OPEN ACCESS
Regressions have been continuously received great attention. However, there are still open issues in regression, and two of the issues is regression with multicollinearity and outlier. Regularization (Ridge, Lasso, and Elastic Net) techniques implement a means to control regression coefficients. The methods can decrease the variance and reduce our sample error for tackle multicollinearity. In robust regression, it is a form of regression method designed to overcome outliers. Robust regression is an important method for analyzing data that are infected with outliers. The data have been interacted on the second order interaction. The data contained 435 different independent interaction variables. The primary focus of this paper is to analyze and compare the impact of three different variable selection techniques regularization regression algorithms for the data seaweed drying. After that, it will be analyzed through robust regression (Tukey BiSquare, Hampel, and Huber). As the result, the LassoHampel was better than others with the MAE (4.09641), RMSE (5.275992), MAPE (7.9962), SSE (182491.2), Rsquare (0.6514791), and Rsquare Adjusted (0.649279). The method of LassoHampel is able to be relied on investigation of the accuracy in big data obtained from regularization and robust regression.
variable selection, regularization regression, robust regression, model selection, 8 selection criteria
Regression methods are algorithms of supervised learning, which are important both Machine Learning and Statistics Learning. The regression methods have been known for a long time because they are many new developments. The regression methods are extending these algorithms significantly [1].
The regression methods are frequently used to calculate an algorithm to forecast future responses. They are aims to investigate relationship between dependent variable (Y) and the independent variables (X) [2].
The regression analyses are often applied most sciences. The regression methods are ones of the main tasks in Machine Learning and Statistics Learning. The regression methods have been successfully applied in many fields such as agriculture and biology for this case using data seaweed drying.
Seaweed should get attention from the Malaysia Government because it has many advantages including lots of nutrients and short growth of only 45 days per cycle. The seaweed is widely cultivated in Sabah because of the environmental and geographical factors which support it. Sabah is very favorable compared to the Malaysian peninsula [3]. The seaweed as an agricultural sector plays an important role in providing a source of food and protein in Malaysia [4].
The abundant supply of seaweed in Malaysia offers promising opportunities to produce and extract such as fucoidan, alginate, agar, and carrageenan. The seaweed is used in various ingredients such as in foods, pharmaceuticals, nutraceuticals, medicals, and other industries.
Seaweed contains beneficial bioactive compounds such as carrageenan powder, agar, or alginate. Seaweed is of great commercial importance as a stabilizer, thickener, gelling agent, and emulsifier. The Malaysia AgroPolicy has developed seaweed as highvalue and valuable commodity that makes seaweed an important industry. Malaysia has great potential to become a significant seaweed supplier in the country, provided Malaysia has fully developed and utilized existing resources [5].
Assessment and comparison of the performance of the available methods are thus important to select the best method with the seaweed drying data and determine when their performance is optimal. Here, we evaluate the relative performance regularization regressions (Ridge, Lasso, and Elastic Net) for selecting variables (to choose the most significant variable from their perspective) and will be analyzed with robust regression (BiSquare, Hampel, and Huber) models.
The methods comprise Ridge, Lasso, and Elastic net regression [617].
Regularization regressions (Ridge, Lasso, and Elastic Net) are applied as a variable selection to select the most significant variables with their perspective. They provide methods for controlling the regression coefficient, which is able to decrease the variance and decrease the sample error to solve the multicollinearity issue. They are applied in various fields of scientific disciplines [18].
Improvement in both Statistics Learning and Machine Learning methoddriven by big data in various disciplines scientificoffers opportunities and challenges for agriculture data analysis (especially the seaweed drying data). Today, in the era of big data, variable selections are a fundamental task in the area of both Statistics Learning and Machine Learning.
In general, the process of variable selection aims to select which are important variables. For example, in regression, it is very useful to select and maintain variables with predictable capabilities.
The aims of variable selection usually are:
(i) To improve predictive model capabilities;
(ii) To avoid multicollinearity problems;
(iii) To provide a more comprehensive understanding of the prediction model by reducing ineffective and unnecessary variables [19].
Both Statistics Learning and Machine Learning aim to build a model that presents the best dataset, these methods involve the task of variable selections. In this paper, a dataset containing 1924 observations will use to study the effect of more 29 different independent variables on the one dependent variable. Then the data will be interacted with in the second interaction. The data contain the effect of 435 different interaction independent variables on the one dependent variable. The more detailed tables for each variable interaction are attached in the Appendix A.
In recent years, agricultural data has increased exponentially with the adoption of automated data collection tools and systems. Data generated from agricultural precision tools has been one of the most significant contributions to this improvement. Due to the fast growth of data, regularization regression (Ridge, Lasso, and Elastic Net) will help to find useful and meaningful in big data, especially in agriculture [20].
In this study, it is to analyse seaweed data with several variables including hourly solar radiation, temperature, humidity, and moisture content.
Big data technology in agriculture has increased adoption rates in precision agriculture and is expected to become more prevalent in the coming years. It is used in the precision agricultural in several aspects of crop production, such as accuracy, agriculture (weather forecasting, yield monitoring, soil conditioning), decisionmaking tool and in enhancing zones of food security. The big data repositories essential knowledge which can be applied to the scientific data, or to give knowledge on interdisciplinary decisions such as economics, politics, or often recently, ‘artificial intelligence of farming’ to enhance food security and potency of agriculture [21].
Regressions continue to get significant appreciation and attention. However, in regressions have still open problems such as multicollinearity and outlier.
The first issue in regression is multicollinearity. Multicollinearity is two or more independent variables with high correlation. It is a common problem which is often encountered in regression methods. It will reduce the accuracy of parameter evaluation in the regression methods [22].
Regularization regressions are applied as a variable selection to select the most significant variables with their perspective. They provide methods for controlling the regression coefficient, which is able to decrease the variance and decrease the sample error to solve the multicollinearity issue. They are applied in various fields of scientific disciplines [18]. So, regularization regression is a regression analysis designed to handle multicollinearity. In this paper, we will use three types of regularization regressions such as Lasso, Ridge, and Elastic Net.
The methods comprise Ridge, Lasso, and Elastic net regression [617]. So, an important property of regularization regressions is respect to multicollinearity in the database (big data).
The second issue in regression is outliers. Outliers are suspicious because they are much larger or much smaller than most of the observations [23, 24]. Outliers are objections that differ significantly from the remaining data. The outliers are also referred to as anomalies, abnormalities, and discordances [25]. The outliers are common in big data and can create severe regression problems. They can lead to model misspecification, inaccurate analysis results and make all evaluation methods meaningless.
So, an important property of robust regressions is method with respect to outliers in big data. Robust regressions are required where the estimated values are not much influenced by much smaller or much larger observations. So, robust regression is a regression method which is designed to address outliers.
Robust regression is an important method for analyzing data which are contaminated outliers [24, 26, 27]. Because ordinary least square (OLS) can be very sensitive to outliers. Robust regressions are applied to detect outliers and provide results that are resistant to the presence of outliers. In this paper, we will use three types of robust regression MEstimation such as BiSquare, Hampel, and Huber.
The methods comprise Tukey BiSquare, Hampel, and Huber regression [2832].
To assess models, we need a model selection. Model selection was also made by different researchers, Abdullah et al. [33] used eight selection criteria (8SC) to obtain the best model among all possible models. Similarly, Javaid et al. used in model selection problem [15, 3436].
Several authors have reviewed 8 Selection Criteria (8SC), but our study is different from their paper. Javaid et al. have made a study of 8 Selection Criteria. They only conducted research on small data. They did not present visualization in comparing models [15, 3638].
The primary focus of this paper is to analyze and compare the impact of three different variable selection techniques regression regularization algorithms (Lasso, Elastic Net, and Ridge) for the data seaweed drying. After that, it will be analyzed through robust regression (Tukey BiSquare, Hampel, and Huber) and to compare the impact of three different regression algorithms for forecast the efficient model, Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Mean Absolute Percentage Error (MAPE)., comparing three regularization and robust regression algorithmsin terms of the best model eight selection criteria (8SC).
2.1 Regularization regression
2.1.1 Lasso
Linear regression equations $\left\{\left(x_{i}, y_{i}\right)\right\}_{i=1}^{N}$ with $N$ samples and independent variables are $p$ dimensional and $y_{i} \in \mathbb{R}$ is dependent variable. The aim is to forecast the dependent variable from the independent variables. Forecast and find independent variables significant play an essential role in regression [39]. The equation assumes:
$y_{i}=\beta_{0}+\sum_{j=1}^{p} x_{i j} \beta_{j}+\varepsilon_{i}$ (1)
$\beta_{0}$ and $\beta_{i}$ are unknown parameters and $\varepsilon_{i}$ is a residual term for $=1, \ldots, p$. The Eq. 2 is a requirement to constrain. For Lasso regression or $\ell_{1}$ regularize regression,
$\min _{\beta_{0}, \beta} \sum_{i=1}^{N}\left(y_{i}\beta_{0}\sum_{j=1}^{p} x_{i j} \beta_{j}\right)^{2}$ subject to $\\beta\_{1} \leq t$ (2)
2.1.2 Ridge
The ridge constrain is $\sum_{j=1}^{p} \beta_{j}^{2} \leq t$ for a positive value $t$. For ridge regression or $\ell_{2}$ regularize regression [39].
$\min _{B_{n} B} \sum_{i=1}^{N}\left(y_{i}\beta_{0}\sum_{j=1}^{p} x_{i j} \beta_{j}\right)^{2}$ subject to $\sum_{J=1}^{P} \beta_{J}^{2} \leq t$ (3)
2.1.3 Elastic net regression
Lasso and ridge regression could be stated with $L_{q}$. Both $q=1$ and $q=2$ are corresponding to lasso and ridge respectively. Eq. (4) can be solved by calculating of type $L_{q}$.
$\arg \min _{\beta}\left\{\boldsymbol{Y}^{T} \boldsymbol{Y}2 \beta^{\boldsymbol{T}} \boldsymbol{X}^{\boldsymbol{T}} \boldsymbol{Y}+\beta^{\boldsymbol{T}} \boldsymbol{X}^{\boldsymbol{T}} \boldsymbol{X} \beta\right\}$subject to $\sum_{j=1}^{k}\left\beta_{j}\right^{q} \leq t$ (4)
Researchers recommend taking $1<q<2$, to choose a compromise between lasso and ridge [40]. The elastic net regression evolves combining between Lasso and Ridge [41]. The elastic net formulation was defined by Zou and Hastie [16] as:
$\sum_{j=1}^{k}\left((1\alpha) \beta_{j}^{2}+\alpha\left\beta_{j}\right\right) \leq d^{2}, \alpha \in[0,1]$ (5)
The elastic net is then used as a penalizing term to obtain the elastic net estimate:
$\hat{\beta}_{\text {Elastic net }}=\arg \min _{\beta}\left\{\boldsymbol{Y}^{\boldsymbol{T}} \boldsymbol{Y}2 \beta^{\boldsymbol{T}} \boldsymbol{X}^{\boldsymbol{T}} \boldsymbol{Y}+\right. \left.\beta^{\boldsymbol{T}} \boldsymbol{X}^{\boldsymbol{T}} \boldsymbol{X} \beta+\lambda \sum_{j=1}^{k}\left((1\alpha) \beta_{j}^{2}+\alpha\left\beta_{j}\right\right)\right\}$ (6)
From the Eq. (7), selecting parameter q is not necessary. We require to select an α value between $0<\alpha<1$. Ridge and Lasso regression could be stated with $\alpha$. Both $\alpha=0$ and $\alpha=1$ are corresponding to ridge and lasso respectively. The elastic net regression evolves combining between Lasso and Ridge.
The elastic net is a method of regularization regression that provides between ridge and lasso [42]. The advantage of the elastic net is achieving stability concerning random sampling [43].
2.2 Robust regression
The Mestimation is general method in robust regression. The M in Mestimation is “Maximum likelihood”. The aim of Mestimation is minimizing error (residual) [44].
The first function regression method, suppose we have a data set of size n such that:
$\begin{aligned} y_{i} &=x_{i}^{T} \beta+e_{i} \\ e_{i}=y_{i} &\hat{y}_{i}=y_{i}\boldsymbol{x}_{i}^{T} \beta \\ e_{i}(\beta) &=y_{i}x_{i}^{T} \beta \end{aligned}$ (7)
Mestimator attempt to minimize the sum of a chosen function $\rho(\cdot)$ which is acting on the residual. Formally defined, Mestimators are given by:
$\hat{\beta}_{M}=\underset{\beta}{\operatorname{argmin}} \sum_{i=1}^{n} \rho\left(e_{i}(\beta)\right) $ (8)
The above form with $\rho$ function is the $\rho$ type Mestimation. Suppose $\sigma$ is known and let the residuals for some estimate $\beta$ be $e_{i}=y_{i}\beta^{T} x_{i}[45]$. Then the regression Mestimate of $\beta$ is the value that minimizes the objective function:
$\sum_{i=1}^{n} \rho\left\{\frac{e_{i}(\beta)}{\sigma}\right\}$ (9)
The $\sigma$ should be estimated robustly. Mestimator of scale $\tilde{\sigma}_{M}$ is found by solution of the equation:
$\frac{1}{n} \sum_{i=1}^{n} \rho\left(\frac{e_{i}}{\sigma}\right)=\frac{1}{n} \sum_{i=1}^{n} \rho\left(\frac{y_{i}\beta^{T} x_{i}}{\sigma}\right)=k $ (10)
When $\beta$ is the $p \times 1$ parameter vector, then $\psi$ type function could be yielding as:
$\sum_{i} \psi\left(e_{i}\right) \frac{\partial e_{i}}{\partial \beta_{i}}$, for $j=1,2, \ldots, p$ (11)
where the derivative function $\psi(e)=\frac{\partial \rho(e)}{\partial(e)}$ is the influence function. Then the weight function could be defined as below:
$w(e)=\frac{\psi(e)}{e}$ (12)
The $\psi(e)$type function becomes:
$\sum_{i} w\left(e_{i}\right) e_{i} \frac{\partial e_{i}}{\partial \beta_{i}}=0$, for $j=1,2, \ldots, p$ (13)
And the object becomes to obtain the following iterated reweighted least square problem:
$\min \sum_{i} w\left(e_{i}^{(k1)}\right) e_{i}^{2}$ (14)
where, k indicates the iterate number [46].
Further, the M robust regression was applied to address the outliers through M–bi square, M–Hampel, and M–Huber. For more detail, we applied in Table 1Formulas for Robust Regression Mestimation.
2.3 Validation models
The metric evaluations are needed to evaluate the appropriateness of a model. They become very important to analyze whether the model is adequate. The metrics including Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), Sum of Square Error (SSE), and Rsquared.
Table 1. Formulas for Robust Regression Mestimation
Methods 
Objective Function 
Weight Function 
BiSquare 
$\rho_{B}=\left\{\begin{array}{c}\frac{k^{2}}{6}\left\{1\left[1\left(\frac{e}{k}\right)^{2}\right]^{3}\right\} \text { for }e \leq k \\ \frac{k^{2}}{6} \text { for }e>k\end{array}\right.$ 
$w_{B}=\left\{\begin{array}{c}{\left[1\left(\frac{e}{k}\right)^{2}\right]^{2} \text { for }e \leq k} \\ 0 \text { for }e>k\end{array}\right.$ 
Huber 
$\rho_{H u}=\left\{\begin{array}{c}\frac{1}{2} e^{2} \text { for }e \leq k \\ ke\frac{1}{2} k^{2} \text { or }e>k\end{array}\right.$ 
$w_{H u}=\left\{\begin{array}{c}1 \text { for }e \leq k \\ \frac{k}{e} \text { for }e<k\end{array}\right.$ 
Hampel 
$\rho_{H a}=\left\{\begin{array}{c}\frac{e^{2}}{2}, \quad 0<e<a \\ ae\frac{e^{2}}{2}, \quad b<e \leq c \\ \frac{a}{2(cb)}(ce)^{2}+\frac{a}{2}(b+ca), b<e \leq c\end{array}\right.$ 

Table 2. Formulas for validation methods
Validation 
Formulation 
Reference 
Mean Absolute Error (MAE) 
$M A E=\sum_{i=1}^{n}\left\frac{Y\hat{Y}_{i}}{\hat{Y}_{i}}\right$ 
[47] 
Mean Square Error (MSE) 
$M S E=\sum_{i=1}^{n}\left(\frac{Y\hat{Y}_{i}}{\hat{Y}_{i}}\right)^{2}$ 
[48] 
Mean Absolute Percentage Error (MAPE) 
$M A P E=\frac{100}{n} \sum_{i=1}^{n}\left\frac{Y\hat{Y}_{i}}{\hat{Y}_{i}}\right$ 
[49] 
Sum of Square Error (SSE) 
$S S E=\sum_{i=1}^{n}\left(Y_{i}\hat{Y}_{i}\right)^{2}$ 
[50] 
Sum of Squared Total (SST) 
$S S T=\sum_{i=1}^{n}\left(\hat{Y}_{i}\bar{Y}\right)^{2}$ 
[50] 
Rsquared 
$R^{2}=\frac{S S R}{S S T}=\frac{S S TS S E}{S S T}=1\frac{S S E}{S S T}$ 
[50] 
The metric evaluations are used to measure the accuracy of the regression model in forecasting the dependent variable within the acceptable rage of accuracies. The formula of the metrics is shown in Table 2.
2.4 Selection models
2.4.1 Phase 1all possible model
$N=\sum_{j=1}^{k} j\left(C_{j}^{k}\right)$
$\# \# 2$ nd Order
where, $N$ is the number of all Possible models, $k$ is total number of independent variables and $j=1,2, \ldots, k$.
A dataset containing 1924 observations will use to study the effect of 29 different independent variables on the one dependent variable. Then the data will be interacted with in the second interaction. The data contain the effect of 435 different interaction independent variables on the one dependent variable. For more detail, the second order inteactions as depicted in Appendix A.
2.4.2 Phase 2selected models
In this paper, we will analyze regression model amongst Lasso, Ridge, and Elastic Net. From these regression model, we will take variable importance. We will take subset of top highest 30 influential variables from each technique and will apply three robust regression algorithms (BiSquare Tukey, Hampel, and Huber).
2.4.3 Phase 3the best model
Regularization such as Lasso, Ridge, and Elastic Net, serve as variable important to take top highest 30 influential variables. After the subset of the influential variable will be followed by robust regression BiSquare, Hampel, and Hubber to determine a regression model.
The next step was to get the best model after a list of selected models was obtained. 8SC were defined for this purpose by Zainodin et al. [34]. 8SC formula can be displayed as shown in Table 3. By using mentioned formulas in Table 3, Akaike information criterion (AIC), RICE, Final Prediction Error (FPE), SCHWARZ (SBC), Generalized Cross Validation (GCV), Sigma square (SGMASQ), SHIBATA, and HannanQuinn (HQ) information on the basis of minimum value obtained from all mentioned criteria.
Table 3. Formula used for 8SC
No 
Methods 
Formulation 
Reference 
1. 
AIC 
$\left(\frac{s s e}{n}\right) e^{\frac{2(k+1)}{n}}$ 
[51] 
2. 
RICE 
$\frac{\left(\frac{s s e}{n}\right)}{\left[1\left(\frac{2(k+1)}{n}\right)\right]} $ 
[52] 
3. 
Final prediction error (FPE) 
$\left(\frac{(S S E)^{2}}{n}\right) \frac{n+(k+1)}{n(k+1)} $ 
[53] 
4. 
Schwarz 
$\left(\frac{s s e}{n}\right) n\left(\frac{k+1}{n}\right)$ 
[54] 
5. 
GCV 
$\frac{\left(\frac{s s e}{n}\right)}{\left[1\left(\frac{k+1}{n}\right)\right]^{2}}$ 
[55] 
6. 
SGMASQ 
$\frac{\left(\frac{s s e}{n}\right)}{\left[1\left(\frac{k+1}{n}\right)\right]}$ 
[55] 
7. 
SHIBATA 
$\left(\frac{s s e}{n}\right)\left(\frac{n+2(k+1)}{n}\right)$ 
[56] 
8. 
HQ 
$\left(\frac{s s e}{n}\right) \ln n^{\frac{2(k+1)}{n}}$ 
[57] 
where, n is total number of observations, k+1 is estimated parameters numbers (including constant), and SSE is sum of square error.
2.4.4 Phase 4goodness fit
The goodness of fit test was performed on the final models selected in phase 3 to check the efficiency of the selected model. Residual data would be gathered by taking into account the difference in real and expected value for the best model in Phase 3 used MAE, RMSE, and MAPE.
3.1 Data
The data was collected from time period of 8.00 am until 5.00 pm starting on 08/04/2017 to 12/04/2017. That is almost four days data. The original data was for each second and then it was converted in hour for data analysis. The variables taken are data contain hourly solar radiation, temperature, humidity, and moisture content. The detailed factor of modelling is shown in Table 4.
In this paper, a dataset containing 1924 observations will be used to study the effect of more 29 different independent variables on the one dependent variable. Significance of interaction terms had also been observed in this study. So, T1*T2 represents the interaction between T1 and T2. Another example H1*PY represents the interaction between H1 and PY. The data contain the effect of 435 different interaction independent variables on the one dependent variable. The more detailed tables for each variable interaction are attached in the Appendix A.
We require to select an α value between 0<α<1. Ridge and Lasso regression could be stated with α. Both α=0 and α=1 are corresponding to ridge and lasso respectively. The elastic net regression evolves combining between Lasso and Ridge.
The elastic net regression, this method evolves in the case of combining Lasso and Ridge. We require to choose an α value between $0<\alpha<1$ because the elastic net regression formulation is $\lambda \sum_{j=1}^{k}\left((1\alpha) \beta_{j}^{2}+\alpha\left\beta_{j}\right\right) \leq d^{2}, \alpha \in[0,1]$.
Table 4. Factors of modelling
Symbols 
Factors 
Definitions 
Y 
Dependent 
Moisture 
H1 
Independent 
Relative Humidity Ambient 
H5 
Independent 
Relative Humidity Chamber 
PY 
Independent 
Solar Radiation 
T1 
Independent 
Temperature (℃) ambient 
T2, T3, T4 
Independent 
Temperature (℃) before enter solar collector 
T5 
Independent 
Temperature (℃)in front of down vGroove (Solar Collector) 
T6, T8 
Independent 
Temperature (℃) in front of up vGroove (Solar Collector) 
T7, T14, T15, T16, T21, T22 
Independent 
Temperature (℃) Solar Collector 
T8, T9, T10, T11, T12 
Independent 
Temperature (℃) behind inside chamber 
T13, T17, T18, T19, T23 
Independent 
Temperature (℃) Infront of (Inside Chamber) 
T20, T23, T24, T25, T28 
Independent 
Temperature (℃) from solar collector to chamber 
Table 5. Results alpha criteria for elastic net
No 
alpha 
Mean Square ErrorMinimum 
Lambda Minimum 
1 
0 
47.2 
0.925 
2 
0.1 
27.2 
0.00925 
3 
0.2 
26.5 
0.00462 
4 
0.3 
25.8 
0.00308 
5 
0.4 
25.5 
0.00231 
6 
0.5 
24.9 
0.00185 
7 
0.6 
25.2 
0.00154 
8 
0.7 
24.8 
0.00132 
9 
0.8 
24.4 
0.00116 
10 
0.9 
24.0 
0.00103 
11 
1 
24.2 
0.000925 
Table 5 shows the estimates on the various value alpha to choose the minimum. The minimum value alpha obtained in step 9 and 10. We select alpha between 0.8 and 0.9 to 0.85 (interpolation) and lambda values between 0.00116 and 0.00103 to 0.001095. It shows that the model that minimized Mean Square Error (MSE) used an alpha of 0.85 and lambda of 0.001095 with the minimum MSE. From Table 5, we will convert to Figure 1. In order to, the selection of alpha value shows the minimized of MSE.
The Figure 1 shows mean square error (MSE) is widely used in model evaluations. Variations of MSE with alpha are portrayed. The Figure 1 depicts alpha (0.85) for minimum MSE (24.1).
Table 6 shows that the values of α=0, α=0.85, α=1 are corresponding to ridge, elastic net, and lasso respectively. The elastic net is a method of regularization regression that provides between ridge and lasso.
In this study, different methods for variable selections are ridge, elastic net and lasso which were performed. The variable selection is the most significant variables with their perspective. Variable selections only provide the rank of highest important variables, which means that techniques didn’t have no rules in selecting the suitable range of variable important [58]. Hence, we choose the 30 highest variable importance. The variable important is shown in Table 7.
Figure 1. Minimized MSE for elastic net regression
Table 6. Choosing values alpha (α)
No 
alpha 
Methods 
1 
0 
Ridge 
2 
0.85 
Elastic Net 
3 
1 
Lasso 
Table 7. The 30 highest of variable importance
No 
Methods 
Variable Importance 
1 
Lasso 
T14, T25, T22, T1*T23, T16, H1, T1*T6, PY, T9*H5, T1*T7, T4*T23, T7*T13, T7*T10, T7*T8, T2*T7, T12*H5, T29, T8, H5, T6, T3, T1, T4, T13, T2, T7, T23 
2 
Elastic Net 
T14, T22, T25, T1*T23, H1, T1*T6, T9*H5, T17*H5, T7*T13, T2*T23, T4*T23, T1*T7, T1*T2, T6*T8, T7*T8, T2*T7, T3*T17, T8*T19, T12*H5, T8, H5, T6, T3, T4, T1, T13, T2, T7, and T23 
3 
Ridge 
T1, T9, T6, T2, T17, T5, T23, T22, T14, T21, T28, T27, T11, T3, T1*T6, T1*T2, T7*T9, T1*T9, T26, T8, T19, H5, T15, T16, T10, T13, T4, T29, T12, and T7 
Table 8. Results for the validation methods
ML 
Robust Regression 
MAE 
MSE 
MAPE 
Sum Square of Error 
Rsquare 
Rsquare Adjusted 
Ridge 
BiSquare 
5.50670320 
87.6181 
10.7123 
167701.12 
0.6797253 
0.674623 
Hampel 
5.46121692 
50.451657 
10.41787 
96564.47 
0.8155816 
0.812643 

Huber 
5.42828203 
50.754504 
10.29807 
97144.12 
0.8144746 
0.811519 

Elastic Net 
BiSquare 
5.87494596 
127.19119 
9.188008 
243443.92 
0.5350719 
0.527665 
Hampel 
5.494431217 
48.762572 
10.26748 
93331.56 
0.8217558 
0.818916 

Huber 
5.41403189 
49.728886 
9.966258 
95181.09 
0.8182236 
0.815328 

Lasso 
BiSquare 
5.518140568 
83.476643 
9.124401 
159774.3 
0.6948638 
0.690002 
Hampel 
5.473598081 
48.411285 
9.17489 
92659.21 
0.8230399 
0.820221 

Huber 
5.395837661 
49.351041 
9.864292 
94457.89 
0.8196048 
0.816731 
It shows subset of 30 variable important that are taken by each technique. Three regression algorithms are applied for this purpose. i.e., Ridge, Elastic Net, and Lasso. They show the final result that was obtained by each variable important ranking technique. All the variable importance was ranked according to their importance score computed by their respective techniques. The more detailed tables for the highest 30 important variables are attached in the Appendix B. In Figure B1 the 30 highest important variables for Ridge Regression, while in Figure B2 the 30 highest important variables for Elastic Net Regression, and Figure B3 the 30 highest important variables for Lasso Regression, respectively.
In order to measure the prediction accuracy, predicted responses with the actual responses are compared of each regressionbased model in terms of the validation methods described in Table 8.
Predefined performances measures for Ridge, Elastic Net, and Lasso sets of data are given in Table 8. All performance measures (MAE, MSE, MAPE, Sum Square of Error, Rsquare, and Rsquare Adjusted) indicate that significantly better results were obtained by LassoHampel in comparison to others. Considering Mean Absolute Error (MAE) values for LassoHampel (5.473598081), MSE (48.411285), MAPE (9.17489), Sum Square of Error (92659.21), Rsquare (0.8230399), and Rsquare Adjusted (0.820221), respectively.
As can be seen in Table 8, In the context of validation (MAE, MSE, MAPE, and Sum Square of Error), the LassoHamper also exhibited the lowest error data which provides the most relevant data of the result. It can be assumed that the method of LassoHamper is able to be relied on investigation of the accuracy in big data obtained from regularization and robust regression.
According to Figure 2, the forecast generated by individual models show that LassoHampel method leads to more accurate forecasts than the other models, because the forecasts by LassoHampel method follow the pattern of actual data better than the other forecast by models used in this study. Based on the accuracy of the MAE, MSE, MAPE, Sum Square of Error, Rsquare, and Rsquare Adjusted, the obtained result is proved by Figure 2.
Figure 2. Accuracy measure for three regularization regressions and three robust regressions
Table 9. Results for 8 Selection Criteria for Ridge, Elastic Net, and Lasso
ML 
Robust regression 
AIC 
GCV 
HQ 
RICE 
SCHWARZ 
SGMASQ 
SHIBATA 
FPE 
RIDGE 
BiSquare 
90.5028 
90.5268 
93.55051 
90.55134 
99.02611 
89.06059 
90.45633 
90.50305 
Hampel 
52.11269 
52.12651 
53.8676 
52.14064 
57.02053 
51.28225 
52.08593 
52.11284 

Huber 
52.42551 
52.43941 
54.19095 
52.45363 
57.36281 
51.59008 
52.39859 
52.42566 

ELASTIC NET 
BiSquare 
131.3787 
131.4136 
135.8029 
131.4492 
143.7517 
129.2852 
131.3113 
131.3791 
Hampel 
50.36799 
50.38135 
52.06415 
50.39501 
55.11152 
49.56535 
50.34213 
50.36814 

Huber 
51.36612 
51.3797 
53.09589 
51.39368 
56.20365 
50.54758 
51.33975 
51.36627 

LASSO 
BiSquare 
86.22497 
86.24784 
89.12862 
86.27122 
94.34541 
84.85093 
86.18069 
86.22521 
Hampel 
50.00514 
50.01841 
51.68908 
50.03197 
54.7145 
49.20828 
49.97947 
50.00528 

Huber 
50.97584 
50.98936 
52.69246 
51.00318 
55.77661 
50.16351 
50.94966 
50.97598 
All possible models have 9 models such as Regularization (Ridge, Lasso, and Elastic Net) and Robust Regression (TukeyBi Square, Hampel, and Hubber). The minimum value for 8SC were found for model LassoHampel meaning that subset the highest 30 variable important from Lasso and continue with Hampel Regression. The results obtained from 8SC are observed in Table 9.
The results show that LassoHampel model provides the best model as compared to other existing methods used in this study. The selection of efficient model needs to deal with all possible models with the second interaction terms. The proposed hybrid (LassoHampel) model is found to be better in terms of MAE, MSE, and MAPE value in comparison to other existing methods. Therefore, the proposed hybrid model LassoHampel can therefore be used for the efficient selection of the model including the interaction terms in it. For future work, each of 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100 highest variable important were selected.
The author would like to thank the University of Sultan Ageng Tirtayasa Banten Indonesia and University Sains Malaysia for the support in this research project.
Appendix A
data.frame(T1, T2, T3, T4, T5, T6, T7, T8, T9, T10, T11, T12, T13, T14, T15, T16, T17, T19, T21, T22, T23, T25, H5, PY, T1*T2, T1*T3, T1*T4, T1*T5, T1*T6, T1*T7, T1*T8, T1*T9, T1*T10, T1*T11, T1*T12, T1*T13, T1*T14, T1*T15, T26, T27, T28, T29, H1, T1*T16, T1*T17, T1*T19, T1*T21, T1*T22, T1*T23, T1*T25, T1*T26, T1*T27, T1*T28, T1*T29, T2*T3, T2*T4, T2*T5, T2*T6, T2*T7, T2*T8, T2*T9, T2*T10, T2*T11, T2*T12, T2*T13, T2*T14, T2*T15, T2*T16, T2*T17, T3*T5, T3*T6, T1*H1, T1*H5, T1*PY, T2*T19, T2*T21, T2*T22, T2*T23, T2*T25, T2*T26, T2*T27, T2*T28, T2*T29, T2*H1, T2*H5, T2*PY, T3*T4, T3*T7, T3*T8, T3*T9, T3*T10, T3*T11, T3*T12, T3*T13, T3*T14, T3*T15, T3*T16, T3*T17, T3*T19, T3*T21, T3*T22, T3*T23, T3*T25, T3*T26, T3*T27, T3*T28, T3*T29, T3*H1, T3*H5, T3*PY, T4*T5, T4*T6, T4*T7, T4*T8, T4*T9, T4*T10, T4*T11, T4*T12, T4*T13, T4*T14, T4*T15, T4*T16, T4*T17, T4*T19, T4*T21, T4*T22, T4*T23, T4*T25, T4*T26, T4*T27, T4*T28, T4*T29, T4*H1, T4*H5, T4*PY, T5*T6, T5*T7, T5*T8, T5*T9, T5*T10, T5*T11, T5*T12, T5*T13, T5*T14, T5*T15, T5*T16, T5*T17, T5*T19, T5*T21, T5*T22, T5*T23, T5*T25, T5*T26, T5*T27, T5*T28, T5*T29, T5*H1, T5*H5, T5*PY, T6*T7, T6*T8, T6*T9, T6*T10, T6*T11, T6*T12, T6*T13, T6*T14, T6*T15, T6*T16, T6*T17, T6*T19, T6*T21, T6*T22, T6*T23, T6*T25, T6*T26, T6*T27, T6*T28, T6*T29, T6*H1, T6*H5, T6*PY, T7*T8, T7*T9, T7*T10, T7*T11, T7*T12, T7*T13, T7*T14, T7*T15, T7*T16, T7*T17, T7*T19, T7*T21, T7*T22, T7*T23, T7*T25, T7*T26, T7*T27, T7*T28, T7*T29, T7*H1, T7*H5, T7*PY, T8*T9, T8*T10, T8*T11, T8*T12, T8*T13, T8*T14, T8*T15, T8*T16, T8*T17, T8*T19, T8*T21, T8*T22, T8*T23, T8*T25, T8*T26, T8*T27, T8*T28, T8*T29, T8*H1, T8*H5, T8*PY, T9*T10, T9*T11, T9*T12, T9*T13, T9*T14, T9*T15, T9*T16, T9*T17, T9*T19, T9*T21, T9*T22, T9*T23, T9*T25, T9*T26, T9*T27, T9*T28, T9*T29, T9*H1, T9*H5, T9*PY, T10*T11, T10*T12, T10*T13, T10*T14, T10*T15, T10*T16, T10*T17, T10*T19, T10*T21, T10*T22, T10*T23, T10*T25, T10*T2, T10*T27, T10*T28, T10*T29, T10*H1, T10*H5, T10*PY, T11*T12, T11*T13, T11*T14, T11*T15, T11*T16, T11*T17, T11*T19, T11*T21, T11*T22, T11*T23, T11*T25, T11*T26, T11*T27, T11*T28, T11*T29, T11*H1, T11*H5, T11*PY, T12*T13, T12*T14, T12*T15, T12*T16, T12*T17, T12*T19, T12*T21, T12*T22, T12*T23, T12*T25, T12*T26, T12*T27, T12*T28, T12*T29, T12*H1, T12*H5, T12*PY, T13*T14, T13*T15, T13*T16, T13*T17, T13*T19, T13*T21, T13*T22, T13*T23, T13*T25, T13*T26, T13*T27, T13*T28, T13*T29, T13*H1, T13*H5, T13*PY, T14*T15, T14*T16, T14*T17, T14*T19, T14*T21, T14*T22, T14*T23, T14*T25, T14*T26, T14*T27, T14*T28, T14*T29, T14*H1, T14*H5, T14*PY, T15*T16, T15*T17, T15*T19, T15*T21, T15*T22, T15*T23, T15*T25, T15*T26, T15*T27, T15*T28, T15*T29, T15*H1, T15*H5, T15*PY, T16*T17, T16*T19, T16*T21, T16*T22, T16*T23, T16*T25, T16*T26, T16*T27, T16*T28, T16*T29, T16*H1, T16*H5, T16*PY, T17*T19, T17*T21, T17*T22, T17*T23, T17*T25, T17*T26, T17*T27, T17*T28, T17*T29, T17*H1, T17*H5, T17*PY, T19*T21, T19*T22, T19*T23, T19*T25, T19*T26, T19*T27, T19*T28, T19*T29, T19*H1, T19*H5, T19*PY, T21*T22, T21*T23, T21*T25, T21*T26, T21*T27, T21*T28, T21*T29, T21*H1, T21*H5, T21*PY, T22*T23, T22*T25, T22*T26, T22*T27, T22*T28, T22*T29, T22*H1, T22*H5, T22*PY, T23*T25, T23*T26, T23*T27, T23*T28, T23*T29, T23*H1, T23*H5, T23*PY, T25*T26, T25*T27, T25*T28, T25*T29, T25*H1, T25*H5, T25*PY, T26*T27, T26*T28, T26*T29, T26*H1, T26*H5, T26*PY, T27*T28, T27*T29, T27*H1, T27*H5, T27*PY, T28*T29, T28*H1, T28*H5, T28*PY, T29*H1, T29*H5, T29*PY, H1*H5, H1*PY, H5*PY)
Appendix B
Figure B1. The 30 highest important variables for Ridge Regression
Figure B2. The 30 highest important variables for Elastic Net Regression
Figure B3. The 30 highest important variables for Lasso Regression
[1] EmmertStreib, F., Dehmer, M. (2019). Highdimensional LASSObased computational regression models: regularization, shrinkage, and selection. Machine Learning and Knowledge Extraction, 1(1): 359383. https://doi.org/10.3390/make1010021
[2] van der Kooij, A.J., Meulman, J.J., Heiser, W.J. (2006). Local minima in categorical multiple regression. Computational Statistics & Data Analysis, 50(2): 446462. https://doi.org/10.1016/j.csda.2004.08.009
[3] M Ali, M.K., Ruslan, M.H., Muthuvalu, M.S., Wong, J., Sulaiman, J., Yasir, S.M. (2014). Mathematical modelling for the drying method and smoothing drying rate using cubic spline for seaweed Kappaphycus Striatum variety Durian in a solar dryer. AIP Conference Proceedings, 1602(1): 113120. https://doi.org/10.1063/1.4882475
[4] Nurhanna, A., Othman, M. (2017). Multiclass support vector machine application in the field of agriculture and poultry: A review. Malaysian Journal of Mathematical Sciences, 11: 3552.
[5] Ali, M.K.M., Fudholi, A., Muthuvalu, M., Sulaiman, J., Yasir, S.M. (2017). Implications of drying temperature and humidity on the drying kinetics of seaweed. AIP Conference Proceedings, 1905(1): 050004. https://doi.org/10.1063/1.5012223
[6] Yahya, W.B., Olaifa, J.B. (2014). A note on ridge regression modeling techniques. Electronic Journal of Applied Statistical Analysis, 7(2): 343361. https://doi.org/10.1285/i20705948v7n2p343
[7] Dorugade, A., Kashid, D. (2010). Alternative method for choosing ridge parameter for regression. Applied Mathematical Sciences, 4(9): 447456. https://doi.org/10.1016/j.jaubas.2013.03.005
[8] Kibria, B., Lukman, A.F. (2020). A new ridgetype estimator for the linear regression model: Simulations and applications. Scientifica, 2021: 9758378. https://doi.org/10.1155/2020/9758378
[9] Tinungki, G. (2019). Orthogonal iteration process of determining K value on estimator of Jackknife ridge regression parameter. Journal of Physics: Conference Series, 1341(9): 092001. https://doi.org/10.1088/17426596/1341/9/092001
[10] Duzan, H., Shariff, N.S.B.M. (2015). Ridge regression for solving the multicollinearity problem: Review of methods and models. Journal of Applied Sciences, 15(3): 392404. https://doi.org/10.3923/jas.2015.392.404
[11] de Vlaming, R., Groenen, P.J. (2015). The current and future use of ridge regression for prediction in quantitative genetics. BioMed Research International, 2015: 143712. https://doi.org/10.1155/2015/143712
[12] Dorugade, A.V. (2014). New ridge parameters for ridge regression. Journal of the Association of Arab Universities for Basic and Applied Sciences, 15: 9499. https://doi.org/10.1016/j.jaubas.2013.03.005
[13] McDonald, G.C. (2009). Ridge regression. Wiley Interdisciplinary Reviews: Computational Statistics, 1(1): 93100. https://doi.org/10.1002/wics.14
[14] Chen, C., Chen, S., Chen, L., Zhu, Y. (2017). Method for solving LASSO problem based on multidimensional weight. Advances in Artificial Intelligence, 2017: 1736389. https://doi.org/10.1155/2017/1736389
[15] Javaid, A., Ismail, M., Ali, M.K.M. (2020). Efficient model selection of collector efficiency in solar dryer using hybrid of LASSO and robust regression. Pertanika Journal of Science & Technology, 28(1): 193210.
[16] Zou, H., Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(2): 301320. https://doi.org/10.1111/j.14679868.2005.00503.x
[17] Algamal, Z.Y., Lee, M.H. (2015). Regularized logistic regression with adjusted adaptive elastic net for gene selection in high dimensional cancer classification. Computers in Biology and Medicine, 67: 136145. https://doi.org/10.1016/j.compbiomed.2015.10.008
[18] Ahrens, A., Hansen, C.B., Schaffer, M.E. (2020). Lassopack: Model selection and prediction with regularized regression in Stata. The Stata Journal, 20(1): 176235. https://doi.org/10.1177/1536867X20909697
[19] Guyon, I., Elisseeff, A. (2003). An introduction to variable and feature selection. Journal of Machine Learning Research, 3: 11571182. https://doi.org/10.1162/153244303322753616
[20] Cebeci, Z., Yildiz, F. (2015). Comparison of kmeans and fuzzy cmeans algorithms on different cluster structures. Agrárinformatika/Journal of Agricultural Informatics, 6(3): 1323. https://doi.org/10.17700/jai.2015.6.3.196
[21] Christensen, A., Srinivasan, V., Hart, J.C., MarshallColon, A. (2018). Use of computational modeling combined with advanced visualization to develop strategies for the design of crop ideotypes to address food security. Nutrition Reviews, 76(5): 332347. https://doi.org/10.1093/nutrit/nux076
[22] Tamura, R., Kobayashi, K., Takano, Y., Miyashiro, R., Nakata, K., Matsui, T. (2017). Best subset selection for eliminating multicollinearity. Journal of the Operations Research Society of Japan, 60(3): 321336. https://doi.org/10.15807/jorsj.60.321
[23] Cousineau, D., Chartier, S. (2010). Outliers detection and treatment: A review. International Journal of Psychological Research, 3(1): 5867. https://doi.org/10.21500/20112084.844
[24] Begashaw, G.B., Yohannes, Y.B. (2020). Review of outlier detection and identifying using robust regression model. International Journal of Systems Science and Applied Mathematics, 5(1): 411. https://doi.org/10.11648/j.ijssam.20200501.12
[25] Salgado, C.M., Azevedo, C., Proença, H., Vieira, S.M. (2016). Noise versus outliers. Secondary Analysis of Electronic Health Records, pp. 163183. https://doi.org/10.1007/9783319437422_14
[26] Alma, Ö.G. (2011). Comparison of robust regression methods in linear regression. Int. J. Contemp. Math. Sciences, 6(9): 409421.
[27] Fox, J., Weisberg, S. (2018). Visualizing fit and lack of fit in complex regression models with predictor effect plots and partial residuals. Journal of Statistical Software, 87(1): 127. https://doi.org/10.18637/jss.v087.i09
[28] Nahar, J., Purwani, S. (2017). Application of robust Mestimator regression in handling data outliers. 4th ICRIEMS Proceedings, pp. 5360.
[29] Riani, M., Cerioli, A., Atkinson, A.C., Perrotta, D. (2014). Monitoring robust regression. Electronic Journal of Statistics, 8(1): 646677. https://doi.org/10.1214/14EJS897
[30] Yu, C., Yao, W. (2017). Robust linear regression: A review and comparison. Communications in StatisticsSimulation and Computation, 46(8): 62616282. https://doi.org/10.1080/03610918.2016.1202271
[31] Muthukrishnan, R., Radha, M. (2010). Mestimators in regression models. Journal of Mathematics Research, 2(4): 2327. https://doi.org/10.5539/jmr.v2n4p23
[32] Alamin, M., Xu, H., Mollah, M., Haque, N. (2020). Robustification of linear regression and its application in genomewide association studies. Frontiers in Genetics, 11: 549. https://doi.org/10.3389/fgene.2020.00549
[33] Abdullah, N., Jubok, Z.H., Ahmed, A. (2011). Improved stem volume estimation using pvalue approach in polynomial regression models. Research Journal of Forestry, 5(2): 5065. https://doi.org/10.3923/rjf.2011.50.65
[34] Zainodin, H., Noraini, A., Yap, S. (2011). An alternative multicollinearity approach in solving multiple regression problem. Trends in Applied Sciences Research, 6(11): 12411255. https://doi.org/10.3923/tasr.2011.1241.1255
[35] Zainodin, H., Khuneswari, G. (2010). Modelbuilding approach in multiple binary logit model for coronary heart disease. Malaysian Journal of Mathematical Sciences, 4(1): 107133.
[36] Ali, M.K.M., Fudholi, A., Muthuvalu, M.S., Sulaiman, J., Yasir, S.M. (2017). Implications of drying temperature and humidity on the drying kinetics of seaweed. In AIP Conference Proceedings, 1905(1): 050004. https://doi.org/10.1063/1.5012223
[37] Sulaiman, J., Ali, M., Tuah, P., Yasir, S., Lee, W. (2017). Productivity cost model in 308 ross chicken poultry systems: Case study of contract farming in rural development cooperative. Malaysian Journal of Mathematical Sciences, 11: 1733.
[38] Lim, H.Y., Fam, P.S., Javaid, A., Ali, M., Khan, M. (2020). Ridge regression as efficient model selection and forecasting of fish drying using vgroove hybrid solar drier. Pertanika Journal of Science & Technology, 28(4): 11791202. https://doi.org/10.47836/pjst.28.4.04
[39] Franklin, J. (2005). The elements of statistical learning: data mining, inference and prediction. The Mathematical Intelligencer, 27(2): 8385. https://doi.org/10.1007/BF02985802
[40] Griva, I., Nash, S.G., Sofer, A. (2009). Linear and nonlinear optimization (Vol. 108). Siam.
[41] Zou. H., Hastie. T., Tibshirani, R. (2006). Journal of Computational and Graphical Statistics. 15(2): 265285. https://doi.org/10.1198/106186006X113430
[42] Hans, C. (2011). Elastic net regression modeling with the orthant normal prior. Journal of the American Statistical Association, 106(496): 13831393. https://doi.org/10.1198/jasa.2011.tm09241
[43] De Mol, C., De Vito, E., Rosasco, L. (2009). Elasticnet regularization in learning theory. Journal of Complexity, 25(2): 201230. https://doi.org/10.1016/j.jco.2009.01.002
[44] Nadia, H., Mohammad, A.A. (2013). Model of robust regression with parametric and nonparametric methods. Mathematical Theory and Modeling, 3(5): 2739.
[45] Ding, J. (2015). An evaluation of some robust estimators of regression coefficients. Faculty of Graduate Studies and Research, University of Regina.
[46] Fox, J., Weisberg, S. (2018). An R Companion to Applied Regression. Sage Publications.
[47] Chai, T., Draxler, R.R. (2014). Root mean square error (RMSE) or mean absolute error (MAE)? – Arguments against avoiding RMSE in the literature. Geoscientific Model Development, 7(3): 12471250. https://doi.org/10.5194/gmd712472014
[48] Rougier, J. (2016). Ensemble averaging and mean squared error. Journal of Climate, 29(24): 88658870. https://doi.org/10.1175/JCLID160012.1
[49] Kim, S., Kim, H. (2016). A new metric of absolute percentage error for intermittent demand forecasts. International Journal of Forecasting, 32(3): 669679. http://dx.doi.org/10.1016/j.ijforecast.2015.12.003
[50] Mo, Z. (2014). An empirical evaluation of OLS hedonic pricing regression on Singapore private housing market. Master thesis, Department of Real Estate and Construction management. Centre of Finance and Banking. https://doi.org/10.13140/RG.2.2.24071.24484
[51] Akaike, H. (1969). Fitting autoregressive models for prediction. Annals of the Institute of Statistical Mathematics, 21(1): 243247. https://doi.org/10.1007/BF02532251
[52] Rice, J. (1984). Bandwidth choice for nonparametric regression. Annals of Statistics, 12(4): 12151230. https://doi.org/10.1214/aos/1176346788
[53] Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6): 716723. https://doi.org/10.1109/TAC.1974.1100705
[54] Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6(2): 461464. https://doi.org/10.1214/aos/1176344136
[55] Golub, G.H., Heath, M., Wahba, G. (1979). Generalized crossvalidation as a method for choosing a good ridge parameter. Technometrics, 21(2): 215223. https://doi.org/10.2307/1268518
[56] Shibata, R. (1981). An optimal selection of regression variables. Biometrika, 68(1): 4554. https://doi.org/10.1093/biomet/68.1.45
[57] Hannan, E.J., Quinn, B.G. (1979). The determination of the order of an autoregression. Journal of the Royal Statistical Society: Series B(Methodological): 41(2): 190195. https://doi.org/10.1111/j.25176161.1979.tb01072.x
[58] GómezVerdejo, V., ParradoHernández, E., Tohka, J. (2019). Signconsistency based variable importance for machine learning in brain imaging. Neuroinformatics, 17(4): 593609. https://doi.org/10.1007/s1202101994153