A modified projective forward-backward splitting algorithm for variational inclusion problems to predict Parkinson's disease

Figures & data

Figure 1. The structure comparison of an inertial projective method and a standard method.

Figure 2. Training-validation accuracy plots of Algorithm 1 which is considered by the RLS$L_1$ ELM mode.

Figure 3. Training-validation loss plots of Algorithm 1 which is considered by the RLS$L_1$ ELM mode.

Figure 4. Training-validation accuracy plots of Algorithm 1 which is considered by the RLS$L_1$-C$L_2$ ELM mode.

Figure 5. Training-validation loss plots of Algorithm 1 which is considered by the RLS$L_1$-C$L_2$ ELM mode.

Figure 6. Training-validation accuracy plots of IMFB (3(3) $\begin{array}{rl}{y}^k& ={\omega }^k+{\sigma }^k({\omega }^k-{\omega }^{k-1}),\\ z^k& =y^k+{\alpha }^k({\omega }^k-y^k),\\ {\omega }^{k+1}& =J_{{\tau }^k}^{\mathbb{G}}(z^k-{\tau }^k\mathbb{F}{z}^k),k\ge 0,\end{array}$(3) ) which is considered by the RLS$L_1$ ELM mode.

Table 1. The overview of PD dataset from UCI Machine Learning Repository.

Attribute Name	Description	Mean	Max	Min	SD	CV
MDVP:Fo (Hz)	Vocal fundamental (mean frequency)	154.2286	260.1050	88.33300	41.390065	0.268368
MDVP: Fhi(Hz)	Vocal fundamental (Max frequency)	197.1049	592.0300	102.1450	91.491548	0.464177
MDVP: Flo (Hz)	Vocal fundamental (Min frequency)	116.3246	239.1700	65.47600	43.521413	0.374138
MDVP:Jitter(%)	Basic frequency variations	0.006220	0.033160	0.001680	0.004848	0.779385
MDVP:Jitter(Abs)	Basic frequency variations	0.000044	0.000260	0.000007	0.000035	0.792145
MDVP:RAP	Basic frequency variations	0.003306	0.021440	0.000680	0.002968	0.897582
MDVHAL:PPQ	Basic frequency variations	0.003446	0.019580	0.000920	0.002759	0.800548
Jitter:DDP	Basic frequency variations	0.009920	0.064330	0.002040	0.008903	0.897519
MDVP:Berkilauan	Basic frequency variations	0.029709	0.119080	0.009540	0.018857	0.634718
MDVP:Berkilau(dB)	Wave variations	0.282251	1.302000	0.085000	0.194877	0.690439
Glitter:APQ3	Wave variations	0.015664	0.056470	0.004550	0.010153	0.648178
Glitter:APQ5	Wave variations	0.017878	0.079400	0.005700	0.012024	0.672532
MDVP:APQ	Wave variations	0.024081	0.137780	0.007190	0.016947	0.703725
Berkilauan:DDA	Wave variations	0.046993	0.169420	0.013640	0.030459	0.648168
NHR	The ratio of noise with regard to total components in voice	0.024847	0.314820	0.000650	0.040418	1.626688
HNR	The ratio of noise with regard to total components in voice	21.88597	33.04700	8.441000	4.425764	0.202219
RPDE	Nonlinear complexity measurements	0.498536	0.685151	0.256570	0.103942	0.208494
DFA	Fractal scaling exponent	0.718099	0.825288	0.574282	0.055336	0.077059
penyebaran1	Methods for calculating fundamental frequency variation	−5.684397	−2.434031	−7.964984	1.090208	−0.191790
penyebaran2	Methods for calculating fundamental frequency variation	0.226510	0.450493	0.006274	0.083406	0.368221
D2	Nonlinear complexity measurements	2.381826	3.671155	1.423287	0.382799	0.160717
PPE	Methods for calculating fundamental frequency variation	0.206552	0.527367	0.044539	0.090119	0.436304
Condition	Health status

Table 2. The accuracy comparison between Elshewey et al. [17] methods and ours.

	Default parameters	BO for optimizing the hyperparameters
SVM	89.6	92.3
RF	87.2	89.7
LR	85.3	87.2
NB	82.1	84.6
RC	80.9	83.3
DT	85.7	88.5
Our ELM	92.86

Table 3. The necessary parameters for our Algorithm 1, IFB (2(2) $\begin{array}{rl}{y}^k& ={\omega }^k+{\sigma }^k({\omega }^k-{\omega }^{k-1}),\\ {\omega }^{k+1}& =J_{{\tau }^k}^{\mathbb{G}}(y^k-{\tau }^k\mathbb{F}{\omega }^k),k\ge 0,\end{array}$(2) ) and IMFB (3(3) $\begin{array}{rl}{y}^k& ={\omega }^k+{\sigma }^k({\omega }^k-{\omega }^{k-1}),\\ z^k& =y^k+{\alpha }^k({\omega }^k-y^k),\\ {\omega }^{k+1}& =J_{{\tau }^k}^{\mathbb{G}}(z^k-{\tau }^k\mathbb{F}{z}^k),k\ge 0,\end{array}$(3) ).

	Optimization model	σ	${\alpha }^k$	${\text{β}}^k$	${\tau }^k$	λ	β
Algorithm 1	RLS$L_1$	0.9999	$\frac{n}{{10}^{6}n+1}$	$\frac{n}{{10}^{6}n+1}$	$\frac{0.999}{\Vert \mathbf{A}{\Vert }^2}$	0.1	–
	RLS$L_2$	0.9999	$\frac{n}{{10}^{6}n+1}$	$\frac{n}{{10}^{6}n+1}$	$\frac{0.999}{\Vert \mathbf{A}{\Vert }^2}$	1000	–
	RLS$L_1$-C$L_1$	0.9999	$\frac{n}{{10}^{6}n+1}$	$\frac{n}{{10}^{6}n+1}$	$\frac{0.999}{\Vert \mathbf{A}{\Vert }^2}$	0.1	61
	RLS$L_1$-C$L_2$	0.9999	$\frac{n}{{10}^{6}n+1}$	$\frac{n}{{10}^{6}n+1}$	$\frac{0.999}{\Vert \mathbf{A}{\Vert }^2}$	0.1	20
	RLS$L_2$-C$L_1$	0.9999	$\frac{n}{{10}^{6}n+1}$	$\frac{n}{{10}^{6}n+1}$	$\frac{0.999}{\Vert \mathbf{A}{\Vert }^2}$	1000	60
	RLS$L_2$-C$L_2$	0.9999	$\frac{n}{{10}^{6}n+1}$	$\frac{n}{{10}^{6}n+1}$	$\frac{0.999}{\Vert \mathbf{A}{\Vert }^2}$	1000	40
IFB (2(2) $\begin{array}{rl}{y}^k& ={\omega }^k+{\sigma }^k({\omega }^k-{\omega }^{k-1}),\\ {\omega }^{k+1}& =J_{{\tau }^k}^{\mathbb{G}}(y^k-{\tau }^k\mathbb{F}{\omega }^k),k\ge 0,\end{array}$(2) )	RLS$L_1$	0.9999	–	–	$\frac{0.2}{\Vert \mathbf{A}{\Vert }^2}$	0.1	–
	RLS$L_2$	0.9999	–	–	$\frac{0.2}{\Vert \mathbf{A}{\Vert }^2}$	0.1	–
IMFB (3(3) $\begin{array}{rl}{y}^k& ={\omega }^k+{\sigma }^k({\omega }^k-{\omega }^{k-1}),\\ z^k& =y^k+{\alpha }^k({\omega }^k-y^k),\\ {\omega }^{k+1}& =J_{{\tau }^k}^{\mathbb{G}}(z^k-{\tau }^k\mathbb{F}{z}^k),k\ge 0,\end{array}$(3) )	RLS$L_1$	0.9999	$\frac{n}{{10}^{6}n+1}$	–	$\frac{0.999}{\Vert \mathbf{A}{\Vert }^2}$	0.1	–
	RLS$L_2$	0.9999	$\frac{n}{{10}^{6}n+1}$	–	$\frac{0.999}{\Vert \mathbf{A}{\Vert }^2}$	1000	–

Figure 7. Training-validation loss plots of IMFB (3(3) $\begin{array}{rl}{y}^k& ={\omega }^k+{\sigma }^k({\omega }^k-{\omega }^{k-1}),\\ z^k& =y^k+{\alpha }^k({\omega }^k-y^k),\\ {\omega }^{k+1}& =J_{{\tau }^k}^{\mathbb{G}}(z^k-{\tau }^k\mathbb{F}{z}^k),k\ge 0,\end{array}$(3) ) which is considered by the RLS$L_1$ ELM mode.

Table 4. The performance by 4 evaluation matrices of our Algorithm 1, IFB (2(2) $\begin{array}{rl}{y}^k& ={\omega }^k+{\sigma }^k({\omega }^k-{\omega }^{k-1}),\\ {\omega }^{k+1}& =J_{{\tau }^k}^{\mathbb{G}}(y^k-{\tau }^k\mathbb{F}{\omega }^k),k\ge 0,\end{array}$(2) ) and IMFB (3(3) $\begin{array}{rl}{y}^k& ={\omega }^k+{\sigma }^k({\omega }^k-{\omega }^{k-1}),\\ z^k& =y^k+{\alpha }^k({\omega }^k-y^k),\\ {\omega }^{k+1}& =J_{{\tau }^k}^{\mathbb{G}}(z^k-{\tau }^k\mathbb{F}{z}^k),k\ge 0,\end{array}$(3) ).

	Optimization model	Number of Iterations	Training Time	Accuracy	Precision	Recall	F1-score
Algorithm 1	RLS$L_1$	433	0.0680	92.86	62.50	100	76.92
	RLS$L_2$	314	0.0631	88.10	37.50	100	54.55
	RLS$L_1$-C$L_1$	449	0.0856	92.86	62.50	100	76.92
	RLS$L_1$-C$L_2$	433	0.0849	92.86	62.50	100	76.92
	RLS$L_2$-C$L_1$	431	0.0818	90.48	62.50	83.88	71.43
	RLS$L_2$-C$L_2$	314	0.0640	88.10	37.50	100	54.55
IFB (2(2) $\begin{array}{rl}{y}^k& ={\omega }^k+{\sigma }^k({\omega }^k-{\omega }^{k-1}),\\ {\omega }^{k+1}& =J_{{\tau }^k}^{\mathbb{G}}(y^k-{\tau }^k\mathbb{F}{\omega }^k),k\ge 0,\end{array}$(2) )	RLS$L_1$	55	0.0170	83.33	12.50	100	22.22
	RLS$L_2$	185	0.0325	80.05	100	19.05	32.00
IMFB (3(3) $\begin{array}{rl}{y}^k& ={\omega }^k+{\sigma }^k({\omega }^k-{\omega }^{k-1}),\\ z^k& =y^k+{\alpha }^k({\omega }^k-y^k),\\ {\omega }^{k+1}& =J_{{\tau }^k}^{\mathbb{G}}(z^k-{\tau }^k\mathbb{F}{z}^k),k\ge 0,\end{array}$(3) )	RLS$L_1$	433	0.0748	92.86	62.50	100	76.92
	RLS$L_2$	314	0.0659	88.10	37.50	100	54.55

Elshewey AM, Shams MY, El-Rashidy N, et al. Bayesian optimization with support vector machine model for Parkinson disease classification. Sensors. 2023;23(4):2085. doi: 10.3390/s23042085

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Data availability statement

Parkinson's disease dataset is available on the UCI website (https://archive.ics.uci.edu/ml/datasets/parkinsons).