Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 405 https://internationalpubls.com Autism Detection Using Fuzzy Ξ²-Neighborhood Based Soft Rough Sets Praba B1, Balambal Suryanarayanan2 1 Department of Mathematics, Sri Siva Subramaniya Nadar College of Engineering, Kalavakkam- 603110; prabab@ssn.edu.in 2Sri Siva Subramaniya Nadar College of Engineering, Kalavakkam- 603110; balambal19013@ece.ssn.edu.in Article History: Received: 14-04-2024 Revised: 29-05-2024 Accepted: 15-06-2024 Abstract: Autism Spectrum Disorder refers to a variety of conditions represented by complications in social skills, limited interests and communication that’s majoritarily non-verbal. The early signs and symptoms of the disorder were found to be noticeable at a young age. However, the clinical tests take longer to diagnose and come at a higher cost. As a result, in recent years, several studies have been made to enable autism detection through early intervention. Rough Set theory is one such efficient mathematical tool for this application. This paper introduces the concept of fuzzy 𝛽-neighborhood based soft rough sets (𝛽 βˆ’ 𝑅𝑆). The major advantage of introducing this concept lies in the construction of fuzzy 𝛽- neighborhood of each object obtained through its 𝛽-reduct. This neighborhood captures those objects having similar characteristics with respect to the significant attributes. The application of this model is highlighted through autism detection by obtaining the neighborhood of the given image. The proposed model also efficiently captures the various levels of autism with great accuracy. The validation is carried out by taking real time data. Keywords: Autism, Fuzzy Sets, Rough Sets, 𝛽 βˆ’neighborhood, Lower Approximation, Upper Approximation. 1. Introduction The uniqueness of human beings as contrasted to other primates can be attributed to the fascinating cognitive functionalities and mysteries of the human brain. At times, damage inflicted to a characteristic part of the brain [18] adds to the brain’s enigma, resulting in intriguing disorders where a person sees sounds, hears shapes, touches emotions, and feels limbs [15]. The difficulties in cognitive, social, and emotional functioning, influenced by genetic and environmental factors give rise to a collection of disorders called neuro-developmental disorders. The most known neuro- developmental disorders include ADHD, autism, mental retardation, cerebral palsy, and control disorders. Of these, autism has continued to enthral and puzzle modern medicine, becoming an unknown frontier like the brain itself [5]. As a spectrum disorder where every person with autism possesses a unique set of strengths and challenges, the prospect of autism going undiagnosed due to mild symptoms or suppressed exhibition through childhood and getting misdiagnosed as post-traumatic stress disorder in cases of trauma- induced autism add on to its mystery. Increasing fascination and research towards the subject have presented a multitude of possibilities to consider in the diagnosis of autism. Autistic individuals exhibit facial signs that are often intuitive [8, 20, 21], resulting in the spilling of expressions inappropriate to Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 406 https://internationalpubls.com a situation they are interacting with. This inability of autistic individuals to convey appropriate signals and emotions in a conversation tends to hinder their capability to appropriately communicate with the external environment, resulting in misunderstandings to arise in a relationship. However, the face markers that engage in the formation of such facial expressions hold the key to understand how an autistic individual differs extensively from their normal counterparts [1], thus offering an insight into understanding the nature of the cues adopted by an autistic person. Studies have been done to upgrade and accelerate the diagnosis of autism spectrum disorder (ASD) using the techniques of Machine Learning. The ultimate objective of all these studies lies in enabling a timely intervention and treatment by aiding in early ASD diagnosis. While Praveena et al. [14] focused on using facial image processing to recognize emotions and predict ASD in children, Vakadkar et al. [17] used Logistic Regression to identify ASD in children during early developmental stages. Mujeeb Rahman et al. [11] used the static features extracted from the photographs of autistic children to identify ASD by utilizing CNN and DNN models. Several studies have also employed facial features for ASD in automated ML models while other studies leveraged data from brain neuroimaging. Boughattas et al. [2] utilized the ABIDE dataset containing functional Magnetic Resonance Images (fMRI) to detect ASD using convolutional neural networks. Moridian et al. [10] reviews several Computer-Aided Diagnosis Systems developed for automated diagnosis of ASD using MRI modalities. Lamani et al. [7] employs a graph convolutional network classifier to determine whether an image depicts normal or autistic characteristics and the research of Emel Koc et al. [3] aims to improve the diagnosis accuracy of ASD by using functional magnetic resonance imaging data. These studies show that an appropriate mathematical model is significant in deriving a crucial conclusion. Specifically, in a growing area of research where one utilizes facial features for the diagnosis of autism, the need for building a model that’s accurate and efficient is imminent [12, 13]. Building on these trends, this paper constructs a fuzzy π›½βˆ’neighborhood based soft rough set to detect autism by utilizing crucial face landmarks. 2. Fuzzy 𝜷 βˆ’Neighborhood Based Soft Rough Sets In this section, the concepts of 𝛽-reduct, fuzzy 𝛽-neighborhood, and fuzzy 𝛽-neighbourhood-based soft rough set of an object are introduced, along with an illustrative example for each. Definition 2.1. Let 𝐼 = (π‘ˆ, 𝐴, 𝐹) represent a covering-based information system where π‘ˆ is a non- empty finite set of objects called the universal set. 𝐴 is the non-empty finite set of attributes with membership function defined by πœ‡π‘Ž: π‘ˆ β†’ [0,1] βˆ€π‘Ž ∈ 𝐴 and 𝐹: 𝐴 β†’ 𝒫(π‘ˆ) defined by 𝑭(𝒂) = {𝒙 ∈ 𝑼 | 𝝁𝒂(𝒙) ≀ πœΉπ’‚} βˆ€π’‚ ∈ 𝑨 F(a) is the set of objects in π‘ˆ possessing the attribute a with respect to a threshold π›Ώπ‘Ž such that ⋃ 𝐹(π‘Ž) = π‘ˆ π‘Žβˆˆπ΄ . The choice of π›Ώπ‘Ž is made following expert insights, thorough experimenting, and the observation of the system under consideration. Definition 2.2. For 𝛽 ∈ [0,1], the 𝛽-reduct, 𝑁𝛽(π‘₯) of an object π‘₯ is defined as π‘΅πœ·(𝒙) = {𝒂 ∈ 𝑨 | 𝝁𝒂(𝒙) β‰₯ 𝜷} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 407 https://internationalpubls.com which represents the set of attributes playing a significant role on π‘₯ with respect to 𝛽. Definition 2.3. For 𝛽 ∈ [0,1], the fuzzy 𝛽-neighborhood, �̃�𝛽(π‘₯) of an object π‘₯ in π‘ˆ is defined as οΏ½ΜƒοΏ½πœ·(𝒙) = {π’š ∈ 𝑼 | 𝝁𝒂(π’š) β‰₯ 𝜷} which represents the set of objects that are in the neighborhood of π‘₯ with respect to the 𝛽-reduct of π‘₯. Definition 2.4. The fuzzy 𝛽-Neighborhood based soft-rough set, 𝛽 βˆ’ 𝑅𝑆(π‘₯) of the object π‘₯ belonging to the universe π‘ˆ is defined by 𝜷 βˆ’ 𝑹𝑺(𝒙) = (οΏ½ΜƒοΏ½πœ·(𝒙)βˆ’, οΏ½ΜƒοΏ½πœ·(𝒙) –) where, οΏ½ΜƒοΏ½πœ·(𝒙)_ = {𝑭(𝒂) | 𝑭(𝒂) βŠ† οΏ½ΜƒοΏ½πœ·(𝒙), 𝒂 ∈ π‘΅πœ·(𝒙)} οΏ½ΜƒοΏ½πœ·(𝒙)βˆ’ = {𝑭(𝒂) | 𝑭(𝒂) ∩ οΏ½ΜƒοΏ½πœ·(𝒙) β‰  βˆ…, 𝒂 ∈ π‘΅πœ·(𝒙)} represent the lower approximation and upper approximation of the fuzzy 𝛽-neighborhood based soft rough set of the object π‘₯. The lower approximation space, �̃�𝛽(π‘₯)_ of the object π‘₯ contain those objects of 𝐹(π‘Ž) that are completely contained in the fuzzy 𝛽-neighborhood of π‘₯ for every π‘Ž that belongs to 𝑁𝛽(π‘₯) and the upper approximation space, �̃�𝛽(π‘₯)βˆ’ of the object π‘₯ contains those objects of 𝐹(π‘Ž) whose intersection with the fuzzy 𝛽-neighbourhood is non empty for every π‘Ž in 𝑁𝛽(π‘₯). Example 2.5. Consider 𝐼 = (π‘ˆ, 𝐴, 𝐹), the covering-based information system. Let π‘ˆ = {π‘₯1, π‘₯2, π‘₯3, π‘₯4} be the universal set and let 𝐴 = {π‘Ž1, π‘Ž2, π‘Ž3, π‘Ž4, π‘Ž5} be the set of attributes. 𝐹: 𝐴 β†’ 𝒫(π‘ˆ) be defined by 𝐹(π‘Ž1) = {π‘₯1, π‘₯4} 𝐹(π‘Ž2) = {π‘₯1, π‘₯3} 𝐹(π‘Ž3) = {π‘₯1, π‘₯2,, π‘₯4} 𝐹(π‘Ž4) = {π‘₯3, π‘₯4} 𝐹(π‘Ž5) = {π‘₯2, π‘₯4} By taking π›Ώπ‘Ž1 = 0.2, π›Ώπ‘Ž2 = 0.1, π›Ώπ‘Ž3 = 0.3, π›Ώπ‘Ž4 = 0.2, and π›Ώπ‘Ž5 = 0.1 respectively. The 𝛽-reduct and fuzzy 𝛽-neighborhood for 𝛽 = 0.2 as given by Table 2. The fuzzy 𝛽-neighborhood based soft rough set of an object π‘₯, 0.2 βˆ’ 𝑅𝑆(π‘₯) is given as in Table 3. Table 1: Information System 𝐼 = (π‘ˆ, 𝐴, 𝐹) U/A a1 a2 a3 a4 a5 x1 0.2 0.1 0.3 0.3 0.3 x2 0.4 0.2 0.3 0.4 0.1 x3 0.3 0.1 0.4 0.2 0.2 x4 0.2 0.3 0.1 0.2 0.1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 408 https://internationalpubls.com Table 2: 0.2-reduct and fuzzy 0.2-neighbourhood of the objects 𝑼 π‘΅πŸŽ.𝟐(𝒙) οΏ½ΜƒοΏ½πŸŽ.𝟐(𝒙) x1 {π‘Ž1, π‘Ž3, π‘Ž4, π‘Ž5} {π‘₯1, π‘₯3} x2 {π‘Ž1, π‘Ž2, π‘Ž3, π‘Ž4} {π‘₯2} x3 {π‘Ž1, π‘Ž3, π‘Ž4, π‘Ž5} {π‘₯1, π‘₯3} x4 {π‘Ž1, π‘Ž2, π‘Ž4} {π‘₯2, π‘₯4} Table 3: Fuzzy 0.2-Neighbourhood Based Soft Rough Set of The Objects 𝑼 𝟎. 𝟐 βˆ’ 𝐑𝐒(π’™π’Š) x1 (𝐹(π‘Ž2), π‘ˆ βˆ’ 𝐹(π‘Ž5)) x2 (βˆ…, 𝐹(π‘Ž3) βˆͺ 𝐹(π‘Ž5)) x3 (𝐹(π‘Ž2), π‘ˆ βˆ’ 𝐹(π‘Ž5)) x4 (𝐹(π‘Ž5), 𝐹(π‘Ž1) βˆͺ 𝐹(π‘Ž3) βˆͺ 𝐹(π‘Ž5)) Remark 2.6. β€’ The 0.2-reduct of π‘₯1, 𝑁0.2(π‘₯1) = {π‘Ž1, π‘Ž3, π‘Ž4, π‘Ž5} consists those significant attributes of π‘₯1 with respect to 𝛽 = 0.2. β€’ The fuzzy 0.2-neighborhood of π‘₯1, {π‘₯1, π‘₯3} contains those objects that are in the neighborhood of π‘₯1 with respect to the 0.2-reduct of π‘₯1. β€’ The fuzzy 0.2-neighborhood based soft rough set of π‘₯1, 0.2 βˆ’ RS(π‘₯1) = (𝐹(π‘Ž2), π‘ˆ βˆ’ 𝐹(π‘Ž5)) consists of the lower and upper approximation spaces of the object π‘₯1. The lower approximation space, οΏ½ΜƒοΏ½0.2(π‘₯1)_ contains 𝐹(π‘Ž2) which is completely contained in the fuzzy 0.2 neighborhood of π‘₯1 where {π‘Ž1, π‘Ž3, π‘Ž4, π‘Ž5} are in 𝑁0.2(π‘₯1). β€’ The upper approximation space, οΏ½ΜƒοΏ½0.2(π‘₯1) – contains all the objects in {𝐹(π‘Ž1), 𝐹(π‘Ž2), 𝐹(π‘Ž3), 𝐹(π‘Ž4)} whose intersection with the fuzzy 0.2 neighborhood is non- empty where {π‘Ž1, π‘Ž2, π‘Ž3, π‘Ž4} are all in 𝑁0.2(π‘₯1). From the given example, it can be observed that the 𝛽-reduct is the set of attributes that primarily characterize the chosen object π‘₯, the fuzzy 𝛽-neighborhood [19] embodies those neighbors of x that resemble the chosen object in terms of the attributes that characterize it. Thus, by making an appropriate choice for 𝛽, one can regulate the lower approximation space to record the neighborhood that closely identifies with the pivotal qualities represented by the chosen object with excellent precision. 3. Implementation of Fuzzy Ξ²-Neighborhood Based Soft Rough Sets to Detect Autism Objective: β€’ To determine whether an object is autistic or otherwise using the 𝛽-RS model by studying the facial features of the object through collected face landmark points. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 409 https://internationalpubls.com β€’ To classify the different levels of autism using the defined π›½βˆ’RS model. Data: A public dataset taken from Kaggle has been utilized for the prediction of autism in individuals. Consider 𝐼 = (π‘ˆ, 𝐹, 𝐴) be the covering-based Information system. Let π‘ˆ = {𝑒1, 𝑒2, 𝑒3, . . . , 𝑒200} be the universal set of face images, 𝐴 = {π‘Ž1, π‘Ž2, π‘Ž3, . . . , π‘Ž15} be the attribute set of pivotal face landmark points of the objects where each attribute is an ordered pair of coordinates π‘Žπ‘– = (π‘Žπ‘₯𝑖 , π‘Žπ‘¦π‘– ). Consequently, the membership value of every object with respect to each attribute is denoted as πœ‡ π‘Žπ‘– (𝑒𝑗) = (πœ‡π‘Žπ‘₯𝑖 (𝑒𝑗), πœ‡π‘Žπ‘¦π‘– (𝑒𝑗)), βˆ€π‘– = 1, 2, . . . , 15 and 𝑗 = 1, 2, . . . , 200 and 𝐹: 𝐴 β†’ 𝒫(π‘ˆ) is given by 𝐹 (π‘Žπ‘–) = {𝑒 ∈ π‘ˆ ∢ |πœ‡π‘Žπ‘– (𝑒) βˆ’ πœ‡π‘ π‘‘π‘‘π‘Žπ‘– (𝑒)| ≀ π›Ώπ‘Žπ‘– } πœ‡π‘ π‘‘π‘‘π‘Žπ‘– (𝑒) is the mean of πœ‡π‘Žπ‘– (𝑒) of neurotypical individuals and π›Ώπ‘Žπ‘– = 0.03 which has been chosen following thorough experimentation and expert knowledge. In Figure 1, the attributes, i.e., the 15 facial features points and their location of an object 𝑒1 is shown. The corresponding membership values of the object 𝑒1 for the 15 facial features is given in Table 4. Table 4: Location of 15 Facial Feature Points of an Object Point Coordinates Name ππ’‚π’Š(𝒙) A (𝐿𝐢π‘₯, 𝐿𝐢𝑦) Left Eye Centre (0.680, 0.852) B (𝑅𝐢π‘₯, 𝑅𝐢𝑦) Right Eye Centre (0.577, 0.750) C (𝐿𝐼π‘₯ , 𝐿𝐼𝑦) Left Eye Inner Corner (0.814, 0.784) D (𝐿𝑂π‘₯ , 𝐿𝑂𝑦) Left Eye Outer Corner (0.505, 0.852) E (𝑅𝐼π‘₯ , 𝑅𝐼𝑦) Right Eye Inner Corner (0.753, 0.739) F (𝑅𝑂π‘₯, 𝑅𝑂𝑦) Right Eye Outer Corner (0.423, 0.682) G (𝐿𝐸𝐼π‘₯ , 𝐿𝐸𝐼𝑦) Left Eyebrow Inner Corner (0.866, 0.943) H (𝐿𝐸𝑂π‘₯ , 𝐿𝐸𝑂𝑦) Left Eyebrow Outer Corner (0.330, 0.977) I (𝑅𝐸𝐼π‘₯ , 𝑅𝐸𝐼𝑦) Right Eyebrow Inner Corner (0.742, 0.886) J (𝑅𝐸𝑂π‘₯, 𝑅𝐸𝑂𝑦) Right Eyebrow Outer Corner (0.278, 0.773) K (𝑁𝐢π‘₯, 𝑁𝐢𝑦) Nose Tip (0.979,0.409) L (𝑀𝐿π‘₯, 𝑀𝐿𝑦) Mouth Left Corner (0.639, 0.205) M (𝑀𝑅π‘₯, 𝑀𝑅𝑦) Mouth Right Corner (0.773, 0.148) N (𝑀𝑇π‘₯, 𝑀𝑇𝑦) Mouth Centre Top Lip (0.907, 0.148) O (𝑀𝐡π‘₯, 𝑀𝐡𝑦) Mouth Centre Bottom Lip (0.907, 0.911) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 410 https://internationalpubls.com Figure 1. Location of 15 Facial Feature Points of an Object The steps involved in the detection of whether an object 𝑒𝑗 is autistic or otherwise are as follows. (1) Compute 𝑁𝛽(𝑒𝑗) and �̃�𝛽(𝑒𝑗) (2) Compute 𝛽 βˆ’ 𝑅𝑆(𝑒𝑗 ) = (�̃�𝛽(𝑒𝑗)_, �̃�𝛽(𝑒𝑗)βˆ’ ) (3) If π›Ώπ‘‘π‘Ž(𝑒𝑗 ) ≀ 𝛼, then 𝑒𝑗 and �̃�𝛽(𝑒𝑗)_ is autistic. Else 𝑒𝑗 and �̃�𝛽(𝑒𝑗)_ is not autistic. where π›Ώπ‘‘π‘Ž(𝑒𝑗 ) = π‘Žπ‘£π‘’π‘Ÿπ‘Žπ‘”π‘’(|πœ‡π‘Žπ‘– (𝑒) βˆ’ πœ‡π‘ π‘‘π‘‘π΄π‘’π‘‘π‘Žπ‘– (𝑒)|) represents the deviation of the object from autistic people, πœ‡π‘ π‘‘π‘‘π΄π‘’π‘‘π‘Žπ‘– (𝑒) is the mean of πœ‡π‘Žπ‘– (𝑒) of the autistic people. Here, 𝛼 = 0.1 following thorough experimentation and expert insights. By comparing π›Ώπ‘‘π‘Ž(𝑒𝑗 ) with the threshold of Ξ±, the system detects whether the object 𝑒𝑗 is autistic or otherwise. If the object 𝑒𝑗 is autistic, then the objects in the lower approximation space of 𝑒𝑗 (�̃�𝛽(𝑒𝑗)_) are also autistic since they are in the closest neighborhood of 𝑒𝑗 . The upper approximation space of the object 𝑒𝑗 represent the objects that might possibly be present in the closest neighborhood of 𝑒𝑗 . Hence, the level of autism might vary. Let us check whether an object 𝑒20 ∈ π‘ˆ is autistic or otherwise. For 𝛽 = 0.4, the corresponding 𝛽- reduct and fuzzy 𝛽-neighborhood are given in Table 5 and the fuzzy 𝛽-neighborhood based rough set of 𝑒20 is given in Table 6 respectively. The deviation of the object 𝑒20 and some of its neighbors in the lower approximation space from the standard autistic object is given in Table 7. Table 5: 𝑁0.4(𝑒20) π‘Žπ‘›π‘‘ οΏ½ΜƒοΏ½0.4(𝑒20) π‘ˆ π‘΅πŸŽ.4(π’–πŸπŸŽ) οΏ½ΜƒοΏ½πŸŽ.4(π’–πŸπŸŽ) 𝑒20 [π‘Ž1, π‘Ž2, π‘Ž3, π‘Ž4, π‘Ž5, π‘Ž6, π‘Ž7, π‘Ž8, π‘Ž9, π‘Ž10, π‘Ž11, π‘Ž12, π‘Ž13, π‘Ž14] [𝑒20, 𝑒21, 𝑒73, 𝑒76, 𝑒77, 𝑒85, 𝑒86, 𝑒89, 𝑒94, 𝑒96, 𝑒99,𝑒188 ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 411 https://internationalpubls.com Table 6: 0.4 βˆ’ 𝑅𝑆(𝑒20) = (οΏ½ΜƒοΏ½0.4(𝑒20)_, οΏ½ΜƒοΏ½0.4(𝑒20)βˆ’) π‘ˆ οΏ½ΜƒοΏ½πŸŽ.πŸ’(π’–πŸπŸŽ)_ οΏ½ΜƒοΏ½πŸŽ.πŸ’(π’–πŸπŸŽ)βˆ’ 𝑒20 [𝑒20, 𝑒73, 𝑒76, 𝑒77, 𝑒85, 𝑒86, 𝑒94, 𝑒188 ] [𝑒2, 𝑒4, 𝑒6, 𝑒8, 𝑒10, 𝑒12, 𝑒18, 𝑒20, 𝑒21, 𝑒76, 𝑒77, 𝑒82, 𝑒83, 𝑒85, 𝑒86, 𝑒89, 𝑒99,𝑒94, 𝑒96𝑒188 ] Table 6: 0.4 βˆ’ 𝑅𝑆(𝑒20) = (οΏ½ΜƒοΏ½0.4(𝑒20)_, οΏ½ΜƒοΏ½0.4(𝑒20)βˆ’) 𝑼 πœΉπ’…π’‚(𝒖𝒋) 𝑒20 0.0891 𝑒85 0.0862 𝑒77 0.0951 𝑒86 0.0590 𝑒188 0.0575 It can be inferred from Table 7 that not only π›Ώπ‘‘π‘Ž(𝑒20) is less than 0.1 but also the deviation of π›Ώπ‘‘π‘Ž(u) for all the images in the neighborhood of 𝑒20 is less than 0.1. Hence, it can be said that all the images in οΏ½ΜƒοΏ½0.4(𝑒20)_ are autistic. Thus, it can be observed that the model has impeccably captured the characteristics of the chosen object along with its neighbors with expected precision. For testing, we consider two unseen images. On providing these inputs, the model successfully returned its decisions in almost few seconds, the results of which are given in Figure 2. For the testobj1, as the deviation π›Ώπ‘‘π‘Ž = 0.047901 ≀ 0.1, the 0.4 βˆ’ 𝑅𝑆 model detects it to be autistic. And for the testobj2, as the deviation π›Ώπ‘‘π‘Ž = 0.1930 β‰₯ 0.1, the 0.4 βˆ’ 𝑅𝑆 model detects it to be non-autistic. Figure 2. Testing Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 412 https://internationalpubls.com The 𝑅𝑆-model can also be used to determine the levels in Autism exhibited by individuals. Diagnostic and Statistical Manual of Mental Disorders describes three levels in the autism spectrum disorder. These levels depend on the severity of the differences [22] in the brain that an individual exhibits and the extent of support such individuals need to continue functioning without interference. Level-I individuals exhibit morphological features like neurotypical subjects and require minimum support when compared to their counterparts, making an attempt to engage in social conventions and create friendships in a community. Level - II autistic subjects require more support than their Level-I counterparts, but lesser than the Level-III subjects. They exhibit an elevated amount of distress when confronted with change, even more than Level-I individuals. Level - III individuals have to rely extensively on external support to manage their activities, and it is in the second and third levels, the attributes become more prevalent. However, a timely diagnosis of the severity of the disorder [[16], [9]] in an individual can help bring about a change in their functionalities, calling for personalized help finding its way to such individuals. To classify autistic individuals according to the severity levels, the deviation 𝛿𝑑𝑛(𝑒𝑗) = π‘Žπ‘£π‘’π‘Ÿπ‘Žπ‘”π‘’(|πœ‡π‘Žπ‘– (𝑒) βˆ’ πœ‡π‘ π‘‘π‘‘π‘Žπ‘– (𝑒)|) of the chosen object 𝑒𝑗 from neurotypical individuals is considered. Given an image 𝑒𝑗 , the 𝛽 βˆ’RS model first detects whether it is autistic or otherwise and also finds its lower approximation space. Then for every element in �̃�𝛽(𝑒𝑗)_ , their deviation 𝛿𝑑𝑛(𝑒𝑗) is calculated. And according to their deviation values the level of autism of the object uj is determined. The decision taken according to values of 𝛿𝑑𝑛(𝑒𝑗) is given in the pseudocode as follows. def findLevels (img): 𝛿𝑑𝑛(𝑒𝑗) = average (|πœ‡π‘Žπ‘–(𝑒𝑗) βˆ’ πœ‡π‘ π‘‘π‘‘π΄(𝑒𝑗)|) countA, countNA=0 for i=1:size(LA): if π›Ώπ‘‘π‘Ž for LA(i)>=0.1: countNA+=1 else: countA+=1 end if π›Ώπ‘‘π‘Ž <= 0.1 & round(π›Ώπ‘‘π‘Ž) = 0 & countNA > count A: decision = Level - I elseif π›Ώπ‘‘π‘Ž <= 0.1 & 𝛿𝑑𝑛 ∈ (0, 0.4] & countNA < countA: decision = Level - II elseif π›Ώπ‘‘π‘Ž <= 0.1 & 𝛿𝑑𝑛 > 0.4 & countNA < countA: decision = Level – III end end function Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 413 https://internationalpubls.com The examples of Level-I, Level-II, Level-III autistic people along their deviation 𝛿𝑑𝑛(𝑒𝑗) detected by the 𝛽 βˆ’ 𝑅𝑆 model is given in Figure 3. Figure 3. Levels of Autism 4. Interpretation of Results Obtained through 𝜷 βˆ’ 𝑹𝑺 model (1) The 𝛽-reduct of the objects aids in attribute reduction and gives significant attribute with respect to Ξ². (2) The 𝛽 βˆ’ 𝑅𝑆 model can be used to detect whether the object is autistic or otherwise, continually implying that �̃�𝛽(𝑒)_ is also autistic. For example, an image 𝑒20 and its neighbors in οΏ½ΜƒοΏ½0.4(𝑒20)_ detected autistic by 0.4 βˆ’ RS model is shown in Figure 4. Figure 4. Autism Detection by Ξ² - RS model (3) By comparing the lower approximation space of an autistic and non-autistic image one can analyze how the facial features of an autistic individual differ from their normal counterparts. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 414 https://internationalpubls.com This is illustrated by taking an example from the lower approximation space of an autistic individual. From the figure, it can be inferred using the bounding boxes that autistic individuals exhibit wider facial markers contrasted to neurotypical individuals. Figure 5. Face Markers: Autistic Vs. Non-Autistic (4) The lower approximation space helps find objects exhibiting similar expressions in the same class. The examples that capture a frown/frustration amongst autistic individuals are presented in Figure 6 and 7. Figure 6. Objects Frowning Figure 7. Objects Expressing Frustration (5) The upper approximation space is advantageous in making decisions about mild autistic individuals who share commonalities in facial features with neurotypical individuals. By analyzing the number of neurotypical individuals in the upper approximation space, one can determine how mildly autistic an individual is. An example of the application of the upper Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 415 https://internationalpubls.com approximation space for an object 𝑒5 along with neurotypical individuals in it is shown in Figure 8. Figure 8. Upper Approximation of Mild Autistic Individual u5 (6) The 𝛽 βˆ’ 𝑅𝑆 model helps to understand the effect of 𝛽 on the rough set of the chosen object in making decision. For example, the effect of 𝛽 on the π›½βˆ’reduct and the lower and upper approximation space of the object u98 is shown in Table 8. Table 8: Effect of 𝛽 in Constructing the Lower and Upper Approximation Spaces for 𝑒98 𝛽 𝑁𝛽(𝑒98) 𝑁𝛽(𝑒98)_ 𝑁𝛽(𝑒98)βˆ’ 0.1 [π‘Ž1, π‘Ž2, π‘Ž3, π‘Ž4, π‘Ž5, π‘Ž6, π‘Ž7, π‘Ž8, π‘Ž9, π‘Ž10, π‘Ž11, π‘Ž12, π‘Ž13, π‘Ž14, π‘Ž15] π‘ˆ π‘ˆ 0.2 [π‘Ž1, π‘Ž2, π‘Ž3, π‘Ž4, π‘Ž5, π‘Ž6, π‘Ž7, π‘Ž8, π‘Ž9, π‘Ž10, π‘Ž11, π‘Ž12, π‘Ž13, π‘Ž14] [𝑒2, 𝑒3, 𝑒4, 𝑒6, … 𝑒174, 𝑒179,𝑒192 , 𝑒197] π‘ˆ 0.4 [π‘Ž1, π‘Ž2, π‘Ž3, π‘Ž4, π‘Ž5, π‘Ž6, π‘Ž7, π‘Ž8, π‘Ž9, π‘Ž11] [𝑒7, 𝑒9, 𝑒11, 𝑒16, 𝑒21, 𝑒34, 𝑒35, 𝑒73, 𝑒76, 𝑒82, 𝑒84, 𝑒89, 𝑒94, 𝑒98, 𝑒99, 𝑒114, 𝑒140] [𝑒6, 𝑒7, 𝑒9, 𝑒11, 𝑒16, 𝑒7, 𝑒9, 𝑒11, 𝑒16, 𝑒20, 𝑒21, 𝑒34, 𝑒35, 𝑒59, 𝑒34, 𝑒65, 𝑒73, 𝑒76, 𝑒77, 𝑒80, 𝑒82, 𝑒84, 𝑒89, 𝑒92, 𝑒94, 𝑒96, 𝑒98, 𝑒101, 𝑒108, 𝑒114, 𝑒139, 𝑒140] 0.7 [π‘Ž5, π‘Ž11] [𝑒6, 𝑒7, 𝑒9, 𝑒11, 𝑒16 𝑒20, 𝑒21, 𝑒98, 𝑒102, 𝑒108] [𝑒6, 𝑒7, 𝑒9, 𝑒11, 𝑒16, 𝑒20, 𝑒101, . . . 𝑒180, 𝑒188] 0.8 [π‘Ž11] [𝑒6, 𝑒7, 𝑒9, 𝑒98] [𝑒6, 𝑒7, 𝑒9, 𝑒11, 𝑒16, 𝑒20, 𝑒98, 𝑒114, 𝑒120] 0.9 [ ] [ ] [ ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 416 https://internationalpubls.com It can be observed that for 𝛽 < 0.4, there is overfitting of the model, as the 𝛽-neighborhood contains those attributes that are less significant and the fuzzy 𝛽-neighborhood contains those objects which have no resemblance with the chosen object. Meaning, the system does not learn any pattern in the data but understands the training data solely for the sake of model building, thus resulting in overfitting. For 𝛽 tending to 1, there is underfitting of the model as the important characterizing attributes of the chosen object are lost and the model begins relying on assumptions about the object for decision making. Hence, an optimal choice of 𝛽 is essential for accurate decision making. (7) The model presented an accuracy of 92.5% for the threshold value Ξ± = 0.1. The confusion matrix of the 0.4 βˆ’ RS model is shown in Figure 9. In comparison to the existing models, this model is extremely advantageous owing to the fact that it makes a decision about the chosen individual by observing the characteristics shared by the individual with its neighbors, thereby improving the speed, accuracy and precision of the prediction. Figure 9. Accuracy of the 0.4 βˆ’ RS model Allowing the mathematical model to work with the electrical activity in the brain recorded using EEGs and focusing on the regions of the brain that contribute to giving rise to unique facial expressions in autistic individuals can help improve its prediction capability by several margins. 5. Conclusion The 𝛽 βˆ’reduct, fuzzy 𝛽-neighborhood, and fuzzy 𝛽-neighborhood based soft rough sets of the objects have been defined. The 𝛽 βˆ’ 𝑅𝑆 model was implemented for autism detection and the model showed an accuracy of about 92.5%. Further, the advantage of the lower approximation space in determining the levels of autism, detecting individuals with similar expression from same class, and differentiation of facial feature of an autistic individual from a neurotypical one are discussed. The significance of the upper approximation space in the case of a mild autistic people has also been shown. Consequently, the main advantage of the model has been found to lie in the fact that it requires no prior information Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 417 https://internationalpubls.com to make decisions about the dataset and utilizes the fuzzy 𝛽-neighborhood of the object for detection and classification of levels of autism. The 𝛽-reduct aids in understanding the behavior of individuals who express themselves similar to a chosen individual, and hence the 𝛽 βˆ’ 𝑅𝑆 model executes the decision-making process in a reduced computational time frame to produce efficient results. The 𝛽 βˆ’ 𝑅𝑆 model can further be tuned by working with equipment that helps understand brain activity to produce improved and reliable results. Funding: This research received no external funding. Acknowledgments: The authors thank the management and Principal, Sri Siva Subramaniya Nadar College of Engineering for their support and encouragement for the successful completion of the work. Conflicts of Interest: The authors declare no conflict of interest. References [1] Aldridge, Kristina, Ian D. George, Kimberly K. Cole, Jordan R. Austin, T. Nicole Takahashi, Ye Duan and Judith H. Miles. β€œFacial phenotypes in subgroups of prepubertal boys with autism spectrum disorders are correlated with clinical phenotypes.” Molecular autism 2 (2011): 1-12. [2] Boughattas, Naouel, and Hanen Jabnoun. β€œAutism Spectrum Disorder (ASD) Detection Using Machine Learning Algorithms.” In International Conference on Smart Homes and Health Telematics, pp. 225-233. Cham: Springer International Publishing, 2022. [3] Koc, Emel, Habil Kalkan, and Semih Bilgen. "Autism Spectrum Disorder Detection by Hybrid Convolutional Recurrent Neural Networks from Structural and Resting State Functional MRI Images." Autism Research and Treatment 2023, no. 1 (2023): 4136087. [4] Ganesan, Srividhya, and J. Senthil. "Prediction of autism spectrum disorder by facial recognition using machine learning." Webology 18, no. Special Issue on Information Retrieval and Web Search (2021): 406- 417. [5] Guenther, Katja. "β€˜It’s all done with mirrors’: VS Ramachandran and the material culture of phantom limb research." Medical history 60, no. 3 (2016): 342-358. [6] Hossain, Md Delowar, Muhammad Ashad Kabir, Adnan Anwar, and Md Zahidul Islam. β€œDetecting autism spectrum disorder using machine learning techniques: An experimental analysis on toddler, child, adolescent and adult datasets.” Health Information Science and Systems 9 (2021): 1-13. [7] Lamani, Manjunath Ramanna, and P. Julian Benadit. β€œAutomatic Diagnosis of Autism Spectrum Disorder Detection Using a Hybrid Feature Selection Model with Graph Convolution Network.” SN Computer Science 5, no. 1 (2023): 126. [8] Loth, Eva, Lucia Garrido, Jumana Ahmad, Ekaterina Watson, Alexa Duff, and Bradley Duchaine. β€œFacial expression recognition as a candidate marker for autism spectrum disorder: how frequent and severe are the deficits?” Molecular autism 9 (2018): 1-11. [9] Miles, Judith H., T. Nicole Takahashi, Julie Hong, Nicole Munden, Nancy Flournoy, Stephen R. Braddock, Rick A. Martin, M. Anne Spence, Richard E. Hillman, and Janet E. Farmer. β€œDevelopment and Validation of a measure of dysmorphology: useful for autism subgroup classification.” American Journal of Medical Genetics Part A 146, no. 9 (2008): 1101-1116. [10] Moridian, Parisa, Navid Ghassemi, Mahboobeh Jafari, Salam Salloum-Asfar, Delaram Sadeghi, Marjane Khodatars, Afshin Shoeibi et al. "Automatic autism spectrum disorder detection using artificial intelligence methods with MRI neuroimaging: A review." Frontiers in Molecular Neuroscience 15 (2022): 999605. [11] Mujeeb Rahman, K. K., and M. Monica Subashini. β€œIdentification of autism in children using static facial features and deep neural networks.” Brain Sciences 12, no. 1 (2022): 94. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 418 https://internationalpubls.com [12] Praba, B; Balambal Suryanarayanan; D. Nagarajan; and Said Broumi. β€œAnalysis of Teaching-Learning Efficiency Using Attribute Based Double Bounded Rough Neutrosophic Set Driven Random Forests.” Neutrosophic Sets and Systems 55, 1 (2023). [13] Praba, B., S. Pooja, and Nethraa Sivakumar. β€œAttribute based Double Bounded Rough Neutrosophic Sets in Facial Expression Detection.” Neutrosophic Sets and Systems 49 (2022): 324-340. [14] Praveena, T. Lakshmi, and NV Muthu Lakshmi. "Perception of Autism Spectrum Disorder Children by Envisaging Emotions from the Facial Images." International Journal of Engineering and Advanced Technology (IJEAT) 10, no. 2 (2020). [15] Ramachandran, Vilayanur S., and Diane Rogers-Ramachandran. "Synaesthesia in phantom limbs induced with mirrors." Proceedings of the Royal Society of London. Series B: Biological Sciences 263, no. 1369 (1996): 377-386. [16] Shanker, Stuart. "Autism and the dynamic developmental model of emotions." Philosophy, Psychiatry, & Psychology 11, no. 3 (2004): 219-233. [17] Weston, Charles SE. "Four social brain regions, their dysfunctions, and sequelae, extensively explain autism spectrum disorder symptomatology." Brain Sciences 9, no. 6 (2019): 130. [18] https://www.spectrumnews.org/news/people-autism-sometimes-give-ambiguous-looks/ [19] https://www.abtaba.com/blog/autism-facial-expressions [20] https://www.medicalnewstoday.com/articles/325106#levels-of-autism https://www.spectrumnews.org/news/people-autism-sometimes-give-ambiguous-looks/ https://www.abtaba.com/blog/autism-facial-expressions https://www.medicalnewstoday.com/articles/325106#levels-of-autism