# On the Information Content of Coarse Data with Respect to the Particle Size Distribution of Complex Granular Media: Rationale Approach and Testing

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## Abstract

**:**

## 1. Introduction

## 2. The Differential Information Equation for the PSD

## 3. Materials and Methods

#### 3.1. Data

#### 3.2. Simulation and Testing

#### 3.2.1. Triplet Description

#### 3.2.2. Simulation Algorithm

- take any ${x}_{0}\in I$ as a starting point,
- choose randomly, with probability ${p}_{i}$, one of the three linear transformations ${\xi}_{i}$, $i=1,2,3$ and calculate the next point in the simulation ${\xi}_{i}\left({x}_{0}\right)={x}_{1}$,
- repeat step (2), obtaining a sequence $\left\{{x}_{k}\right\}$ with ${x}_{k}={\xi}_{i}\left({x}_{k-1}\right)$ with probability ${p}_{i}$, chosen randomly, $i=1,2,3$.

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PSD | Particle Size Distribution |

IE | Information Entropy |

KS | Kolmogorov-Smirnov |

USDA | United States Department of Agriculture |

## References

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**Figure 2.**Actual (continuous line) and simulated (dots) PSD for soil 44. Top row shows the simulation using triplet ${I}_{1}=[0.59-1.46]$, ${I}_{2}=[1.46-1406.77]$ and ${I}_{3}=[1406.77-3473.45]$ $\mathsf{\mu}$m. On the left the x scale is the particle diameter in $\mathsf{\mu}$m, while on the right, for visualization purposes, it is on the logarithmic scale. For the bottom row, the input triplet used was ${I}_{1}=[0.59-37.84]$, ${I}_{2}=[37.84-1174.13]$ and ${I}_{3}=[1174.13-3473.45]$ $\mathsf{\mu}$m. In both cases, the maximum allowed distance for the acceptance of the KS test at a 0.05 level was 0.28.

**Figure 3.**Representation of the KS distance, ${D}_{n}$, for all possible triplets, in the $({\alpha}_{1},{\alpha}_{2})$ plane, for soil 44. The values of ${\alpha}_{i}$ are in the log scale. The horizontal plane, at height, 0.28, is the limit value for ${D}_{n}$ for the acceptance region at the 0.05 level. Blue points, below the plane, are the ones that passed the test, while orange ones do not pass it. For this soil, 403 triplets (37.28%) pass the test.

**Figure 4.**Red dots represent the percentage of samples that pass the KS test for a given triplet (${\alpha}_{1}$,${\alpha}_{2}$). For visualization purposes, the projection of the percentages on the plane have been added (blue dots).

**Figure 5.**Heatmap for the percentage of samples that pass the KS test for a given triplet (${\alpha}_{1}$,${\alpha}_{2}$).

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**MDPI and ACS Style**

García-Gutiérrez, C.; Martín, M.Á.; Pachepsky, Y.
On the Information Content of Coarse Data with Respect to the Particle Size Distribution of Complex Granular Media: Rationale Approach and Testing. *Entropy* **2019**, *21*, 601.
https://doi.org/10.3390/e21060601

**AMA Style**

García-Gutiérrez C, Martín MÁ, Pachepsky Y.
On the Information Content of Coarse Data with Respect to the Particle Size Distribution of Complex Granular Media: Rationale Approach and Testing. *Entropy*. 2019; 21(6):601.
https://doi.org/10.3390/e21060601

**Chicago/Turabian Style**

García-Gutiérrez, Carlos, Miguel Ángel Martín, and Yakov Pachepsky.
2019. "On the Information Content of Coarse Data with Respect to the Particle Size Distribution of Complex Granular Media: Rationale Approach and Testing" *Entropy* 21, no. 6: 601.
https://doi.org/10.3390/e21060601