Elemental compositional modeling of magnetic ordering temperature for spinel ferrite magnetocaloric compounds using intelligent algorithms

Figures & data

Table 1. Analysis of data sample

Data sample contents	Mean	Standard deviation	Minimum	Maximum	Correlation coefficient
MOT	435.1034	181.1642	127.0000	705.0000	1.0000
A (pm)	84.0517	5.3358	68.5000	88.0000	−0.2128
x	0.4681	0.2184	0.2000	1.0000	0.4108
C (pm)	30.6552	43.4283	0.0000	109.0000	0.4874
y	0.0784	0.1210	0.0000	0.4000	0.5571
D (pm)	89.1379	7.2837	83.0000	115.0000	0.2460
z	0.5224	0.2378	0.1000	1.0000	−0.3293
H(T)	3.5414	1.5790	1.3000	5.0000	0.5566

Figure 1. Computational details of hybrid support vector regression and genetic algorithm.

Figure 2. Combinatory influence of the utilized genetic algorithm (a) penalty factor at various iteration (b) Epsilon as a function of iteration (c) kernel mapping parameter as number of iteration progresses (d) error convergence at different iteration.

Table 2. Results of the genetic algorithm (SVR parameters)

Tuning parameters in SVR	Global value
Kernel parameter	0.2107
Penalty factor	553.796
Kernel function	Gaussian
Epsilon	0.001
Population size	20
Hyper-plane lambda	0.0000001

Figure 3. Computational flowchart of ELM-based model for MOT prediction.

Table 3. Extreme learning machine weights for model reproducibility

$\mathit{p}$	${\mathit{\chi }}_1$	${\mathit{\chi }}_2$	${\mathit{\chi }}_3$	${\mathit{\chi }}_4$	${\mathit{\chi }}_5$	${\mathit{\chi }}_6$	${\mathit{\chi }}_7$	$\mathit{\eta }$	$\mathit{\delta }$
1	−0.51442	0.151525	−0.43281	−0.86704	0.55957	−0.5095	0.777457	0.223514	76.4672
2	−0.80917	−0.45068	0.718078	−0.12048	0.374293	−0.53945	0.031748	0.541308	20.52116
3	−0.41783	0.341106	0.225585	0.208956	0.663016	0.774093	−0.67649	0.236643	−15.3892
4	−0.38577	0.633167	−0.19743	0.957932	−0.34641	0.78404	0.770397	0.906057	120.7967
5	−0.47773	−0.52775	−0.48743	−0.57335	−0.1234	−0.4297	−0.12157	0.941593	−98.9796
6	−0.59868	0.320187	−0.70865	0.070804	0.107112	−0.24257	0.356221	0.776397	−60.2309
7	−0.79962	−0.20243	0.577352	−0.56484	0.350181	−0.86789	−0.6626	0.264603	−18.4983
8	0.270824	−0.16955	0.3856	0.575908	−0.1193	−0.24123	−0.45716	0.195132	−75.3344
9	0.340729	−0.37352	0.654177	−0.86508	−0.96217	0.295099	−0.0482	0.58689	195.644
10	−0.07051	−0.9425	−0.93752	0.634082	0.955,957	0.80182	−0.01961	0.12487	−75.8461
11	0.369909	−0.18364	0.378473	−0.03254	−0.50891	0.102341	−0.39589	0.104899	−105.307
12	−0.77867	−0.7751	0.832865	0.529901	0.146851	0.994273	0.024252	0.793454	227.9922
13	0.821016	0.85932	0.964453	0.361826	−0.69734	−0.96171	−0.04577	0.992594	−24.3423
14	−0.08973	0.299174	−0.51289	−0.64904	−0.64072	0.507575	−0.21206	0.909765	−23.0427
15	−0.92027	0.999584	0.383714	0.869967	0.58757	−0.56553	−0.60631	0.54698	41.32443
16	−0.04494	−0.74659	0.165421	−0.45317	0.098467	0.518194	−0.6411	0.015117	73.46305
17	0.622783	−0.37899	−0.07266	−0.46926	0.663773	−0.3549	−0.68931	0.997319	−88.6559
18	−0.78973	0.070096	0.410547	−0.35925	0.452651	0.131605	0.70523	0.768356	15.5778
19	−0.38237	0.647427	−0.15006	0.653853	−0.61099	−0.37736	0.012908	0.013378	−134.387
20	0.977435	−0.81774	0.666434	0.630454	−0.93585	0.483203	0.633249	0.522898	−126.741
21	0.313783	0.010106	0.283035	0.515528	0.926921	0.262038	0.825067	0.12139	−54.4741
22	−0.60014	−0.67071	0.879881	−0.8095	0.587198	0.300808	−0.25754	0.263307	46.22696
23	−0.17224	0.384147	0.727237	−0.57865	−0.5857	0.06782	−0.13936	0.958459	−152.68
24	−0.22987	−0.79605	−0.20304	0.606628	0.822933	0.447162	0.295455	0.760732	60.34544
25	−0.45422	−0.04456	0.394616	0.060541	0.53102	−0.42409	−0.95641	0.468824	−4.36292
26	0.284952	−0.83335	0.739757	0.59883	−0.69984	−0.26852	−0.03739	0.247054	−13.5968
27	0.940503	−0.78427	−0.77965	0.189996	0.840038	−0.35564	0.804232	0.09899	−66.2988
28	0.208398	0.958938	0.072635	−0.1644	−0.79962	−0.01393	−0.73616	0.201595	113.3039
29	0.141952	−0.82512	0.140164	0.44286	−0.08406	−0.00586	−0.70423	0.971506	11.02041
30	0.212846	−0.85562	−0.06854	0.694439	−0.22971	0.962336	0.759488	0.814683	−279.042
31	−0.41665	−0.25776	−0.9756	0.447226	0.325068	−0.39841	0.287847	0.36057	89.63651
32	0.853431	0.95307	−0.58287	0.338494	0.579737	0.954724	0.366302	0.738951	−54.9099
33	−0.8989	−0.75724	0.265736	−0.07829	0.398651	0.720636	0.671711	0.470602	83.22384
34	0.116969	−0.97595	−0.19522	−0.2985	−0.30889	−0.84822	0.825154	0.836717	−49.9442
35	−0.83074	0.875486	−0.88185	0.61555	0.523008	0.320069	−0.01012	0.837285	22.48595
36	−0.08125	−0.66036	−0.07005	0.513692	0.20266	0.143828	0.865389	0.306878	5.458474
37	0.970573	−0.85927	0.084475	−0.1649	0.206746	0.926972	0.21228	0.138601	−99.1811
38	−0.29104	0.8238	0.260729	0.046355	−0.10938	0.183163	−0.33061	0.293205	−49.9669
39	0.117424	−0.7014	0.964604	−0.28035	0.507089	0.602329	−0.50693	0.498643	340.9157
40	−0.86686	−0.88254	−0.85346	−0.42886	−0.34353	0.412891	−0.29265	0.017486	−81.3404
41	0.746414	−0.92241	0.490665	0.288544	0.520277	0.979305	−0.71502	0.416498	−149.643
42	−0.53127	−0.1721	0.271789	−0.40229	−0.86941	0.03631	0.385583	0.078356	−38.6198
43	−0.64338	0.787192	−0.69247	−0.73417	−0.07329	−0.01149	0.325306	0.409607	−29.9596
44	−0.83909	0.313927	0.721478	0.449674	−0.13722	−0.21391	0.13829	0.249443	−40.6912
45	0.55663	0.23352	−0.65654	0.032435	−0.86543	−0.44689	−0.68503	0.356767	55.05174
46	−0.45897	−0.01062	−0.16238	0.646557	0.550094	−0.98171	−0.06412	0.893158	−111.963
47	0.558745	0.86732	0.444558	0.307026	−0.12374	0.944949	0.446797	0.001712	139.8847
48	−0.47612	−0.12164	0.923099	−0.37132	0.433339	−0.14079	−0.59591	0.532258	−51.384
49	0.235091	−0.17192	0.200231	−0.44194	0.026407	−0.40198	−0.59422	0.49563	77.20547
50	−0.98094	−0.49147	0.705473	−0.84763	0.336172	0.380735	0.44793	0.284686	26.19464
51	−0.68787	0.855684	0.412611	−0.67918	0.869583	0.440092	0.933794	0.814636	98.97161
52	−0.79538	−0.40033	−0.88197	0.97166	−0.22275	−0.28991	0.860131	0.825306	133.2404
53	−0.57416	0.665569	0.812562	0.150751	0.081135	0.857895	0.120364	0.111805	−119.48
54	0.62067	−0.12834	0.762917	0.828662	0.510371	0.517299	−0.5704	0.50425	53.34696
55	−0.76749	0.492631	−0.97983	0.296692	0.362353	0.711477	−0.30019	0.944011	−127.381
56	0.849134	0.314621	−0.22608	−0.77793	0.902875	−0.39672	−0.05598	0.839121	4.440011
57	0.386008	0.461785	0.279021	−0.3529	−0.89524	−0.39555	−0.99684	0.01459	47.2274
58	−0.90204	0.213319	−0.85119	−0.40301	−0.34397	0.682762	0.376978	0.431022	4.936711
59	0.24445	0.758876	−0.71051	0.310961	−0.36849	−0.62696	0.680268	0.775376	−51.891
60	−0.55873	0.226296	−0.4886	−0.33622	0.196823	−0.6504	0.088063	0.071867	14.9209
61	0.969918	0.017161	0.084454	0.241556	−0.64874	−0.96111	−0.58014	0.206049	−48.2017
62	0.915395	−0.58482	0.792565	−0.2735	−0.99951	−0.67445	−0.22295	0.994204	−1.50322

Figure 4. Predictive capacity of developed models (a) using RMSE for training dataset (b) using MAE for training dataset (c) using RMSE for testing dataset (d) using MAE for testing dataset.

Table 4. Performances of the developed intelligent models

	dataset	CC	RMSE	MAE
GBSVRA	Training sample	1	0.001	0.001
	Testing sample	0.8697	97.6035	76.9928
ELM-SG	Training sample	0.895881	78.44218	52.26572
	Testing sample	0.838834	91.4621	74.78362
ELM-SN	Training sample	1	1.38E-12	1.15E-12
	Testing sample	0.946512	55.99322	47.3012
Superiority of ELM-SN over GBSVRA	Testing sample	8.115267	42.63196	38.56413
Superiority of ELM-SN over ELM-SG	Testing sample	11.37635	38.77987	36.74925

Figure 5. Predictions of the developed intelligent models for different spinel ferrite magnetocaloric compounds [S1 = Zn_0.6Cu_0.4Fe₂O₄ (F. Hcini et al., 2021), S2 = Zn_0.4Ni_0.2Cu_0.4Fe2O₄ (F. Hcini et al., 2021), S3 = Zn_0.250Ni_0.250Mg_0.5Fe₂O₄ (F. Hcini et al., 2021), S4 = Ni_0.7Zn_0.3Fe₂O₄ (Bahhar et al., 2021), S5 = Ni_0.5Zn_0.5Fe2O₄ (Oumezzine et al., 2015), S6 = Ni_0.3Zn_0.7Fe₂O₄ (Oumezzine et al., 2015), and S7 = Cu_0.4Zn_0.6Fe2O₄ (Bahhar et al., 2021)].

Figure 6. Comparison of the estimates of the developed intelligent models for classes of spine ferrite magnetocaloric compounds [S8 = Cu_0.2Zn_0.8Fe₂O₄ (Oumezzine et al., 2015), S9 = Ni_0.4Zn_0.6Fe₂O₄ (Oumezzine et al., 2015), S10 = Ni_0.35Zn_0.65Fe₂O₄ (Oumezzine et al., 2015), S11 = Ni_0.3Zn_0.7Fe₂O₄ (Oumezzine et al., 2015), S12 = Ni_0.25Zn_0.75Fe₂O₄ (Oumezzine et al., 2015), S13 = MnCeFeO₄ (Bahhar et al., 2021), S14 = CoCeFeO₄ (Bahhar et al., 2021)].

Figure 7. Magnetic ordering temperature of spinel ferrite magnetocaloric compounds obtained from developed intelligent models [S15 = Ni_0.4Mg_0.3Cu_0.3Fe₂O₄, S16 = Cu_0.3Zn_0.7Fe₂O₄ (Bahhar et al., 2021), S17 = Mg_0.5Zn_0.5Fe₂O₄ (Fortas et al., 2020), S18 = Mg_0.3Zn_0.7Fe₂O₄ (Bouhbou et al., 2022),S19 = Mg_0.2Zn_0.8Fe₂O₄ (Bouhbou et al., 2022), S20 = Zn_0.7Ni_0.1Cu_0.2Fe₂O₄ (Bouhbou et al., 2022), S21 = Mg_0.35Zn_0.65Fe₂O₄ (Bouhbou et al., 2022)].

Figure 8. Estimated MOT for different spinel ferrite magnetocaloric compounds [S22 = Zn_0.375Ni_0.125Mg_0.5Fe₂O₄ (Fortas et al., 2020), S23 = Zn_0.250Ni_0.250Mg_0.5Fe₂O₄ (Fortas et al., 2020), S24 = Mg_0.6Cu_0.2Ni_0.2Fe₂O₄ (Fortas et al., 2020),S25 = Mg_0.6Cu_0.4Fe₂O₄ (Fortas et al., 2020), S26 = Ni_0.4Cd_0.3Zn_0.3Fe₂O₄ (Fortas et al., 2020),S27 = Ni_0.6Cd_0.2Zn_0.2Fe₂O₄ (Fortas et al., 2020), S28 = Zn_0.7Ni_0.3Fe₂O₄ (Oumezzine et al., 2015), S29 = Zn_0.7Ni_0.2Cu_0.1Fe₂O₄ (Fortas et al., 2020)].

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Data availability statement

section 3.2 contained the details of the raw data utilized for these findings with their references.