Statistical Estimation of the Shannon Entropy

In this paper, we introduced an advanced discussion of the 3D morphology of TiO2 coatings deposited on ITO substrate by electrodeposition under different deposition times. Atomic force microscopy was applied for obtaining topographic images of the samples. The images were processed using the MountainsMap 8.0 commercial software according to ISO 25178‐2:2012. Moreover, fractal theory was applied to study the surface microtexture of coatings. The morphology was affected by the deposition time, where the grain size decreased with the increase of the time, making film's surfaces smoother. In addition, the surface roughness exhibited a random behaviour, but does not presented significant difference between samples. The fractal dimension showed similar values for all coatings. In contrast, surface texture isotropy also exhibited random behaviour. However, advanced fractal parameters revealed that when the deposition time increased, the coatings microtexture has become uniform and less porous. Furthermore, all coatings presented high topographic uniformity, regardless of deposition time. These results revealed that the morphology and microtexture of TiO2‐based coatings can be controlled by the deposition time. Lay Description An advanced characterization on the micromorphology of 3D morphology, using AFM images, of Titanium dioxide (TiO2) coatings deposited on ITO substrate by electrodeposition under different deposition times. TiO2 is one of the most studied semiconductors to make photovoltaic devices. The versatility of this semiconductor is associated with low toxicity, high photochemical stability, abundance, and the facility to obtain by conventional synthesis routes. The obtention of a homogeneous and stable layer in the semiconductor TiO2 film deposition is a crucial stage in the assembly of sensitized photovoltaic devices. Atomic Force Microscopy (AFM) is a technique which can magnify up to a billion times and it uses a tip or probe which touches the sample surface point by point. The tip deflection is interpreted as the surface topography by the software, producing 2D or 3D images that generate several tribological parameters such as roughness in respect to a scanned area, has been a technique widely reported in the morphological characterization, determination of thickness, roughness, and particle size in thin films. Therefore, in this paper, the morphology was studied by atomic force microscopy using MountainsMap commercial software. The main goal was to study the influence of the deposition time on the morphology and microtexture of the material. New parameters such as surface entropy, fractal succolarity and fractal lacunarity were obtained for studying coatings microtexture's complexity.

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“…These patterns were provided by the Shannon's entropy, as described by Bulinski and Dimitrov 44 as follows:

H^{((), 2)} = - \sum_{i = 1}^{N} \sum_{j = 1}^{N} p_{i j} l o g p_{i j},

where p ij is the probability of having or not discrepant pixels in the universe of heights.…”

Section: Methodsmentioning

confidence: 99%

Nanoscale morphology and fractal analysis of TiO₂ coatings on ITO substrate by electrodeposition

Amâncio

Pinto

Matos

et al. 2021

Journal of Microscopy

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“…Shannon entropy [27] and minimum entropy [28] are effective tools to depict the statistical characteristics of random numbers. Shannon entropy can quantitatively evaluate the effective information in a random sequence, and the independence and uncertainty of each bit.…”

Section: A Select the Least Significant Bitsmentioning

confidence: 99%

Fast True Random Number Generator Based on Chaotic Oscillation in Self-Feedback Weakly Coupled Superlattices

et al. 2020

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Random numbers play a vital role in communications and cryptography. However, most existing true random number generators have difficulty in satisfying the requirements of high-speed communications due to their complexity and bulkiness, or low speed limitations due to equipment bandwidth. Then, the all-electron true random number generator was presented based on GaAs/Al 0.45 Ga 0.55 As superlattices conducted under direct current bias at room temperature, which not only possesses the characteristics of miniaturization but also generates random numbers at rates up to the Gbit/s. However, the bit rate of random number generators based on superlattices is still much slower compared to chaotic laser random number generators. In order to generate higher-rate random numbers, we modified a DC-excited superlattice and then redirected the signal generated by the superlattice back into itself, thus introducing a self-feedback, and achieved a self-feedback superlattice true random number generator. This improvement makes the superlattice have a more detailed signal shape, a more effective signal amplitude, and with lower power consumption. Therefore, these advances made the self-feedback superlattice more suitable for generating random numbers. Moreover, we propose a new post-processing method, called the adjacent bits reversal exclusive-or. This method can reduce the sequence bias and correlation without discarding any random bit. The random number obtained by the self-feedback superlattice at a sampling rate of 10 GS/s passed the triple standard deviation test and the random number standard test (NIST SP 800-22), indicating that it possessed good statistical properties as a miniaturized random number generator.

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“…Let X ∼ N (a, Σ) where a ∈ R d and Σ > 0. It is easily seen that, for any ε > 0, one has E| log f X (X)| ε < ∞ (see, e.g., [10]). Since, for x ∈ R d and y ∈ M ,…”

Section: Now We Show That Uniformly Formentioning

confidence: 99%

“…Applying the expressions obtained for nominator and denominator of the latter fraction in (70) and taking into account that a function P(X ∈ B(x, t)) is continuous in (x, t) ∈ R d × R + (see, e.g., Lemma 1 in [10]) we get, for each n ∈ N (n > 1), k ∈ {1, . .…”

Section: Now We Show That Uniformly Formentioning

confidence: 99%

“…There are a number of papers devoted to various estimates of (unconditional) entropy. In this regard we indicate the recent work [10] where the estimates of the Shannon differential entropy are studied and where one can find further references. The scheme (4) under consideration comprises the famous logistic model widely used in the classification problems (see, e.g., [24]).…”

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confidence: 95%

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Statistical estimation of conditional Shannon entropy

Bulinski

Kozhevin

2019

ESAIM: PS

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The new estimates of the conditional Shannon entropy are introduced in the framework of the model describing a discrete response variable depending on a vector of d factors having a density w.r.t. the Lebesgue measure in R d . Namely, the mixed-pair model (X, Y ) is considered where X and Y take values in R d and an arbitrary finite set, respectively. Such models include, for instance, the famous logistic regression. In contrast to the well-known Kozachenko -Leonenko estimates of unconditional entropy the proposed estimates are constructed by means of the certain spacial order statistics (or k-nearest neighbor statistics where k = k n depends on amount of observations n) and a random number of i.i.d. observations contained in the balls of specified random radii. The asymptotic unbiasedness and L 2 -consistency of the new estimates are established under simple conditions. The obtained results can be applied to the feature selection problem which is important, e.g., for medical and biological investigations.

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Statistical Estimation of the Shannon Entropy

Cited by 28 publications

References 23 publications

Nanoscale morphology and fractal analysis of TiO₂ coatings on ITO substrate by electrodeposition

Nanoscale morphology and fractal analysis of TiO₂ coatings on ITO substrate by electrodeposition

Fast True Random Number Generator Based on Chaotic Oscillation in Self-Feedback Weakly Coupled Superlattices

Statistical estimation of conditional Shannon entropy

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Statistical Estimation of the Shannon Entropy

Cited by 28 publications

References 23 publications

Nanoscale morphology and fractal analysis of TiO2 coatings on ITO substrate by electrodeposition

Nanoscale morphology and fractal analysis of TiO2 coatings on ITO substrate by electrodeposition

Fast True Random Number Generator Based on Chaotic Oscillation in Self-Feedback Weakly Coupled Superlattices

Statistical estimation of conditional Shannon entropy

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Product

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Nanoscale morphology and fractal analysis of TiO₂ coatings on ITO substrate by electrodeposition

Nanoscale morphology and fractal analysis of TiO₂ coatings on ITO substrate by electrodeposition