Bienvenue Kouwayè
email: kouwaye2000@yahoo.fr

Regression Trees and Random forest based feature selection for malaria risk exposure prediction

Keywords: Regression trees, random forest, cross-validation, variables selection, prediction

This paper deals with prediction of anopheles number, the main vector of malaria risk, using environmental and climate variables. The variables selection is based on an automatic machine learning method using regression trees, and random forests combined with stratified two levels cross validation. The minimum threshold of variables importance is accessed using the quadratic distance of variables importance while the optimal subset of selected variables is used to perform predictions. Finally the results revealed to be qualitatively better, at the selection, the prediction, and the CPU time point of view than those obtained by GLM-Lasso method.

Introduction

Generally, studies about disease like chikungunya, aids, and malaria provide data set containing a high number of variables and a small number of observations. When it is important to perform prediction on the risk of these diseases, the goal is to provide a consistent heuristic to select the probable candidate predictor and perform prediction for the study and also where only explanatory data are available. Generally, experts in medicine, epidemiology, genetic, perform treatment on variables before analysis, operations of selection, and forecast. Based on their knowledge, they decide to transform some variables in classes, to fix interactions between some variables, etc. In epidemiology context, the aim of this work is to provide an automatic algorithm for variables selection based on regression trees and random forest. This procedure must overcome the treatment done by experts, generate automatically a stable, and optimal subset of predictors, and perform prediction for an other area where the target variable is not available. There are a lot of statistical modeling approach, such as linear model (LM), linear mixed model (LMM), generalized linear (GLM), generalized linear mixed (GLMM) for selection or prediction. However, these models fail when p > n, the number of variables (p) is more important than the number of observations (n). Experts also assume independence among explanatory variables. A lot of methods of variables selection provide a subset for prediction but not stable, not consistent or more demanding in computation time like Wrapper, embedded, filter, ranking, and their variant [START_REF] Guyon | An introduction to variable and feature selection[END_REF][START_REF] Bontempi | Structural feature selection for wrapper methods[END_REF][START_REF] Ramn | Gene selection and classification of microarray data using random forest[END_REF]. In recent works, we proposed one method of variables selection based on combination of Lasso and GLM through a double cross-validation (LOLO-DCV) named GLM-Lasso [START_REF] Kouwaye | Lasso based feature selection for malaria risk exposure prediction[END_REF][START_REF] Kouwaye | Sélection de variables par le glm-lasso pour la prédiction du risque palustre[END_REF]. The present work combines the stratified double cross validation (LOLO-DCV), and regression trees or random forest. This implies two methods : LOLO-DCV combined with regression trees, and LOLO-DCV combined with random forest. For malaria risk prediction, four strategies of variables selection LDRT, LDCT, LDRF, and LDCF are implemented. These strategies use some criteria such as : the mean, the quadratic risk, the absolute risk of the predictions, and the CPU time of algorithm computation. Each strategy is applied on four groups of variables (original, original with village, recoded, recoded with village). Most of the algorithms implemented in our work are based on [START_REF] Kouwaye | Lasso based feature selection for malaria risk exposure prediction[END_REF][START_REF] Kouwaye | Sélection de variables par le glm-lasso pour la prédiction du risque palustre[END_REF][START_REF] Friedman | Lasso and elastic-net regularized generalized linear models[END_REF][START_REF] Goeman | L1 penalized estimation in the cox proportional hazards model[END_REF][START_REF] Zou | Regularization and variable selection via the elastic net[END_REF]. We first provided a threshold of variable importance measurement for each strategy. This threshold is very important because it puts out the importance or not of predictors. The second step is to make prediction through a double cross validation loop. The last step is to select predictors according to their frequency and make consistent prediction with the remain predictors. The results are compared to those obtained by reference method. The results obtained by such procedure are clearly better improved compared to those obtained by GLM-Lasso [START_REF] Kouwaye | Lasso based feature selection for malaria risk exposure prediction[END_REF][START_REF] Kouwaye | Sélection de variables par le glm-lasso pour la prédiction du risque palustre[END_REF] taken as the reference method. The improvement is about all properties such as the selection power, the selection accuracy, the sparsity of the best subset of variables, and the prediction. Moreover, the CPU time used to display our program is smaller than the one required by the reference method and only few climate and environmental variables are the main factors associated to the malaria risk exposure with an improved accuracy.

Materials

In this section, we briefly recall the description of the study area, the mosquito collection and identification as well as the data, and related variables. For more details, see [START_REF] Cottrell | Modeling the Influence of Local Environmental Factors on Malaria Transmission in Benin and Its Implications for Cohort Study[END_REF].

Study area

The study was conducted in the district of Tori-Bossito (Republic of Benin), from July 2007 to July 2009. Tori-Bossito is on the coastal plain of Southern Benin, 40 kilometers north-east of Cotonou. This area has a subtropical climate and during the study, the rainy season lasted from May to October. Average monthly temperatures varied between 27 • C and 31 • C. The original equatorial forest has been cleared and the vegetation is characterized by bushes with sparse trees, a few oil palm plantations, and farms. The study area contained nine villages (Avamé centre, Gbédjougo, Houngo, Anavié, Dohinoko, Gbétaga, Tori Cada Centre, Zébè, and Zoungoudo). Tori Bossito was recently classified as mesoendemic with a clinical malaria incidence of about 1.5 episodes per child per year [START_REF] Damien | Malaria infection and disease in an area with pyrethroid-resistant vectors in southern benin[END_REF]. Pyrethroid-resistant malaria vectors are present [START_REF] Damien | Malaria and infection disease in an erea with pyrethroid-resitant vectors in southern Benin[END_REF].

Mosquito collection and identification

Entomological surveys based on human landing catches (HLC) were performed in the nine villages every six weeks for two years (July 2007 to July 2009). Mosquitoes were collected at four catch houses in each village over three successive nights (four indoors and four outdoors, i.e. a total of 216 nights every six weeks in the nine villages). Five catch sites had to be changed in the course of the study (2 in Gbedjougo, 1 in Avamè, 1 in Cada, 1 in Dohinoko) and a total of 19 data collections were performed in the field from July 2007 to July 2009. In total, data from 41 catch sites are available. Each collector caught of predictional mosquitoes landing on the lower legs and feet between 10 pm and 6 am. All mosquitoes were held in bags labeled with the time of collection. The following morning, mosquitoes were identified on the basis of morphological criteria [START_REF] Gillies | The Anophelinae of Africa south of the Sahara)[END_REF][START_REF] Gillies | A supplement to the Anophelinae of Africa south of the Sahara (Afrotropical region)[END_REF]. All An. gambiae complex and An. funestus mosquitoes were stored in individual tube with silica gel and preserved at 220 • C. P. falciparum infection rates were then determined on the head and thorax of individual anopheline specimens by CSP-ELISA [START_REF] Wirtz | Comparative testing of monoclonal antibodies against Plasmodium falciparum sporozoites for ELISA development[END_REF].

Environmental and behavioral data

Rainfall was recorded twice a day with a pluviometer in each village. In and around each catch site, the following information was systematically collected:

(1) type of soil (dry lateritic or humid hydromorphic)assessed using a soil map of the area (map IGN Benin at 1/200 000 e , sheets NB-31-XIV and NB-31-XV, 1968) that was georeferenced and input into a GIS; (2) presence of areas where building constructions are ongoing with tools or holes representing potential breeding habitats for anopheles; (3) presence of abandoned objects (or ustensils) susceptible to be used as oviposition sites for female mosquitoes; (4) a watercourse nearby; (5) number of windows and doors; (6) type of roof (straw or metal); (7) number of inhabitants; (8) ownership of a bed-net or (9) insect repellent; And (10) normalized difference vegetation index (NDVI) which was estimated for 100 meters around the catch site with a SPOT 5 High Resolution (10 m colors) satellite image (Image Spot5, CNES, 2003, distribution SpotImage S.A) with assessment of the chlorophyll density of each pixel of the image. Due to logistical problems, rainfall measurements are only available after the second entomological survey. Consequently, we excluded the first and second survey (performed in July and August 2007 respectively) from the statistical analyses.

Variables

The dependent variable was the number of Anopheles collected in a house over the three nights of each catch and the explanatory variables were the environmental factors, i.e. the mean rainfall between two catches (classified according to quartile), the number of rainy days in the ten days before the catch (3 classes [01], [24], >4 days), the season during which the catch was carried out (4 classes: end of the dry season from February to April; beginning of the rainy season from May to July; end of the rainy season from August to October; beginning of the dry season from November to January), the type of soil 100 meters around the house (dry or humid), the presence of constructions within 100 meters of the house (yes/no), the presence of abandoned tools within 100 meters of the house (yes/no), the presence of a watercourse within 500 meters of the house (yes/no), NDVI 100 meters around the house (classified according to quartile), the type of roof (straw or Sheet metal), the number of windows (classified according to quartile), the ownership of bed nets (yes/no), the use of insect repellent (yes/no), and the number of inhabitants in the house (classified according to quartile). These pre-treatments based on the knowledge of experts in entomology, and medicine operated on some original variables generate a second type of variables called recoded variables. The original and recoded variables are described in Table 4. Two types of variables set are used : the first set, the original variables with all variables obtained by interactions; The second set, the recoded variables with all variables obtained by interactions. For knowing the effect of the village on the selection method and prediction, four groups of variables are considered : Group 1 (original variables), Group 2 (original variables with village as fixed effect), Group 3 (recoded variables), and Group 4 (recoded variables with village as fixed effect) 3 Methodology

Variables importance

For individual tree, the variable importance is defined used the out-of-bag sampling like :

V I T (X p ) = (err OOB p t -errOOB t ) (1)
The naive importance measure in tree-based ensemble methods is to merely count the number of times each variable is selected by all individual tree in the group of trees. There also exist the "Gini importance" measurement used in random forest for classification. The more advanced variable importance measurement in random forest is the "permutation accuracy importance" defined as:

V I RF (X p ) = 1 ntree ntree t=1 (err OOB p t -errOOB t ) (2)
where V I T is variable importance of a tree, V I RF is variable importance of a random forest, X p is the p-th variable, OOB t the out-of-bag of tree t, OOB p t the sample obtained by randomly permuting the value of X p in OOB t and errOOB t the out-of-bag error for the tree t, ntree the number of regression trees in the random forest.

The Gini importance, and the permutation accuracy importance measures are employed as variable selection criteria in many recent study in various disciplines. The effects induced by the differences in scale level of the predictors are more pronounced for the randomForest function, where variable selection in the individual tree is biased, than the one with cforest function where the individual trees are unbiased [START_REF] Strobl | Bias in random forest variable importance measures: Illustrations, sources and a solution[END_REF][START_REF] Archer | Empirical characterization of random forest variable importance measures[END_REF]. It has been also shown that if cforest function is used with bootstrap sampling, the variables selection frequencies of the categorical predictors still depend on their number of categories. Variable importance has a sensitivity to the number of observations and the number of variables. This sensitivity is reduced with increasing number of true variables. Variable importance has also sensitivity to mtry the minimum number of observations at a node for splitting, and ntree the maximal number of trees in forest. It has been shown that for a fixed number of observations and variables, the effect of taking a larger value for mtry is evident. Indeed, the magnitude of variable importance is more double starting from mtry = 14 to mtry = 100, and it again increases with mtry = 200. The effect of ntree is less visible but taking ntree = 2000 leads to better stability [START_REF] Genuer | Variable selection using random forests[END_REF] A lot of strategies have been developed for variable selection. The recursive elimination of feature based on variable importance developed by Avlarez de André runs like this. They first compute random forest variable importance. Then, at each step they eliminate the 20% of variables having the less importance and build a new forest with the remaining variables. They finally select the set of variables leading to the smallest OOB error rate of a forest defined by

errOOB = 1 n Card{i ∈ {1, . . . , n}|y i = ŷi } ( 3
)
where ŷi is the most frequent label predict by trees t for which (x i , y i ) is in the OOB t sample [START_REF] Ramn | Gene selection and classification of microarray data using random forest[END_REF]. But this proposition of variables elimination is arbitrary, and the method does not depend on the data. Robin Genuer proposed an other method of variable selection based on variable importance stratified in two steps [START_REF] Genuer | Variable selection using random forests[END_REF]. The step 1 is a preliminary ranking which consisting in sorting the variables in decreasing order of Random forest scores of importance, and canceling the variables of small importance, m is the number of remaining at the second step, he selected the variables involved in the model leading to the smallest OOB error and at the end constructed an ascending sequence of Random forest models by invoking, and testing the variables stepwise. The variables of the last model are selected. But this method lacks of precision because in the step 1 this strategy is sensible when it exist irrelevant variables, and at last step, variables invoking or testing can be sensitive to high correlation among variables. The method to access variable importance proposed by Daz-Uriarte [START_REF] Daz-Uriarte | Variable selection using random forest[END_REF] in scaled, unscaled, and Gini version is only available when the dependent variable is a factor.

Performance and accuracy in variables selection for strategies

This part of the work is based on simulated data. It is necessary to show the power of each strategy to reduce effectively the number of variables and select the right variables in the optimal subset for prediction. Let V R and V W the set of the real, and wrong variables respectively, S R , S W the set of the real, and wrong selected variables respectively. Let V = V R ∪ V W and S = S R ∪ S W .

Selection power

It is defined as the ratio of the number of variables selected on the number of total variables (real and wrong). This quantity gives an idea of percentage of elimination of variables. The selection power is noted SP and defined as :

SP = Card(S) Card(V) (4)

Selection accuracy

It is defined as the ratio of the number of real variables selected on the number of total variables selected. This quantity gives the accuracy selection of variables. It is noted SA and defined as :

SA = Card(S R ) Card(S) (5)

Strategy parameters construction

The parameters used in the strategies for variables selection are, the minimum threshold of variables importance, the minimum number of observations at each node before splitting in trees, and the maximum number of trees in forest building.

Heuristic of variable importance measurement

The strategy of variable importance measurement proposed in this paper is based on a minimum threshold. For any model of regression trees and random forest, if any variable has importance greater than this threshold, it is considered as important variable in the model. One of the difficulties in this study is the decision of the minimum of importance of variable. We have a lot of techniques to check this number. The strategy proposed by Genuer et al [START_REF] Genuer | Variable selection using random forests[END_REF] is sensible when it exist irrelevant variables. A classical alternative is to select the threshold according to some elbow finding strategy on the variable importance mean curve.

In this paper we propose a new strategy running like this : we run the full model using the whole data frame n r times (default n r =100) with the default parameters. The matrix of variables important noted M V I is a n r × q-matrix defined as :

M V I =      V I 11 V I 12 . . . V I 1nr V I 21 V I 22 . . . V I 2nr . . . . . . . . . V I q1 V I q2 . . . V I qnr      V I ij is the importance of i th variable at j th repetition, 1 ≤ i ≤ q and 1 ≤ j ≤ n r . Let M V I = (V I .1 , V I .2 , . . . , V I .nr ) if V I .i = (V I 1i , V I 2i , . . . , V I qi ) t then σ i = min{V I .i , V I .i = θ, 1 ≤ i ≤ n r } and σ = (σ 1 , σ 2 , . . . , σ nr )
where A t is the transposed of the matrix A, and θ is the null vector. The minimum threshold of variables importance noted V I min is defined as :

V I min = min(σ) + sd(σ) (6)

Parameters mtry and ntree accessing

These parameters are accessed through a simple cross validation process. The data set is divided into two parts : E A the learning set, and E T the test set. On E A , it has been performed one kind of variable selection method varying one specific parameter. For the regression tree, the parameter concerned mtry, the minimum number of observations that must exist at a node in order for a split to be attempted. For the forest, this parameter is ntree, the maximal number of trees in the forest. These parameters will be noted m for simplification of notations, 1 ≤ m ≤ n obs where n obs is the number of observations in E A . For each value of m, the corresponding regression tree or the random forest provides a vector of importance V I . m . Let V I m the vector of mean of the vectors V I . m . The quadratic distance is defined as :

d( V I m , V I . m ) =   nvar j=1 ( V I m -V I j m ) 2   1/2 , 1 ≤ m ≤ n obs (7)
where n var is the number of variables The parameter m is determined by optimizing the quadratic distance of importance. Let define :

H = Arg min m d( V I m , V I . m ) (8)
If Card(H) = 1 then H = {h 0 } and

mtry = ntree = h 0 (9)
If Card(H) ≥ 2 then mtry = min{H} and ntree = max{H} (10)

A regression tree with this value of mtry or a Random forest (RF) with this value of ntree will perform prediction on E T . All this process will be repeated until prediction is computed for all observations.

Algorithm of variables selection and prediction

This algorithm is similar to the one developed in a recent work [START_REF] Kouwaye | Sélection de variables par le glm-lasso pour la prédiction du risque palustre[END_REF][START_REF] Kouwaye | Lasso based feature selection for malaria risk exposure prediction[END_REF].

Variables selection strategies

Four strategies of variables selection are implemented and compared to the reference method GLM-Lasso developped in [START_REF] Kouwaye | Sélection de variables par le glm-lasso pour la prédiction du risque palustre[END_REF][START_REF] Kouwaye | Lasso based feature selection for malaria risk exposure prediction[END_REF]. The first strategy, LDRT is a combination of LOLO-DCV (Leave-one-out-double-cross-validation), and regression tree (RT) [START_REF] Gey | Model selection for cart regression trees, Information Theory[END_REF]20]; The second LDCT is a combination of LOLO-DCV and conditional tree (CT) [START_REF] Hothorn | A laboratory for recursive partytioning[END_REF]; The third LDRF, a combination of LOLO-DCV ii. Tree or forest model M m is computed on E A using m iii. The importance of variables V I . m . is accessed. iv. Predictions are performed using a M m model on E T 4. The step (3c) is repeated until predictions are performed for all observations, and a matrix M V I of importance is recorded.

5. The vector mean V I (representative) of M V I is determined.

6. The selection of each variable is done by V I using V I min .

and Random forest (RF) [START_REF] Archer | Empirical characterization of random forest variable importance measures[END_REF][START_REF] Biau | Analysis of random forests model[END_REF][START_REF] Liaw | Breiman and cutler's random forests for classification and regression[END_REF]; And the last LDCF is a combination of LOLO-DCV and conditional forest (CF) [START_REF] Hothorn | A laboratory for recursive partytioning[END_REF]. The selection power, the selection accuracy of each strategy is determined on simulated data. The model construction parameter for each strategy is accessed based on the quadratic distance between importance and the mean of importance. Each strategy based on the threshold of the minimum of variable importance selects an optimal subset of variables.

Results

First of all, we show the numerical convergence of V I min in Equation ( 6) for each strategy on simulated data. The results are shown in table 1 and figure 2. We also show the power of each strategy to reduce the number of variables, and select the true variables in the optimal subset for prediction. The results are presented in table 1 4

.1 Simulated study

We simulated a data base of n observations and p true variables. We generated p-explanatory variables X, and the target variable Y knowing that (Y |X) ∼ P(E(Xβ)) where β is the vector of coefficients of X, and P(E(Xβ)) is a Poisson distribution of parameter E(Xβ). We also generated another p-variables Z which don't participate to the determination of Y . One of the strength of the algorithm is its capacity to avoid in selection the wrong variables at most possible. The final number of variable in learning is 2 × p the set X of generic explanatory variables contains : Gaussian variable, X p N ∼ N (µ p , σ 2 p ); discrete variable which values are in range (1, 10), categorical variables with at most 10 modalities, and variables following Poisson distribution of parameter λ X P . For illustration, β ∼ N (0, 1), µ p ∈ {-1, 0, 1}, σ p = 1, and λ X P = 1

Application to malaria data

The results of application of strategies LDRT, LDCT, LDRF, and LDCF on malaria data are shown in tables 2, and 3. The strategy LDRT do not converge. The threshold of variable importance is null for LDCF. So any variable with non null importance will be importante. The threshold of variable importance is very high for LDCT, table 2. The strategies LDCT, LDCF, and LDRF have a mean in prediction which is equal to the mean of observations. LDRF has the low quadratic risk, abolute risk, the low computation time, and the most sparse subset of remained variables but its mean in prediction is greater than the one of observations, table 3.

Discussion

The table 1 shows that computation is not compiled for LDRT if the number of variables is greater than 40 approximatively. This is due to the non convergence of the rpart function in the package rpart for construction of regression tree. For Random forest, the convergence of the percentage of selected variables is not ensured. But the convergence is obtained when we combined LOLO-DCV Selection Power q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q LDCT LDCF LDRT LDRF

Threshold of variable importance

Number of variables

Minimum of VI q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q LDRT LDCT LDCF LDRF

Conclusion

In this work, we implemented an algorithm for the prediction of malaria risk using environmental and climate variables. We performed the variables selection using an automatic machine learning by a method combining regression trees or random forest, and stratified two levels cross validation. The minimum threshold of variable importance is computed. variables selected by each strategy are used to perform prediction. The results obtained with this method is clearly improved by those obtained with the combination of Lasso, and LOLO-DCV (GLM-Lasso) taken as reference method. The improvement concerned all properties such as the quality of the selection, and prediction. Moreover, this method didn't need interaction between variables, the pre-treatments of experts were overcome, and the CPU time used to display our program is smaller than the one required by the reference method. The optimal subset of variables for prediction contained season, mean rain fall, and vegetation index.

Apendix

Algorithme 3 . 1

 31 LOLO-DCV-Tree-Forest 1. Determination of V I min 2. The data are separated in N -folds 3. A each step of the first level (a) The folds are regrouped in two part : E A and E T , E A : the learning set which contained the observations of (N -1)-folds, E T : the test set, contained the observations of the last fold. (b) Holding-out E T (c) The second level of cross-validation i. A full cross validation is computed on E A for determination of the first model construction parameter m (mtry or ntree).

Figure 1 :

 1 Figure 1: Selection power of strategies according to number of variables. Each line shows the trajectory of the selection for each strategy on simulated data.

Figure 2 :

 2 Figure 2: Threshold of variable importance measurement. Each line shows the trajectory of the variable importance for each strategy on simulated data.

Table 1 :

 1 Summary

	on results of selection power and selection accuracy
	for different strategies. MVI= Minimum variable importance, SP= selection
	power, SA= Selection accuracy
	number of variables		250	300	350	400	500	80	600	800
	LDRT	SP	-	-	-	-	-	-	-	-
		SA	-	-	-	-	-	-	-	-
		MVI	-	-	-	-	-	-	-	-
	LDCT	SP	38.00	45	46	47.00	38.0 44.0 46.00	36.00
		SA	26.88	26.11	26.5	15.33	9.75	9	6.83	5.71
		MVI	2.00	2	2.20	2.2	2.0 2.00	2.00	2
	LDRF	SP 20.62 21.67 17.00	15.00 11.75 9.00 9.33	8.57
		SA	38.12	33.33	33	23	14.75	12	8.83	7.14
		MVI	1.00	1.0	1	1.00	1 1.00	1.00	1.0
	LDCF	SP	48 62.00	5.0	67.00	88	5 5.00	5.00
		SA	30	34.44	2.5	22.33	22	1	0.83	0.71
		MVI	0	0	0	0	0	0	0	0
	with Random Forest (LDRF). The percentage of remained variables is around
	5%. The results are also shown in figure 1. It is evident that the convergence

Table 2 :

 2 Threshold of variable importance measurement on real data.

		LDRT	LDCT LDRF LDCF
	VI min	-5.12507	1.1	0

Table 3 :

 3 Summary on results of selection power, selection accuracy, and minimum of variable importance on malaria data The figure2confirms the non convergence of LDRT over approximatively 40 variables. Even if the algorithm did not converge for all number of variable, the minimum is attempted, and the trajectory is convex. For the strategies LDCT, LDRF, and LDCF, the minimum of variable importance converges, and the results are shown in the line "MVI" of table 1, and in figure2for each strategy. This denotes that the algorithm can compute correctly the threshold of importance of variable for any number of variable but not for LDRT. The results of application about malaria data are shown in table3. The minimum of variable importance that we got for each method are noted in table 2. Unfortunately this minimum for LDCF is null. It means that for any positive value, the variable is important but four variable are selected at the end. For LDCT, the threshold of importance is very high, table 2 nevertheless four variables are remained in the final model, tabe 3. It denotes that only few variables are important in the model. The methods which have the mean in prediction equals to the mean of observations are LDCT, and LDCF. LDCF is the best in selection accuracy. LDRF has the lowest quadratic risk, the lower absolute risk but it isn't the best prediction. LDRF is the most sparse method with three variables. Unlike LDRF, LDCF, and LDCT which are more time consuming table 3.

	Method	Mean	QR Absolute risk Remain variables Time CPU
	Observations	3.74	-	-	-	-
	GLM-Lasso	3.74	54.54	3.669	3	25786.87
	LDRT	-	-	-	-	-
	LDCT	3.74	50.56	3.217	4	24378.38
	LDRF	3.75	49.56	2.876	3	6715.09
	LDCF	3.74 49.987	3.001	4	24830.34
	of LDCF is not stable because the alternative high and low percentage of re-
	mained variables.

Table 4 :

 4 Table of variables Description The authors have declared that no competing interests exist. Description of variables. Variables with star are recoded.

		Nature	Number of modalities Modalities
	Repellent	Non-numeric	2	Yes/ No
	Bed-net	Non-numeric	2	Yes/ No
	Type of roof	Non-numeric	2	Sheet metal/ Straw
	Utensils	Non-numeric	2	Yes/ No
	Presence of constructions	Non-numeric	2	Yes/ No
	Type of soil	Non-numeric	2	Humid/ Dry
	Water course	Non-numeric	2	Yes/ No
	Majority class *	Non-numeric	3	1/2/3
	Season	Non-numeric	4	1/2/3/4
	Village *	Non-numeric	9
	House *	Non-numeric	41
	Rainy days before mission * Non-numeric	3	Quartile
	Rainy days during mission	Numeric	Discrete	0/1/• • • /3
	Fragmentation index *	Non-numeric	4	Quartile
	Openings *	Non-numeric	4	Quartile
	Nber of inhabitants *	Non-numeric	3	Quartile
	Mean rainfall *	Non-numeric	4	Quartile
	Vegetation *	Non-numeric	4	Quartile
	Total Mosquitoes	Numeric	Discrete	0/• • • /481
	Total Anopheles	Numeric	Discrete	0/• • • /87
	Anopheles infected	Numeric	Discrete	0/• • • /9


Zhuang Li

professor En-Gang Wang
email: egwang@mail.neu.edu.cn

Yu Xu

Lin Xu

Fei Li

An-Yuan Deng

Study of interfacial fluctuation of molten steel and liquid slag in continuous casting mould with vertical electromagnetic brake

Keywords: vertical electromagnetic brake, VOF model, interfacial fluctuation, continuous casting mold

A 3-D mathematical model has been developed to investigate the behavior of molten steel and liquid slag interfacial fluctuation considering the effects of vertical electromagnetic brake (V-EMBr) in slab continuous casting mold. The Volume of fluid (VOF) model is used to describe the interfacial behavior of molten steel and liquid slag with different current density of coils. The results show that the V-EMBr can effectively reduce the interfacial fluctuation caused by the changing of nozzle submergence depth and port angle For a given casting speed, as increasing of magnetic induction intensity the height of interfacial fluctuation (the maximum value of wave height) is significantly reduced and its position gradually approached to the narrow face of the mold, that is beneficial to reduce the possibility of mold flux entrapment.

Introduction

In the process of continuous casting, the phenomenon of mold flux entrapment is easily to be caused with the increasing of casting speed. The developed of electromagnetic brake technique can effective control the flow of molten steel, stable the fluctuation of steel/slag interfacial to reduce the probability of mold flux entrapment and promotes the removal of inclusions, thus both the surface and internal quality of slab can be improved. However the effect of traditional electromagnetic brake is obviously influenced by process and electromagnetic parameters, such as SEN port angle, distance between SEN port and magnetic pole position (Level Magnetic Field) [1] , current density match of upper and lower magnetic pole (FC-Mold) [2] and so on. An unreasonable parameters match will lead to an unexpected brake effect for traditional electromagnetic brake.

In this study, a new type electromagnetic brake device that called vertical electromagnetic brake (V-EMBr) is proposed to control the molten steel and liquid slag interfacial fluctuation in slab continuous casting process, and the effect of different current density of coils on the behavior of molten steel and liquid slag interfacial fluctuation in V-EMBr mold is studied using a mathematical model.

Mathematical Model

In order to ease the complexity of establishing mathematical model, certain assumptions were made in this model. The molten steel flow in the mold is a 3D transient state, incompressible and viscosity flow process, the effects of solidification shell and oscillation of the mold on the molten steel flow are ignored, only the liquid slag layer is considered, other state slag layers are ignored.

Fluid flow equations

Continuity equation

: =0 u     (1)
Momentum equation:

2 eff m = - u u u p u g F t                          ( ) (2)
where,

slag steel steel steel (1 )        
, kg/m3; p is pressure, Pa; eff  is effective viscosity which is determined by the standard k-ε two-equation turbulence model as proposed by Launder and Spalding [3] ; m F  is induced electromagnetic force components, N/m 3 .

Interface fluctuation model equation

In this study, VOF model is used to describe the interfacial behavior of molten steel and liquid slag. The volume fraction of molten steel should satisfy the flowing continuous function:

steel steel 0 u t          ( 3
)
Where t is time, s; Volume fraction steel  is volume fraction of steel; u  is vector of velocity, m/s.

Electromagnetic field equations

The induced current density equation is given by:

1 j B       (4) With 0 B B b     
, 0 B  is externally imposed magnetic field; b  is induced magnetic field due to fluid motion. The induced magnetic field b  can be solved by flowing equation [START_REF]Fluent 6.1-Mannual: Fluent[END_REF] :

2 0 0 1 ( ) (( ) ) ( ) b u b b B b u u B t                         (5)
Electromagnetic force equation:

F j B      (6)
Where t is time, s; j  is the current density;  is the electrical conductivity of the steel, S/m; u 1. The scalar magnetic potential method is used to solve the electromagnetic field in fluid zone for different current intensity of coils with commercial software ANSYS. The coupling calculation of magnetic field and flow field is performed in FLUENT computational fluid dynamics software. The pressure-velocity coupling algorithm is PISO algorithm. In order to obtain the most accurate interface shape, the geometric reconstruction approach is adopted, and the continuum surface force model (CSF) is employed to describe the effect of surface tension [START_REF] Haiqi | [END_REF] .

 is vector of velocity, m/s; F  is induced electromagnetic force, N/m 3 .

Boundary conditions and process of solution

Results and Discussion

Flow characteristic in mold with vertical electromagnetic brake

Effect of current density on behavior of steel/slag interface profile

Fig. 3 shows the steel/slag interface profiles with different current density in vertical EMBr mold. It can be found that the wave height can be obviously depressed by V-EMBr, the value of wave height decreases with the increasing of current density, which the maximum value of wave height decreases from17.9mm (without V-EMBr) to 1.2mm (current density is 350A). It also can be seen from Fig. 3 that the position (along X axis direction) with maximum value of wave height gradually approached to the narrow face of the mold from 33mm to the narrow face reduce to 22mm with the increasing of current density. The maximum value of wave height approached to the narrow face is beneficial to promote the melting of slag and to reduce the possibility of mold flux entrapment.

Effect of current density on surface velocity and vertical velocity

Fig. 4 shows the surface velocity at steel/slag interface and vertical velocity Vy near narrow face of mold with different current density. We can see from Fig. 4(a), the surface velocity at steel/slag interface obviously decreased with the increasing of current density, and the maximum value of surface velocity decreased from 0.23m/s to 0.025m/s. The suppression effect of V-EMBr (current density is 350A) on the surface velocity at steel/slag interface is close to the suppression effect of FC-Mold [6] . Fig. 4

Conclusions

1) The velocity of molten steel in upward backflow zone can be obviously suppressed by V-EMBr.

2) The use of V-EMBr can decline the height of wave with a low current density (350A), while the brake effect of V-EMBr is closed to FC-Mold with a high current density.

3) For given SEN submerged depth, casting speed and SEN port angle, the brake effect of V-EMBr is enhanced with the increasing of current density of coils. With V-EMBr applied, not only can reduce the surface velocity, but also can reduce vertical velocity near the narrow side of mold.

1 ).

 1 Inlet velocity: The inlet velocity is determined by steel flow rate equilibrium, the initial values of k-ε at the inlet are: d inlet is the diameter of the SEN. 2) Wall: Velocity and the current density components normal to the wall are set to zero and all walls are set to be electrically insulated. 3) Symmetry faces: The velocity and the current density components normal to the symmetry faces are set to zero and all other variables normal gradients are set to zero. 4) Outlet: At the outlet of the mold, normal gradients of all the variables are set to zero. Schematics of geometrical model and hexahedral structured mesh of fluid zone are shown in Fig.1, and the detailed process and simulation parameters are shown in Table

Fig. 1 :

 1 Schematics of geometrical model (a) Refined mesh; (b) Fluid zone; (c) Hexahedral structured mesh of fluid zone

Fig. 2 Fig. 3 :Fig. 2 :

 232 Fig. 2 shows the velocity vectors at half thickness symmetric plane of mold with and without V-EMBr. Without V-EMBr applied, the jet flow of molten steel impinges on the narrow face of the mold with high impinging intensity and forms two opposite directional re-circulation zone. The value of wave height is increased because of the strong upward flow and the maximum value ups to 17.9mm. While with V-EMBr applied, the magnetic field successfully suppressed the flow intensity and velocity of molten steel in upward backflow zone and meniscus zone which cause the value of wave height decreased from 17.9mm to 8.43mm. Obviously it is mainly because of the magnetic field which produced by V-EMBr can cover upward backflow zone and meniscus zone at the same time. Therefore, the flow intensity and velocity of molten steel in different zones are successfully suppressed by V-EMBr.

  (b) shows the vertical velocity of molten steel nearby the narrow face of mold. It can be seen that the value of velocity vector in the vicinity of impact point are all declined due to the V-EMBr applied. The value of vertical velocity decreased from 0.3m/s to 0.18m/s in the upward backflow zone. The reduction of the vertical velocity in upward backflow zone is helpful to decrease the height of molten steel and liquid slag interfacial fluctuation and even to reduce the possibility of mold flux entrapment.

Fig. 4 :

 4 Fig. 4: Surface velocity at steel/slag interface and vertical velocity Vy near narrow face of mold

Table 1

 1 Process and physical parameters

	Parameters	Value	Parameters	Value
	Mold size, mm	1200 x 200	SEN submergence depth, mm	180
	SEN port angle, °	-15	Casting speed, m/min	1.8
	Molten steel density, kg/m 3	7200	Molten steel viscosity, Pa•s	0.006
	Liquid slag density, kg/m 3	2700	Liquid slag viscosity, Pa•s	0.2
	Thickness of Slag layer, mm	30	Tension coefficient, N/m	1.6
	Contact angle, °	60	Current density, A	0, 100, 200, 350
	Electric conductivity, S/m	7.14 x 10 5	Magnetic conductivity, H/m	1.257 x 10 -6

Acknowledgement

The authors would like to gratefully acknowledge the financial supports from the National Nature Science Foundation of China (No.51574083 and 51474065) and the Program of Introducing Talents of Discipline to Universities (the 111 Project of China, No. B07015)


M Spezzapria

E Beridze

M Breda

S Bruschi

I Calliari

C Gennari

M Forzan

F Michieletto

Investigation of Electro-Plastic Effect on Ferritic Stainless Steel AISI 430

Keywords: Electroplastic Effect, Ferritic Stainless Steel, Metal Forming, Electrically Assisted Manufacturing

In many modern methods of metal processing, the materials are softened by Joule effect induced by electric current; in this way, the formability is enhanced and especially the flow stress is significantly reduced. In addition, when electric current is involved, changes in other aspects have been observed, including rebound, improved elongation and microstructural features, depending on materials and microstructures. Electric current was found to affect the flow stress on the basis of the so-called Electro-Plastic Effect (EPE), enhancing the material flow properties and especially its formability.

Introduction

The realization of final and intermediate products frequently requires bulk deformation stages, which are commonly accomplished in the hot-working temperature range or through cold-working steps followed by annealing treatments (incremental forming). An alternative method developed for materials forming is represented by Electrically Assisted Manufacturing (EAM), a fabrication technique based on the Electro-Plastic Effect (EPE) induced by the electric current on the material flow properties [START_REF] Jones | Empirical modeling of the stress-strain relationship for an upsetting process under direct electrical current[END_REF][START_REF] Magargee | Analysis and observations of current density sensitivity and thermally activated mechanical behavior in electrically-assisted deformation[END_REF][START_REF] Salandro | Electrically Assisted Forming -Modeling and Control[END_REF][START_REF] Liu | Experimental study of electroplastic effect on advanced high strength steels[END_REF][START_REF] Tang | Experimental study of electroplastic effect on stainless steel wire 304[END_REF][START_REF] Mai | Experimental study of electrical resistivity and flow stress of stainless steel 316L in electroplastic deformation[END_REF][START_REF] Breda | Experimental study on electroplastic effect in AISI 316L austenitic stainless steel[END_REF][START_REF] Salandro | Electrically Assisted Forming -Modeling and Control[END_REF]. Several studies have pointed out the real effectiveness of this method, especially in forging, allowing an enhanced material formability by reducing the required stresses and improving the strains at rupture [1-8-9]. However, not all the examined materials have exhibited advantageous influences upon current application; moreover, it must be noticed that EPE may be hindered by the unavoidable rise in temperature as a consequence of the Joule heating. The effect of electricity within materials has been object of several studies in the past years [1-2-8-9-10-11], leading to the development of theoretical and experimental relationship explaining EPE during plastic deformation. Three main mechanisms were found to induce EPE in the materials -localized resistive heating, electron-induced kinetic energy (electron-wind force) and the presence of an additional quantity of electrons in the material -facilitating dislocations motion within crystal lattices and resulting in an enhancement in materials workability [START_REF] Salandro | Electrically Assisted Forming -Modeling and Control[END_REF][START_REF] Liu | Experimental study of electroplastic effect on advanced high strength steels[END_REF][START_REF] Salandro | Electrically Assisted Forming -Modeling and Control[END_REF]. In metals, EPE was found to be more significant in BCC materials respect to FCC ones [START_REF] Liu | Experimental study of electroplastic effect on advanced high strength steels[END_REF], even though a recent work by Salandro et al. [START_REF] Salandro | Electrically Assisted Forming -Modeling and Control[END_REF] has pointed out that any relationship between crystal structure and a current threshold seems to not exist. Many metals were found to be positively influenced by electricity during deformation, but the occurring of EPE was frequently evaluated by measuring the reduction in flow stress at a fixed strain, without bringing the specimens to fracture and without separating the thermal contribution induced by the Joule effect. In a previous work by the authors [START_REF] Breda | Experimental study on electroplastic effect in AISI 316L austenitic stainless steel[END_REF], the presence of EPE in uniaxial tension has been investigated on an AISI 316L austenitic stainless steel, but any favourable variations in material properties respect room-temperature and thermal tests were not observed. In this work, the results concerning the study of EPE on an AISI 430 ferritic stainless steel in tensile deformation are presented. The material was strained in uniaxial tension applying continuous current and in thermal chamber at the same temperature in order to separate the current effect from the heating one.

Materials and Methods

A sheet of AISI 430 ferritic stainless steel has been used, it was preliminary examined by means of a Leica DMRE Optical Microscope after chemical etching (solution composed of 75% aqua regia in ethanol), for microstructural characterization in the Rolling Direction (RD). All samples were tested along RD on a MTS uniaxial tensile machine in strain-controlled conditions adopting a strainrate of 10 -1 s -1 (E-01). The tensile tests were conducted in thermal and current regimes, and a proper tooling setup was developed in order to avoid interferences between machine and specimen upon current application. Moreover, a cooling device aimed to limit specimens heating was employed. An electrical DC-current power supply was adopted for continuous electric current application, and the related rise in temperature was monitored using a laser pyrometer (accuracy of ±0.5°C), in order to reproduce thermal tensile tests for the separation of temperature-related effects to those induced by current.

Results and Discussions

One of the most critical aspect in EPE tests is the separation of the electroplastic effect from the temperature dependence of the flow stress. In fact, the heat generated in the samples is strongly correlated to the material resistivity, since it is induced by the Joule effect, but also depends on the thermal conductivity of the material. In the case of AISI 430, the high electrical resistivity, the low thermal conductivity and the type of cooling device did not allow the use of high current densities without increasing the temperature exceeding the cold-working regime. For this reason, the maximum current density reached in this work has been set to 21 A/mm 2 . Fig. 1: Microstructure of the base material, RD is vertical (LH) and engineering curves in current regime (RH).

The as-received steel was constituted of fine ferritic grains roughly oriented toward RD, denoting a partial annealing performed after the forming operations (Fig. 1). The material was tested along RD, and the tensile test results as a function of the applied current density are reported in Fig. 1, together with the reached temperatures. Electricity caused an unavoidable temperature rise -partially limited by the cooling device -that modified the material mechanical response by reducing the elongation at rupture and by lowering the Ultimate Tensile Strength (UTS). Respect to the base material curve (baseline), any enhancement in formability was not observed and the employed current densities caused a progressive reduction in mechanical properties. Therefore, owing to not negligible workpieces heating, EPE cannot be clearly observed by solely consider the test performed in current regime. In the present experimental study, the separation of the electrical contribution from the thermal one was achieved by reproducing thermal tests at the temperatures recorded upon current application and by comparing the relative variations of total elongation and UTS respect to the baseline (relative strain and relative UTS). Two examples of thermal and current stress/strain curves are reported in fig. 2: the differences of the material behaviour respect to the baseline are clearly evident. Here, by comparing thermal and electrical results, it can be noticed that electricity acted on the material properties by mainly influencing the elongation at rupture, whereas UTS was not substantially altered; in any case, the flow properties were always reduced if compared to the baseline. As can be seen from Fig. 3, where relative strain and UTS are plotted against the registered temperature, this behaviour was observed in the entire experimental current density range, denoting an effective contribution of electricity in improving formability of the steel under study. However, this enhancement was reduced by applying increased current densities, since important thermal contributions in current regime were not completely avoided by the employed cooling device.

Fig. 3: Relative variations of engineering strain (LH) and UTS (RH) against the testing temperature.

Conclusions

In the present work, an experimental investigation of EPE on AISI 430 ferritic stainless steel has been presented. In the considered steel, the application of electric current from 5 to 21 A/mm 2 during uniaxial tensile deformation reduced the mechanical properties respect to the room-temperature test; however, an improvement in strain at rupture respect to the thermal counterparts was observed, whereas no noticeable effects on UTS were revealed.

A similar experimental study on an AISI 316L austenitic stainless has been recently performed by the authors [START_REF] Breda | Experimental study on electroplastic effect in AISI 316L austenitic stainless steel[END_REF], employing the same current densities and the same cooling device, and a reduction of the mechanical properties respect to the thermal tests has been observed as the current density was increased. Therefore, considering the same electrical resistivity for the two stainless steels grades, the enhanced response of AISI 430 can be ascribed to its different crystal structure, as also noted by Liu et al. [START_REF] Liu | Experimental study of electroplastic effect on advanced high strength steels[END_REF]. Nevertheless, in both cases, the materials formability was always reduced respect to the baselines, making electricity in continuous regime not effective for AISI 430 as for other metals such as titanium or aluminium [START_REF] Perkins | Metallic forging using electrical flow as an alternative to warm/hot working[END_REF].

Fig. 2 :

 2 Fig. 2: Comparison among the baseline and the engineering curves obtained in current and thermal regime.