# Water Particles Monitoring in the Atacama Desert: SPC Approach Based on Proportional Data

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## Abstract

**:**

## 1. Introduction

## 2. The Data

## 3. Methodology

#### 3.1. The Unit-Lindley Distribution

`expint`package [28] in R.

`pracma`package [30] in R.

#### 3.2. Proposed Unit-Lindley Chart

## 4. Statistical Performance

#### 4.1. In-Control Processes

#### 4.2. Out-of-Control Processes

#### 4.3. Comparison with Some Standard Control Charts

## 5. Application

## 6. Concluding Remarks and Future Prospects

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Statistics of the cloud occurrence over the Atacama region (panels B and C), shaded from high probability of cloud occurrence in dark-red to low probability of cloud occurrence in beige. The bottom left-hand map represents Terra MODIS (panel B), and Aqua MODIS (panel C) is in the bottom center map. The transaction of the dark-red area, which occurs mainly during the dawn and morning and is most associated with the Camanchaca increasing the humidity of the Chilean third region coast up to the beginning of the highlands, turns into the low scale of humidity right in the afternoons, represented by the full red map. Two bays are to be noticeable: in Copiapó and Huasco, as convergence points. In panel A, the digital elevation model for part of the Copiapó watershed is presented. Our data acquisition came from a weather station located at the University of Atacama in Copiapó, Chile (in the top left-hand picture).

**Figure 3.**Unit-Lindley density function for the different parameter values considered in this study.

**Figure 4.**Humidity variation, collected per hour, from Copiapó (Chile) in the last five years. Panel A presents the histogram of the time series (TS), and panel B shows the dynamic of this series in light blue. Also, the solid line represents the TS average using a LOESS (an acronym for “Locally Estimated Scatterplot Smoothing”) smoothing method [42].

**Figure 5.**Histogram of the minimum (

**left-hand panel**) and maximum (

**right-hand panel**) observations, after aggregating the daily humidity representation of the data in day periods (parts 1–4), followed by a black solid line representing the estimated PDF of the unit-Lindley distribution.

**Figure 6.**A visualization of the dynamic of Phase 1 (minimum and maximum TS), considering the observation points as four periods of the day, one in each graphic. Top-left panel: the day period part 1 (from midnight to 5:59 a.m. UTC); top-right panel: the day period part 2 (from 6:00 a.m. to 11:59 a.m. UTC); bottom-left panel: the day period part 3 (from midday to 5:59 p.m. UTC); bottom-right panel: the day period part 4 (from 6:00 p.m. to 11:59 p.m. UTC). Thus, the red lines represent three tolerance ($\alpha $) levels (15% as thick dashed line, 10% as thin dashed line, and 1% as solid line) for each estimated control limit (UCL for the maximum TS, and LCL for the minimum TS).

**Figure 7.**SDA bivariate control chart for the daily humidity in Phase II monitoring (records from 2021). Through the 3D plot, the z-axis represents the maximum upper bound and the y-axis the minimum lower bound from the daily humidity (aggregated per periods), adopting a certain tolerance level ($\alpha =0.15$ or $15\%$), whereas the x-axis is related to the time observation points (as dots). The estimated control limits are represented as a shaded box, observed out-of-control points are highlighted as red points and their projections placed in the control chart projections. Thus, the TS projection on the bottom (x- and y-axes) is the control chart related to the minimum daily humidity, and the TS projection on the background (x- and z-axes) is the control chart of the maximum daily humidity.

$\mathit{\alpha}=0.1$ | $\mathit{\alpha}=0.01$ | $\mathit{\alpha}=0.0027$ | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{\mu}$ | LCL | CL | UCL | LCL | CL | UCL | LCL | CL | UCL |

0.08 | 0.0048 | 0.08 | 0.2190 | 0.0005 | 0.08 | 0.3303 | 0.0001 | 0.08 | 0.3802 |

0.12 | 0.0079 | 0.12 | 0.3124 | 0.0008 | 0.12 | 0.4428 | 0.0002 | 0.12 | 0.4965 |

0.16 | 0.0115 | 0.16 | 0.3954 | 0.0011 | 0.16 | 0.5320 | 0.0003 | 0.16 | 0.5846 |

0.20 | 0.0158 | 0.20 | 0.4688 | 0.0016 | 0.20 | 0.6038 | 0.0004 | 0.20 | 0.6530 |

0.24 | 0.0208 | 0.24 | 0.5335 | 0.0021 | 0.24 | 0.6623 | 0.0006 | 0.24 | 0.7072 |

0.28 | 0.0269 | 0.28 | 0.5905 | 0.0027 | 0.28 | 0.7107 | 0.0007 | 0.28 | 0.7512 |

0.32 | 0.0341 | 0.32 | 0.6407 | 0.0035 | 0.32 | 0.7512 | 0.0009 | 0.32 | 0.7873 |

0.36 | 0.0428 | 0.36 | 0.6851 | 0.0044 | 0.36 | 0.7854 | 0.0012 | 0.36 | 0.8174 |

0.40 | 0.0534 | 0.40 | 0.7244 | 0.0055 | 0.40 | 0.8146 | 0.0015 | 0.40 | 0.8429 |

0.44 | 0.0662 | 0.44 | 0.7592 | 0.0070 | 0.44 | 0.8397 | 0.0019 | 0.44 | 0.8646 |

0.48 | 0.0819 | 0.48 | 0.7902 | 0.0088 | 0.48 | 0.8616 | 0.0024 | 0.48 | 0.8834 |

0.52 | 0.1012 | 0.52 | 0.8179 | 0.0112 | 0.52 | 0.8807 | 0.0030 | 0.52 | 0.8997 |

0.56 | 0.1250 | 0.56 | 0.8426 | 0.0142 | 0.56 | 0.8975 | 0.0039 | 0.56 | 0.9139 |

0.60 | 0.1545 | 0.60 | 0.8648 | 0.0183 | 0.60 | 0.9124 | 0.0050 | 0.60 | 0.9265 |

0.64 | 0.1912 | 0.64 | 0.8848 | 0.0240 | 0.64 | 0.9256 | 0.0066 | 0.64 | 0.9377 |

0.68 | 0.2366 | 0.68 | 0.9029 | 0.0319 | 0.68 | 0.9375 | 0.0089 | 0.68 | 0.9477 |

0.72 | 0.2927 | 0.72 | 0.9193 | 0.0433 | 0.72 | 0.9481 | 0.0122 | 0.72 | 0.9566 |

0.76 | 0.3612 | 0.76 | 0.9341 | 0.0607 | 0.76 | 0.9578 | 0.0174 | 0.76 | 0.9647 |

0.80 | 0.4433 | 0.80 | 0.9477 | 0.0881 | 0.80 | 0.9665 | 0.0260 | 0.80 | 0.9720 |

0.84 | 0.5393 | 0.84 | 0.9600 | 0.1339 | 0.84 | 0.9744 | 0.0417 | 0.84 | 0.9786 |

0.88 | 0.6475 | 0.88 | 0.9713 | 0.2151 | 0.88 | 0.9817 | 0.0739 | 0.88 | 0.9847 |

0.92 | 0.7642 | 0.92 | 0.9817 | 0.3645 | 0.92 | 0.9883 | 0.1522 | 0.92 | 0.9902 |

**Table 2.**Statistical summary description, per year, highlighting in bold the highest values per statistic.

Min. | 1st Quartile | Median | Mean | 3rd Quartile | Max. | ||
---|---|---|---|---|---|---|---|

Minimum | 2016 | 0.33 | 0.398 | 0.525 | 0.544 | 0.66 | 0.81 |

2017 | 0.10 | 0.43 | 0.58 | 0.581 | 0.74 | 0.98 | |

2018 | 0.072 | 0.418 | 0.57 | 0.575 | 0.741 | 0.965 | |

2019 | 0.015 | 0.408 | 0.557 | 0.563 | 0.722 | 0.963 | |

2020 | 0.059 | 0.413 | 0.571 | 0.571 | 0.74 | 0.957 | |

2021 | 0.295 | 0.39 | 0.567 | 0.559 | 0.731 | 0.873 | |

Maximum | 2016 | 0.50 | 0.61 | 0.77 | 0.726 | 0.81 | 0.89 |

2017 | 0.24 | 0.69 | 0.81 | 0.774 | 0.87 | 0.98 | |

2018 | 0.182 | 0.674 | 0.816 | 0.771 | 0.878 | 0.973 | |

2019 | 0.079 | 0.652 | 0.808 | 0.755 | 0.866 | 0.972 | |

2020 | 0.216 | 0.668 | 0.815 | 0.77 | 0.873 | 0.973 | |

2021 | 0.449 | 0.686 | 0.805 | 0.753 | 0.847 | 0.958 |

**Table 3.**The p-values from the Kolmogorov-Smirnov goodness-of-fit test for some continuous distributions defined on the range $(0,1)$ adjusted to the minimum and maximum daily relative humidity data.

Distribution | Minimum | Maximum |
---|---|---|

Unit-Lindley | 0.769 | 0.797 |

Beta | 0.104 | 0.012 |

Simplex | 0.089 | 0.038 |

Kumaraswamy | 0.176 | 0.015 |

**Table 4.**Control limits of the unit-Lindley chart for the minimum and maximum daily humidity monitoring.

Tolerance | Minimum | Maximum | ||||
---|---|---|---|---|---|---|

($\mathit{\alpha}$) | LCL | CL ($\widehat{\mathit{\mu}}$) | UCL | LCL | CL ($\widehat{\mathit{\mu}}$) | UCL |

0.15 | 0.197 | 0.584 | 0.840 | 0.447 | 0.760 | 0.927 |

0.10 | 0.142 | 0.584 | 0.856 | 0.361 | 0.760 | 0.934 |

0.01 | 0.017 | 0.584 | 0.906 | 0.061 | 0.760 | 0.958 |

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**MDPI and ACS Style**

Fonseca, A.; Ferreira, P.H.; Nascimento, D.C.d.; Fiaccone, R.; Ulloa-Correa, C.; García-Piña, A.; Louzada, F.
Water Particles Monitoring in the Atacama Desert: SPC Approach Based on Proportional Data. *Axioms* **2021**, *10*, 154.
https://doi.org/10.3390/axioms10030154

**AMA Style**

Fonseca A, Ferreira PH, Nascimento DCd, Fiaccone R, Ulloa-Correa C, García-Piña A, Louzada F.
Water Particles Monitoring in the Atacama Desert: SPC Approach Based on Proportional Data. *Axioms*. 2021; 10(3):154.
https://doi.org/10.3390/axioms10030154

**Chicago/Turabian Style**

Fonseca, Anderson, Paulo Henrique Ferreira, Diego Carvalho do Nascimento, Rosemeire Fiaccone, Christopher Ulloa-Correa, Ayón García-Piña, and Francisco Louzada.
2021. "Water Particles Monitoring in the Atacama Desert: SPC Approach Based on Proportional Data" *Axioms* 10, no. 3: 154.
https://doi.org/10.3390/axioms10030154