# Robust Nonparametric Methods of Statistical Analysis of Wind Velocity Components in Acoustic Sounding of the Lower Layer of the Atmosphere

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Procedure of Outlier Detection and Selection

#### 2.1. Adaptive Pendular Truncation Algorithm

_{n}≤ 1 is a monotonically decreasing function of n.

#### 2.2. Adaptive Pendular Truncation Algorithm

- Calculate ${\overline{T}}_{n}({\overrightarrow{x}}_{n})=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\mathsf{\phi}({x}_{i})}$,
- Calculate ${T}_{i}({x}_{i})=(\mathsf{\phi}({x}_{i})-{\overline{T}}_{n}({\overrightarrow{x}}_{n}))$,
- Sort the variables ${t}_{i}(n)=\left|{T}_{i}({x}_{i})\right|$, ${t}_{(1)}(n)<{t}_{(2)}(n)<\dots <{t}_{(n)}(n)$,
- Calculate ${S}_{n}=\frac{1}{n-1}{\displaystyle \sum _{j=1}^{n}{({T}_{i}({x}_{i}))}^{2}}$,
- Calculate ${L}_{n}=\frac{{S}_{n}}{{S}_{N}}$,
- Find the first-order differences ${\Delta}_{n}^{1}={L}_{n}-{L}_{n-1}$,
- Find the second-order differences ${\Delta}_{n}^{2}(n)={\Delta}_{n}^{1}(n)-{\Delta}_{n-1}^{1}(n)$,
- Remove the observation ${x}_{{i}_{0}}$ corresponding to ${t}_{(n)}(n)$ from the sample,
- Execute the above cycle from item 1 to item 9 for $n=N,N-1,\dots ,[\raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$2$}\right.]$.

#### Generalization of the Algorithm

## 3. Simulation

#### 3.1. Remote Outliers

#### 3.2. Asymmetry

#### 3.3. Correlation

_{S}= 0.93, and the sample correlation coefficient with outliers was R

_{S}= 0.42. The independence criterion based on the statistic ${T}_{obs}={R}_{S}\cdot \sqrt{N-2}/\sqrt{1-{R}_{S}^{2}}$ at the significance level $\alpha =0.01$ for the critical value ${T}_{crit}=2.88$ demonstrates that with outliers, ${T}_{obs}=2.04<{T}_{crit}=2.88$, and the zero hypothesis is accepted; without outliers, ${T}_{obs}=7.61>{T}_{crit}=2.88$, and the zero hypothesis is rejected.

_{S}= 0.91, and the criterion unambiguously rejects the zero hypothesis, but in the presence of two outliers, R

_{S}decreased by more than twice, down to R

_{S}= 0.42, and the criterion unambiguously accepts the zero hypothesis. Figure 5 shows the results of application of the APT algorithm for N = 18 + 2 outliers depending on the number of truncated observations n

_{1}. From Figure 5c, it can be seen that the algorithm detects and selects 2 outliers.

## 4. Statistical Analysis of Vertical Profiles of Wind Velocity Components from Results of Minisodar Measurements using the Pendular Truncation Algorithm

_{i}, in m/s (a), variances σ

_{i}

^{2}, in m

^{2}/s

^{2}(b), skewnesses K

_{i sc}(c), and kurtoses K

_{i kurt}(d), where i = x, y, z.

^{2}/s

^{2}. The red curves here show the results of calculations for the full sample, and the black curves show the results of calculations for the truncated sample using the APTA.

_{x}for individual sounding cycles (individual vertical profiles) becomes uncorrelated. Here, the influence of atmospheric turbulence and noise becomes pronounced.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Singal, S.P. Acoustic Remote-Sensing Applications; Springer-Verlag: Berlin, Germany, 1997; p. 585. [Google Scholar]
- Kallistratova, M.A.; Kon, A.I. Radioacoustic Sounding of the Atmosphere; Nauka: Moscow, Russia, 1985; p. 197. (In Russian) [Google Scholar]
- Krasnenko, N.P. Acoustic Sounding of the Atmosphere; Nauka: Novosibirsk, Russia, 1986; p. 168. (In Russian) [Google Scholar]
- Krasnenko, N.P. Acoustic Sounding of the Atmospheric Boundary Layer; Vodolei: Tomsk, Russia, 2001; p. 279. (In Russian) [Google Scholar]
- Bradley, S. Atmospheric Acoustic Remote Sensing: Principles and Applications; CRC Press Taylor & Francis Group: Boca Raton, FL, USA, 2007; p. 296. [Google Scholar]
- Simakhin, V.A.; Cherepanov, O.S.; Shamanaeva, L.G. Spatiotemporal dynamics of the wind velocity from minisodar measurement data. Russ. Phys. J.
**2015**, 58, 176–181. [Google Scholar] [CrossRef] - Hampel, F.; Ronchetti, E.; Rausseu, P.; Shtael, V. Robustness in Statistics. Approach Based on Influence Functions; MIR: Moscow, Russia, 1989; p. 512, (Russian translation). [Google Scholar]
- Shulenin, V.P. Methods of Mathematical Statistics; Publishing House of Scientific and Technology Literature: Tomsk, Russia, 2016; p. 260. (In Russian) [Google Scholar]
- Muthukrishnan, R.; Poonkuzhali, G. A comprehensive survey on outlier detection methods. Am. -Eurasian J. Sci. Res.
**2017**, 12, 161–171. [Google Scholar] - Chandola, V.; Banerjee, A.; Kumar, V. Anomaly detection: A survey. ACM Comput. Surv.
**2009**, 41, 1–83. [Google Scholar] [CrossRef] - Hodge, V.; Austin, J. A survey of outlier detection methodologies. Artif. Intell. Rev.
**2004**, 22, 85–126. [Google Scholar] [CrossRef] - Grubbs, F.E. Sample criteria for testing outlying observations. Ann. Math. Stat.
**1950**, 21, 27–58. [Google Scholar] [CrossRef] - Tietjen, G.L.; Moore, R.H. Some Grubbs-type statistics for the detection of several outliers. Technometrics
**1972**, 14, 583–597. [Google Scholar] [CrossRef] - Rosner, B. On the detection of many outliers. Technometrics
**1975**, 17, 221–227. [Google Scholar] [CrossRef] - Ferguson, T.S. On the rejection of outliers. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA, 20–30 July 1961; Volume 1, pp. 253–287. [Google Scholar]
- Orlov, A.I. Instability of parametric methods of rejection of sharply allocated observations. Zavod. Lab.
**1992**, 7, 40–42. (In Russian) [Google Scholar] - Rocke, D.M.; Woodruff, D.L. Identification of outliers in multivariate data. J. Am. Stat. Assoc.
**2012**, 91, 1047–1061. [Google Scholar] [CrossRef] - Shevlyakov, G.L.; Vilchevski, N.O. Robustness in Data Analysis: Criteria and Methods; VSP: Utrecht, The Netherlands, 2002; p. 315. [Google Scholar]
- Fedorov, V.A. Measurements with the “Volna-3” sodar of the parameters of radial components of wind velocity vector. Atmos. Ocean. Opt.
**2003**, 16, 151–155. [Google Scholar] - Simakhin, V.A.; Cherepanov, O.S. Detection and selection of signal outliers. In Proceedings of the XIX International Symposium “Atmospheric and Oceanic Optics. Atmospheric Physics”, Barnaul, Russia, 1–3 July 2013; pp. С221–С224. (In Russian). [Google Scholar]
- Huber, P.J. Robust Statistics; Willey: New York, NY, USA, 1981; p. 308. [Google Scholar]
- Simakhin, V.A. Robust Nonparametric Estimates; Lambert Academic Publishing: Saarbrücken, Germany, 2011; p. 292. [Google Scholar]
- Krasnenko, N.P.; Tarasenkov, M.V.; Shamanaeva, L.G. Spatiotemporal dynamics of the wind velocity from data of sodar measurements. Russ. Phys. J.
**2014**, 57, 1539–1546. [Google Scholar] [CrossRef]

**Figure 1.**Results of the application of the adaptive pendular truncation algorithm to distributions without outliers: (

**a**) Dependence of the statistic ${L}_{n}$ on the number n

_{1}of truncated observations, (

**b**) dependence of the statistic ${\Delta}_{n}^{1}$ on the number of truncated observations, and (

**c**) dependence of the statistic ${\Delta}_{n}^{2}$ on the number of truncated observations.

**Figure 2.**Results of application of the adaptive pendular truncation algorithm to distributions with asymmetric outliers: (

**a**) Dependence of the statistic ${L}_{n}$ on the number of truncated observations, (

**b**) dependence of the statistic ${\Delta}_{n}^{1}$ on the number of truncated observations, and (

**c**) dependence of the statistic ${\Delta}_{n}^{2}$ on the number of truncated observations.

**Figure 3.**Nonparametric estimates of the distribution density (

**a**) in the presence of internal and remote outliers and (

**b**) in the presence of internal outliers.

**Figure 4.**Results of application of the adaptive pendular algorithm to truncation of internal outliers: (

**a**) Dependence of the statistic L

_{n}on the number of truncated observations, (

**b**) dependence of the statistic ${\Delta}_{n}^{1}$ on the number of truncated observations, and (

**c**) dependence of the statistic ${\Delta}_{n}^{2}$ on the number of truncated observations.

**Figure 5.**Results of application of the algorithm of adaptive pendular truncation of outliers to correlation analysis: (

**a**) Dependence of the statistic ${\Delta}_{n}^{1}$ on the number of truncated observations, (

**b**) dependence of the statistic ${\Delta}_{n}^{2}$ on the number of truncated observations, and (

**c**) dependence of the statistic of the sample correlation coefficient R

_{S}on the number of truncated observations.

**Figure 6.**Vertical profiles of four moments of the x-component of the wind velocity ${V}_{x}$ retrieved from minisodar measurements in the morning (from 07:00 till 07:10, local time) using the standard minisodar data processing algorithm [23] (solid curves) and the adaptive pendular truncation algorithm (dashed curves): (

**a**) Average V

_{x}values, in m/s; (

**b**) variances, in m

^{2}/s

^{2}; (

**c**) skewnesses; and (

**d**) kurtoses.

**Figure 7.**Vertical profiles of four moments of the y-component of the wind velocity ${V}_{y}$ retrieved from minisodar measurements in the morning (from 07:00 till 07:10, local time) using the standard minisodar data processing algorithm [23] (solid curves) and the adaptive pendular truncation algorithm (dashed curves): (

**a**) Average ${V}_{y}$ values, in m/s; (

**b**) variances, in m

^{2}/s

^{2}; (

**c**) skewnesses; and (

**d**) kurtoses.

**Figure 8.**Vertical profiles of four moments of the z-component of the wind velocity ${V}_{z}$ retrieved from minisodar measurements in the morning (from 07:00 till 07:10, local time) using the standard minisodar data processing algorithm [23] (solid curves) and the adaptive pendular truncation algorithm (dashed curves): (

**a**) Average ${V}_{z}$ values, in m/s, (

**b**) variances, in m

^{2}/s

^{2}, (

**c**) skewnesses, and (

**d**) kurtoses.

**Figure 9.**Dependences of the autocorrelation functions retrieved using the APTA from the data of minisodar measurements of the x-component of the wind velocity ${V}_{x}$ at altitudes of 45 m (

**a**) and 180 m (

**b**) from 08:00 till 08:10, local time, on the lag τ.

**Figure 10.**Dependences of the structure functions of the x-component of the wind velocity ${V}_{x}$ retrieved using the APTA from the data of minisodar measurements at altitudes of 35 m (

**a**) and 175 m (

**b**) from 08:00 till 08:10, local time, on the lag τ.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Krasnenko, N.; Simakhin, V.; Shamanaeva, L.; Cherepanov, O.
Robust Nonparametric Methods of Statistical Analysis of Wind Velocity Components in Acoustic Sounding of the Lower Layer of the Atmosphere. *Symmetry* **2019**, *11*, 961.
https://doi.org/10.3390/sym11080961

**AMA Style**

Krasnenko N, Simakhin V, Shamanaeva L, Cherepanov O.
Robust Nonparametric Methods of Statistical Analysis of Wind Velocity Components in Acoustic Sounding of the Lower Layer of the Atmosphere. *Symmetry*. 2019; 11(8):961.
https://doi.org/10.3390/sym11080961

**Chicago/Turabian Style**

Krasnenko, Nikolay, Valerii Simakhin, Liudmila Shamanaeva, and Oleg Cherepanov.
2019. "Robust Nonparametric Methods of Statistical Analysis of Wind Velocity Components in Acoustic Sounding of the Lower Layer of the Atmosphere" *Symmetry* 11, no. 8: 961.
https://doi.org/10.3390/sym11080961