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Computational Intelligence and Neuroscience
Volume 2015 (2015), Article ID 760403, 7 pages
Research Article

Application of -Number Based Modeling in Psychological Research

1Department of Computer Aided Design, Azerbaijan State Oil Academy, Azerbaijan
2Department of Psychological Counselling and Guidance, Near East University, 99138 Lefkoşa, Northern Cyprus, Mersin 10, Turkey

Received 5 December 2014; Revised 6 February 2015; Accepted 17 February 2015

Academic Editor: Rahib H. Abiyev

Copyright © 2015 Rafik Aliev and Konul Memmedova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Pilates exercises have been shown beneficial impact on physical, physiological, and mental characteristics of human beings. In this paper, Z-number based fuzzy approach is applied for modeling the effect of Pilates exercises on motivation, attention, anxiety, and educational achievement. The measuring of psychological parameters is performed using internationally recognized instruments: Academic Motivation Scale (AMS), Test of Attention (D2 Test), and Spielberger’s Anxiety Test completed by students. The GPA of students was used as the measure of educational achievement. Application of Z-information modeling allows us to increase precision and reliability of data processing results in the presence of uncertainty of input data created from completed questionnaires. The basic steps of Z-number based modeling with numerical solutions are presented.

1. Set of Problem

The investigation of the effect of physical activities on motivation, attention, anxiety, and educational performances of students is one of the important research areas in psychology. Hannaford in [1] noticed that “Thinking and learning are not all in our head. Our movements that not only express knowledge and facilitate greater cognitive function, they actually grow the brain as they increase in complexity. Our entire brain structure is intimately connected to and grown by the movement mechanism within our body.” Recently some research works, demonstrating the achievement of the academic performances through physical activities, have been published in different sources. In [2] is given a review showing the effectiveness of aerobic, resistance, and multimodal exercise interventions on a wide range of outcome measures, including cognition, general physical function, mobility, strength, balance, flexibility, quality of life. It was shown [3] that the students with the highest fitness level performed better on standardized tests and students with the lowest fitness level performed lower in class grades. In [46] it was found that the physical activity improves a cognitive function, learning, and academic achievement. Salmon in [7] presented cross-sectional and longitudinal studies analyzing the effect of physical exercises on anxiety, depression, and sensitivity to stress. Increased physical activity therefore reduces premature mortality [8] and the establishment and maintenance of exercises’ habits has become target for clinical psychologists [9].

The above research studies consider impact of psychological activities on performances of students using different psychological parameters. Considering and modeling the effect of physical activities on basic psychological parameters of humans acquire great importance. In this paper, we suggest the model that takes the parameters—motivation, attention, and anxiety—into account and improves educational achievement of students through Pilates exercises.

The structure that we have designed for modeling the effect of Pilates exercises on motivation, attention, anxiety, and educational achievement is presented in Figure 1.

Figure 1

There is broad agreement in the literature that the physical exercises and particularly Pilates exercises are associated with the motivation, attention, anxiety, and educational achievement [1015]. The classical statistical approach is used today for finding the relationship between Pilates and above mentioned psychological parameters. Application of statistical techniques allows estimating the relationship between input (effect of Pilates) and output (achievement) from probabilistic point of view. This approach embraces only statistical uncertainty, but not fuzzy uncertainty inherent in psychological phenomena. Motivation of the application of -number theory to solve the presented problem is as follows.(1)The concepts in the human brain for perceiving, recognizing, and categorizing natural phenomena are vague and imprecise. The boundaries of these parameters (such as low, moderate, or high levels of anxiety, motivation, and attention) are not exactly defined. Therefore, claiming that this emerges from them also becomes vague.(2)-number theory allows us to estimate the relationship between input and output by using the concept of fuzzy information and partial reliability.(3)-number based approach enables us to use uncertainty measures to quantify the ambiguity associated with prediction of psychological variables.

We will start with short introduction of basic terms: Pilates, motivation, attention, and anxiety.

2. Introduction

2.1. Benefits of Pilates

In developing his method, Pilates (founder of Pilates exercises) combines both Eastern and the Western concepts [10] by including mental focus and specific breathing of yoga with the Western physical exercise systems.

The mind-body approach is a basic of Pilates principles: centering, concentration, control, precision, flow, and breath [10, 11].

Pilates has the following physical, psychological, and social impacts [1115].(1)Pilates exercise improves muscular strength, balance, posture, flexibility, and bone density and decreases back pain. The development of musculoskeletal fitness with long term resistance training is associated with enhanced cardiovascular function and musculoskeletal metabolism.(2)Pilates increases a cognitive function—Pilates is different than many other forms of exercise because it requires the mind to pay attention to what you are doing. Research shows that when required to think about how you are moving, your brain cells grow at a faster rate and your nervous system creates better connections throughout your body.(3)Pilates increases brain neurotransmitters, brain-derived neurotrophins that support neuronal differentiation and survival in the developing brain. Neurotrophins assure the survival of neurons in areas responsible for learning, memory, and higher thinking and, as an overall result, improve the overall quality of life.(4)Pilates increases blood flow and oxygen flow to the brain and raises levels of serotonin and endorphins—all of which help to reduce stress, anxiety, and fatigue and improve mood, motivation, and achievement.(5)Beyond academic achievement, many researchers connect Pilates to absenteeism, drop-out rate, and social communications of students.

2.2. Motivation

Motivation is defined as the process that initiates, guides, and maintains goal-oriented behaviors. It involves the biological, emotional, social, and cognitive forces that activate behavior. Motivation has been shown to positively influence study strategy and academic performance. Self-determination theory describes human motivation on a continuum categorized by amotivation, instincts, and extrinsic motivations [16].

Amotivation is the state of lacking the intention to act, whether resulting from not valuing the activity, not feeling capable of performing the task, or not expecting it to produce a desired outcome.

Extrinsic motivations are those that arise from outside of the individual and often involve rewards such as trophies, money, social recognition, or praise.

Intrinsic motivations are those that arise from within the individual, for the inherent satisfaction of the activity itself.

Pilates has effect on instinct motivation through the following mechanisms: improving total mood, body energy, self-esteem, psychological well-being, and vitality, reducing stress and anxiety, releasing certain neurotransmitters that alleviate physical and mental pain, and satisfying the basic psychological needs as competence, autonomy, and relatedness.

2.3. Anxiety

Anxiety is one of the major psychological variables which is considered as an important part of personality development. Anxiety is a psychological and physiological state characterized by somatic, emotional, cognitive, and behavioural components [17]. Anxiety generally helps in improving the performance of an individual. However, when anxiety becomes overwhelming, it may fall under the classification of anxiety disorder. This means anxiety should not cross its threshold value; otherwise it will reach up to an abnormal level.

American and European studies have found a negative correlation between anxiety and academic achievement.

Pilates has effect on anxiety through the following mechanisms: increasing body energy, sleep quality, attention, and concentration, releasing negatively thinking, improving blood and oxygen circulated to the brain, and relaxing muscles.

2.4. Attention

Attention was originally defined as processing one out of what seem several simultaneously possible objects or trains of thought. It implies withdrawal from some things in order to deal effectively with others [18].

Attention can be considered as a filter, in which many pieces of information come into the brain, but only one of these pieces of information is processed.

Pilates has effect on attention through the following mechanisms: concentration and precision are two main principles of Pilates; controlling a body movement by the brain; increasing sleep quality; relaxing of body; releasing negative thinking; decreasing the stress and anxiety.

3. Application of Fuzzy Logic and -Number Theory in Psychology Researches

The applications of fuzzy logic in psychology researches have been started since mid-1980 [19]. Averkin and Tarasov in [20] examined application of fuzzy modeling relation in psychology. Hesketh et al. considered [21] application of fuzzy graphical rating scale to the psychology. In [22] fuzzy logic based model of emotion is given. The role of fuzzy logic in psychological researches was examined in [23]. The researchers studied the relationship between motivation and anxiety using fuzzy logic and concluded that the fuzzy logic method has more advantages over statistical analysis to control uncertainties in data.

In this paper, we put a new approach for modeling of psychology processes through the -valuation concept. -information approach includes value of variable of interest and its reliability and has a major advantage for modeling such type of concepts. This approach enables us to use uncertainty measures to quantify the ambiguity associated with prediction of psychological parameters.

3.1. Preliminaries

A discrete -number [24] is an ordered pair , where is a discrete fuzzy number playing a role of a fuzzy constraint on values that a random variable may take:and is a discrete fuzzy number with a membership function , playing a role of a fuzzy constraint on the probability measure of :A concept of a discrete -number is closely related to the concept of a discrete -number. Given a discrete -number -number is a pair consisting of a fuzzy number, , and a random number :where plays the same role as it does in a discrete -number and plays the role of the probability distribution , such that .

4. Problem Solving

Flowchart diagram of -number based modeling is given in Figure 2.

Figure 2: Flowchart diagram of -number modeling.

Input raw data is created from the questionnaires (tests of motivation, attention, and anxiety) completed by students.

This data is imprecise and involves uncertainty related with process of completing (filling) of questionnaires. The results of measuring are processed as fuzzy variables with the different fuzzy subsets.

The relationships between educational achievement and the above mentioned psychological variables are presented as Z-if…then rules (Table 1) by the following linguistic variables: H—high; L—low; M—medium; G—good; E—excellence; U—usually; P—plausible; R—rare.

Table 1: Z-if…then rules.

-rule base concept plays pivotal role in economics, decision making, forecasting, and other human centric systems functioning in -information environment. The -rule base is complete when for all the possible observations there exists at least one rule whose -antecedent part overlaps the current antecedent -valuation, at least partially. Otherwise, the -rule base is incomplete. In case that there is incomplete (sparse) -rule base, the classical reasoning methods based on compositional rule of inference (Zadeh [25], Mamdani [26], and R. A. Aliev and R. R. Aliev [27]) or Takagi and Sugeno [28] reasoning approach are not so effective to adapt generating an output for the observation covered by none of the rules. Consequently, we will use inference techniques which in the lack of matching rules can perform an approximate reasoning, namely, -interpolated methods.

A problem of -interpolation is an interpolation of of fuzzy is given below [24, 29].

Given the following -rules:and the fact thatfind the -value of .

The idea underlying the suggested interpolation approach is that the ratio of distances between the conclusion and the consequent parts is identical to ones between the observation and the antecedent parts. For -rules interpolation we havewhere dist is the distance between -numbers. As dist, method suggested in [29] can be used.

Let us consider the special case of the considered problem of -rules interpolation.

Given the -rulesand the fact that is , find the value of .

For this case, formula (6) is reduced to

For example, using supremum metric of fuzzy numbers will be

Taking into account that is a scalar and applying the approach to multiplication of a -number by a scalar described in [30] we will have

Let the knowledge base of 26 -rules of the following form be given:

The considered -rules were described in terms of linguistic labels of , given in Table 1.

Consider a problem of reasoning within the given -rules base by using -interpolation approach. Let the current input information be described by the following -numbers :

-interpolation approach based reasoning consists of two main stages.

(1) For each rule compute distance between the current input -information and -antecedents of -rules base as follows: where is the supremum metric:with

Consider computation of for 1st and 15th rules. -antecedents of the 1st rule are -numbers :

Thus, we need to compute according to (13), where are computed on the base of (14). We have obtained the results:

Thus, the distance for 1st rule is

The inputs of the 15th rule are

Analogously, we computed the distance for 15th rule as

The distances computed for the rest of the rules are

(2) Computation of the aggregated output for -rules base by using linear -interpolation are as follows:

Thus, we need to compute convex combination of outputs of the rules base. The outputs of the -rules base are as follows:

Therefore, the aggregated output is defined as

We have obtained the following result:

In accordance with codebook we have the following. Achievement is “medium” with the reliability “usually.”

The analysis shows that there are no significant differences between the mean of results obtained by conventional statistical method and -number based modeling. The standard deviations of the outputs estimated by statistical method and -number based modeling show significant differences of results. The less variance in the -number based modeling method is related with the ability of this method to control uncertainty in data.

5. Conclusion

In this paper we have suggested a new approach for modeling of the effect of the Pilates exercises on students’ motivation, attention, anxiety, and educational achievement. The uncertainty of data related with cognitive measuring of psychological parameters and their partial reliability have been promoted for the first time application of “Z-if…then rules” for modeling the considered relationship.

We used an inference techniques for approximate reasoning based on -interpolation method suggested by Zadeh to embrace incomplete and lack of matching rules.

Computer simulation proves adequacy of the model.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


  1. C. Hannaford, Smart Moves: Why Learning Is Not All in Your Head, Creat River Books, Salt Lake City, Utah, USA, 2nd edition, 2005.
  2. B. B. Fox, B. Hodgkinson, and D. Parker, “The effects of physical exercise on functional performance, quality of life, cognitive impairment and physical activity levels for older adults aged 65 years and older with a diagnosis of dementia: a systematic review,” The JBI Database of Systematic Reviews and Implementation Reports, 2014. View at Publisher · View at Google Scholar
  3. D. P. Coe, J. M. Pivarnik, C. J. Womack, M. J. Reeves, and R. M. Malina, “Health-related fitness and academic achievement in middle school students,” Journal of Sports Medicine and Physical Fitness, vol. 52, no. 6, pp. 654–660, 2012. View at Google Scholar · View at Scopus
  4. J. U. Edwards, L. Mauch, and M. R. Winkelman, “Relationship of nutrition and physical activity behaviors and fitness measures to academic performance for sixth graders in a Midwest city school district,” Journal of School Health, vol. 81, no. 2, pp. 65–73, 2011. View at Publisher · View at Google Scholar · View at Scopus
  5. T. J. H. Keeley and K. R. Fox, “The impact of physical activity and fitness on academic achievement and cognitive performance in children,” International Review of Sport and Exercise Psychology, vol. 2, no. 2, pp. 198–214, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. P. D. Tomporowski, C. L. Davis, P. H. Miller, and J. A. Naglieri, “Exercise and children’s intelligence, cognition, and academic achievement,” Educational Psychology Review, vol. 20, no. 2, pp. 111–131, 2008. View at Publisher · View at Google Scholar
  7. P. Salmon, “Effects of physical exercise on anxiety, depression, and sensitivity to stress: a unifying theory,” Clinical Psychology Review, vol. 21, no. 1, pp. 33–61, 2001. View at Publisher · View at Google Scholar · View at Scopus
  8. P. H. Fentem, “Benefits of exercise in health and disease,” British Medical Journal, vol. 308, no. 6939, pp. 1291–1295, 1994. View at Publisher · View at Google Scholar · View at Scopus
  9. R. S. Paffenbarger Jr. and R. T. Hyde, “Exercise adherence, coronary heart disease and longevity,” in Exercise Adherence: Its Impact on Public Health, R. K. Dishman, Ed., pp. 41–73, Human Kinetics Books, Champaign, Ill, USA, 1988. View at Google Scholar
  10. J. Pilates and J. Robbins, Pilates' Return to Life Through Contrology. Revised Edition for the 21st Century, 2012, (Originally published by Joseph Pilates in 1945).
  11. A. Ungaro, Pilates: Body in Motion, Dorling Kindersley Publishing, London, UK, 2002.
  12. K. Rodenroth, A study of the relationship between physical fitness and academic performance [Partial fulfillment of the requirements for the degree doctor of education], Liberty University, 2010.
  13. A. R. Aladro-Gonzalvo, M. Machado-Díaz, J. Moncada-Jiménez, J. Hernández-Elizondo, and G. Araya-Vargas, “The effect of Pilates exercises on body composition: a systematic review,” Journal of Bodywork and Movement Therapies, vol. 16, no. 1, pp. 109–114, 2012. View at Publisher · View at Google Scholar · View at Scopus
  14. K. Caldwell, M. Harrison, M. Adams, and N. Travis Triplett, “Effect of Pilates and taiji quan training on self-efficacy, sleep quality, mood, and physical performance of college students,” Journal of Bodywork and Movement Therapies, vol. 13, no. 2, pp. 155–163, 2009. View at Publisher · View at Google Scholar · View at Scopus
  15. C. H. Hillman, K. I. Erickson, and A. F. Kramer, “Be smart, exercise your heart: exercise effects on brain and cognition,” Nature Reviews Neuroscience, vol. 9, no. 1, pp. 58–65, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. E. L. Deci and R. M. Ryan, “Self-determination theory: a macrotheory of human motivation, development, and health,” Canadian Psychology, vol. 49, no. 3, pp. 182–185, 2008. View at Publisher · View at Google Scholar · View at Scopus
  17. M. S. Chapell, Z. B. Blanding, M. Takahashi et al., “Test anxiety and academic performance in undergraduate and graduate students,” Journal of Educational Psychology, vol. 97, no. 2, pp. 268–274, 2005. View at Publisher · View at Google Scholar · View at Scopus
  18. P. Castle and S. Buckler, “What Was I Saying? Concentration and Attention,”
  19. M. Smithson, “Applications of fuzzy set concepts to behavioral sciences,” Mathematical Social Sciences, vol. 2, no. 3, pp. 257–274, 1982. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. A. N. Averkin and V. B. Tarasov, “The fuzzy modelling relation and its application in psychology and artificial intelligence,” Fuzzy Sets and Systems, vol. 22, no. 1-2, pp. 3–24, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  21. T. Hesketh, R. Pryor, and B. Hesketh, “Application of a computerized fuzzy graphic rating scale to the psychological measurement of individual differences,” International Journal of Man-Machine Studies, vol. 29, no. 1, pp. 21–35, 1988. View at Publisher · View at Google Scholar · View at Scopus
  22. M. S. El-Nasr, J. Yen, T. R. Loerger, and Flami, Fuzzy Logic Adaptive Model of Emotions Autonomous Agents and Multi-Agent Systems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
  23. G. S. Kushwaha and S. Kumar, “Role of the fuzzy system in psychological research,” Europe's Journal of Psychology, vol. 5, no. 2, pp. 123–134, 2009. View at Publisher · View at Google Scholar
  24. L. A. Zadeh, “A note on Z-numbers,” Information Sciences, vol. 181, no. 14, pp. 2923–2932, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965. View at Google Scholar · View at MathSciNet
  26. E. H. Mamdani, “Applications of fuzzy logic to approximate reasoning using linguistic systems,” IEEE Transactions on Computers, vol. 26, no. 12, pp. 1182–1191, 1977. View at Publisher · View at Google Scholar · View at Scopus
  27. R. A. Aliev and R. R. Aliev, Soft Computing and Its Application, World Scientific, River Edge, NJ, USA, 2001.
  28. T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116–132, 1985. View at Google Scholar · View at Scopus
  29. L. T. Kóczy and K. Hirota, “Approximate reasoning by linear rule interpolation and general approximation,” International Journal of Approximate Reasoning, vol. 9, no. 3, pp. 197–225, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. R. A. Aliev, O. H. Huseynov, R. R. Aliyev, and A. A. Alizadeh, The Arithmetic of Z-Numbers, Theory and Applications, Word Scientific, 2015.