Kinetic energies in random vectors - Sciendo

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 Kinetic energies in random vectors

 Alexandru DAIA
 The Bucharest University of Economic Studies, Bucharest, Romania
 alexandru130586@yandex.com

 Stelian STANCU
 The Bucharest University of Economic Studies, Bucharest, Romania
 stelian.stancu@csie.ase.ro

 Om SUCHAK
 Princeton University, Princeton, NJ, USA
 om.suchak@gmail.com

Abstract. This paper uses the Onicescu Coefficient concept, which is the sum of squared probabilities,
through mathematical formulations to show the kinetic energy in random vectors. The paper has a brief
introduction section that addresses the background of the study, the purpose of the study, and its
objective. The literature review section provides an in-depth industrial application of mathematical
formulations in solving real-life problems. The paper contains a methodology section highlighting the
study design, data collection methods, and analysis section, highlighting some of the keywords used in
locating resources and significant databases that provided the study with information. Later, the study
takes the result and discussion sectional approach to present the experiments' findings backed with facts
from previous studies by other scholars in the same field. Lastly, the paper concludes with a section
recapping the critical points of this study and study application in real-life, concluding with a list of
references utilized by this study.

Keywords: Kinetic Energy, Random Vectors, Machine Learning

Introduction
Solving real-life problems using mathematical formulas remains the most effective way to
strategize and address the phenomenon.
 The transformation of real-life problems into mathematical equations often demands
an in-depth understanding of the problem. Traditionally, mathematical formulations were
elusive to most scholars that did not have any interest in mathematics. Today, more
familiarized math formulations are available to different fields purposely to assist in solving
different problems and gather facts and explanations around scenarios.
 Octav Onicescu discovered kinetic Energy, and it is described simply as the sum of
squared probabilities. Octav Onicescu discovered kinetic Energy, and it is described simply
as the sum of squared probabilities (Iosifescu, 2018). Octav Onicescu was a world-renowned
Romanian mathematician who was part of the Romanian Academy and is known for founding
the Romanian School of Probability Theory and Statistics (Daia, Stancu & Popescu, 2019).
Onicescu was born on August 20, 1892, in Botosani, Romania, and died on the evening of his
91st birthday, in Bucharest on August 19, 1983 (Agop, Gavriluţ, & Rezuş, 2018). This paper

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uses the Onicescu Coefficient, defined as the sum of squared probabilities to study random
vectors. In particular, these coefficients will be used to investigate the kinetic energy in these
random vectors.
 This study aims to determine the usefulness of mathematical formulation in
determining the kinetic energy in random vectors through a series of experiments.

Literature review
In this section of the paper, a brief review of past studies in math and that of the mathematical
formulation is completed. Most of the scholars’ work revolved around deriving mathematical
formulations out of real-life problems. First, the section begins by highlighting mathematical
formulations and how the concept came into existence.
 Informational energy is a concept inspired by the kinetic expression of classical
mechanics. From the informational theory point of view, the formulas informational energy
is a measure of uncertainty or randomness of a probability system and was introduced and
studied for the first time by Onicescu in the mid-1960s.
 According to Rivera, Afsar & Prins (2016) mathematical formulations can be used to
solve critical problems revolving around machines through the substitution of basic factors
and figure to derive formulas. In their case, the used mathematical formulations to solve a
problem with the multitrip cumulative capacitated single-vehicle routing.
 Singh et al, (2008) affirm that mathematical formulations can be used as diagnostic
tools for complex problem. In their explanations, they used mathematical formulations for
dynamic multiple fault diagnosis. Their experiments involved mostly vehicles and it was
handful in solving performance problems, fuel consumption and engine efficacy.
 Last but not least, Archetti et al, (2014) imply that mathematical formulations can also
be used to solve inventory problems to improve customer service delivery. Their explanation
benefits this paper as it offers proof of the application of mathematical formulations in
solving real-life problems.

Methodology
In this section, the data collection methods, data analysis, and presentations are highlighted.
First, the study adopts a combination of qualitative and some quantitative methods to utilize
data and come up with useful conclusions that address the research objective.
 This study used a series of desktop methods to retrieve secondary data in
preparedness for the analysis process. For history purposes, the study used secondary data
retrieved from databases and keywords to gather relevant sources that would purposely
serve this study's need. Due to the coronavirus pandemic, the study was limited to secondary
data, virtual meetings, and desktop strategies. In the experiments section, the study used
coding from the Open Source Python Programming Language to develop the results after
crating the mathematical formulations.

Results and discussions
After a series of experiments, secondary data sources, and analysis, the study came up with
valuable conclusions that address the study objective. In particular, most of the experiments'
findings are explained using formulas and mathematical expressions, as will see in this
section.

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 The informational energy and entropy are both measures of randomness, but they
describe distinct features (Alipour & Mohajeri, 2018). This study deals with the informational
energy in the framework of the statistical models. The chapter contains the informational
energy's main properties, its first and second variation, relation with entropy, and numerous
worked-out examples.

Definitions and examples
Considering the statistical model exposed below
S={ = ( ; ) = ( 1 , , , , , ) ∈ } (1)

 The informational energy on S is a function I:E→ defined by,
I( )=∫ 2(x, )dx. (2)

 Observe the energy is convex and invariant under measure preserving
transformations, properties similar to those of entropy.

 In the finite discrete case, when = { 1 , … . , }, formula is replaced by
I( )= ∑ =1 2(xk, ). (3)
 However, if = ,we have the following result

 Let ( ) be a probability destiny on satisfying:
 (i) ( )is continuous
 (ii) ( ) → 0 → ±∞.

 The informational energy of p is finite, i.e... The following integral convergent
 
I(p)=∫−∞ 2(x)dx 0 ℎ ℎ ( ) < | | > . This
follows the fact that ( ) ↘ 0 | | → ∞ Writing
 − ∞
 I(p)= ∫−∞ 2 ( ) + ∫− 2(x)dx+∫ 2(x)dx (5)

 We note that
 − − 
∫−∞ 2 ( ) < ∫−∞ ( ) = ( ) < (6)

 ∞ ∞
∫ 2 ( ) < ∫ ( ) = (7)

 Where Ϝ( ) denotes the distribution function of ( ) since the function ( ) is
continuous, it reaches its maximum on the interval [− , ], denoted by , then we have the
estimation
 
 ∫− 2 ( ) < ∫− ( ) = ( ( ) − (− )) < 2 (8)

 It follows that ( ) ≤ 2 + 2 < ∞, which ends the proof.

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Kinetic Energy
We describe kinetic Energy, or information energy, for random vectors as being analogous to
kinetic Energy from physics in probability. Some say it is entropy, similar to Shannon
entropy, for measuring bits of information to determine uncertainty (Alexandru, 2019).
 It is indeed entropy, but the correct way to think about it is to think of it as ½ ∗ m ∗ v2
of a random vector. As it is the sum of squared probabilities, The kinetic Energy is similar to
physics, except here we lack the (1/2)∗ mass because these probabilities are floating things.
Since they are floating, either the sum of their mass is 1, which is not very plausible, or their
mass is 0 if you think of them as floating inertial bodies with no mass.
Kinetic-Based Machine Learning models:
 We implemented the notion of kinetic energies into Deep Learning. This was done for
Neural Networks by using the new correlation coefficient as a performance metric instead of
cross-entropy. Kinetic Energy in neural networks has a wide variety of use cases. For
example, this implementation can work well in a convolutional neural network for computer
vision or a deep neural network for concluding large quantities of data. In the case of genetic
algorithms, we used the kinetic energies as a fitness function.

Correlation coefficients
Let’s suppose that we have these two random vectors:
 X=x1,x2,..xN (9)
 Y=y1,y2,...Yn (10)
 There are 2 main coefficients:
 The informational correlation IC (this does not make use of kinetic energy)
 C(x,y) = Cx1,x2,…xN,y1,y2,….,yN = ∑ℕi=0 p(xi)∗p(yi) (11)
The IC is bounded between [0,1], being totally 0 if both vectors are zero as the system is total
indifferent.
 The informational correlation coefficient
 Similar to other statistical correlation coefficients the IC could suffer normation using
kinetic energy resulting in:
 O(X,Y)=(∑ℕi=0 p(xi)∗p(yi))/(∑ℕi=0 xi2∗∑ℕi=0 yi2) (12)
Notice the denominator is exactly the kinetic energy, so the ICC coefficient could be written
as:
 O(X,Y)=IC/kinetic(X)∗kinetic(Y) (13)

 In order to compute the dot product, it is necessary that the 2 random vectors have
the same cardinality of unique events, or classes.
For example if x=1,2,1 and y=4,2,5 then IC:=1∗4+2∗2+1∗5 but what if the 2 random vectors
look like: x=1,2,1,4 and y=7,5,1 and they had different shapes from one another, the
correlation coefficient will not work.
Kinetic PCA. Changing the correlation matrix that uses Pearson’s R correlation or in some
cases the covariance, with correlation matrix based on the Onicescu Correlation Coefficients.
We found that it brought more space into the distance between the clusters of the species of
the Plant species dataset below.

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 Figure no. 1. Principal component analysis
 Source: Authors’ processings

 Notice that for Versicolor and Virginica there is practically no space between clusters
of the 2 groups, causing a machine learning algorithm to over-fit by creating a too rigid curve
without space for generalization.
 After testing the kinetic correlation on another toy data set against the Pearson
coefficient, we plotted the results:

 Figure no. 2. TFI Pearson correlation
 Source: Authors’ processings

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 Figure no. 3. TFI Kinetic correlation
 Source: Authors’ processings

 The kinetic correlation is much higher than the Pearson one, as expected since
Pearson’s Correlation can only detect linear relations in data.

Conclusion
At the beginning of the study, the primary objective was to determine the usefulness of
mathematical formulation in determining the kinetic energy in random vectors through
experiments.
 Throughout the study design, approaches and experiments, this objective has been
fully met. The Open Source Python Programming Language software played a significant part
in examining the mathematical formulation from the results and discussion section. As a
result, it is evident that using the Onicescu Coefficient remains important when describing
random vectors and requires recognizing its strengths when dealing with sets of random
vectors to be used for Machine Learning. In particular, this study shows the correlation
between real-life problems and how mathematical formulations can solve these problems.

References
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Agop, M., Gavriluţ, A. and Rezuş, E. (2018). Implications of Onicescu's informational energy
 in some fundamental physical models. [online] Worldscientific.com. Available at:
 https://www.worldscientific.com/doi/abs/10.1142/S0217979215500459
 [Accessed 29 Jun. 2018].

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Alexandru, D., 2019. Experimented Kinetic Energy As Features For Natural Language
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