Support vector machine based emissions modeling using particle swarm optimization for homogeneous charge compression ignition engine

. 2021 Nov 9;24(2):536–551. doi: 10.1177/14680874211055546

Algorithm 2: PSO based RBF kernel SVM algorithm
Result: HCCI emission model: $y (u)$ training data set: ${u, z}$ splitting data set: training ${u_{tr}, z_{tr}}$ , cross-validation ${u_{cv}, z_{cv}}$ , and test ${u_{ts}, z_{ts}}$ Random hyperparameters: $C_{LSVM}, ϵ_{LSVM}, σ$ Run Quadratic Programming of equation (11) to calculate $α^{-}$ and $α^{+}$ Calculate support vector sets based on equation (13) Calculate model based on random hyperparameters using Equations (15) and (16) Set PSO options: $MaxIteration s^{1}$ , $MaxStallIteration s^{2}$ , $FunctionToleranc e^{3}$ , and $SwarmSiz e^{4}$ ; whilei∈ $MaxIterations$ or $d_{5}$ over $MaxStallIterations$ ≥ $FunctionTolerance$ do Calculate cost function: $\begin{matrix} J (C_{LSVM}, ϵ_{LSVM}, σ) = (\frac{1}{n_{tr}} \sum_{i = 1}^{n_{tr}} (z_{tr, i} - y_{tr, i})^{2} + \\ \frac{1}{n_{cv}} \sum_{i = 1}^{n_{cv}} (z_{cv, i} - y_{cv, i})^{2}) \end{matrix}$ Run PSO algorithm to minimize $J (C_{LSVM}, ϵ_{LSVM}, σ)$ and find hyperparameters Update hyperparameters: $C_{LSVM}, ϵ_{LSVM}, σ$ Run Quadratic Programming of equation (11) to calculate $α^{-}$ and $α^{+}$ Calculate support vector sets based on equation (23) Calculate model based on Updated hyperparameters using equations (25) and (26) $i = i + 1$ end 1. Maximum number of iterations for optimization ( $= 600$ ), 2. Positive integer ( $= 20$ ), 3. Non- negative scalar: Iterations end when the relative change in cost function value over the last MaxStallIterations iterations is less than FunctionTolerance $= 1 e - 6$ , 4. Number of particles in the swarm $= 200$ , 5. Relative change

Algorithm 2: PSO based RBF kernel SVM algorithm

Result: HCCI emission model:

y (u)

training data set:

{u, z}

splitting data set: training

{u_{tr}, z_{tr}}

, cross-validation

{u_{cv}, z_{cv}}

, and test

{u_{ts}, z_{ts}}

Random hyperparameters:

C_{LSVM}, ϵ_{LSVM}, σ

Run Quadratic Programming of equation (11) to calculate

α^{-}

and

α^{+}

Calculate support vector sets based on equation (13)
Calculate model based on random hyperparameters using
Equations (15) and (16)
Set PSO options:

MaxIteration s^{1}

MaxStallIteration s^{2}

FunctionToleranc e^{3}

,
and

SwarmSiz e^{4}

;
whilei∈

MaxIterations

d_{5}

over

MaxStallIterations

≥

FunctionTolerance

do
Calculate cost function:

\begin{matrix} J (C_{LSVM}, ϵ_{LSVM}, σ) = (\frac{1}{n_{tr}} \sum_{i = 1}^{n_{tr}} (z_{tr, i} - y_{tr, i})^{2} + \\ \frac{1}{n_{cv}} \sum_{i = 1}^{n_{cv}} (z_{cv, i} - y_{cv, i})^{2}) \end{matrix}

Run PSO algorithm to minimize

J (C_{LSVM}, ϵ_{LSVM}, σ)

and find hyperparameters
Update hyperparameters:

C_{LSVM}, ϵ_{LSVM}, σ

Run Quadratic Programming of equation (11) to calculate

α^{-}

and

α^{+}

Calculate support vector sets based on equation (23)
Calculate model based on Updated hyperparameters
using equations (25) and (26)

i = i + 1

end
1. Maximum number of iterations for optimization (

= 600

), 2. Positive integer (

= 20

), 3. Non- negative scalar: Iterations end when the relative change in cost function value over the last MaxStallIterations iterations is less than FunctionTolerance

= 1 e - 6

, 4. Number of particles in the swarm

= 200

, 5. Relative change