# Statistical Distribution of TSS Event Loads from Small Urban Environments

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Monitoring Sites and Data

^{2}, 65 events), a high traffic street (HT, 2.5 ha, 16 events), a parking lot (PL, 2350 m

^{2}, 46 events) and a residential catchment (RC, 9.4 ha

^{2}, 23 events) are available. A summary of descriptive statistics is given in Table 1. Furthermore, Figure 1 depicts the distribution of site-specific TSS event loads as empirical cumulative distribution functions and box-plots, respectively.

#### 2.2. Theoretical Distribution Functions

#### 2.3. Distribution Fitting and Goodness-Of-Fit Assessment

_{i}the ith observation of variable X (i.e., TSS event loads), n is the total number of observations and f(|θ) the density function of the theoretical distribution function used. Parameters to be optimized are denoted by θ.

_{0}(“The sample follows a specified distribution”) can be accepted or must be rejected at a specified significance level. Alternatively, hypothesis H

_{A}is defined as “the sample does not follow a specified distribution”. Critical values for the acceptance decision of the KS test are calculated according to Equation (4) for sample sizes > 35. For sample sizes below 35, critical values are obtained from Reference [31].

#### 2.4. Monte-Carlo Resampling Strategy to Determine Minimum Sample Size

- Estimating parameters of lognormal distribution function by maximum likelihood taking all samples into account.
- Sampling k ($k\in \mathbb{N},0k\le n$) events from all events n with 1000 repetitions. If less than 1000 repetitions are possible, all possible combinations are taken into account (Equation (3)).$$repetitions=MIN\left(\left(\begin{array}{c}n\\ k\end{array}\right),1000\right)$$
- Computing of KS distance between empirical cumulative distribution function of sample and theoretical distribution function with estimated parameters for all repetitions.
- Computing of mean, standard deviations of KS distances for all repetitions.

## 3. Results

#### 3.1. Distribution Fitting

_{0}gets rejected). These two functions are not able to sufficiently reflect the initially steep gradient and subsequent moderate gradient of the empirical distribution (cf. Figure 1).

^{−2}, 2014: 0.8 gm

^{−2}, 2015: 1.34 gm

^{−2}). The optimized parameters of the Lognormal distribution for both sites highlight the individuality of each year as they strongly vary. This is also confirmed by the spread of goodness-of-fit values.

#### 3.2. Minimum Sample Size

_{0}(“The data follow the Lognormal distribution”) generally requires Kolmogorov-Smirnov’s D

_{n}to be approximately below the μ + 2σ threshold, which is satisfied for minimum sample sizes of about 40 at site FR and of roughly 30 at site PL. It can be legitimately assumed that simulated KS statistics follow a normal distribution which according to the empirical rule (The empirical rule states that for a normal distribution 99.7% of the data fall within three standard deviations, 95% are within two standard deviations and 68% fall within one standard deviation [31]) consequently implies that more than approximately 95% of samples lead to KS statistics lower than 0.188 at site FR and 0.211 at site PL. Narrowing the uncertainty range to the upper limit of μ + σ threshold results in KS statistics of 0.159 at site FR and 0.176 at site PL (approx. more than 68% of samples are within this range).

## 4. Discussion

#### 4.1. Distribution Fitting

#### 4.2. Minimum Sample Size

_{0}with high probability, it is suggested to choose at least the minimum of 40 samples because of (i) the chance of having a sample which can be statistically represented by the Lognormal distribution is high (>95%) and (ii) the mean of KS statistic in this case only slightly differs from the optimal value taking all samples into account (0.131 > 0.099 at site FR and 0.122 > 0.12 at site PL). However, the choice of criteria remains subjective and might be adapted as further data becomes available.

## 5. Conclusions

- The Lognormal distribution function is most expressive to approximate empirical TSS event load distributions at all experimental sites.
- Successfully derived and fitted distribution functions provide a closed characterization of TSS event load distributions allowing to intra- and extrapolate of probabilistic event characteristics not observed.
- A robust fitting should prioritize sample size over sampling period.
- Roughly 40 events are required to reasonably fit the Lognormal distribution. Using more samples potentially improves the goodness-of-fit but subsequently requires to extend the duration of cost-intensive monitoring campaigns.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Empirical cumulative distribution functions and boxplots of site-specific monitored Total Suspended Solids (TSS) event loads (FR: Flat Roof, HT: High Traffic Street, PL: Parking Lot, RC: Residential Catchment).

**Figure 2.**Approximation of empirical TSS event load distribution function with lognormal distribution function at all sites.

**Figure 3.**Approximation of empirical TSS event load distribution function grouped by year with lognormal distribution function at sites FR and PL.

**Figure 4.**Mean and regions of one and two standard deviations of Kolmogorov-Smirnov’s statistic as function of sample size from Monte-Carlo-based sampling for sites Flat Roof (FR) and Parking Lot (PL). Critical values for 90% confidence are indicated as black solid line.

**Table 1.**Descriptive statistics (min, 0.1-, 0.25-, 0.5-, 0.75-, 0.9-percentiles, max, mean, standard deviation) of site-specific Total Suspended Solids (TSS) event loads (FR: Flat Roof, HT: High Traffic Street, PL: Parking Lot, RC: Residential Catchment).

Site | n | TSS Event Loads (g m^{−2}) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

min | 0.1-Perc | 0.25-Perc | 0.5-Perc | 0.75-Perc | 0.9-Perc | Max | Mean | sd | ||

FR | 65 | 0.001 | 0.002 | 0.008 | 0.024 | 0.169 | 0.492 | 1.942 | 0.174 | 0.358 |

HT | 16 | 0.164 | 0.313 | 0.361 | 0.795 | 1.357 | 2.916 | 4.746 | 1.255 | 1.275 |

PL | 46 | 0.011 | 0.046 | 0.086 | 0.126 | 0.257 | 0.633 | 1.109 | 0.230 | 0.255 |

RC | 23 | 0.014 | 0.027 | 0.065 | 0.093 | 0.349 | 0.735 | 0.935 | 0.261 | 0.295 |

Name (Abbreviation) | Formula | Parameter |
---|---|---|

Exponential (exp) | $F\left(x\right)=\{\begin{array}{cc}\hfill 0,& \hfill x\le 0\\ \hfill 1-{e}^{-\alpha x},& \hfill x>0\end{array}$ | α (rate) |

Gamma (gamma) | $F\left(x\right)=\{\begin{array}{cc}\hfill 0,& \hfill x\le 0\\ \hfill \frac{{b}^{p}}{\mathsf{\Gamma}\left(p\right)}\times {{\displaystyle \int}}_{0}^{x}{t}^{p-1}{e}^{-bt}dt,& \hfill x>0\end{array}$ | p (shape), b (rate) |

Lognormal (lnorm) | $F\left(x\right)=\{\begin{array}{cc}\hfill 0,& \hfill x\le 0\\ \hfill \frac{1}{\sigma \sqrt{2\pi}}\times {{\displaystyle \int}}_{0}^{x}\frac{1}{t}{e}^{-\frac{1}{2}\left(\frac{\mathrm{ln}t-\text{}\mu}{\sigma}\right)}dt,& \hfill x0\end{array}$ | μ (meanlog), σ (sdlog) |

Weibull (weibull) | $F\left(x\right)=\{\begin{array}{cc}\hfill 0,& \hfill x\le 0\\ \hfill 1-{e}^{-\alpha {x}^{\beta}},& \hfill x>0\end{array}$ | α (scale), β (shape) |

**Table 3.**Goodness-of-fit statistics used to evaluate the fitting (F

_{n}denotes the empirical distribution function, F represents the fitted theoretical distribution function, sup abbreviates supremum which indicates the least element of x that is greater than or equal to all elements of x (“least upper bound”)).

Statistic (Abbreviation) | Formula | |
---|---|---|

Kolmogorov-Smirnov (KS) | ${D}_{n}=\begin{array}{c}sup\\ x\end{array}\left|{F}_{n}\left(x\right)-F\left(x\right)\right|$ | (2) |

Anderson-Darling (AD) | ${A}^{2}=n{{\displaystyle \int}}_{-\infty}^{\infty}\frac{{\left({F}_{n}\left(x\right)-F\left(x\right)\right)}^{2}}{F\left(x\right)\left(1-F\left(x\right)\right)}dF\left(x\right)$ | (3) |

**Table 4.**Results of fitting empirical TSS load distribution functions to theoretical distribution functions (FR: Flat Roof, HT: High Traffic Street, PL: Parking Lot, RC: Residential Catchment, LL: LogLikelihood, AD: Anderson-Darling statistic A

^{2}, KS: Kolmogorov-Smirnov statistic D

_{n}; bold values indicate best-fit).

Site | Distr. | Goodness-Of-Fit | Parameter Estimates (Standard Error) | ||||||
---|---|---|---|---|---|---|---|---|---|

LL | AD | KS | Rate | Shape | Meanlog | Sdlog | Scale | ||

FR | exp | 48.66 | 29.074 | 0.442 * | 5.747 (0.713) | - | - | - | - |

gamma | 88.29 | 2.254 | 0.186 * | 1.994 (0.504) | 0.347 (0.049) | - | - | - | |

lnorm | 89.9 | 0.806 | 0.099 | - | - | −3.69 (0.301) | 2.429 (0.213) | - | |

weibull | 92.05 | 1.123 | 0.131 | - | 0.484 (0.046) | - | - | 0.077 (0.021) | |

HT | exp | −19.64 | 0.379 | 0.153 | 0.797 (0.199) | - | - | - | - |

gamma | −19.25 | 0.394 | 0.136 | 1.068 (0.412) | 1.341 (0.428) | - | - | - | |

lnorm | −18.18 | 0.192 | 0.128 | - | - | −0.19 (0.228) | 0.912 (0.161) | - | |

weibull | −19.46 | 0.382 | 0.137 | - | 1.121 (0.208) | - | - | 1.316 (0.312) | |

PL | exp | 21.69 | 1.168 | 0.126 | 4.356 (0.642) | - | - | - | - |

gamma | 22.03 | 1.279 | 0.157 | 5.093 (1.175) | 1.169 (0.218) | - | - | - | |

lnorm | 25.31 | 0.398 | 0.116 | - | - | −1.96 (0.146) | 0.987 (0.103) | - | |

weibull | 21.72 | 1.203 | 0.137 | - | 1.030 (0.111) | - | - | 0.233 (0.035) | |

RC | exp | 7.91 | 1.011 | 0.222 | 3.833 (0.799) | - | - | - | - |

gamma | 8.1 | 0.681 | 0.189 | 3.283 (1.120) | 0.857 (0.219) | - | - | - | |

lnorm | 9.07 | 0.38 | 0.131 | - | - | −2.03 (0.259) | 1.243 (0.183) | - | |

weibull | 8.23 | 0.586 | 0.174 | - | 0.882 (0.142) | - | - | 0.244 (0.061) |

_{0}.

**Table 5.**Results of fitting empirical TSS load distribution functions grouped by year to lognormal distribution function (FR: Flat Roof, PL: Parking Lot, LL: LogLikelihood, AD: Anderson-Darling statistic A

^{2}, KS: Kolmogorov-Smirnov statistic D

_{n}).

Site | Year | n | Distr. | Goodness-Of-Fit | Parameter Estimates (Standard Error) | |||
---|---|---|---|---|---|---|---|---|

LL | AD | KS | Meanlog | Sdlog | ||||

FR | all years | 65 | lnorm | 89.9 | 0.806 | 0.099 | −3.69 (0.301) | 2.429 (0.213) |

2015 | 25 | lnorm | 24.54 | 0.64 | 0.138 | −2.99 (0.359) | 1.80 (0.254) | |

2014 | 17 | lnorm | 41.63 | 0.288 | 0.142 | −5.04 (0.786) | 3.24 (0.556) | |

2013 | 23 | lnorm | 32.52 | 0.365 | 0.12 | −3.45 (0.388) | 1.86 (0.274) | |

PL | all years | 46 | lnorm | 25.31 | 0.398 | 0.116 | −1.96 (0.146) | 0.987 (0.103) |

2014 | 30 | lnorm | 23.76 | 0.616 | 0.167 | −2.08 (0.161) | 0.88 (0.114) | |

2013 | 16 | lnorm | 2.93 | 0.243 | 0.105 | −1.72 (0.281) | 1.12 (0.199) |

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**MDPI and ACS Style**

Leutnant, D.; Muschalla, D.; Uhl, M.
Statistical Distribution of TSS Event Loads from Small Urban Environments. *Water* **2018**, *10*, 769.
https://doi.org/10.3390/w10060769

**AMA Style**

Leutnant D, Muschalla D, Uhl M.
Statistical Distribution of TSS Event Loads from Small Urban Environments. *Water*. 2018; 10(6):769.
https://doi.org/10.3390/w10060769

**Chicago/Turabian Style**

Leutnant, Dominik, Dirk Muschalla, and Mathias Uhl.
2018. "Statistical Distribution of TSS Event Loads from Small Urban Environments" *Water* 10, no. 6: 769.
https://doi.org/10.3390/w10060769