athabasca River near Windfall), 07BE001 (Athabasca River at Athabasca), and 07DA001 (Athabasca River below Fort McMurray).

**Figure 5.**RSCD Results for warm months using Growth Rate (

**a**), Relative Difference (

**b**), and a 30-year (1961–1990) period for calculating µ and σ, and ε thresholds from Table 2. The streamflow (m

^{3}/s) is shown on the y-axis in each panel. The use of red and blue shades aids in distinguishing between the different regimes.

**Figure 6.**RSCD Results for cold months using Growth Rate (

**a**), Relative Difference (

**b**), and a 30-year (1961–1990) period for calculating µ and σ, and ε thresholds from Table 2. The streamflow (m

^{3}/s) is shown on the y-axis in each panel. The use of red and blue shades aids in distinguishing between the different regimes.

**Figure 7.**RSCD Results for warm and cold months using Growth Rate (

**a**), Relative Difference (

**b**), and a 30-year (1961–1990) period for calculating µ and σ, and ε thresholds from Table 2. The streamflow (m

^{3}/s) is shown on the y-axis in each panel. The use of red and blue shades aids in distinguishing between the different regimes.

Set | ID | Name | Considered Period for This Study | Gross Drainage Area (km^{2}) | Elevation (m) |
---|---|---|---|---|---|

07AA002 | Athabasca River near Jasper | 1960–2021 | 3870 | 1041 | |

07AD002 | Athabasca River at Hinton | 1960–2021 | 9760 | 963 | |

Train | 07AE001 | Athabasca River near Windfall | 1960–2021 | 19,600 | 735 |

07BE001 | Athabasca River at Athabasca | 1960–2021 | 74,600 | 513 | |

07DA001 | Athabasca River below Fort McMurray | 1960–2021 | 133,000 | 246 | |

07AH001 | Freeman River near Fort Assiniboine | 1965–2020 | 1660 | 661 | |

07BB002 | Pembina River near Entwistle | 1960–2015 | 4400 | 727 | |

Test | 07BK005 | Saulteaux River near Spurfield | 1969–2015 | 2600 | 585 |

07BK007 | Driftwood River near the Mouth | 1968–2020 | 2100 | 569 | |

07DA006/S38 ^{1} | Steepbank River near Fort McMurray | 1972–2021 | 1320 | 277 |

^{1}WSC and RAMP (The Regional Aquatics Monitoring Program (RAMP), 2022) dataset for this station were fused.

**Table 2.**ε thresholds separated by cold and warm sets of months (seasons) for each hydrometric station. At the bottom part of the table, general $\epsilon $ thresholds are separated by cold and warm sets of months (seasons) for all hydrometric stations from the ARB region.

ID | Season | Growth Rate ε | Relative Difference ε |
---|---|---|---|

07AA002 | Cold | 0.322 | 0.130 |

07AA002 | Warm | 0.215 | 0.135 |

07AD002 | Cold | 0.231 | 0.160 |

07AD002 | Warm | 0.257 | 0.202 |

07AE001 | Cold | 0.293 | 0.197 |

07AE001 | Warm | 0.193 | 0.174 |

07BE001 | Cold | 0.288 | 0.202 |

07BE001 | Warm | 0.285 | 0.174 |

07DA001 | Cold | 0.271 | 0.200 |

07DA001 | Warm | 0.350 | 0.230 |

General | Cold | 0.288 | 0.197 |

General | Warm | 0.257 | 0.174 |

**Table 3.**Accuracy test by various metrics for SARIMA modeling (07AD002 and 07BE001) and RFR modeling (07AA002 and 07AE001). The ideal value for Explained Variance Score (EVS) and the coefficient of determination (R2) is a number close to one, while the ideal value for the Mean Absolute Error (MAE) and the Mean Squared Error (MSE) is a number close to zero.

Metrics | 07AA002 | 07AD002 | 07AE001 | 07BE001 |
---|---|---|---|---|

Train: EVS | 1.00 ± 2.32 × 10^{−4} | 0.901 | 0.99 ± 1.50 × 10^{−3} | 0.789 |

Test: EVS | 0.98 ± 2.20 × 10^{−3} | 0.96 ± 4.15 × 10^{−2} | ||

Train: MAE | 2.89 ± 9.64 × 10^{−2} | 30.692 | 9.74 ± 1.01 × 10^{0} | 104.842 |

Test: MAE | 7.60 ± 6.00 × 10^{−1} | 27.83 ± 8.83 × 10^{0} | ||

Train: MSE | 23.16 ± 2.01 × 10^{0} | 2982.181 | 294.80 ± 8.40 × 10^{1} | 30,943.826 |

Test: MSE | 159.69 ± 2.07 × 10^{1} | 2563.86 ± 2.70 × 10^{3} | ||

Train: R2 | 1.00 ± 2.32 × 10^{−4} | 0.901 | 0.99 ± 1.50 × 10^{−3} | 0.788 |

Test: R2 | 0.98 ± 2.21 × 10^{−3} | 0.95 ± 4.31 × 10^{−2} |

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## Share and Cite

**MDPI and ACS Style**

Dastour, H.; Gupta, A.; Achari, G.; Hassan, Q.K.
A Robust Regime Shift Change Detection Algorithm for Water-Flow Dynamics. *Water* **2023**, *15*, 1571.
https://doi.org/10.3390/w15081571

**AMA Style**

Dastour H, Gupta A, Achari G, Hassan QK.
A Robust Regime Shift Change Detection Algorithm for Water-Flow Dynamics. *Water*. 2023; 15(8):1571.
https://doi.org/10.3390/w15081571

**Chicago/Turabian Style**

Dastour, Hatef, Anil Gupta, Gopal Achari, and Quazi K. Hassan.
2023. "A Robust Regime Shift Change Detection Algorithm for Water-Flow Dynamics" *Water* 15, no. 8: 1571.
https://doi.org/10.3390/w15081571