# Water Consumption Pattern Analysis Using Biclustering: When, Why and How

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## Abstract

**:**

## 1. Introduction

- overview of notorious contributions in the literature contemplating the opportunities and limitations of clustering water time series data;
- taxonomy for a structured view, principled application, and critical assessment of biclustering water consumption data;
- novel methodology for the correct application of coclustering and biclustering methods to water consumption data analysis;
- empirical validation and comprehensive discussion using a real-world case study from a WDS corresponding to a large tourist and residential resort.

#### Related Work

## 2. Background

#### 2.1. Time Clustering

**Definition 1.**

#### 2.2. Subspace Clustering

#### 2.2.1. Biclustering

**Definition 2.**

#### 2.2.2. Coclustering and Subspace Clustering Variants

**Definition 3.**

## 3. Solution: Biclustering for Water Consumption Pattern Mining

- biclustering-based paradigms on water consumption data (Section 3.1);
- biclustering settings (coherence, structure, quality, statistical significance) and their impact (Section 3.2);
- principles for guiding the development of biclustering-based pattern mining on time series water consumption data (Section 3.3).

#### 3.1. Major Subspace-Clustering Paradigms

#### 3.2. Biclustering Properties and Their Impact on the Pattern Mining Water Consumption Data

#### 3.2.1. Biclustering Coherence

**Definition 4.**

**Definition 5.**

**Definition 6.**

#### 3.2.2. Biclustering Structure

#### 3.2.3. Biclustering Quality

**Definition 7.**

#### 3.2.4. Biclustering Statistical Significance

#### 3.3. Principles for Biclustering-Based Time Series Analysis on Water Consumption Data

## 4. Case Study: Water Distribution Network of Quinta Do Lago

**RQ1. Are clustering approaches adequate for water consumption profiling from time series data? What are their major limitations?****RQ2. Does coclustering, as a more flexible clustering approach, aid the clustering analysis of water consumption data?****RQ3. Is biclustering able to retrieve novel actionable water consumption patterns? Can biclustering address the established shortcoming of clustering and co-clustering tasks?****RQ4. Which principles should be placed on the design and application of biclustering approaches for an effective descriptive and predictive analysis of water consumption profiles?**

#### 4.1. Dataset

^{2}of land, varying from 2000 to 14,000 inhabitants in winter and summer, respectively, creating a relevant water demand seasonal variation. The WDN, managed by InfraQuinta, supplies 1.7 mm

^{3}/year of water mainly to domestic consumers and hotels. The consumption data was measured by a telemetry system every hour at each of the around 2170 end-users., during the entire year of 2017. Figure 7 shows and overview of Quinta do Lago’s WDN.

#### 4.2. Experimental Setting

#### 4.3. Data Preprocessing

#### 4.4. Clustering Analysis (RQ1)

- Consumption behaviour is grouped across the entire time axis, neglecting local patterns;
- Sensitive to noise and outliers requiring data transformations and cleaning procedures which are frequently not sufficient;
- Method-specific parameterization needs that considerably impact the clustering analysis, e.g., manually specifying the number of clusters in the case of K-means;
- Limited to constant relationships between time series, not considering other meaningful coherent consumption profiles explained by shifting, scaling and lagged factors.

#### 4.5. Coclustering Analysis (RQ2)

- Coclustering approaches generally disregard temporal dependencies within and across consumption signals, thus penalizing misalignments between coherent profiles as well as the inherent consumption variability along time. It further discards temporal contiguity, and as a result, water consumption patterns are generally grouped under non-sequential periods, limiting the interpretability and actionability of the gathered patterns;
- Coclustering guarantees the discovery of subspaces that can be evaluated according to a homogeneity measure, meaning that coclusters with low homogeneity can be filtered before analysis. Nevertheless, there is the need to manually specify the number of coclusters;
- Coclustering can discover groups of users with coherent consumption behavior under some periods, not limiting the search for global consumption patterns. However, coclustering assumes that each user is only associated with one consumption pattern, disregarding the possibility of associating multiple patterns with an user’s consumption profile. In addition, the partitioning of the time axis is restricting, preventing the discovery of flexibly positioned subspaces with arbitrarily-high overlaps along the time dimensions.

#### 4.6. Biclustering Analysis (RQ3)

**Constant consumption patterns**

**Noise robustness**

**Coherent patterns with consumption shifts**

**Time-lagged consumption patterns**

**Statistically significance consumption patterns**

#### 4.7. Guiding Biclustering Principles for Water Consumption Tasks (RQ4)

- Detection of local consumption profiles, surpassing the limitation of traditional time clustering methods that only unveil global patterns;
- Efficient search for patterns with multiple coherence assumptions and quality, instead of only assuming constant relationships between time series;
- Retrieval of well-defined consumption patterns with solid guarantees of coherence and quality, in contrast with high variability of clustering consumption profiles;
- Flexible pattern-based search that can be customized to guide and restrict the search, preventing redundant consumption patterns and ensuring efficient searches.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

WDS | Water distribution system |

WDN | Water distribution network |

SVM | Support vector machine |

SOM | Self-organizing map |

HAC | Hierarchical agglomerative clustering |

DTW | Dynamic time warping |

DBA | Dynamic time warping barycenter averaging |

DWT | Discrete wavelet transform |

PAA | Piecewise aggregate approximation |

PLA | Piecewise linear approximation |

SAX | Symbolic aggregate approximation |

LCSS | Longest common sub-sequence |

MODH | Modified hausdorff |

HMM | Hidden markov model |

SSE | Sum of squared error |

CD | Distance between clusters index |

IQR | Interquartile range |

CCC | Contiguous column coherent biclustering |

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**Figure 1.**Time series clustering approaches (adapted from [24]).

**Figure 3.**Illustrative forms e of subspace coherence: (

**a**) constant values, (

**b**) constant values on rows (pattern on columns), (

**c**) constant values on columns (pattern on rows), (

**d**) coherent values (additive model), (

**e**) coherent values (multiplicative model), (

**f**) overall coherent evolution (order-preserving model), (

**g**) coherent evolution on the rows, (

**h**) coherent evolution on the columns [12].

**Figure 4.**Taxonomy of Biclustering-based data analysis on water demand data: structured view on the major biclustering paradigms, biclustering aspects affecting the analysis, and principles to design and assess biclustering-based approaches.

**Figure 6.**Pattern-centric transformation to map time series data onto multivariate data, obtained by comparing end-users against the found biclusters.

**Figure 7.**Water Distribution System of Quinta do Lago. (Adapted from [61]).

**Figure 9.**Frequency of the flow rate measurements at InfraQuinta, 2017. (

**a**) Absolute consumption values. (

**b**) Normalized consumption values.

**Figure 10.**Hierarchical Clustering (Sensors Dimension) Dendrogram of daily consumption at InfraQuinta, 2017.

**Figure 11.**Hierarchical Clustering (Time Dimension) Dendrogram of scaled daily consumption at InfraQuinta, 2017.

**Figure 12.**Optimal K for K-means clustering of scaled daily consumption at InfraQuinta, 2017. (

**a**) Distortions for each K (Elbow Method). (

**b**) Average silhouettes scores for each K.

**Figure 14.**K-means 13th cluster and barycenter for the scaled daily consumption at InfraQuinta, 2017.

**Figure 16.**Homogeneity and number of coclusters (N) for the spectral Coclustering of scaled daily, weekly, and monthly consumption at InfraQuinta, 2017.

**Figure 17.**Size of coclusters and number of coclusters (N) for the spectral Coclustering of scaled daily, weekly, and monthly consumption at InfraQuinta, 2017.

**Figure 18.**Rearranged daily, weekly, and monthly consumption data matrices to reveal the coclustering solutions ($N=5,N=10,N=10$) at InfraQuinta, 2017. Each of the identified coclusters are highlighted in red. Note that for the weekly and monthly datasets, we only highlight the valid coclusters, as the algorithm did not find coclusters for all the users.

**Figure 19.**Illustration of the selected coclusters (Cocluster 0, Cocluster 2, and Cocluster 1) found on the daily, weekly, and monthly datasets at InfraQuinta, 2017.

**Figure 20.**Coclusters and barycenters for the daily, weekly, and monthly dataset at InfraQuinta, 2017.

**Figure 21.**Illustration of the selected constant biclusters (Bicluster 9964, Bicluster 245, and Bicluster 210) found on the daily, weekly and monthly dataset. Consumption patterns on the first row, the user consumption scaled time series on the second row, and the scaled data heatmap on the third row.

**Figure 23.**Illustration of the selected constant biclusters allowing noise (Bicluster 197684, Bicluster 33405, and Bicluster 412) found on the daily, weekly and monthly dataset. This figure shows the consumption patterns on the first row, the user consumption scaled time series on the second row, and the scaled data heatmap on the third row.

**Figure 24.**Illustration of the selected biclusters assuming shifting factors (Bicluster 141, Bicluster 478, and Bicluster 239) found on the daily, weekly and monthly dataset. This figure shows the consumption patterns on the first row, the user consumption scaled time series on the second row, and the scaled data heatmap on the third row.

**Figure 25.**Illustration of the selected biclusters allowing time-lagged patterns (Bicluster 8476, Bicluster 965, and Bicluster 120) found on the daily, weekly and monthly dataset. This figure shows the consumption patterns on the first row, the user consumption scaled time series on the second row, and the scaled data heatmap on the third row.

**Figure 26.**Statistical significance vs. size of biclusters found assuming constant patterns at InfraQuinta, 2017.

**Figure 27.**Number of patterns found for each user when combining the biclustering solutions with different pattern assumption at InfraQuinta, 2017.

**Figure 28.**Number of patterns found for each user when combining the biclustering solutions obtained from datasets of different granularity at InfraQuinta, 2017.

Dataset | ID | #Users | #Time Points (First, Last) |
---|---|---|---|

Daily | 0 | 161 | 88 (0, 87) |

Weekly | 2 | 147 | 15 (14, 28) |

Monthly | 1 | 142 | 3 (4, 8) |

**Table 2.**Properties of the biclustering solutions found assuming constant patterns at InfraQuinta, 2017.

Solution | Post-Processed | ||||||
---|---|---|---|---|---|---|---|

Dataset | (min #Users,min #Time Points) | #bics | $\mathbf{\mu}\left|\mathit{I}\right|\pm \mathbf{\sigma}\left|\mathit{I}\right|$ | $\mathbf{\mu}\left|\mathit{J}\right|\pm \mathbf{\sigma}\left|\mathit{J}\right|$ | #bics | p-Value< 0.05 | p-Value< $1\times {10}^{-2}$ |

Daily | (20, 7) | 18,666 | 65.2 ± 43.5 | 26.0 ± 20.7 | 655 | 655 | 655 |

Weekly | (20, 4) | 1310 | 69.8 ± 65.9 | 9.4 ± 5.8 | 263 | 168 | 133 |

Monthly | (20, 3) | 221 | 50.7 ± 51.5 | 4.5 ± 1.5 | 94 | 23 | 10 |

Dataset | ID | #Users | #Time Points (First, Last) | p-Value |
---|---|---|---|---|

Daily | 9964 | 20 | 7 (331, 337) | 0.0029 |

Weekly | 245 | 27 | 13 (31, 43) | 1.09 × 10${}^{-8}$ |

Monthly | 210 | 21 | 9 (1, 9) | 5.17 × 10${}^{-5}$ |

**Table 4.**Properties of e-CCC biclustering solutions with tolerance to noise under a constant pattern assumption at InfraQuinta, 2017.

Solution | Post-Processed | ||||||
---|---|---|---|---|---|---|---|

Dataset | (min #Users,min #Time Points) | #bics | $\mathbf{\mu}\left|\mathit{I}\right|\pm \mathbf{\sigma}\left|\mathit{I}\right|$ | $\mathbf{\mu}\left|\mathit{J}\right|\pm \mathbf{\sigma}\left|\mathit{J}\right|$ | #bics | p-Value< 0.05 | p-Value1 × 10${}^{-2}$ |

Daily | (20, 7) | 786,232 | 72.3 ± 44.8 | 28.9 ± 21.7 | 2347 | 839 | 744 |

Weekly | (20, 4) | 55,073 | 70.1 ± 63.3 | 11.7 ± 6.6 | 4304 | 279 | 160 |

Monthly | (20, 3) | 6441 | 57.5 ± 59.8 | 5.6 ± 1.8 | 942 | 18 | 6 |

Dataset | ID | #Users | #Time Points (First, Last) | p-Value |
---|---|---|---|---|

Daily | 197,684 | 20 | 65 (206, 270) | 2.86 × 10${}^{-125}$ |

Weekly | 33,405 | 47 | 25 (19, 43) | 1.83 × 10${}^{-9}$ |

Monthly | 412 | 22 | 10 (1, 10) | 8.40 × 10${}^{-6}$ |

**Table 6.**Properties of the biclustering solutions found assuming shifted factors at InfraQuinta, 2017.

Solution | Post-Processed | |||||||
---|---|---|---|---|---|---|---|---|

Dataset | (min #Users,min #Time Points) | L-Shift | #bics | $\mathbf{\mu}\left|\mathit{I}\right|\pm \mathbf{\sigma}\left|\mathit{I}\right|$ | $\mathbf{\mu}\left|\mathit{J}\right|\pm \mathbf{\sigma}\left|\mathit{J}\right|$ | #bics | p-Value< 0.05 | p-Value< $1\times $ 10${}^{-2}$ |

Daily | (20, 7) | 1 | 38,933 | 87.0 ± 51.8 | 23.6 ± 16.4 | 625 | 593 | 588 |

2 | 46,308 | 111.4 ± 64.6 | 25.4 ± 17.1 | 383 | 340 | 330 | ||

3 | 32,669 | 124.6 ± 63.8 | 29.6 ± 21.9 | 367 | 332 | 323 | ||

4 | 16,033 | 114.8 ± 66.6 | 37.6 ± 26.5 | 345 | 310 | 301 | ||

Weekly | (20, 4) | 1 | 2828 | 76.7 ± 74.0 | 8.2 ± 4.8 | 404 | 193 | 178 |

2 | 2743 | 96.2 ± 90.6 | 8.3 ± 5.0 | 369 | 153 | 131 | ||

3 | 1677 | 109.4 ± 97.5 | 10.0 ± 6.7 | 360 | 144 | 123 | ||

4 | 1391 | 84.5 ± 82.13 | 10.9 ± 7.0 | 357 | 141 | 121 | ||

Monthly | (20, 3) | 1 | 372 | 56.3 ± 57.0 | 4.1 ± 1.4 | 113 | 33 | 21 |

2 | 318 | 65.2 ± 69.4 | 4.2 ± 1.4 | 108 | 27 | 18 | ||

3 | 251 | 60.3 ± 65.7 | 4.5 ± 1.5 | 108 | 30 | 21 | ||

4 | 245 | 55.4 ± 55.8 | 4.5 ± 1.6 | 108 | 30 | 21 |

Dataset | ID | #Users | #Time Points (First, Last) | p-Value |
---|---|---|---|---|

Daily | 141 | 23 | 8 (358, 364) | 0.002 |

Weekly | 478 | 26 | 14 (30, 43) | 2.27 × 10${}^{-9}$ |

Monthly | 239 | 21 | 9 (1, 9) | 5.17× 10${}^{-5}$ |

**Table 8.**Properties of the biclustering solutions found assuming unbounded time lagged patterns at InfraQuinta, 2017.

Solution | Post-Processed | ||||||
---|---|---|---|---|---|---|---|

Dataset | (min #Users,min #TimePoints) | #bics | $\mathbf{\mu}\left|\mathit{I}\right|\pm \mathbf{\sigma}\left|\mathit{I}\right|$ | $\mathbf{\mu}\left|\mathit{J}\right|\pm \mathbf{\sigma}\left|\mathit{J}\right|$ | #bics | p-Value< 0.05 | p-Value< $1\times $ 10${}^{-2}$ |

Daily | (20, 7) | 15,844 | 56.1 ± 51.3 | 26.5 ± 21.8 | 1471 | 1471 | 1471 |

Weekly | (20, 4) | 1738 | 60.6 ± 61.9 | 10.2 ± 6.3 | 393 | 393 | 393 |

Monthly | (20, 3) | 243 | 61.3 ± 66.2 | 4.8 ± 1.6 | 99 | 98 | 98 |

Dataset | ID | #Users | #Time Points (First, Last) | p-Value |
---|---|---|---|---|

Daily | 8476 | 32 | 112 | 0 * |

Weekly | 965 | 44 | 29 | 1.26× 10${}^{-121}$ |

Monthly | 120 | 25 | 9 | 1.65 × 10^{−9} |

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## Share and Cite

**MDPI and ACS Style**

Silva, M.G.; Madeira, S.C.; Henriques, R.
Water Consumption Pattern Analysis Using Biclustering: When, Why and How. *Water* **2022**, *14*, 1954.
https://doi.org/10.3390/w14121954

**AMA Style**

Silva MG, Madeira SC, Henriques R.
Water Consumption Pattern Analysis Using Biclustering: When, Why and How. *Water*. 2022; 14(12):1954.
https://doi.org/10.3390/w14121954

**Chicago/Turabian Style**

Silva, Miguel G., Sara C. Madeira, and Rui Henriques.
2022. "Water Consumption Pattern Analysis Using Biclustering: When, Why and How" *Water* 14, no. 12: 1954.
https://doi.org/10.3390/w14121954