Fish freshness is a key attribute for the quality of fish, which is a highly perishable product. The fishing industry is an important contributor to many economies in the world. One of the senses used by consumers to determine the freshness of fish is the smell. Indeed, the volatilome of fish changes rapidly according to the product degree of freshness, and that is why sensory analysis are used by consumers and industrialists to assess fish quality. Then, the key volatile compounds that contribute to this characteristic odor can be measured and used as quality indicators    . These characteristic aromatic volatile compounds are generated by different biological pathways including the lipid autoxidation, the action of spoilage organisms and autolytic enzymes.
Recently, Duflos et al.  studied the spoilage of whiting at five stages of ice storage by comparing the analysis of volatile compounds obtained by solid phase microextraction (SPME) coupled to the combination of gaz chromatography/mass spectrometry (GC/MS) and SPME with two sensory analysis methods. Two separate steps of statistical multidimensional approaches were used to identify volatile compounds and characterize fish freshness assessed by two different indices. In the first step, control charts were used to control the daily progression of freshness and spoilage indices. The second step begins by reducing the dimension of the data set (excluding the two indices variables) to two principal components via the application of Principal Component Analysis (PCA) method.
Then, a hierarchical clustering approach and a heuristic variable selection were used for clustering the fish samples on three classes and to identify the volatile compounds that respectively characterize these classes. However, the indices (or response variables) were not directly taken into account in the later procedure.
Recently, Sidi et al.  applied stability selection and randomization techniques in norm penalized quantile regression on the same data set. These approaches highlighted volatile compounds that are more relevant for the evaluation of fish freshness throughout its storage, so, are assumed to influence more the fish freshness and quality.
This paper aims at using Ma and Zhang  approach to select a reduced subset of volatile compounds which can be used to explain whiting spoilage during its conservation. This approach allows a robust and model-free feature screening based on quantile correlation proposed by Li et al.  .
The lines below are organized as follow: The methodology is briefly presented in section 2 and section 3 is dedicated to the experimental framework. Finally, the results are discussed in section 4 followed by concluding remarks in section 5.
This section is dedicated to the following methods: Quantile Correlation, Sure Independence Screening via Quantile Correlation, Composite Quantile Correlation and Sure Independence Screening via Composite Quantile Correlation.
2.1. Quantile Correlation
As advocated in Li et al.  , quantile correlation is a novel measure used to examine the linear relationship between any two random variables Y and X for a given quantile. So,
is the conditional quantile of Y and
Moreover, if X is independent of Y, the; else (X and Y are corre- lated), the.
2.2. Sure Independence Screening via Quantile Correlation
Consider Y as the dependent variable and be the p-dimensional independent variables. Let with
Sure Independent Screening via Quantile Correlation method selects the first d independent variables with largest; where
with the sample estimate of.
2.3. Composite Quantile Correlation
Composite Quantile Correlation(CQC) is motivated by the fact that previous quantile correlation cannot characterize the entire relationship between X and Y. So, the composite quantile correlation is defined by:
2.4. Sure Independence Screening via Composite Quantile Correlation
The CQC screening is based on the vector with components
Sub models are selected based on decreasing values of. Furthermore, as advo- cated by Ma and Zhang  , when using screening techniques, the number of selected
variables is often set to be or the integer part of.
3. Materials and Methods
The sample preparation and sensory evaluation methods are briefly presented in this section. More details about the full experimental procedure can be found in  .
3.1. Sample Preparation
As advocated by Duflos et al.  , the sample considered is based on two different catches of respectively 20 and 15 fish. These catches were stored in crushed ice at 4˚C in self-draining polystyrene boxes for 7 days. Fresh crushed ice was added daily. Sensory evaluation and volatile analysis were performed on seven different fish on days 1, 2, 3, 4 and 7.
3.2. Sensory Evaluation
According to Duflos et al.  , two methods were used for the sensory evaluation of fish. These methods lead to freshness and quality indices which represent two response variables for our selection process.
4. Results and Discussion
The empirical results of the analysis of freshness and quality indices influenced by a great number of volatile compounds are presented below.
The sample size and the number of predictors (volatile compounds) are respectively and. So, the number of predictors is higher than the sample size.
In order to perform variables selection, screening methods are applied on whiting data set using QC-SIS package available for R software.
The tuning parameter d used to select covariates with significant effect on each
response variable can be set to or the integer part of.
Furthermore, Figure 1 and Figure 2 display the Pearson correlation matrix through bivariate scatter plots for each index with corresponding selected compounds. These figures have been made using Performance Analytics package available for R software.
Table 1. Selected volatile compounds ranked by decreasing weights for freshness index
Table 2. Selected volatile compounds ranked by decreasing weights for quality index.
Figure 1. Correlation matrix chart for freshness index and nine(9) related compounds: On top; the (absolute) value of the correlation with significance levels. The distribution of each variable is represented on the diagonal and at bottom, the bivariate scatterplots, with a fitted line. Components of the vector are respectively tagged with symbols corresponding to the associated p-values: “***” (p-value ≤ 0.001), “**” (p-value ≤ 0.01), “*” (p-value ≤ 0.05), “.” (p-value ≤ 0.1), “ ” (p-value ≤ 1).
Figure 2. Correlation matrix chart for quality index and ten(10) related compounds: On top; the (absolute) value of the corre- lation with significance levels. The distribution of each variable is represented on the diagonal and at bottom, the bivariate scatterplots, with a fitted line. Components of the vector are respectively tagged with symbols corresponding to the associated p-values: “***” (p-value ≤ 0.001), “**” (p-value ≤ 0.01), “*” (p-value ≤ 0.05), “.” (p-value ≤ 0.1), “ ” (p-value ≤ 1).
4.1. Results for Freshness Index
According to Table 1, Quantile Correlation and Composite Quantile Correlation Sure Independent Screening methods select the same subset of volatile compounds for freshness index.
These compounds have been previously identified as spoilage markers.
For example, the compounds Ethanol, 3-Methyl-1-butanol, 2,3-Butanedione, 2-Me- thyl-1-butanol, 3-Methyl butanal, 2-Methyl-1-propanol and Ethylacetate are identified as correlated to the second principal component axis in Duflos et al.  .
The previous seven compounds were included in the eight compounds (with limonene) that characterized category 2/3 (intermediate category between freshness and spoilage) in Duflos et al.  .
Moreover, compounds Trimethylamine and 1-Butanol were not identified as corre- lated to the first principal component axis in Duflos et al.  .
Finally, considering Freshness index, only Trimethylamine, 1-Butanol and 2-Methyl- 1-butanol were not selected by QR and other randomization approaches high- lighted in Sidi et al.  .
4.2. Results for Quality Index
According to Table 2, Quantile Correlation and Composite Quantile Correlation Sure Independent Screening methods do not select the same subset of volatile compounds for quality index.
The compounds Ethanol, 3-Methyl-1-butanol, 2,3-Butanedione, 2-Methyl-1-butanol, 3-Methyl butanal and 2-Methyl-1-propanol are identified as correlated to the second principal component axis in Duflos et al.  .
The previous six compounds were included in the eight compounds (with limonene) that characterized category 2/3 (intermediate category between freshness and spoilage) in Duflos et al.  .
The compounds 1-Butanol, 3-Pentanol, 3-Pentanone and 3-Hydroxy-2-butanone are not identified as correlated to the first principal component axis in Duflos et al.  .
For the quality index, only Ethanol, 3-Methyl butanal, 3-Methyl-1-butanol and 2- Methyl-1-butanol were selected in Sidi et al.  .
Choosing includes freshness markers like Propanal, Hexanal, 1-Penten-3-ol, Pentanal, 2,3-Pentanedione, 1-Penten-3-one, Heptanal, (E)-2-Pentenal, (Z)-2-Penten-1-ol, 1-Pentanol, Butanal, Octanal, 1-hexanol and 4,4-Dimethyl-1,3- dioxane.
5. Concluding Remarks
Sure Independence Screening via Quantile Correlation and Composite Quantile Corre- lation methods highlighted relevant volatile compounds influencing freshness and quality indices during whiting conservation.
The selected compounds include Ethanol, 3-Methyl-1-butanol, 2,3-Butanedione, 2- Methyl-1-butanol, 3-Methyl butanal, 2-Methyl-1-propanol and Ethylacetate, previously identified as spoilage markers.
For future investigation on whiting data, it will be very interesting to explore the following issues:
1) Simultaneous model selection in multiple quantile regression 
2) Selection of groups of highly correlated compounds 
3) Quantile regression models and inference processes based on  and  .
The authors wish to thank Pr Abdallah Mkhadri (Cadi Ayyad University of Morroco), François Leduc (ANSES), The Nord-Pas-de-Calais Regional Council (France) and ANSES (France). This support is greatly appreciated.