0Estimation of Curie temperature of manganite-based materials for magnetic refrigeration application using hybrid gravitational based support vector regression
Taoreed O. Owolabi, Kabiru O. Akande, Sunday O. Olatunji, Abdullah Alqahtani, and Nahier Aldhafferi
Citation: AIP Advances 6, 105009 (2016); doi: 10.1063/1.4966043 View online: http://dx.doi.org/10.1063Z1.4966043 View Table of Contents: http://aip.scitation.org/toc/adv/6/10 Published by the American Institute of Physics
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Estimation of Curie temperature of manganite-based materials for magnetic refrigeration application using hybrid gravitational based support vector regression
Taoreed O. Owolabi,1,2,a Kabiru O. Akande,3 Sunday O. Olatunji,4 Abdullah Alqahtani,4 and Nahier Aldhafferi4
1 Physics Department, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
2 Physics and Electronics Department, Adekunle Ajasin University, Akungba Akoko, Ondo State, Nigeria
3 Institute for Digital Communications, School of Engineering, University of Edinburgh, United Kingdom
4College of Computer Science and Information Technology, University of Dammam, Dammam, Saudi Arabia
(Received 8 July 2016; accepted 7 October 2016; published online 18 October 2016)
Magnetic refrigeration (MR) technology stands a good chance of replacing the conventional gas compression system (CGCS) of refrigeration due to its unique features such as high efficiency, low cost as well as being environmental friendly. Its operation involves the use of magnetocaloric effect (MCE) of a magnetic material caused by application of magnetic field. Manganite-based material demonstrates maximum MCE at its magnetic ordering temperature known as Curie temperature (TC). Consequently, manganite-based material with TC around room temperature is essentially desired for effective utilization of this technology. The TC of manganite-based materials can be adequately altered to a desired value through doping with appropriate foreign materials. In order to determine a manganite with TC around room temperature and to circumvent experimental challenges therein, this work proposes a model that can effectively estimates the TC of manganite-based material doped with different materials with the aid of support vector regression (SVR) hybridized with gravitational search algorithm (GSA). Implementation of GSA algorithm ensures optimum selection of SVR hyperparameters for improved performance of the developed model using lattice distortions as the descriptors. The result of the developed model is promising and agrees excellently with the experimental results. The outstanding estimates of the proposed model suggest its potential in promoting room temperature magnetic refrigeration through quick estimation of the effect of dopants on TC so as to obtain manganite that works well around the room temperature. © 2016Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/L4966043]
I. INTRODUCTION
The environmental danger posed by the conventional gas compression refrigeration system (CGCRS) is of serious concern which warrants alternative means of refrigeration. The recent advent of magnetic refrigeration (MR) technology opens up ways through which CGCRS can be conveniently replaced with less pollutant and more efficient system of refrigeration with relatively low cost.1-4 Effective deployment and utilization of MR technology requires magnetic material that is characterized with ambient temperature magnetocaloric effect (MCE).5-13 Manganite-based
Correspondence: Owolabi Taoreed Olakunle, Physics Department, King Fahd University of Petroleum and Minerals, Dhahran, Kingdom of Saudi Arabia. Tel: +966556202152. Email: owolabitaoreedolakunle@gmail.com
2158-3226/2016/6(10)/105009/12
6, 105009-1
© Author(s) 2016
material with chemical formula Ri_X AXMnO3 (where R and A represent trivalent and divalent cations respectively) has high MCE and thereby suitable as magnetic refrigerant. The prominent physical means of tuning MCE and Curie temperature (TC) of manganite based material to the desired ambient value is chemical substitution of foreign materials called dopants into the crystal lattice structure of manganite.5'14 In order to experimentally develop a manganite-based material that can be used as magnetic refrigerant in MR technology, the manganite is subjected to doping so as to determine the concentration and the nature of dopant that shifts the TC of manganite-based material towards the ambient temperature. Definitely, this requires intensive experimental procedure which takes appreciable time and other valuable resources. This research work proposes gravitational search algorithm based support vector regression (GSA-SVR) model, an efficient computational technique, for determining the influence of dopants on TC of manganite-based materials. Implementation of the proposed GSA-SVR model would be of immense significance in identifying suitable dopants that shift the TC of manganite to the desired ambient value.
Support vector regression is a robust computational intelligence technique that has demonstrated excellence in generalization and prediction due to its unique features which include generalization of error bound and non-convergence to local minimal. It has several applications especially in condensed mater physics where experimental data is limited.15-17 Its grand breaking applications include the estimation of superconducting properties,18-20 estimation of the properties of perovskite21 and estimation of surface properties that are experimentally difficult to measure.22-25 SVR successfully handles non-linear problems through kernel trick which allows transformation of data-points into high dimensional space with the use of mapping function called kernel function. The predictive strength of SVR is governed by the proper selection of its hyper-parameters which include regularization or penalty factor that maintains the tradeoff between flatness of the decision function and the level to which deviation lager than the specified threshold (epsilon) is tolerated. This work hybridizes the gravitational search optimization algorithm with SVR for developing a robust model that estimates the TC of manganite based material with the aid of crystal lattice parameters as descriptors. The crystal lattice parameters influence many factors such as Mn-O-Mn bond angle, ionic size mismatch as well as bond length that have direct impact on the Curie temperature of manganite based materials.5 Gravitational search algorithm is a recently proposed population based optimization algorithm that works on the principle of Newton gravitational law and laws of motion.26 It has demonstrated excellence in many practical applications.27,28 Its hybridization with SVR in this work ensures optimum performance of the developed GSA-SVR model in estimating TC of manganite-based materials.
Operation of MR depends on MCE which measures the response of magnetic material to the applied field. When external magnetic field is applied to a magnetic material, the electronic spins of the material align along the axis of externally applied magnetic field and consequently leads to lower entropy of the system since entropy measures the degree of disorderliness. In an attempt to compensate this response of electronic spins to the magnetic field, atoms of the material begins to vibrate and lead to increase in the thermal energy as well as the temperature of the magnetic material.29 With the reduction in the strength of applied field (or switched off), the electronic spins return to their random orientation which lowers the vibration of atoms due to increase in entropy of the system and the temperature of the system becomes lower. This rising and lowering in the temperature of the system as a result of the variation in the strength of the applied magnetic field (phenomenon known as magneto caloric effect) lead to magnetic refrigeration. Large MCE is desired for achieving efficient MR. Large MCE of manganite based material as well as ease with which its TC can be altered to a desired value near room temperature through doping stands it out among other magnetic refrigerants.
Results of our modeling and simulation during the development and implementation of the proposed GSA-SVR model, on the basis of correlation coefficient on the testing set of data show accuracy of 89.4 %. The developed model also characterized with a small value of root mean square error and mean absolute error. The estimated TC of several manganite -based materials investigated agree excellently with the experimental results which further suggest excellent predictive power of the developed GSA-SVR model.
I. DESCRIPTION OF SUPPORT VECTOR REGRESSION
SVR is a machine learning tool which is capable of forming unique pattern between descriptors and desired targets using statistical learning theory developed from structural risk minimization principle.30'31 It uses e - insesitive loss function for searching a global minimum and in locating possible error within certain distance of the training data.32 Utilization of kernels contributes to its uniqueness and further distinguishes SVR from the support vector machine which is meant for classification problems. Equation (1) describes a decision function which allows the descriptors x = {a, b, c} to be mapped into high dimensional feature space with the help of a non-linear Gaussian mapping function.
f (w, x) = (w, x) + z w € N and z €%
where (w, x) represents the inner product
SVR algorithm ensures flatness of the generated function through Euclidian norm minimization, that is 1 ||w||2 subjected to condition presented in equation (2) as described in Ref. 33.
Tq - (w, Xi)- z < s (w, Xi) + z - Tq < s
where TCi stands for Curie temperature
Incorporation of Slack variables , becomes significant purposely to create room for infeasible constraint and the optimization problem is modified accordingly and presented in equation (3).
Subjected to
- ||w||2 + CY (Ç + £) 2 i=i
Tq - (w, Xi)- z < S + Çi (w, Xi) + z - Tq < S + Ç*
where C stands for the regularization factor and n is the number of data-points
The optimization problem is well handled using Lagrange multiplier as described in Ref. 23, the decision function including kernel function ^ (xt .x) is presented in equation (5).
f (x) = X (®i - a*) * (XiX) + z
where a and a* are langrage multipliers
In the course of optimizing SVR hyper-parameters using GSA, several kernel function were tested and the non-linear mapping function that gives optimum performance to the proposed model is Gaussian kernel function presented in equation (6)
*(Xi, Xj ) = exp
\Xi Xj I
where a is the kernel parameters that would be tuned using GSA algorithm
The value of SVR hyper-parameters such as the loss functione, regularization or penalty factor
C and Gaussian kernel option influence the estimation accuracy of the model.32 Kernel option is the parameter of kernel function which controls the transformation of data-points to high dimensional space while lambda hyper-parameter sorts for hyper-plane in which minimum error is ensured.23 The optimal values of these hyper-parameters are obtained using GSA.
A. Description of the optimization algorithm (GSA)
Gravitational search algorithm (GSA) is a search heuristic algorithm developed few years back.26 The working principle of the algorithm is based on gravitational law and equation of motion proposed by the famous physicist, Isaac Newton. It considers the objects in Newtonian description as agents whose mass determines the actual performance. In its operational principle, the gravitational force results into global movement towards massive agents. Each of the agent is characterized with position, active gravitational mass, inertial mass, and passive gravitational mass. Active gravitational mass (Ma) measures gravitational field strength resulted from a particular agent in the population while inertial mass (Mi) measures the resistance to change of state of motion due to applied force. On the other hand, passive gravitational mass (MP) measures the strength of the interaction of the agent with the gravitational field. The inertial masses of each of the agent are determined using the fitness function and the corresponding positions of the agents are updated accordingly.27,28 The entire GSA can be considered as an isolated system of masses governed by law of gravitation and Newton law of motion. Equation (7) defines the position of kth agent in a system of N number of agents where each of the agents has D dimension.
Xk = (xk,.....,xf), where k = 1,2,..., N (7)
The gravitational attractive force between mass MPk and mass Maj is presented in equation (8) including time dependent gravitational constant illustrated in equation (9)
n Mpk (t)Maj (t) n n
FD (t) = G(t) ,,) +j {xf (t) - xf (t)} (8)
kj rkj (t) + e* j k
MP and Ma represent the passive and active gravitational mass respectively.
' to \ P
(IT "<>
G(t) = G(to) f P<1 (9)
where rkj (t) = ||Xk (t),Xj (t)|| represents the Euclidian distance and e* stands for a constant value
The value of the initialized gravitational constant decreases with time. Stochastic feature is imposed on the algorithm through equation (10) and the acceleration akD (t) of the kth agent in the dimension fth at time t is presented in equation (11).
Ff (t) = ^ randjFf (t) (10)
j=1j*k
where randj represents a random number spanning [0,1] interval. Its enforces randomness to the search procedure and ensures the ultimate location of the optimum SVR hyper-parameters.
af(,)=M^ (11)
Mik = inertial mass of kth agent
The acceleration of an agent coupled with some fraction of its present velocity gives its next velocity. Hence, the velocity and position are updated in accordance to equation (12) and (13) respectively.
vf (t + 1) = randkvf (t) + af (t) (12)
xf (t + 1) = xf (t) + vf (t + 1) (13)
The fitness function helps in determining the inertial and gravitational masses and a more efficient agent (that is the one with heavier mass) moves sluggishly due to its large attractive gravitational force. With the assumption contained in equation (14), the inertial and gravitational mass are updated each time in accordance to equation (15) and (16)
Mak = Mpk = Mik, k = 1,2,..., N (14)
Ff (t)
Us fitk (t) - worst(t)
mk (t)= t 777\-77^ (15)
best(t) - worst(t)
mk (t)
Mk (t) = (16)
2 mj (t)
where fitk(t) = fitness of kth agent at time t
B. Parameters that measure the predictive strength of the developed GSA-SVR model
The potential of the proposed GSA-SVR model to generalize to unseen dataset was evaluated using root mean square error (RMSE), correlation coefficient (CC) and mean absolute error (MAE). Equations (17), (18) and (19) respectively describe the formulation of the parameters. Low RMSE, low MAE and high CC characterize model with high precision.
RMSE = -
-Ye* (17)
= (Tq(exp) - T£(exp)) (TQ(est) - Tc(eSt))
CC = , '=1 (18)
■ " - T' ^2
V« . , 2 n . . 2
'=1 (TQ(exp) - T'c^exp)) £ \Tc'(est) - T(C(est)j
-Vje'l (19)
MAE le.
Where ei and n respectively represent error and the number of data points. TCi(exp) and TCi(est) respectively stand for the experimental and the estimated Curie temperature while T'C( ) and T'C(est) represent their mean value.
III. DEVELOPMENT OF GSA-SVR MODEL A. Description of data set
The distorted lattice parameters of manganite-based materials are used as the descriptors for determining the Curie temperature of the manganite. The proposed GSA-SVR model was developed by using seventy-five experimental lattice parameters as well as their corresponding Curie temperature using test-set cross validation method. The experimental data was extracted from the literature.9-11,29,34-38 The outcomes of the statistical analysis performed on the dataset are presented in Table I.
The average of the content of the dataset as well as the disparities in the dataset can be deduced from the results of mean and standard deviations presented in the table. The variation in c-crystal lattice parameter can be attributed to the orthorhombic nature of most of the doped manganite based materials. Table I also shows the Correlation coefficient between each of the descriptors and the Curie temperature. Correlation coefficient measures the extent of linear relationship between crystal
TABLE I. Results of the statistical analysis.
a (Â) b(Â) c (Â) Tc(K)
Mean 5.482 6.570 7.943 235.55
Standard deviation 0.032 1.255 3.317 95.92
Maximum 5.543 7.760 13.597 370.000
Minimum 5.429 5.438 5.419 18.52
Correlation coefficient (%) 53.18 -1.3 19.3
lattice constants and Curie temperature. Lattice parameter b shows negative correlation with TC while crystal lattice constants a and c are positively correlated with the TC. The results of the statistical analysis give us an insight to the data and its strength for the proposed estimation.
B. Computational methodology involved in developing GSA-SVR model
The proposed GSA-SVR model was developed within MATLAB computing environment. The data was randomized and partitioned into training dataset (lattice constants and their corresponding Curie temperatures for training the model) and testing dataset (lattice constants and their corresponding Curie temperatures for testing the model) in the ratio of eight to two respectively. Randomization means random partitioning of the data-points. The initial population for optimizing SVR hyperparameters using gravitational search algorithm was generated and the RMSE for each of the generated agent was evaluated. The best value (which corresponds to the minimum error), G0 and worse of the population (which corresponds to the maximum error) were updated, followed by the calculation of respective mass and acceleration of each of the agents. The inertial and gravitational masses are updated each time until the global minimum error is achieved. The optimized values of SVR hyperparameters were fed into SVR coupled with the training dataset to generate support vectors needed for future estimation. The lattice parameters from the testing dataset partition were then fed into the trained SVR and the corresponding Curie temperatures were estimated. The estimation accuracy of the developed GSA-SVR model was assessed through evaluation of RMSE, MAE and CC. A complete description of the working principle of the algorithm is illustrated in Algorithm 1. The optimized SVR-hyper-parameters are presented in Table II.
ALGORITHM 1. Flow chart illustrating the development of GSA-SVR model.
TABLE II. Optimum values of SVR-hyper-parameters obtained using GSA.
SVR hyper-parameter Optimum value
Regularization factor 691.563
Lambda 0.0000001
Epsilon 0.5063
Kernel option 0.0057
Kernel function Gaussian function
IV. RESULTS AND DISCUSSION
A. Performance sensitivity of the proposed model to the initial population of the agents
The performance sensitivity of the proposed GSA-SVR model to the number of agents was investigated and presented in Fig.1. Model performance is measured on the basis of RMSE. When the number of agents is 10 as depicted by the graph, the model shows minimum error which may be regarded as a local minimum. The global minimum was achieved when an initial population of 50 agents was used as depicted by the graph.
Table III shows the measure of the estimation accuracy of the developed GSA-SVR and their corresponding values using 50 initial populations of agents. The CC for training and testing dataset was respectively 100% and 89.4%. This shows little or no disparities between the experimentally reported Curie temperature and estimated values using the developed GSA-SVR model.
The low values of RMSE and MAE as presented in Table III are direct indications of model excellent estimation accuracy. After the development of the model, the developed model was subjected to further validation and implementation. In this case, the developed GSA-SVR was only supplied
FIG. 1. Sensitivity of GSA-SVR model to the number of initial population.
TABLE III. Measure of estimation accuracy of the developed GSA-SVR models.
SVR-based (training) SVR-based SVR (testing)
CC (%) 100.0 89.4
RMSE (K) 0.5 43.2
MAE (K) 0.5 29.9
Computational time (s) 1426.7
with lattice parameters of some doped manganite based materials and the model utilized its generated support vectors in estimating the unknown Curie temperature.
1. Implementation of the developed GSA-SVR model
In figure 2 to 5, we have used neither the training nor the testing datasets rather what we have done is to further assess the generalization and predictive capacity of the GSA-SVR model in order to demonstrate its general applicability. The results of those figures were obtained by deploying the developed GSA-SVR model to other manganite based materials with different doping materials. The fundamental principle involved in SVR is that it develops, in conventional terms, a set of support vectors during training phase and these support vectors are essentially the developed model. The testing phase is done to validate the accuracy of the proposed model in generalizing to unseeing data, the testing dataset in this case. Hence, further application of the developed model (as shown in figures 2 to 5) has no relation to any data-points in the training or testing dataset as they are independent of each other. The results only correspond to further testing of the developed model which was carried out to test its robustness.
FIG. 2. Effect of strontium on the curie temperature of La07SrXBi03-XMnO3.
FIG. 3. Effect of samarium on the curie temperature of (La1_XSmX)o.67Pbo.33MnÜ3.
Transition metal(X)
FIG. 4. Effect of transition metals on the curie temperature of Pr0.7Ca0.3Mn0.9X0.1O3.
B. Estimation of Curie temperature for lanthanum based manganite materials using the developed GSA-SVR model
We present the effect of doping on the TC of lanthanum based manganite in Fig.2 and Fig.3. Incorporation of strontium (Sr) into lattice structure of La0.7SrXBi0 3-XMnO3 manganite raises the Curie temperature as depicted in Fig.2. Without doping, the Curie temperature was reported as 115K which is far less than the room temperature (-300K).39 In an effort to raise its Curie temperature to the room temperature, introduction of 0.1 concentration (in %) of strontium leads to increase in its TC to 195K (also below the room temperature) and further increase in the concentration of strontium raises the TC to around 342K. With around 0.15% concentration of strontium, this material can serve as a magnetic refrigerant around the room temperature in accordance to the presented results. The results of the developed GSA-SVR model agree excellently with the experimentally reported values.39
In the same vein, the effect of samarium on the TC of (La 1-XSm X)0 67Pb0 33MnO3 manganite is also presented in Fig.3. Samarium (Sm) lowers the Curie temperature of (La 1-XSm X)0 67Pb0 33MnO3 manganite towards the room temperature. For instance, 0.3 concentration of Sm lowers the Tc to about
FIG. 5. Effect of praseodymium on the curie temperature of Smo.55-XPrXSro.45MnÜ3.
286K which is very close to room temperature. The results of the developed model show excellent agreement with the experimental values.8
1. Effect of transition metals on the Curie temperature of praseodymium based manganite materials using GSA-SVR model
The effect of transition metals such as nickel (Ni), cobalt (Co) and chromium (Cr) on the TC of Pro.7Cao.3Mno.9Xo.1O3 manganite is presented in Fig.4. The results of the presented modeling and simulation agree with experimental data.9 It can be deduced from the graph that the transition metals lower the Curie temperature on this class of manganite.
2. Estimation of Curie temperature for samarium based manganite materials using GSA-SVR model
Similarly, the influence of praseodymium doping on the TC of Sm045Pr01Sr045MnO3 manganite is presented in Fig.5. It is seen that the TC of manganite increases as the concentration of praseodymium increases and GSA-SVR estimates agree well with the reported experimental values.40
The main point to be derived from the figures (figure 2 to 5) is that GSA-SVR has accurately captured the general trend of the relationship existing between lattice parameters and Tc and as such, given any set of parameters, it is able to estimate the corresponding Tc with a high degree of accuracy. Indeed, this is the aim of this research which is to show that GSA-SVR model, an efficient computational technique, can serve as alternative to the time-consuming, expensive and often cumbersome experimental approach.
C. Estimation of Curie temperature of manganite based materials with no experimental data using GSA-SVR model
The proposed GSA-SVR model was further deployed for estimating the effects of dopants on some manganite based materials whose Curie temperatures are not yet in the literature. The model utilized the crystal lattice parameters of these compounds for the estimation of their Curie temperature. Fig.6 shows the influence of yttrium on the Curie temperature of La0.5-xYxSr0.5MnO3 manganite as obtained using the developed model. The figure shows an increase in the Curie temperature as the concentration of yttrium increases, reaches a maximum value and begins to decrease. The effect of strontium on the Curie temperature of Sm1-xSrxMnO3 manganite was also investigated and the results of the investigation are presented in fig.7. The figure shows that strontium doping lowers the Curie temperature of Sm1-xSrxMnO3 manganite.
Concentration of yttrium(x) FIG. 6. Effect of yttrium doping on the Curie temperature of La0.5-X YXSr0.45MnO3 manganite.
Concentration of strontium(x) FIG. 7. Effect of strontium doping on the Curie temperature of SMi_xSrxMnO3 manganite.
V. CONCLUSION AND RECOMMENDATION
This work employs a hybrid of support vector regression with gravitational search algorithm for developing a robust computational technique model, GSA-SVR. The developed model is applied to the estimation of Curie temperature of manganite-based material for improving room temperature magnetic refrigeration. Seventy-five experimental values of Curie temperature for different classes of manganite based materials and the corresponding lattice constants were incorporated in the model development using test-set-cross validation method. The results of the developed GSA-SVR model agree excellently with the reported experimental values. The developed GSA-SVR was then deployed to estimate Curie temperature of some manganite based materials whose experimental values are not yet reported in the literature. The results of this research work would be of immense significance for quick estimation of the effect of different dopants on the Curie temperature of manganite refrigerant without loss of precision. By implementing this model, manganite with magnetic ordering temperature around room temperature can be quickly identified and put into use for enhancing magnetic refrigeration. This will lead to significant reduction in the use of harmful ozone-depleting conventional gas compression refrigeration.
ACKNOWLEDGMENTS
The support received from University of Dammam is acknowledged.
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