**Thiago Rafael da Silva Moura ^{*}**

Facult of Sciences, Federal University of Pará, Interdisciplinary Innovation Laboratory, Raimundo Santana Street, Salinópolis - PA, 68721-000, Brazil

^{*}**Corresponding author:** Thiago Rafael da Silva Moura, Facult of Sciences, Federal University of Pará,
Interdisciplinary Innovation Laboratory, Raimundo Santana Street, Salinópolis -
PA, 68721-000, Brazil. Tel: +5591983675044: Email: trsmoura@yahoo.com.br

**Received Date:** 29 May, 2018; **Accepted Date:** 15
June, 2018; **Published Date:** 21 June,
2018

The absence of biomarkers hinders the early diagnosis of
children with Autism Spectrum
Disorders (ASDs) applied in routine examinations in newborns, for example. At moments, post-birth years, difficulties
in social interaction and other emerging factors
are used for the evaluation of children with ASD. We have reported numerical
results that analyze
the social interaction and the repetitive behavior of children with and
without ASD. To represent the
interaction, by analogy, we used two sets of random walkers. These sets are
labeled in two ways, a group, e.g., the first set of
random walkers represents children with autism, characterized by persistent and change-resistant microscopic dynamics. The second set of walkers, which we labeled as healthy, is represented as a set that presents diffusive regimes,
this group is represented by the Elephant Random Walks (ERW) model. We have analyzed, from the perspective of
information entropy, how the two sets of random
walkers influence the entropy variations of each other. Walkers interact
in pairs; the impact of this interaction is analyzed in the entropy variations
of each of the sets. Therefore, we have reported
the entropy variations for the ERW model
and its variations when the interaction becomes stronger with probability *f*.
We have
reported that, surprisingly, *f *changes do not provoke changes in entropy for the group of random walkers who represent
the group of children with autism.

**2.
****Keywords:*** *Autism;
Entropy; Interaction; Invasive Disorder; Random Walks

**1.
****Introduction**

We have reported the use of discrete random walks as a diagnostic tool for the analysis of developmental disorders. Developmental disorders are classified into two subtypes: specific disorder and pervasive disorder. Delays in development in one or more specific areas is a characteristic of developmental disorder. The other subtype, the pervasive disorder, presents deficiencies in basic functions in multiple contexts that include socialization and communication. Pervasive De- Velopmental Disorders (PDD’s) are part of the group of disorders of the Autistic Spectrum (ASD), known as neurodevelopmental disorder. Symptoms accompanying individuals with ASD are the delay in verbal and non-verbal communication; resistance to routine change; restricted and persistent interests in relation to an activity, topic, object, speech, idiosyncratic phrases, etc.; abnormalities in eye contact and body expression; difficulties in starting and maintaining social relations. Each of these symptoms may vary from mild, moderate or severe and are part of the diagnostic criteria of the autistic spectrum disorders used in routine examinations [1-3]. Inspired by these symptoms and the diagnostic criteria for ASDs, we constructed a model using discrete random walks as a diagnostic tool for autism. Our starting point is to define a model to compare with autistic behavior. The model we chose as the standard of comparison is the Elephant Random Walks Model (ERW). The ERW presents pertinent characteristics to be chosen as the standard model: their diffusive regimes are well known, the model is accurate and several variations have been proposed with the objective of exploring distinct search mechanisms in memory, the loss of in- formation and its impact on the diffusive process, among other characteristics [4-7]. Further details can be found in [8-15].

**2.
****The
model**

Our model consists of pairs of random walkers. The ﬁrst
walker, we call” Professor”; the second one,” Student”. We consider that in
neurological exams in children the analysis of communication and social
interaction in multiple contexts is part of the routine, including repetitive
patterns of behavior, the examiner (physician) has knowledge of the patient’s
actions; therefore, in the context of our problem, the Professor has knowledge
of the student’s actions, but he/she is not inﬂuenced by these actions. In this
case, by analogy, Physician is the Professor and the patient is the Student.
However, what happens in a context where the Professor is also inﬂuenced by the
Student? This problem can be analyzed as follows: The Professor can learn (not
learn) from Student’s actions with probability fP (1-fP), similarly, the
Student can learn (not learn) with the Professor probability fA (1-fA). Random
walker’s pairs can learn from each other, both can learn Professor and Student
labels. For reasons of order, let’s call the ERW model “Professor” and the
other one “Student”.

**2.1.
****The Professor**

We deﬁned the” Professor” as the ERW model. Nevertheless,
we have described the relevant aspects of Schu¨tz and Trimper, their main
characteristics and diﬀusion regimes. The model of random walks with uniform
memory proﬁle considers a discrete random walk so that each decision, taken at
every moment of time, depends on the spectrum of decisions taken in every
previous history of the walker in an equitable manner. Recovering each past
decision in an equable manner is a characteristic of a random walk with uniform
memory proﬁle. The probability that an action taken in the past will be
remembered is 1/t, to t the current time. The random walk is recorded at all
times, this aspect attributes to the process a non-Markovian characteristic.
The stochastic dynamics of the process occur in the following way: the walker
starts in time t0 in a position x0, at each instant of time the walker walks
one step to the right or step to the left, with probability p and (1-p),
respectively. For this process, the stochastic evolution equation is given by

*X _{t}*

for the time t + 1. The variable σt+1 assumes the value +1 when the walker walks one step to the right and -1 when the walker walks one step to the left. The memory consists of a set of random variables σtj for the time t prime < t that the walker can recover. This process occurs as follows:

(a) in time t+1 a number t0 of the set 1,2,..,t is chosen randomly with uniform probability 1/t.

(b) σt+1 is determined stochastically by following the following relations, σt+1 = σt0 with probability p and σt+1 = -σt0 with probability 1-p.

The ﬁrst step at time t = 1, is taken according to the following dynamics: the walker is in the position X0 and moves to the right with probability q or to the left with probability 1-q, i.e., σ1 = +1 with probability q and σ1 = -1 with probability 1-q. The stochastic evolution equation is

The parameter p is the probability of the walker repeating an action from the
past in a time t^{j}. When (p
> 1/2) the walker presents a persistent behavior, this is characterized by the repetition of past actions.
For (p < 1/2)
the walker takes an action contrary
to the action that was selected, presenting an anti- persistent behavior. For the value of (p = 1/2) the random
walk is Markovian. In the boundary regions
of (p = 0) and (p = 1) two extreme
behaviors arise. In (p
= 0) the maximum of the anti-persistent behavior occurs, while in (p = 1) the
maximum persistent behavior
occurs; in the latter case, the movement is ballistic, characterized by the typical
value of the Hurst exponent (H = 1) [7]. The diffusive
behavior changes with the probability values p, when (p < 3/4) the
second moment depends
linearly on t and diffusion is ordinary (H = 1/2),
para (p > 3/4) diffusion is characterized as super
diffusive, the point (p = 3/4) separates the diffusive and super
diffusive regimes. Additional information about the ERW model, such as the study of its limits
and other models
that are variants
of it, are found in [7,16,17].

**4.2 ****The Student**

The microscopic dynamics
of the student occurs in a manner
similar to that of the Professor. The Student
initiates his/her movement at the
position *Y*_{0}, at time *t*_{0} = 0, at every instant of time the random walker walks
one step to the left or to the right. The stochastic equation that quantifies
Student’s steps is given by

*Y _{t}*

*v _{t}*

(a)in time *t *+
1 a number *t ^{j} *of the set 1

(b)the variable *v _{t}*

At the moment *t *= 1, the walker is in position
*Y*_{0} accepting (rejecting) the
decisions of the ERW model with
probability *f _{A} *(1

**4.3
****The Professor and Student Interaction**

The core of our model
is to describe the interaction between random walkers and their consequences. The Professor can learn from the student’s microscopic decisions. Therefore, the Professor’s microscopic
dynamics will be influenced by the Student with probability *f _{P}
*and not influenced with probability 1

The probability values *f _{P} *= 0 and

and *v _{t}*

The Student’s learning, in relation to the Professor’s
microscopic dynamics, occurs as follows: The Student accepts (rejects) the
Professor’s decisions with probability *f _{A} *(1

For the maximum non-learning limit, *f _{A}
*= 0, the Student recalls actions solely from its history, its stochastic equation is given by (4), another extreme,

The characteristic described above, which deals with the quantification of learning of random walkers, we defined as the process of Professor-Student interaction.

There are two cases for the
learning consideration of Professor-Student interaction: the symmetric case
and the non-symmetrical case. In the
symmetrical case, the
Professor and Student
present the same
probability of learning (*f _{P} *=

Let’s discuss the case in which (*f _{P} *=

For the case of minimal
learning, *f *= 0, the models
recover their ordinary results without mutual learning, with stochastic equations (4) and (2) for the Student and for the ERW model, respectively. The maximum learning occurs when *f *= 1, at this point, Professor and Student are at the maximum of mutual
influence.

We analyze the variation of the Shannon
entropy in the Professor-Student interaction process. The Shannon
entropy, also
known as information entropy, was formulated by Claude Shannon in a context different from what we are applying, in the context of
telecommunications applications to analyze the
economy expectancy related to the statistical structure in sending
and receiving information [19]. We will use Shannon’s entropy
in the context of discrete
random walks for the problems
of Professor-Student interaction. The interaction depends on two control
parameters: the probability of interaction between walkers *f *and the probability of
the random walker *p *to repeat an
action from the past in an instant *t ^{j}*. The information entropy associated with
the diffusion of walkers is given by equation

And ∆*S _{Sh} *is the Shannon
entropy and

∆*S *provides
the variation of relative entropy associated with the diffusion process. The
highest value of entropy, according to equation 12, is ∆*S *= 1,
that occurs when ∆*S _{Shannon} =* ∆

For the Professor, the variations in the control
parameters exert influence
under the entropy variations. We will
focus on the proper description of entropy for the Professor. We measured the entropy using the
controllable parameters *p *and *f *,
its variations allow us to quantify the entropy variations related to the ERW model, which was the model observed
for the construction of ours, as well
as the obtaining of quantitative values of the entropy that differ substantially

from those observed for the ERW model. In (Figure 1) the resulting diagram of Professor-Student interaction in the representation of information entropy
is presented. We present our
results at the point *f *= 0, at the point of zero inter-
action, in which there is no Professor-Student interaction, the typical entropy
variations in the ERW model are displayed.
At this point, *f *= 0, controlling the parameter *p *in the range 0 ™ *p *™ 1,
the entropy variations are approximately the same in the range 0 ™ *p *™ 3*/*4, above *p > *3*/*4 the variance of the entropy associated with the ERW model is larger the greater
the values of *p*. In the range
0 ™ *f *™
0*.*1, there has been a rapid growth of
entropy followed by a smoother decay
in the range 0*.*1 *< f *“ 0*.*6.
Approximately in the curve *f *“ 0*.*6, of all the curves in which *f *=
0, is the one with the smallest entropy variation with the lowest entropy
variation of the point *f *“ 0*.*6 and *p *= 0. The greatest entropy variations occur in the curves where,
first, *p *= 1 and 0 ™ *f *™ 1; secondly, when we consider
the maximum interaction curve between the random walkers
*f *= 1 and 0 ™ *p *™ 1, the
greatest entropy variation occurs at the point p* *= 1 and *f *=
1 of the surface.

(Figure 2) is a color map of Figure 1. The variation of the intensity of the colors is
related to the entropy variations, the regions with red tonalities are associated with larger
entropy variations, the darker tones
are related to smaller variations. We emphasize, on the map, the curve in which *f *=
1 of greater interaction between random walkers, accompanied by greater entropy variations, ∆*S *= 1. Contour curves are drawn by connecting points on the surface at
which the entropy is invariant. We highlighted, centered on the left side
of the Figure 2, the contour with
entropy variation ∆*S *= 0*.*6, immediately above, to the extent
that *p *grows, the curve with ∆*S *= 0*.*8,
the entropy varies more strongly. Just below the curve with ∆*S *= 0*.*6,
in the direction *f *decreases, we highlighted the curve of equal entropy with
typical measurement ∆*S *= 0*.*4.

Another area that also should
be highlighted is the area 0 ™ *p *™ 1 and 0 ™ *f *™ 0*.*1.
In Figure 3 we highlighted the
entropy variations in the ERW model, *f *=
0, its variations in the extent that *f *grows. For this value of the
interaction parameter, *f *= 0, the entropy variations are
greater the greater they are values
of the parameter *p*. Even though it is
accompanied by a rapid variation of entropy,
this region, 0 ™ *p *™ 1 and
0 ™
*f *™ 0*.*1, has a
surface of equal entropy ∆*S *= 0*.*4, highlighted in
the figure map 2. The typical entropy measures are smaller for *f *=
0, to the extent that *f *increases in entropy variations are
observed for each *p*. The increase
of the parameter *p*, also, accompanies greater variations of entropy.

Autism spectrum disorders
have inspired us to construct theoretical models that represent characteristics of social interaction. An
analogy with the deficiencies of social
interaction and the microscopic dynamics
of random walkers, for the discrete case, allows us
to make typical measurements of physical observables typical of the diffusive
process of random walkers. We used two sets of random walkers to study Autism,
which we call Professor and Student.
The set of random walkers labeled Professor are inspired by the ERW
model. The set of random
walkers labeled Student,
by definition, exhibits persistent microscopic behavior. The probability of Professor and Student
learning, *f _{P} *and

Our results focus on the presentation of typical measures
of entropy for the Professor. The Student, however, presents
typical measures of the invariant entropy for the values of the
probabilities 0 ™ *p *™ 1 and 0 ™ *f *™
1, we observed for these values that
∆*S *= 0. The entropy for the Student
does not change with the interaction with the Professor. The same does not
occur for the Professor, we can
summarize our results in the following way:

(a)the Professor does not interact with the Student to *f *=
0. Typical entropy values are those of the ERW model;

(b)we observed a rapid growth of entropy in the range 0 ™ *f *™
0*.*1, accompanied by a decrease in the range 0 ™ *f *“0*.*6 followed by entropy
growth in 0*.*6 “*f *™ 1;

(c)in the color map of Figure
2, we observed the existence of regions where the
entropy, related to the diffusive process,
is invariant. The curves of entropy invariant
with equal measures are highlighted, ∆*S *= 0*.*4, ∆*S *= 0*.*6,

∆*S *= 0*.*8 and ∆*S *= 1*.*0.

(d)comparing the entropy variations between Professor and Student, the Professor shows greater entropy variations than the Student.

For the Student
the entropy changes
are always null,
∆*S *= 0, not susceptible
to variations of the interaction parameter *f *and the parameter *p*. The Professor
interacts with the Student with probability *f*, shows
variations in the entropy
as *f *varies. Increases with *f *growth.
We observed that the Professor is
also susceptible to variations of *p*, showing larger
entropy changes, the greater the p When the Professor
and Student do not interact they make decisions based only on their own memory. This limit is observed in the
Professor-Student interaction process, which occurs with probability *f*.
The behavior of the ERW model and the
maximum of restricted interests for the Professor and the Stu- dent,
respectively, occurs when the probability of interaction is null *f *=
0. We observed that for
Professor-Student interaction, *f > *0, the Professor presents greater variations in the measures
of information entropy, ∆*S > *0, than the Stu- dent. The greater the probability of
interaction with the Student, the greater the variation of Professor entropy. Therefore, the greater the
interaction with the Student, the greater the influence that the Professor
suffers from his/her restricted interests. The Student’s entropy
remains invariant
to the transformations of probability of interaction *f *and variations in *p*, ∆*S *= 0.

**Figure 1:** Behavior of information entropy for the stochastic process
labeled Professor. Typical values of the entropy variation
are plotted as a function of the interaction parameter *f
*and of the probability *p *in the ranges 0 ™ *f *™ 1 and 0 ™
*p *™
1, respectively.

**Figure
2:** Color map for the quantitative values of the
information entropy, obtained from the data of Figure 1, arranged according to
the parameters f and p, for the stochastic process labeled Professor. Contour
curves are drawn in the range 0 ™ f ™ 1 and 0 ™ p ™ 1, to highlight regions of
equal probability ∆S = 0.4, ∆S = 0.6, ∆S = 0.8 and ∆S = 1.0.

**Figure
3:** We showed values of the entropy variation of Professor
information for the ranges 0 ™ p ™ 1 and 0 ™ f ™ 0.1. The chosen probability of
interaction values are f = 0, 0.001, 0.0025, 0.005, 0.01, 0.025 and 0.1,
entropy grows in the sense that f and p grow.

**6.
**Tavar´e S, Giddings BW (1989) Mathematical Methods for DNA Sequences (ed. Waterman M. S.) 117-132.
CRC Press, Boca Raton.

**Citation:** Moura
TRS (2018) Information Entropy for The Clinical Evaluation of Autism Spectrum
Disorders. Res Adv Brain Disord Ther: RABDT-106. DOI: 10.29011/RABDT-106. 100006

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