# Spatial Hurst–Kolmogorov Clustering

^{*}

## Definition

## 1. Introduction

#### 1.1. HK Clustering

#### 1.2. Stochastic Analysis in the Scale Domain

## 2. Methodology

## 3. Illustrative Applications

#### 3.1. One-Dimensional Turbulence

^{6}records, and after a data-homogenization procedure, we were able to analyze them as if they were recorded at the same location and with similar initial conditions (for more details on the homogenization, see [12]). In Figure 4, we show the standardized second-order metrics of the climacogram, autocovariance and variogram, i.e., c(h) − c(0), and power spectrum, where it can be seen that the variogram and the climacogram exhibit the smallest uncertainty at the long-range scales and, in particular, on the double logarithmic slope that is attributed to the strength of the long-range dependence (i.e., the magnitude of the Hurst parameter). However, since the variogram has the limitation that the double logarithmic slope tends to zero, only from the climacogram is it possible to robustly estimate the Hurst parameter (for comparison between the two metrics, see [32]).

#### 3.2. Benchmark Analysis of Two-Dimensional Art Paintings

#### 3.3. Two-Dimensional Rock-Formations

#### 3.4. Spatio-Temporal Wind Speed of a Hurricane

#### 3.5. Spatio-Temporal Evolution of Clustering

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## Entry Link on the Encyclopedia Platform

## References

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**Figure 1.**(

**a**) In 1951, H.E. Hurst discovered the clustering behavior in nature while (

**b**) A.N. Kolmogorov proposed a decade before a stochastic process that describes this clustering behavior. Source: Wikipedia.

**Figure 2.**Hurst–Kolmogorov (HK) dynamics present in the annual minimum water level of the Nile River as a result of the perpetual change of Earth’s climate, and as compared to a roulette timeseries resembling a white noise process. Source: [14] (supplementary material).

**Figure 3.**(

**a**,

**c**) Examples of data series with different statistical characteristics in one dimension. (

**b**,

**d**) Standardized climacograms of the data series shown in (

**a**,

**c**).

**Figure 4.**Standardized second-order stochastic metrics for the longitudinal velocity of the grid-turbulence dataset: (

**a**) climacogram; (

**b**) autocovariance; (

**c**) variogram; (

**d**) power spectrum. Source: [56].

**Figure 5.**Benchmark image analysis; (

**a**) white noise; (

**b**) image with clustering; (

**c**) art painting. Source: [35].

**Figure 6.**(

**a**) Climacograms of the benchmark images; (

**b**) standardized climacograms (i.e., divided by the variance of the image) of the benchmark images. Source: [35].

**Figure 7.**Images of sandstone as seen (

**a**) from the SEM (50 μm); (

**b**) from a polarizing microscope, (3.5 mm); (

**c**) from a hand specimen (with length approximately 5 cm); and (

**d**) from a field outcrop (1 m). For more information on the source, description and processing of the images, see [32].

**Figure 8.**Standardized climacograms of sandstone images depicted at four different scales and the fitted 2D HK model (source: [32]).

**Figure 9.**The image on the left shows Hurricane Sandy from a satellite view (source: www.nhc.noaa.gov/data; accessed on 15 January 2013), on which wind velocity and directions measurements are based. The images on the right represent the wind speed magnitude (in m/s) across a 150 × 150 grid, with 10 km resolution, centralized to the eye of the hurricane observed on (from upper left to lower right) 23, 25, 27 and 29 October 2012 (more details in [40]).

**Figure 10.**Climacograms (i.e., aggregated variance of brightness vs. spatial scale in units k × 10 km) for the (

**a**) 23rd, (

**b**) 25th, (

**c**) 27th and (

**d**) 29th day of October 2012. Source: [40].

**Figure 11.**Benchmark of image analysis, the evolution of the universe [68]: (

**a**) 500 million years after Big Bang image with faint clustering, average brightness 0.45; (

**b**) 5000 million years after Big Bang image with clustering, average brightness 0.37; (

**c**) 10,000 million years after Big Bang image with intense clustering, average brightness 0.33. Source: [39].

**Figure 12.**Example of the stochastic analysis of a 2D picture, in escalating spatial scales, as shown in red color on the left. The variability of the grouped pixels at different scales are used to construct the climacogram’s images; (

**a**–

**c**) correspond to different time frames as given in Figure 11. Source: [39].

**Figure 13.**(

**a**) Climacograms of the benchmark images; (

**b**) standardized climacograms of the benchmark images. A standardized climacogram is not helpful to evaluate the range of the evolution of clustering but is helpful to estimate the curves’ slope for further analysis. Source: [39].

**Figure 14.**(

**a**) Cumulative areas underneath each climacogram for each scale; (

**b**) rate of alteration. Source: [39].

**Figure 15.**Example of image analysis, evolution of the universe (zoomed view): Images of millions of years after the Big Bang (

**a**) 1000; (

**b**) 2000; (

**c**) 3000; (

**d**) 6000. Source: [39].

**Table 1.**Climacogram definition and expressions for a 2D continue process, a discretized one, a common estimator for the climacogram and the estimated value, based on this estimator.

Type | 2D Climacogram | |
---|---|---|

continuous | $\gamma \left({k}_{1},{k}_{2}\right):=\frac{\mathrm{Var}\left[{{\displaystyle \int}}_{{t}_{1}}^{{t}_{1}+{k}_{1}}{{\displaystyle \int}}_{{t}_{L}}^{{t}_{L}+{k}_{L}}\underset{\_}{x}\left({\xi}_{1,}{\xi}_{2}\right)\mathrm{d}{\xi}_{1}\mathrm{d}{\xi}_{2}\right]}{{\left({k}_{1}{k}_{2}\right)}^{2}}$ | (T1-1) |

discretized | ${\gamma}^{\left(\Delta \right)}\left({\kappa}_{1}{\Delta}_{1},{\kappa}_{2}{\Delta}_{2}\right):=\gamma \left({k}_{1},{k}_{2}\right)$ | (T1-2) |

typical estimator | $\underset{\_}{\widehat{\gamma}}\left({\kappa}_{1}{\Delta}_{1},{\kappa}_{2}{\Delta}_{2}\right)=\frac{{\kappa}_{1}{\kappa}_{2}}{{n}_{1}{n}_{2}}{\displaystyle {\displaystyle \sum}_{{r}_{2}=1}^{[{n}_{2}/{\kappa}_{2}]}}{\displaystyle {\displaystyle \sum}_{{r}_{1}=1}^{[{n}_{1}/{\kappa}_{1}]}}\left(\frac{1}{{\kappa}_{1}{\kappa}_{2}}\right({\displaystyle {\displaystyle \sum}_{{i}_{2}={\kappa}_{2}({r}_{2}-1)+1}^{{\kappa}_{2}{r}_{2}}}{\displaystyle {\displaystyle \sum}_{{i}_{1}={\kappa}_{1}({r}_{1}-1)+1}^{{\kappa}_{1}{r}_{1}}}{\underset{\_}{x}}_{{i}_{1},{i}_{2}}^{\left({\Delta}_{1},{\Delta}_{2}\right)})-\frac{{{\displaystyle \sum}}_{{i}_{2}=1}^{{n}_{2}}{{\displaystyle \sum}}_{{i}_{1}=1}^{{n}_{1}}{\underset{\_}{x}}_{{i}_{1},{i}_{2}}^{\left({\Delta}_{1},{\Delta}_{2}\right)}}{{n}_{1}{n}_{2}}),\mathrm{where}[{n}_{l}/{\kappa}_{l}]\mathrm{is}\mathrm{the}\mathrm{floor}\mathrm{of}{n}_{l}/{\kappa}_{l}$ | (T1-3) |

expected value of estimator | $\mathrm{E}\left[\underset{\_}{\widehat{\gamma}}\left({\kappa}_{1}{\Delta}_{1},{\kappa}_{2}{\Delta}_{2}\right)\right]=\gamma \left({\kappa}_{1}{\Delta}_{1},{\kappa}_{2}{\Delta}_{2}\right)-\gamma \left({n}_{1}{\Delta}_{1},{n}_{2}{\Delta}_{2}\right)$ | (T1-4) |

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**MDPI and ACS Style**

Dimitriadis, P.; Iliopoulou, T.; Sargentis, G.-F.; Koutsoyiannis, D. Spatial Hurst–Kolmogorov Clustering. *Encyclopedia* **2021**, *1*, 1010-1025.
https://doi.org/10.3390/encyclopedia1040077

**AMA Style**

Dimitriadis P, Iliopoulou T, Sargentis G-F, Koutsoyiannis D. Spatial Hurst–Kolmogorov Clustering. *Encyclopedia*. 2021; 1(4):1010-1025.
https://doi.org/10.3390/encyclopedia1040077

**Chicago/Turabian Style**

Dimitriadis, Panayiotis, Theano Iliopoulou, G.-Fivos Sargentis, and Demetris Koutsoyiannis. 2021. "Spatial Hurst–Kolmogorov Clustering" *Encyclopedia* 1, no. 4: 1010-1025.
https://doi.org/10.3390/encyclopedia1040077