# Assessing Inhomogeneities in Extreme Annual Rainfall Data Series by Multifractal Approach

^{1}

^{2}

^{*}

## Abstract

**:**

_{q}and the multifractal spectra f(α) were obtained, and their main parameters were used to decide whether or not a break point existed.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Source

^{2}, and is located in the Tiber river basin, which crosses the region from the north to the south-west. Its landscape is partially mountainous due to the presence of the Apennine Mountains that reach up to 2000 m a.s.l., and partially hilly in the central and western areas with altitudes ranging from 100 to 800 m a.s.l.

^{2}). For this study, only the stations with at least 50 continuous years of extreme 24hour duration rainfall data have been considered (Table 1): Bastia Umbra, Bevagna, Nocera Umbra, Petrelle, Ponte Nuovo di Torgiano, Spoleto, Terni and Todi. Figure 1 shows the location of the eight sites considered in the Umbria region.

#### 2.2. Methodology

#### 2.2.1. Homogeneity Tests

_{1}, y

_{2}, …, y

_{n}, with n the number of data, with specific characteristics. Some tests are known as parametric because they assume that the analyzed variable is normally distributed. Those that do not make this assumption are called non-parametric tests.

_{i}(i = 1, …, n) of the testing variable y are independent and identically distributed and the series is considered as homogeneous. Under the alternative hypothesis, the series shows a break in the mean and is considered as inhomogeneous. Moreover, there are tests that are able to give information about the year the break occurred, whereas those that assume the series is not randomly distributed under alternative hypothesis cannot provide this information.

- (1)
**Buishand Range test**This test is based on the adjusted partial sums or cumulative derivations from the mean of the n data points, $\overline{y}$:$${S}_{0}^{*}=0;{S}_{k}^{*}={\displaystyle \sum}_{i=1}^{k}({y}_{i}-\overline{y}),k=1,\text{}\dots ,\text{}n$$For a homogeneous record, ${S}_{k}^{*}$ values fluctuate around zero. The values of ${S}_{k}^{*}$ can be rescaled by the sample standard deviation, ${D}_{y}$:$${S}_{k}^{**}=\frac{{S}_{k}^{*}}{{D}_{y}},k=0,\dots ,\text{}n$$The Q, which is sensitive to departures from homogeneity, can then be obtained as:$$Q=ma{x}_{0\le k\le n}\left|{S}_{k}^{**}\right|$$For different significance levels, the critical values for the test statistics (Qc) depend on the number of data and can be found in [30].- (2)
**Standard Normal Homogeneity test**This test [31] compares the mean of the first k years of data with the last (n − k) years by using the T_{k}of the statistics:$${T}_{k}=k\overline{{z}_{1}}+(n-k)\overline{{z}_{2}},k=1,2,\dots n$$$$\overline{{z}_{1}}=\frac{1}{k}\text{}{\displaystyle \sum}_{i=1}^{k}\frac{{y}_{i}-\overline{y}}{s}\text{},\overline{{z}_{2}}=\frac{1}{n-k}\text{}{\displaystyle \sum}_{i=k+1}^{n}\frac{{y}_{i}-\overline{y}}{s}\text{},and\text{}s=\frac{1}{n}\text{}{\displaystyle \sum}_{i=1}^{n}{({y}_{i}-\overline{y})}^{2}$$The year k shows a break if the value of T_{k}is the maximum. If the ${T}_{0}={\mathrm{max}}_{1\le k\le n}{T}_{k}$ is greater than the critical values [32], the null hypothesis is rejected.- (3)
**Pettitt test**The Pettitt test [33] is based on the ranking r_{i}of the y_{i}values. The ranks r_{i}are obtained by ordering the data in crescent order, so that the smallest one gets ranked 1 and the highest gets the n-rank. The U_{k}is then obtained as:$${U}_{k}=2{\displaystyle \sum}_{i=1}^{k}{r}_{i}-k(n+1)k=1,\text{}2,\text{}\dots ,\text{}n$$If a break occurs in the year K, then the U_{k}is:$${U}_{K}=ma{x}_{1\le k\le n}\left|{U}_{k}\right|$$The statistical significance of the break point is checked by comparing the value of ${U}_{K}$ (Equation (7)) to its theoretical value:$${U}_{K}{}_{\alpha}={(-\frac{Ln(\alpha ({n}^{3}+{n}^{2}))}{6})}^{1/2}$$- (4)
**Sequential Mann–Kendall test**This test is usually applied to evaluate trends in data series and also to check the moment when they start being significant [34,35,36]. It calculates two series of statistic values: one (U(t)) for the progressive temporal data set (y_{1}, y_{2}, …, y_{n}) and the other (U’(t)) for the corresponding regressive data set (y_{n}, y_{n−1}, …, y_{1}), both with an average value of zero and a standard deviation of one. Usually, the ranks of the y_{i}values are preferred.The t_{k}has to be obtained by:$${t}_{k}={\displaystyle \sum}_{i=1}^{k}{n}_{i},(2\le k\le n)$$_{i}being the number of cases for which ${y}_{i}>{y}_{j}\text{}(j=1,\text{}2,\text{}\dots ,\text{}i-1).$Under the null hypothesis, it is assumed that no trend exists, and t_{k}follows a normal distribution with average and variance values given by:$$E\left[{t}_{k}\right]=\frac{k(k-1)}{4}Var\text{}\left[{t}_{k}\right]=\frac{k(k-1)(2k+5)}{72}(2\le k\le n)$$The sequential values of U(t_{k}) are finally obtained as:$$U({t}_{k})=\frac{{t}_{k}-E\left[{t}_{k}\right]}{\sqrt{Var({t}_{k})}},\text{}(k=2,\text{}3,\text{}\dots ,\text{}n)$$A positive value of U(t_{k}) shows a positive trend, whereas a negative trend appears if the U(t_{k}) value is negative.The null hypothesis is rejected if $\left|U({t}_{k})\right|>{Z}_{(1-\frac{\alpha}{2})}$ where ${Z}_{(1-\frac{\alpha}{2})}$ is the critical value of the typified normal distribution with a probability higher than $\frac{\alpha}{2}$. For a significance level α = 5%, the critical value of ${Z}_{(1-\frac{\alpha}{2})}$ is 1.9604.The values of U’(${t}_{k}$) are also obtained from regressive temporal data series, t_{k’}, E[t_{k’}], Var[t_{k’}] by using Equations (9)–(11), and finally:$${U}^{\prime}({t}_{k})=-U({t}_{{k}^{\prime}})with\text{}{k}^{\prime}=n+1-j(j=1,\text{}2,\text{}\dots ,\text{}n)$$By plotting the curves of the U(t) and U’(t) as a function of the year, a break point is detected where the curves cross and diverge.- (5)
**Mann–Whitney U test**This test can detect a step change in a time series by assessing the difference in the means of two sub-series arising from splitting the complete data series. The original time series (y_{i}, i = 1, 2, …, n) is broken into two subseries, one from y_{1}to y_{n1}and the other from y_{n1+1}to y_{n}, of sizes n_{1}and n_{2}(n − n1 + 1), respectively. A new data series, z_{t}(t = 1, …, n) is obtained by rearranging the original time series y_{i}in increasing order of magnitude. The Mann–Whitney test statistic is obtained by:$${Z}_{c}=\frac{{{\displaystyle \sum}}_{i=1}^{{n}_{1}}R({y}_{i})-{n}_{1}({n}_{1}+{n}_{2}+1)/2}{{\left[{n}_{1}{n}_{2}({n}_{1}+{n}_{2}+1/12\right]}^{0.5}}$$_{i}) is the rank of observation y_{i}in the series z_{t}. If $\left|{Z}_{c}\right|>{Z}_{(1-\frac{\alpha}{2})}$ the hypothesis of equal means in both subseries is rejected, being ${Z}_{(1-\frac{\alpha}{2})}$ the (1 − α/2) quantile of the normal distribution.- (6)
**Cumulative Sum test**The distribution-free cumulative sum technique was proposed by [37] to detect if a significant step change occurred in a given time series at a significance level α = 0.05 [38]. The null hypothesis considers that no step change exists. For the given data series y_{i}(i = 1, …, n), the test statistics V_{k}is obtained as:$${V}_{k}={\displaystyle \sum}_{i=1}^{k}sign\text{}({y}_{i}-{y}_{median})$$_{median}is the median of the time series, and the sign function is:$$\mathrm{sign}\text{}({y}_{i}-{y}_{median})=\{\begin{array}{c}1if{y}_{i}{y}_{median}\\ 0if{y}_{i}={y}_{median}\\ -1if{y}_{i}{y}_{median}\end{array}$$A step-change point consists of any year when the maximal or minimal V_{k}falls outside the 95% confidence limit of ±1.36$\sqrt{n}$ [38].

#### 2.2.2. Multifractal Characterization

_{q}, also known as the Rénji spectrum [42] by:

_{q}for q = 0 is known as the box-counting dimension or the fractal dimension of the system. For q = 1 the information entropy is obtained. The so-called correlation dimension is obtained for q = 2.

_{q}. The relation between the Rénji spectrum (D

_{q}) and τ

_{q}is obtained as:

## 3. Results

#### 3.1. Break Point Detection

_{c}, and an inhomogeneity is detected in 1977. The SNH test results show that no inhomogeneity is present in any of the data series analyzed, considering that all the statistics T

_{0}are below their critical values T

_{0c}.

_{c}only for the Spoleto extreme data series. A break point is then detected in Spoleto in 1969. For the rest of the data series analyzed in the Umbria region, no inhomogeneities were detected by the PT test.

_{k}) curve. The progressive U(t

_{k}) and regressive U’(t

_{k}) curves cross and diverge in 1980 where a break point is detected; the average extreme annual rainfall for the 1959 to 1980 period is 70.65 mm while for the period from 1981 to 2017 it is 60.21 mm. After 1981, the trend is decreasing and is significant starting from 2003 when the U(t

_{k}) values become lower than the confidence level limit considered ($\alpha =0.05$).

_{k}values obtained for both stations data series are always below the confidence limit considered, being close for year 1969 in Bastia Umbra and for 1960 and 1962 in the Terni station.

#### 3.2. Multifractal Behavior

_{0}is equal to 1. This value is related to the number of boxes needed to cover the fractal object or system under analysis. For all the sites and series, the value of D

_{0}= 1 fills the entire 1D domain.

_{1}, describes the degree of heterogeneity of the measure and characterizes the distribution and intensity of singularities with respect to the mean [43,44]. For the complete extreme rainfall data sets, the most heterogeneous is the one at the Spoleto station, followed by the Petrelle, Nocera Umbra and Terni stations. This behavior can also be analyzed through the value of (D

_{0}-D

_{1}). The greater the value, the less uniform is the data set. The most uniform data series are those at Nocera Umbra and Terni stations, where fewer differences in the data than those in Petrelle and Spoleto are expected.

_{q}for several q values (Rénji spectrum) is shown in Figure 5. The multifractal spectra f(α) were also obtained for all the available data series (Figure 6) and several multifractal parameters are shown in Table 6.

## 4. Discussion

_{0}-D

_{1}), more uniform data are found before the break point than after it. Moreover, the former data set is also more predictable than the latter, according to the (D

_{0}-D

_{2}) values.

_{1}, D

_{2}, (D

_{0}-D

_{1}) and (D

_{0}-D

_{2}) are found, showing similar uniformity and degree of predictability in the data sets.

_{0}-D

_{1}) values, and more predictable than the one after the break point according to the (D

_{0}-D

_{2}) values.

_{1}, D

_{2}, and their differences with D

_{0,}are found for the complete data set of Spoleto, and those obtained with the extreme rainfall data set after 1980. Completely different values are found for both data series obtained before and after the break point in 1969, and those obtained for the complete data set.

_{q}for the complete data series is almost coincident with the one obtained for the rainfall data series before the break point. Different values were obtained for the rainfall data series after the break point. The similarity in the behavior of the complete data series and the one before the potential break point means that no real break point exists since the addition of the data after it does not change the general pattern.

_{q}values are considered. Nevertheless, for negative q values, similar D

_{q}functions are obtained for the data sets before and after the break points in 1969 and 1980. These similitudes are maintained for positive q values for D

_{q}functions of data sets before both break points. Nevertheless, D

_{q}functions are different for positive q values for data sets after the break points, with the one after the inhomogeneity at 1980 being the closest to the D

_{q}function for the complete data series. This behavior means that the break point took place in 1969.

_{q}function are found for the complete data series and those before and after the break point year of 1957, respectively. For the former series, similar values are found for q positive values whereas for the latter the similarities arise for negative q values. Different D

_{q}functions are obtained for those series derived from the break point in1962. This behavior implies that the inhomogeneity is placed at 1962 in the Terni extreme rainfall data series.

_{min}, α

_{max}, a

_{s}and w) were obtained when analyzing both the complete data set and the one before the break point.

_{s}), whereas the complete data series shows a greater influence of less extreme events, with a positive value of a

_{s}. The width of the spectra (w) is also different with a loss of richness (lower multifractality) in the complete data set compared to the one before the break point, maybe due to the influence of the data after the break point, where the richness has the lowest value.

_{s}), which implies the dominance of extreme events in all the data series. The values closer to the αmin and αmax parameters are found for the complete data series and for the series before and after 1957, and thus a similar influence by extreme and smooth events can be expected for all the series. Similar complex processes are found according to the values of w. This behavior is not present for data series before and after the break point of 1962, where a loss of multifractality is found after the break point compared to the data series before it. This last data series also shows the strong influence of extreme events according to both the values of a

_{s}and αmin.

_{0}and w from the multifractal spectrum to assess changes in climate data series. For all the data sets studied in this work, the values of α

_{0}are always equal to 1 and thus no possible changes can be explained based on these values. Nevertheless, there are differences in the values obtained for the parameter that describes the degree of multifractality of the data series, w. Thus, the differences in the values of this multifractal parameter will be of great importance in deciding if a break point can really be identified in a data series.

_{1}and D

_{2}. Thus, the most important inhomogeneity is the one detected by the statistical tests in 1969. For the Terni station, the greatest differences in multifractal properties are found between the complete data series and those obtained considering the inhomogeneity in 1962, especially for the value of w, D

_{1}and D

_{2}. Therefore, no break point should be considered in 1957.

## 5. Conclusions

_{1}, the correlation fractal dimension D

_{2}, and the complexity of the data set given by w provide the bases for the decision criteria. If the complete data series maintained the values of D

_{1}, D

_{2}or w with respect to any of the split series before or after the break point, it was considered that no inhomogeneity existed. The multifractal behavior of the split series was also found in the complete series. On the contrary, if the values of the multifractal parameters were different among the data series, the break point under consideration was accepted.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Step changes detected by the Mann–Whitney U test at the Nocera Umbra, Spoleto, Petrelle and Terni stations.

**Figure 5.**Multifractal dimensions functions D

_{q}for the complete data series and also for the rainfall series before and after the break points detected in the Nocera Umbra, Petrelle, Spoleto and Terni stations.

**Figure 6.**Multifractal spectra f(α) for the complete rainfall data series and also for the rainfall data series before and after the break points detected in the Nocera Umbra, Petrelle, Spoleto and Terni stations.

Station. | Period (Years) Considered |
---|---|

Bastia Umbra | from 1966 to 2017 |

Bevagna | from 1967 to 2017 |

Nocera Umbra | from 1961 to 2017 |

Petrelle | from 1951 to 2017 |

Ponte Nuovo di Torgiano | from 1954 to 2017 |

Spoleto | from 1949 to 2017 |

Terni | from 1952 to 2017 |

Todi | from 1948 to 2017 |

**Table 2.**Results obtained for the parametric Buishand range (BR) and Standard Normal Homogeneity (SNH) tests; the statistics Q and T

_{0}and their critical values Q

_{c}and T

_{0c}are also shown.

Station | BRT | SNHT | ||
---|---|---|---|---|

Q | Q_{c} | T_{0} | T_{0c} | |

Bastia Umbra | 0.862 | 1.271 | 6.689 | 8.480 |

Bevagna | 0.746 | 1.270 | 2.626 | 8.456 |

Nocera | 0.650 | 1.273 | 3.094 | 8.586 |

Petrelle | 0.978 | 1.277 | 3.970 | 8.768 |

Ponte | 0.615 | 1.276 | 2.182 | 8.717 |

Spoleto | 1.295 | 1.277 | 7.546 | 8.784 |

Terni | 1.088 | 1.276 | 7.505 | 8.735 |

Todi | 0.739 | 1.278 | 3.794 | 8.814 |

Station | Pettitt Test | |
---|---|---|

K | K_{0.05} | |

Bastia Umbra | 213.0 | 267.5 |

Bevagna | 170.0 | 259.9 |

Nocera | 194.0 | 306.7 |

Petrelle | 322.0 | 390.4 |

Ponte | 152.0 | 356.1 |

Spoleto | 435.0 | 399.1 |

Terni | 276.0 | 373.1 |

Todi | 237.0 | 461.8 |

Station | Parametric Tests | Non-Parametric Tests | ||||
---|---|---|---|---|---|---|

BR | SNH | PT | SQMK | MWU | CUSUM | |

Bastia Umbra | - | - | - | - | - | - |

Bevagna | - | - | - | - | - | - |

Nocera Umbra | - | - | - | - | 1968 | - |

Petrelle | - | - | - | - | 1994 | - |

Ponte | - | - | - | - | - | - |

Spoleto | 1977 | - | 1969 | 1980 | 1969 | - |

Terni | - | - | - | 1957 | 1962 | - |

Todi | - | - | - | - | - | - |

**Table 5.**Main results obtained for the Rénji spectrum of the extreme rainfall data series for which break points were detected by the statistical tests.

Station | Complete Data Series | Before Break Point Series | After Break Point Series | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

D_{1} | D_{2} | D_{0}-D_{1} | D_{0}-D_{2} | D_{1} | D_{2} | D_{0}-D_{1} | D_{0}-D_{2} | D_{1} | D_{2} | D_{0}-D_{1} | D_{0}-D_{2} | ||

Nocera Umbra | 1968 | 0.980 | 0.958 | 0.020 | 0.042 | 0.977 | 0.953 | 0.023 | 0.047 | 0.984 | 0.967 | 0.016 | 0.033 |

Petrelle | 1994 | 0.988 | 0.975 | 0.012 | 0.025 | 0.985 | 0.971 | 0.015 | 0.029 | 0.988 | 0.977 | 0.012 | 0.023 |

Terni | 1957 | 0.980 | 0.960 | 0.020 | 0.040 | 0.980 | 0.959 | 0.020 | 0.041 | 0.979 | 0.955 | 0.021 | 0.045 |

1962 | 0.965 | 0.930 | 0.035 | 0.070 | 0.986 | 0.972 | 0.014 | 0.028 | |||||

Spoleto | 1969 | 0.991 | 0.981 | 0.009 | 0.019 | 0.989 | 0.977 | 0.011 | 0.023 | 0.989 | 0.978 | 0.011 | 0.022 |

1980 | 0.989 | 0.977 | 0.011 | 0.023 | 0.990 | 0.981 | 0.010 | 0.019 |

**Table 6.**Main results obtained for the multifractal spectrum of the extreme rainfall data series for which break points were detected by the statistical tests.

Station | Complete Data Series | Before Break Point Series | After Break Point Series | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

α_{min} | α_{max} | a_{s} | w | α_{min} | α_{max} | a_{s} | w | α_{min} | α_{max} | a_{s} | w | ||

Nocera Umbra | 1968 | 0.74731 | 1.16829 | −0.43561 | 0.42098 | 0.74697 | 1.14838 | −0.56416 | 0.40142 | 0.81372 | 1.14083 | −0.29407 | 0.32712 |

Petrelle | 1994 | 0.85847 | 1.18429 | 0.09650 | 0.32581 | 0.82446 | 1.18982 | −0.08529 | 0.36536 | 0.88417 | 1.11864 | −0.09868 | 0.23448 |

Terni | 1957 | 0.80952 | 1.15864 | −0.17081 | 0.34913 | 0.79642 | 1.17564 | −0.30561 | 0.37922 | 0.77523 | 1.15035 | −0.30485 | 0.37513 |

1962 | 0.71493 | 1.26671 | −0.16686 | 0.55178 | 0.79981 | 1.13520 | −0.38638 | 0.33539 | |||||

Spoleto | 1969 | 0.87427 | 1.11828 | −0.13999 | 0.24401 | 0.84608 | 1.16869 | 0.02276 | 0.32261 | 0.84776 | 1.14035 | −0.17251 | 0.29258 |

1980 | 0.84608 | 1.16869 | 0.02276 | 0.32261 | 0.88881 | 1.13704 | −0.00022 | 0.24823 |

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**MDPI and ACS Style**

García-Marín, A.P.; Estévez, J.; Morbidelli, R.; Saltalippi, C.; Ayuso-Muñoz, J.L.; Flammini, A.
Assessing Inhomogeneities in Extreme Annual Rainfall Data Series by Multifractal Approach. *Water* **2020**, *12*, 1030.
https://doi.org/10.3390/w12041030

**AMA Style**

García-Marín AP, Estévez J, Morbidelli R, Saltalippi C, Ayuso-Muñoz JL, Flammini A.
Assessing Inhomogeneities in Extreme Annual Rainfall Data Series by Multifractal Approach. *Water*. 2020; 12(4):1030.
https://doi.org/10.3390/w12041030

**Chicago/Turabian Style**

García-Marín, Amanda P., Javier Estévez, Renato Morbidelli, Carla Saltalippi, José Luis Ayuso-Muñoz, and Alessia Flammini.
2020. "Assessing Inhomogeneities in Extreme Annual Rainfall Data Series by Multifractal Approach" *Water* 12, no. 4: 1030.
https://doi.org/10.3390/w12041030