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A Gaussian function model for simulation of complex environmental sensing
Complex Adaptive Systems Modeling volumeÂ 3, ArticleÂ number:Â 3 (2015)
Abstract
Background
Sensors can be used to sense not only simple behavior but also complex ones. Previous work has demonstrated how agentbased modeling can be used to model sensing of complex behavior in Complex Environments.
Findings
Here, we propose a mathematical model using Gaussian function for the previously developed Agent Based Model (ABM) for Sensing of Emergent behavior in Complex Adaptive System (SECAS). The goodness of the fitted curve was observed by using standard tools, e.g. by determining SSE, SSM, ASSM and RMSE.
Conclusions
Our proposed model provides a good fit for data obtained from the earlier model. Also the developed model provides a bench mark against the data obtained from a former Agent Based Model.
Findings
Wireless sensors are a growing area of research focus. In previous works Niazi and Hussain (2011a, b), a formal model of wireless sensor networks has been given along with an agentbased simulation model. The idea was to use sensing to examine and identify complex behavior such as flocking. The papers also presented a formal specification model is based on the Z formal specification language. While the idea was interesting, these papers did not present a traditional mathematical model. In the current paper, I expand the ideas presented in the earlier papers and present an alternative mathematical model in the form of a Gaussian model for the results of sensing presented earlier in Niazi and Hussain (2011a).
Background
Mathematical modeling of curve fitting
Curve fitting provides an ample opportunity to capture nicely the trend in the data by assigning a single function across the entire range. There are many possible ways to do this e.g. using Gaussian function, smoothing spine, sum of the trigonometric function or weibull function etc. Ordinary and partial differential equations are also helpful in determining the trend beautifully. In this paper, I have used Gaussian function.
Gaussian model
This is a method which has been frequently adopted in various kinds of curve fitting e.g. in (Jonsson and Eklundh 2002). Gaussian function arises by composing the exponential functions and is given by:
where a stands for the amplitude, b for centroid and c represents the peak width. In order to fit a curve using this function, one needs to optimize the parameters involved in the function.
Methodology
In order to fit the data, mathematical model was developed using Gaussian function as given in Eq.Â 1. All the simulation work was performed using (MatlabÂ 2011;Â Curve Fitting Toolbox 2011) . The validation data was taken on the basis of the original simulation model presented in (Niazi and Hussain 2011a). To find the optimal values of the coefficients of the given equation, Trustregion algorithm was used. This algorithm is not only very useful in the evaluation of constraint coefficients but also in handling the complex nonlinear problems more efficiently than the other algorithms. The goodness of the fitted curve was observed by determining of the sum of the squares of residuals (SSE), summed squares about the mean (SSM), adjusted summed squares about the mean (ASSM) and root mean squared error (RMSE). SSE represents the sum of squares due to the error of the fit where a value in the vicinity of zero indicates a fit which is more useful for prediction. SSM represents the square of the correlation between response and predicted response values, where ASSM depends on the adjusted residual degrees of freedom. RMSE is quite commonly used and requires no debate on it.
Results and discussion
Mathematical modeling is an important area of research work where curve fitting along with the determination of optimal set of parameters provides an opportunity to find a bench mark against some experimental data. In this paper, a mathematical model using Gaussian function was developed to fit a curve against the data obtained from a former Agent Based Model Niazi and Hussain (2011a). The goodness of the fitted curve was observed by determining SSE, SSM, ASSM and RMSE. It was noted that with the increase in the value of n, the values of SSE, SSM, ASSM and RMSE towards the fit were improved, e.g., for the value of nÂ =Â 6 using data I; SSE is 9.335eÂ +Â 4, SSM is 0.8769, ASSM is 0.8748, and RMSE is 9.745, whereas for the value of nÂ =Â 8; SSE is 3.648eÂ +Â 4, SSM is 0.9519, ASSM is .9508, and RMSE is 6.11 which is much improvement as compared to the case when nÂ =Â 6. Therefore, the value of nÂ =Â 8 was chosen to fit the data. Similar analysis was performed for the data II. The fitted curves for both the data are shown in Figs.Â 1 and 2, where the fitted values of these parameters for nÂ =Â 8 along with the 95Â % confidence bounds are given in the TableÂ 1.
Conclusion and future work
In this paper, I have presented a mathematical model using Gaussian function for Sensing of Emergent behavior in Complex Adaptive System (SECAS). This work is actually an extension of the formal model presented earlier by Niazi and Hussain for the sensing of complex behavior. The lack of a traditional mathematical model motivated me to do this. In the future, it would be interesting to couple formal models with formal specification models in other domains such as in the domain of the Internet of Things, networks of consumer electronics and more.
Abbreviations
 ABM:

Agent Based Model
 SECAS:

sensing of emergent behavior in Complex Adaptive System
 SSE:

sum of the squares of residuals
 SSM:

summed squares about the mean
 ASSM:

adjusted summed squares about the mean
 RMSE:

root mean squared error
References
Curve Fitting Toolbox, The MathWorks Inc. MATLAB 7.12.0.635 (R2011a), 2011
Jonsson P, Eklundh L (2002) Seasonality extraction by function fitting to timeseries of satellite sensor data. IEEE Trans Geosci Remote Sens 40:1824â€“1832
Niazi MA, Hussain A (2011a) Sensing emergence in complex systems. Sens J IEEE. 11:2479â€“2480
Niazi MA, Hussain A (2011b) A novel agentbased simulation framework for sensing in complex adaptive environments. Sens J IEEE 11:404â€“412
The MathWorks Inc. MATLAB 7.12.0.635 (R2011a), 2011
Acknowledgements
The author of this paper acknowledge the support given by University of Engineering and Technology, Lahore. However, UET has no role in writing this paper.
Competing interests
The author declares that they have no competing interests.
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Chaudhry, Q.A. A Gaussian function model for simulation of complex environmental sensing. Complex Adapt Syst Model 3, 3 (2015). https://doi.org/10.1186/s4029401500090
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DOI: https://doi.org/10.1186/s4029401500090
Keywords
 Complex Adaptive System
 Environmental sensing
 Gaussian function
 Mathematical model