domain: stat phys sde

text:
Sémou Diouf
email: diouf.semou3@ugb.edu.sn

El Hadji Dème
email: elhadji.deme@ugb.edu.sn

Abdoulaye Dème
email: abdoulaye.deme@ugb.edu.sn

Imputation methods for missing values: the case of Senegalese meteorological data

Keywords: Missing values, Meteorological data, Senegal, missForest, imputeTS

Climate change studies require comprehensive databases to analyze the climate signal, to monitor its evolution, and to predict more accurately future changes. Since complete observation of any continuous process is almost impossible, it is then inevitable to encounter missing information in meteorological databases. The aim of this work is to evaluate the performance of five (5) imputation methods: missForest, k-nn, ppca, mice and imputeTS. The results show that missForest is the best performing method to handle missing temperature data. In the case of precipitation data, the imputeTS method is the preferred one.

Introduction

Dealing with missing data is an essential step in almost all studies which make use of a database (for example in biology [START_REF] Yu | Probabilistic principal component analysis with expectation maximization (PPCA-EM) facilitates volume classification and estimates the missing data[END_REF]), meteorology [START_REF] Moron | Trends of mean temperatures and warm extremes in Northern Tropical Africa (1961-2014) from observed and PPCAreconstructed time series[END_REF]), social sciences (Aurélien and Groothuis-Oudshoorn (2016)) and medicine [START_REF] Che | Recurrent Neural Networks for Multivariate Time Series with Missing Values[END_REF]). Indeed, the presence of gaps in a database can lead to a loss of information or introduce bias into the results of analysis [START_REF] Lotsi | Comparison of Imputation Methods for Missing Values in Longitudinal Data Under Missing Completely at Random (MCAR) Mechanism[END_REF]). In the process of data acquisition, missing data issues remains widespread and almost inevitable. Several factors can explain the absence of observations in time series [START_REF] Schafer | Missing Data: Our View of the State of the Art[END_REF]. As example, one can mentioned: the malfunctioning of measuring instruments, the human error as a result of manual measurement, the maintenance of weather stations, etc. [START_REF] Beaulieu | Synthèse des techniques d'homogénéisation des séries climatiques et analyse d'applicabilité aux séries de précipitations[END_REF]). In a context of climate change marked by an increase in extreme weather events such as heat waves, dust storms, etc., having a complete and reliable database is of capital importance. In order to carry out a climate study, while taking into account the importance of trends, it is essential to have long time series of at least 30 years with a minimum of missing values. However, the meteorological data at our disposal show gaps of more than 10% for most of the stations studied in Senegal over the period 1973 -2020. This is common in meteorology and is problematic. Therefore, it is necessary to find a solution to these missing data before conducting a study. This prompted us to look for appropriate methods to fill the gaps in the various databases.

However, there are several methods dealing with missing values in time series data. The most common method used by default in many statistical software is to restrict the study to individuals with complete information (the complete cases method). This method shows its limits when the percentage of missing values exceeds 5% (we can cite [START_REF] Davey | Correcting for selective non response in the National Longitudinal Survey of Youth using multiple imputation[END_REF], [START_REF] Niass | Analysis of missing data in sereo-epidemiologic studies[END_REF]). Other techniques that fill in the gaps with a plausible value or values are used to solve the problem of missing data. These methods, known as "imputation" are numerous and varied. Depending on the field of application, some methods are considered more effective than others. The main objective of our study is to evaluate the performance of five (5) methods among the most commonly used imputation techniques in the literature dealing with missing data.

Traditionally, precision measures such as root mean square error (RM SE), mean absolute error (M AE), Akaike information criterion (AIC), etc., are used separately as a tool to evaluate the performance of time series technical reconstruction (see [START_REF] Niass | Analysis of missing data in sereo-epidemiologic studies[END_REF], [START_REF] Lotsi | Comparison of Imputation Methods for Missing Values in Longitudinal Data Under Missing Completely at Random (MCAR) Mechanism[END_REF], [START_REF] Sattari | Assessment of different methods for estimation of missing data in precipitation studies[END_REF], [START_REF] Fadoua | Collection of historical weather data: issues with missing values[END_REF], [START_REF] Bousri | Validation d'une méthode d'imputation de données manquantes pour la reconstitution des séries de température[END_REF], [START_REF] Dixneuf | A computational study on imputation methods for missing environmental data[END_REF]). An approach based on the Taylor diagram [START_REF] Taylor | Summarizing multiple aspects of model performance in a single diagram[END_REF]), combining several complementary statistical methods, allowed to quantify the degree of correspondence between the simulated and observed behaviour. To the best of our knowledge, this study constitutes a first attempt to use the Taylor diagram to evaluate the performance of imputation techniques on meteorological data. In addition to the use of this diagram, we also suggest the Kling-Gupta Efficiency (KGE) (Gupta et al. (2009)) which is a metric generally used in the evaluation of hydrological models. This paper is organised as follows. First, we describe in Section 2 the typology of missing data. In Section 3 we present the data used for the study and then describe the statistical tools applied. The analysis and discussion of the results is given in Section 4, followed by the conclusion in Section 5.

Typology of missing data

The classification established by [START_REF] Little | Statistical analysis with missing data[END_REF] is a reference in the literature dealing with the problem of missing data (MD). This classification highlights 3 categories of MD: a. A data called to be M CAR (Missing Completely At Random) if the probability of having missing data for a variable is independent of the values of the variable itself and of the other variables measured. Explicitly, let X = (x ij ) ∈ R n×p a climate database with n observations x 1• , . . . , x n• and p variables x •1 , . . . , x •p :

X = (X 1 , X 2 , . . . , X p ) =       x 11 . . . x 1p
x 21 . . . . . . . . .

x n1 x np      
The data set is incomplete if there is at least one variable with a missing value. Consider M = (m ij ), the missing value indicator matrix. With

m ij = 1 if x ij is missing 0 if x ij is observed
According to M , our base can be written as: X = (X obs , X miss ), where X obs = XI {M =0} is the observed part, and X miss = XI {M =1} is the missing one in the incomplete data set.

Let (x ij ) miss = x i j be the missing value associated to the i th observation and to the j th variable of the database.

It is clear that:

(x ij ) miss = {x i j | m i j = 0, j < j } . (1)
Formally, the missing data mechanism is characterized by the conditional distribution of M knowing X given by P(M |X).

In the case of the M CAR data, the absence of data does not depend on the values of X so:

P(M | X) = P(M ). (2)
b. If the probability of missing data depends on the values taken by other measures in our data set but not on the missing values of the variable itself, then the missing data process is called M AR (Missing At Random). This case is formally defined by:

P(M | X) = P (M | X obs ) . (3) c.
A data is M N AR ( Missing Not At Random) if the probability of absence depends on the variable in question.

Unfortunately, the underlying mechanisms for the lack of observations in the databases cannot generally be known from the observations alone.

Data and methods

Data

The data of our study concerns the daily temperature (maximum, minimum and mean) and the rainfall databases from (11) stations in Senegal (see figure 1) over the period 1973 -2020. These observations are extracted from the Global Surface Summary of the Day (GSOD) database produced by the National Climatic Data Center (NCDC). GSOD consists of 18 daily surface weather variables. The data can be accessed from the link https://catalog.data.gov/dataset/global-surfacesummary-of-the-day-gsod or directly from the R software with the GSODR package (Global Surface Summary Daily Weather Data in R software). This databases contents different percent of missing values. As example, we note between 1% to 20% of missing values in the daily temperature and the rainfall databases for all the stations under study. As illustration, we represent in figure 1 all the stations under study with there percent of missing values in the case of daily maximum temperature. This can be also applied to all other weather variables in the data set.

In the next section, we use different methods and there evaluation criterion in order to complete the missing values on the data set under the study.

Treatment of missing data

In recent decades, several imputation methods have been developed dealing with the problem of missing data. There exist 2 imputation approaches:

-Simple imputation (SI): it concerns to replace a missing value by a single plausible value.

-Multiple imputation (MI): which consists of successively imputing several values to each missing data item, i.e. generating several possible values for each missing data item and then combining the results. The interest of this approach is to minimise the error due to imputation.

Five imputation methods widely used in the literature on missing data were tested: k-nearest neighbors (k-nn) [START_REF] Kowarik | Imputation with the R package VIM[END_REF]), time series missing value imputation (imputeTS) [START_REF] Moritz | imputeTS: Time Series Missing Value Imputation in R[END_REF]), probabilistic principal components analysis (ppca) [START_REF] Josse | missMDA: A Package for Handling Missing Values in Multivariate Data Analysis[END_REF]), multiple imputation by chained equations (mice) [START_REF] Buuren | mice: Multivariate imputation by chained equations in R[END_REF]) and random forest [START_REF] Stekhoven | Using the missForest package[END_REF]) on daily climate databases of temperature and rainfall.

As the databases are all incomplete, we used the k-nearest neighbor method (k-nn) to create our complete database (reference matrix) in order to compare the different imputation techniques (see Section 4). We randomly choose a rate of missing values in the range of 5%, 10%, 20%, 30% until 40% (five (5) values of percentage). This methodology was replicated on the 11 stations in Senegal. We generate and complete 55 = 5 × 11 matrices of missing values for all these stations.

• k-nearest neighbors (k-nn):

The k-nearest neighbors method is an imputation method based on the notion of proximity between the observations of a variable [START_REF] Troyanskaya | Missing Value Estimation Methods for DNA Microarrays[END_REF], [START_REF] Kowarik | Imputation with the R package VIM[END_REF]). This proximity is often materialised by the definition of a distance. Many methods for calculating this distance have been proposed in the literature. In this work, we use the Euclidean or the Manhattan distance [START_REF] Faisal | Nearest neighbor imputation for categorical data by weighting of attributes[END_REF]), since it has the most used metric in the statistical software.

The imputation is carried out in 4 steps:

Algorithm 1 Impute missing values with k-nn 1. Select an integer k:

1 ≤ k ≤ n 2. Calculate distances d (X i , X i ) , i = 1, . . . , n 3.
Choose the k observations X (i1) , . . . , X (i k ) associated to the smallest distances 4. Assign to missing values the average of the k-nearest neighbors values :

(x ij ) miss = x i * j * = 1 k X (i1) + • • • + X (i k ) .
• Time series missing value imputation (imputeTS): ImputeTS is a specialised algorithm for imputing univariate time series. Most imputation methods are related on correlations between attributes, whereas imputeTS uses temporal dependence serials. According to [START_REF] Moritz | imputeTS: Time Series Missing Value Imputation in R[END_REF], for long time series and structural complex data (with trend and seasonality), it is always wise to try the function "na.kalman" on the package which very often gives the best results. This imputation method uses Kalman filters to operate on state space models of the shape:

X t = Z t α t + ε t , ε t ∼ N (0, H t ) α t+1 = T t α t + R t , η t ∼ N (0, Q t ) α 1 ∼ N (a 1 , P 1 ) (4)
X t : observed series (possibly with missing values) α t : unobserved series ε t and η t are disturbance vectors considered as Gaussian white noise H t and Q t are respectively the variance-covariance matrices of the white Gaussian noise of observations ε t and η t Z t and R t are the matrices of the system a 1 and P 1 are the initial values of the mean and variance respectively.

The Kalman's filter uses an algorithm to find the best estimated values of α t . Once the Kalman's filter has been applied to the whole time period, we obtain the best estimates of a t , P t at t = 1, 2, . . . , T .

For more details, one can refer to [START_REF] Koopman | Continuous Time State Space Modelling with an Application to High-Frequency Road Traffic Data[END_REF].

• Multiple imputation by chained equations (mice):

Mice is a multiple imputation method based on a Monte Carlo Markov Chain (MCMC) algorithm.

Multiple imputation was first proposed by [START_REF] Rubin | Multiple imputation for interval estimation from simple random samples with ignorable nonresponse[END_REF]. It consists of performing m > 1 imputations in order to obtain m values for each missing data item, and then combining the independently calculated statistics on the m data sets. Multiple imputation method is based on the M AR assumption. According to [START_REF] Rubin | Multiple Imputation After 18+ Years, with discussion[END_REF], this imputation technique can produce good results even if one deviates slightly from M AR. By default, the predictive mean matching (pmm) approach is used for quantitative data, logistic regression for binary data, polytomous regression imputation for unordered qualitative data and proportional chance model for ordered qualitative data [START_REF] Dixneuf | A computational study on imputation methods for missing environmental data[END_REF]). Imputation with mice is carried out in 6 steps [START_REF] Azur | Multiple imputation by chained equations: what is it and how does it work[END_REF]):

Algorithm 2 Impute missing values with mice 1. Perform a simple imputation, such as imputing the mean for each missing value in the data set. These mean imputations can be considered as "place holders". 2. The mean imputations "place holders" for a variable ("var") are restored as missing.

3. The observed values of the variable "var" in step 2 are regressed on the other variables in the imputation model, which may or may not consist of all the variables in the data set. 4. Missing values for "var" are then replaced by predictions (imputations) from the regression model. 5. Steps 2 to 4 are then repeated for each variable for which data is missing. The cycle through each of the variables constitutes an iteration. At the end of a run, all missing values have been replaced with predictions from regressions that reflect the relationships observed in the data. 6. Steps 2 to 4 are repeated for a number of cycles, with the allocations being updated in each cycle.

• Probabilistic principal components analysis (ppca):

This method allows for the handling of missing values in the analysis of multivariate data [START_REF] Josse | missMDA: A Package for Handling Missing Values in Multivariate Data Analysis[END_REF]). The values of the missing entries are replaced by plausible values. This method takes into account both the similarity between individuals and the relationship between variables. It has been designed under the assumption of missing at random (M AR). The data matrix X ∈ R n×p is assumed to be generated according to a fully connected ppca model, i.e. by factoring the loading matrix B ∈ R r×p and r latent variables grouped into the matrix W ∈ R n×r ,

X = 1α + W B + , with        W = (W 1• | . . . |W n• ) T , with W i• ∼ N (0 r , Id r×r ) ∈ R r , B of rank r < min{n, p}, α ∈ R p and 1 = (1, . . . , 1) T ∈ R n , = ( 1• | . . . | n• ) T , with i• ∼ N 0 p , σ 2 Id p×p ∈ R p , (5)
for σ 2 et r known. Noting respectively by x •j and x i• the column j and the row i of X, the rows of X are identically distributed, ∀i ∈ {1, . . . , n},

x i• ∼ N α, B T B + σ 2 Id p×p .
For more details, refer to [START_REF] Sportisse | Estimation et imputation en analyse probabiliste en composantes principales avec données manquantes non aléatoires[END_REF].

• Random Forest (missForest):

Missforest is a non-parametric imputation technique based on the principle of random forests. This method can be applied to virtually any type of data. MissForest works iteratively, continuously updating the variably imputed matrix, and evaluates its performance between iterations. This evaluation is done by considering the difference(s) between the previous imputation result and the new imputation one. As soon as this difference (in the case of one type of variable) or these differences (in the case of variables of mixed type) increase, or when the maximum number of iterations specified by the user is reached, the algorithm stops [START_REF] Stekhoven | Using the missForest package[END_REF]).

For an arbitrary variable X j containing missing values at the inputs i j miss ⊆ {1, . . . , n}, with j = 1, . . . , p, we can separate the data set into 4 categories: 1. The observed data of the variable X j which we note y j obs . 2. The missing data of X j which we note y j miss . 3. Variables other than X p with observations i p obs = {i, . . . , n}\i p miss are denoted x j obs . 4. Variables other than X j with observations i j miss are denoted by x j miss .

Let's sort according to the number of missing values, starting with the lowest.

The algorithm below gives a representation of the missForest method. It requires some tools of the imputation matrix X and the stopping criterion ζ.

Performance evaluation criteria

Taylor's Diagram

The Taylor's diagram described by [START_REF] Taylor | Summarizing multiple aspects of model performance in a single diagram[END_REF] is used to display the quality of the predictions of the imputation methods with respect to the observed values. This diagram has the advantage of representing both 3 complementary statistics which are the standard deviation of the simulation from the observation (σ), the centered root mean square error (RM SE c ) and the correlation coefficient between the observation and the simulation (r). It is frequently used by meteorologists and atmospheric scientists to evaluate the performance of climate models.

• Standard deviation (σ): measures the dispersion of a data set from its mean. More the standard deviation is lower, less the observations are dispersing around their mean. For a given variable x •j , j ∈ {1 . . . p}, we define the complete series O •j associated with x •j obtained by the k-nearest neighbors method. As mentioned above, this reference series allows us to establish a comparison with the other imputation methods. Thus, the standard deviation of the actual observed values is:

σ O j = 1 n n i=1 O ij -O •j 2 , ( 6
)
where

O •j = 1 n n i=1
O ij , the average of the observed series.

We denote by E •j , the test variable estimated from another imputation method.

The standard deviation of the estimated values is given by

σ E j = 1 n n i=1 E ij -E •j 2 , (7)
where

E •j = 1 n n i=1
E ij , the mean of the estimated series.

• Correlation coefficient (r): is used to measure the degree of linkage or dependence among variables. It is between -1 and 1. The best method is that which maximises the correlation coefficient.

To find r, we first calculate the covariance given by the following formula:

Cov(O •j , E •j ) = 1 n n i=1 O ij -O •j E ij -E •j . (8)
The linear correlation coefficient between the observed and estimated values is calculated as follows:

r(O •j , E •j ) = Cov(O •j , E •j ) σ O j σ E j . (9)
• Centred root mean square error (RM SE c ): indicates how the data are concentrated around the line of best fit (regression line). The best method is the one that minimises the RM SE c . According to [START_REF] Twumasi-Ankrah | Efficiency of imputation techniques in univariate time serie[END_REF], the RM SE c is an appropriate error measure for data with very high seasonality, as well as data sets with no trend and no seasonality. It is calculated as follows:

RM SE c = 1 n n i=1 O ij -O •j -E ij -E •j 2 1/2 . ( 10
)
RM SE 2 c = 1 n n i=1 O ij -O •j 2 + n i=1 E ij -E •j 2 -2 n i=1 O ij -O •j E ij -E •j . (11)
These 3 statistics are related on a Taylor diagram by the following equation:

RM SE 2 c = σ O j 2 + σ E j 2 -2σ O j σ E j r(O •j , E •j ). ( 12
)
The construction of the diagram (with the correlation given by the cosine (cos) of the azimuth's angle) is based on the similarity of the equation ( 12) and the law of Cosines:

c 2 = a 2 + b 2 -2ab cos φ. ( 13
)
Where a , b and c are the lengths of the sides of the triangle, and φ is the angle between the sides a and b.

Kling-Gupta Efficiency (KGE)

The Kling-Gupta efficiency (KGE) is a metric traditionally used in hydrology to summarise the performance of models. This metric developed by Gupta et al. (2009) combines three components: the correlation, the bias, and the ratio of variances or variation coefficients. A value of KGE = 1, indicates a perfect agreement between simulations and observations. We used a slightly modified version of the KGE statistic proposed by [START_REF] Kling | Runoff conditions in the upper Danube basin under an ensemble of climate change scenarios[END_REF] to measure the agreement between the estimated values and the observations.

KGE = 1 -(r -1) 2 + (β -1) 2 + (γ -1) 2 , ( 14
) with β = E•j O•j , γ = σ E j /E•j σ O j /O•j
, KGE represents the modified KGE statistic and r is the correlation coefficient between the observed and estimated data. The parameter β is the bias ratio while γ represents the variability rate.

Results and discussion

In this paper, the daily temperature (maximum, minimum and mean) and the rainfall databases from 11 stations in Senegal have been evaluated with different percentage of missing values (5%, 10%, 20%, 30% and 40%). Using five imputation methods among the most commonly used in the literature for missing values, these missing databases has been completed. The study of the performance of these different imputation methods is based on the Taylor's diagram approach on the one hand and the Kling-Gupta efficiency (KGE) on the other. Three complementary statistics are represented on the Taylor's diagram: i) the centred mean square error (green contours) is proportional to the distance of the point on the x-axis identified as the "observations"; ii) the correlation coefficient (grey dashed lines) measures the similarity between the observed and simulated values; iii) the standard deviation is proportional to the radius distance from the origin (blue contours). The five methods to be compared are represented by points of different colours on the diagram. If a coloured point is close to the black point (observations), then we infer that the estimated data are in good agreement with the observations. Quantitatively, this translates into an r close to 1, an RM SE c as small as possible and σ E j very close to σ O j . The Taylor's diagrams presented in Figures 2,3,4, and 5 provide a summary of the relative performance of the five imputation methods in estimating daily maximum temperatures for four stations in Senegal that are fairly representative of the study area (Dakar, Kedougou, Matam and Ziguinchor). The results show that temperatures are particularly well estimated on almost all methods with correlations greater than 0.9. The RM SE c values are less than 1C, for low to medium percentages of missing data (between 5% and 20%). They exceed 1C when the percentage of missing data becomes important (≥ 30%) (Figures 5, and6). The statistical parameters represented through the Taylor's diagrams show that the missForest method is the most performance one for the temperature series under study. [START_REF] Stekhoven | Missforest-Non-parametric missing value imputation for mixed-type data[END_REF] found the same conclusion by comparing missForest to the k-nn, mice and missPALasso methods on the 11 real databases. [START_REF] Dixneuf | A computational study on imputation methods for missing environmental data[END_REF] showed also that missForest is the preferred imputation method for handling missing data in the environmental studies. Again, [START_REF] Bousri | Validation d'une méthode d'imputation de données manquantes pour la reconstitution des séries de température[END_REF] showed the efficiency of the missForest imputation method over mice, k-nn and ppca, in the case of missing temperature data. However, the performance of the missForest method decreases as the number of missing data increases (see Figure 6). This is well confirmed through the Kling-Gupta Efficiency (KGE) approach, whose it's average values around 0.80 and 0.95, respectively for 40% and 5% of missing temperature data (see Tables 1 and5).

The k-nn method makes estimates very close to the observation in the overall behaviour for temperatures. Among the least efficient imputation methods we find ppca with relatively high RM SE c values. It should be noted that all the imputation methods, except the mice one, underestimate the variability of temperature because they all fall within the black contour (standard deviation of the observed series). Similar results are obtained by repeating the analyses on minimum and mean temperatures (figures not shown). For the precipitation, it is less well simulated, but the correlation with observations is clearly positive in all scenarios of missing values (Figures 7,8,9,and 10). The simulated variance and RM SE c are, on the other hand, very high compared to the observations, especially from 30% of missing data (Figures 10 and11). The imputeTS method is the preferred imputation technique for filling in the gaps of precipitation time series (see the KGE values calculated in Tables 1, 2, 3, 4 and5). In the second place, we find the ppca method with KGE values very close to those of imputeTS. In summary, the imputation methods are generally successful in estimating the observed temperature data, regardless of the percentage of missing values. This is largely due to the fact that temperatures have a very little spatiotemporal variation and also continuous variables.

Based on these two performance indicators (Taylor's diagram and KGE), missForest is the method of choice for imputation on the temperature time series data and imputeTS is those for the precipitation data.

Conclusion

There are a lot of missing values in the meteorological data from the GSOD database for all stations in Senegal studied over the period 1973 -2020. These missing values produce a bias on the result in a climate study. Anticipating the climate impact of climate change, which is a major challenge for humanity, requires a robust monitoring and tracking best meteorological data. These data will allow researchers to identify and reach populations vulnerable to climate-related hazards on the one hand and for climate applications studies on the other. It is then necessary to find a solution in order to solve the mistake on these times series in the meteorological data. This prompted us to look for suitable techniques to reconstruct these missing data. Five imputation methods widely used in the literature on missing data have been tested (k-nn, ppca, imputeTS, mice and missForest) with missing data scenarios of 5%, 10%, 20%, 30% and 40%. The performance of these methods is compared graphically using the Taylor's diagram and quantitatively with the KGE approach. These two metrics combine several complementary statistics at once. The Taylor's diagram shows how the correlation coefficient (r), standard deviation (σ) and root mean square error (RM SE c ) vary simultaneously. The Kling-Gupta Efficiency (KGE) combines correlation, bias and variability. The results show that when the percentage of missing data is low, the performance is almost the same for the 5 above imputation methods. However, missForest is the preferred imputation technique to complete the temperature databases. The results show also that imputation of univariate series using imputeTS method was the most successful method for reconstructing precipitation time series data. col="red",cex.axis=0.7,font.lab=1,pos.cor=TRUE,show.gamma=TRUE,normalize=F,ngamma=3,gamma.col=8,sd.arcs=1,ref.sd=T,sd.method="sample",mar=c(5,5,5,5),xlab="Standard deviation (C)" ylab="Standard deviation(C)",cex.main=0.8, pcex=1, pch=19) # Representation of the k-nn method on the Taylor diagram taylor.diagram(as.numeric(Obs$Tmax),as.numeric(imp_knn$Tmax), add=TRUE,col="#0000FF",normalize=F,pcex=1, pch=19) # Representation of the mice method on the Taylor diagram taylor.diagram(as.numeric(Obs$Tmax),as.numeric(imp_mice$Tmax), add=TRUE,col="orange",normalize=F,pcex=1, pch=19) # Representation of the missForest method on the Taylor diagram taylor.diagram(as.numeric(Obs$Tmax),as.numeric(imp_miss$Tmax), add=TRUE,col="yellow",normalize=F,pcex=1, pch=19) # Representation of the ppca method on the Taylor diagram taylor.diagram(as.numeric(Obs$Tmax),as.numeric(imp_ppca$Tmax), add=TRUE,col="#1FFED8",normalize=F,pcex=1, pch=19) # Ajouter une légende legend(1.3,1.5,legend=c("imputeTS","knn","mice","missForest","ppca"), pch=19,col=c("red","#0000FF","orange","yellow","#1FFED8"),cex=0.6) # Repeat the same procedure for the other 3 variables, i.e. # Tmin, Tmoy and Prcp b.) Kling-Gupta Efficiency (KGE) library(hydroGOF) KGE(Obs$Tmax,imp_ImputeTS$Tmax,method="2012") KGE(Obs$Tmax,imp_knn$Tmax,method="2012") KGE(Obs$Tmax,imp_mice$Tmax,method="2012") KGE(Obs$Tmax,imp_miss$Tmax,method="2012") KGE(Obs$Tmax,imp_ppca$Tmax,method="2012")

Figure 1 :

 1 Figure 1: Presentation of the study area (right), geographical location of the 11 meteorological stations of Senegal and percentage of there missing data (left).

Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :

 23456 Figure 2: Evaluation of the performance of imputation methods with the Taylor diagram for the maximum temperature in the 5% deficit scenario.

Figure 7 :Figure 8 :Figure 9 :Figure 10 :Figure 11 :

 7891011 Figure 7: Evaluation of the performance of imputation methods with the Taylor diagram for rainfall in the 5% deficit scenario.

  Algorithm 3 Impute missing values with MissForest 1. Make an initial estimate of missing values 2. k ← vector of sorted indices of columns in X 3. while not ζ do: 4. X imp old ← store previously imputed matrix 5. for j in k do: 6. Fit a random forest predicting non-missing values of X p : y

		(j) obs ∼ x	(j) obs
	7. Predict y miss using x (j) (j) miss 8. X imp new ← update imputed matrix, using predicted y	(j) miss
	9. end for
	10. update ζ
	11. end while
	12. return X imp (final imputed matrix)

Table 1 :

 1 Kling-Gupta Efficiency values for 5% of missing data

	Stations	Methods	Tmax	Tmin	Tmoy	Rainfall
		imputeTS	0.9865404	0.993449	0.9945015	0.9791523*
		knn	0.9913801**	0.9968463**	0.9983326**	0.9511953
	Dakar	mice	0.98926	0.9944122	0.9961835	0.9523733
		missForest	0.9918108*	0.9968882*	0.9986567*	0.9467742
		ppca	0.9609837	0.9633031	0.9631444	0.9786464**
		imputeTS	0.9835714	0.9831029	0.9824979	0.9787246**
		knn	0.9872866**	0.9922031**	0.9958493*	0.9533001
	Diourbel	mice	0.9820596	0.9857274	0.9929311	0.9075048
		missForest	0.9882202*	0.9922049*	0.9957484**	0.9780226
		ppca	0.9614748	0.9631652	0.9651033	0.9802646*
		imputeTS	0.9857776	0.9831615	0.9815469	0.9816835**
		knn	0.9898338**	0.9934953*	0.9958361*	0.9423991
	Kaolack	mice	0.9830087	0.9899196	0.993429	0.9588845
		missForest	0.9901377*	0.993473**	0.9957497**	0.9821197*
		ppca	0.9643667	0.9635497	0.9630784	0.980103
		imputeTS	0.9866526	0.9801753	0.9801625	0.976709*
		knn	0.9931234**	0.9966309*	0.9973178**	0.9540655
	Kedougou	mice	0.9871749	0.9933905	0.9948324	0.9495064
		missForest	0.9933127*	0.9961542**	0.9987452*	0.9712559
		ppca	0.9674373	0.9648475	0.9619325	0.973956**
		imputeTS	0.9866557	0.9824853	0.9803995	0.9764007*
		knn	0.9917396*	0.9960442**	0.9970255*	0.9617717
	Kolda	mice	0.9845515	0.993894	0.9953118	0.9703627
		missForest	0.991047**	0.9965738*	0.9969733**	0.9759152**
		ppca	0.9667154	0.9632489	0.9632126	0.972823
		imputeTS	0.9864733	0.9837789	0.9828467	0.9769688*
		knn	0.9937659**	0.9947636**	0.9975958*	0.9578589
	Linguere	mice	0.990216	0.9925821	0.9961286	0.9681812
		missForest	0.9938736*	0.9954301*	0.9975107**	0.9737952
		ppca	0.9623165	0.9630348	0.9630211	0.9762678**
		imputeTS	0.9874799	0.9861619	0.9874464	0.9713865**
		knn	0.9945376**	0.9958507*	0.9972229**	0.9405689
	Matam	mice	0.9908603	0.993387	0.9958952	0.9647158
		missForest	0.9948816*	0.995802**	0.9974089*	0.9752687*
		ppca	0.9635225	0.9630854	0.9628401	0.9706273
		imputeTS	0.9858443	0.9824782	0.9855335	0.9752229**
		knn	0.9932766	0.9965744**	0.9979183*	0.9427042
	Podor	mice	0.9913298	0.9950445	0.9953457	0.9635889
		missForest	0.9934186**	0.9968041*	0.9978009**	0.9752141
		ppca	0.9935254*	0.9950492	0.9977413	0.9767951*
		imputeTS	0.9821086	0.9896404	0.9922694	0.9745764*
		knn	0.9877594**	0.9934764*	0.9964357**	0.952844
	Saint-louis	mice	0.9801834	0.9887986	0.993268	0.9643499
		missForest	0.9879188*	0.9934561**	0.9964775*	0.965519
		ppca	0.9626198	0.962475	0.9644535	0.974359**
		imputeTS	0.9905763	0.9859656	0.9905723	0.9649452*
		knn	0.9934872**	0.9926156**	0.9962607**	0.9444012
	Tamba	mice	0.9903511	0.9901326	0.9946909	0.9226171
		missForest	0.9938854*	0.9937758*	0.9966363*	0.9615129
		ppca	0.9643679	0.9662771	0.9645768	0.9634039**
		imputeTS	0.9852339	0.9860766	0.9813073	0.9721407*
		knn	0.9868091*	0.9903742**	0.9914166**	0.9559325
	Ziguinchor	mice	0.9797848	0.9857397	0.9860004	0.9632859
		missForest	0.986755**	0.9909888*	0.9917572*	0.9713052 **
		ppca	0.9657004	0.9622084	0.9624807	0.9685283
	* : Most efficient method
	** : Second most efficient method

Table 2 :

 2 Kling-Gupta Efficiency values for 10% of missing data

	Stations	Methods	Tmax	Tmin	Tmoy	Rainfall
		imputeTS	0.9742983	0.9864979	0.9909158	0.9512005*
		knn	0.9835933*	0.9913948*	0.9962769*	0.8886353
	Dakar	mice	0.973679	0.9847564	0.9913977	0.9401131
		missForest	0.983133**	0.990773**	0.9961248**	0.9365231
		ppca	0.9246999	0.9264849	0.9254661	0.951091**
		imputeTS	0.9690529	0.969751	0.9631348	0.9440101*
		knn	0.973288**	0.9807138*	0.9882758**	0.9103046
	Diourbel	mice	0.9567858	0.9713056	0.9819863	0.870513
		missForest	0.9756013*	0.9795403**	0.9884941*	0.9287741
		ppca	0.9244631	0.9277226	0.9266385	0.9400512**
		imputeTS	0.9712198	0.9654686	0.9625625	0.935039*
		knn	0.9749044**	0.9836308*	0.9876659**	0.8935629
	Kaolack	mice	0.964391	0.9740824	0.9821262	0.9175003
		missForest	0.9765756*	0.9836033**	0.9877323*	0.9163366
		ppca	0.9235411	0.9253951	0.9273283	0.9303957**
		imputeTS	0.9738781	0.9581549	0.9610708	0.9435184*
		knn	0.9846228**	0.9914743*	0.9939307*	0.9194818
	Kedougou	mice	0.9657183	0.9816317	0.9876904	0.916222
		missForest	0.9846293*	0.9909756**	0.9931505**	0.9318037
		ppca	0.9275517	0.9251908	0.925359	0.9357381**
		imputeTS	0.9723472	0.9659642	0.9631914	0.9602853*
		knn	0.9805787*	0.9888951**	0.9919841**	0.9249435
	Kolda	mice	0.9657158	0.9804727	0.9844629	0.9100413
		missForest	0.9805072**	0.9901115*	0.9919874*	0.9561171**
		ppca	0.9253652	0.9244662	0.925865	0.9520566
		imputeTS	0.9742215	0.9654038	0.9679253	0.9553972*
		knn	0.9836709**	0.9853011**	0.9923785**	0.8995276
	Linguere	mice	0.9756955	0.9790786	0.9881078	0.9033303
		missForest	0.9846965*	0.9868898*	0.9934569*	0.9410949
		ppca	0.9252903	0.9252601	0.9266173	0.9530688**
		imputeTS	0.974245	0.9720889	0.9774673	0.9457648*
		knn	0.9845535**	0.9886217*	0.9935792**	0.8725973
	Matam	mice	0.9770006	0.9826105	0.9899651	0.7967105
		missForest	0.9855236*	0.9884442**	0.9939428*	0.9508656
		ppca	0.9258248	0.9248583	0.927853	0.9447019**
		imputeTS	0.9727686	0.9689566	0.9728239	0.7507307**
		knn	0.9849382**	0.9911875**	0.9936215**	0.7497202
	Podor	mice	0.978146	0.9857734	0.990697	0.745625
		missForest	0.9853519*	0.9913851*	0.9946925*	0.754012*
		ppca	0.9848894	0.9887638	0.9947152	0.7471575
		imputeTS	0.9672531	0.9806488	0.9812949	0.9305436**
		knn	0.9706623*	0.9823451**	0.9869663**	0.8698441
	Saint-louis	mice	0.9548545	0.9736495	0.9801286	0.701353
		missForest	0.9702862**	0.9830735*	0.9872416*	0.9294397
		ppca	0.9282804	0.9247957	0.9240631	0.9308417*
		ImputeTS	0.9815279	0.9710154	0.9800543	0.9579458*
		knn	0.9834133**	0.9816936**	0.9904125**	0.916992
	Tamba	mice	0.9725739	0.9711661	0.9842544	0.9370051
		missForest	0.9843444*	0.9822263*	0.9909438*	0.9498022
		ppca	0.9256659	0.9299564	0.9272146	0.9539049**
		imputeTS	0.969303	0.9750733	0.9660592	0.9249013*
		knn	0.9686838**	0.9787809*	0.9798637*	0.9085949
	Ziguinchor	mice	0.9540606	0.9672709	0.9680786	0.8829261
		missForest	0.9697603*	0.978204**	0.9796364**	0.9210572**
		ppca	0.9216131	0.9289487	0.9267662	0.9153979
	* : Most efficient method
	** : Second most efficient method

Table 3 :

 3 Kling-Gupta Efficiency values for 20% of missing data

	Stations	Methods	Tmax	Tmin	Tmoy	Rainfall
		imputeTS	0.950391	0.9705658	0.9849741	0.8967307*
		knn	0.9607593**	0.9750774*	0.9857613**	0.7181958
	Dakar	mice	0.9441618	0.9611632	0.976273	0.72424
		missForest	0.9637968*	0.9744868**	0.9862538*	0.8539797
		ppca	0.8432188	0.8423417	0.8429205	0.8900698**
		imputeTS	0.9340079	0.9354445	0.9242161	0.8862514*
		knn	0.9389426**	0.9562784*	0.9729727**	0.8003527
	Diourbel	mice	0.904265	0.9263232	0.9524887	0.8779328
		missForest	0.9436502*	0.9553339**	0.9740977*	0.8407321
		ppca	0.8432292	0.8435042	0.8431742	0.8765237**
		imputeTS	0.9395557	0.9255107	0.924127	0.8669699*
		knn	0.9444442**	0.9614671*	0.9720787*	0.7978136
	Kaolack	mice	0.9110506	0.9322998	0.9526561	0.8150224
		missForest	0.9458438*	0.9563822**	0.9687265**	0.8243893
		ppca	0.8459225	0.841268	0.8482069	0.8550712**
		imputeTS	0.9416419	0.9125086	0.9233509	0.884596*
		knn	0.9594384**	0.974097**	0.9807917**	0.850061
	Kedougou	mice	0.9217074	0.9463948	0.9634472	0.8555764
		missForest	0.960714*	0.9786135*	0.9829236*	0.8620142
		ppca	0.8432254	0.8400276	0.8437931	0.8692026**
		imputeTS	0.9405598	0.9284303	0.9246609	0.8415878**
		knn	0.9484852**	0.9695288**	0.9750071**	0.8248824
	Kolda	mice	0.9164409	0.9391756	0.9611785	0.8201832
		missForest	0.9531797*	0.9768731*	0.9793212*	0.8456972*
		ppca	0.8421375	0.8412171	0.8424865	0.8251458
		imputeTS	0.9436239	0.9229241	0.9308659	0.8870548**
		knn	0.9546697**	0.954274**	0.9693137**	0.7910099
	Linguere	mice	0.9382864	0.946674	0.9681658	0.7549926
		missForest	0.961255*	0.9661018*	0.9820246*	0.8904868*
		ppca	0.8437567	0.8387873	0.8446292	0.8767768
		imputeTS	0.9503072	0.9412743	0.9485721	0.8910474*
		knn	0.9590702**	0.9605299**	0.9740292**	0.8237312
	Matam	mice	0.9416208	0.9522463	0.9716193	0.8723237
		missForest	0.9632512*	0.9718421*	0.9833947*	0.8880761**
		ppca	0.8486919	0.8421232	0.8445653	0.8826692
		imputeTS	0.9410155	0.9289479	0.9404773	0.8886997*
		knn	0.9613556**	0.9702743**	0.9769638**	0.7469454
	Podor	mice	0.9388928	0.9556975	0.97219993	0.8633523
		missForest	0.9622411*	0.9732124*	0.9815529*	0.8340305
		ppca	0.9400722	0.9515189	0.9742363	0.8853863**
		imputeTS	0.9222525	0.9600159*	0.9661419**	0.837998**
		knn	0.9283175*	0.9455097	0.9623939	0.7466041
	Saint-louis	mice	0.8942903	0.9316363	0.9475551	0.8380671*
		missForest	0.9280098**	0.9552623**	0.9681688*	0.7872174
		ppca	0.8432123	0.841914	0.8477032	0.8344962
		ImputeTS	0.9626397*	0.9291145	0.9587488	0.8212936*
		knn	0.9559973	0.9353456**	0.9706511**	0.7770106
	Tamba	mice	0.9304526	0.9218699	0.9576508	0.7972693
		missForest	0.9583325**	0.9472575*	0.9734317*	0.781293
		ppca	0.8457186	0.838419	0.842911	0.8110117**
		imputeTS	0.9386362*	0.9416573**	0.9297139	0.8835026*
		knn	0.9272946	0.9271487	0.9433288**	0.8210452
	Ziguinchor	mice	0.89517	0.9166376	0.9192091	0.8076997
		missForest	0.9324179**	0.9440498*	0.9485731*	0.854701
		ppca	0.8476705	0.8416495	0.8455593	0.8616168**
	* : Most efficient method
	** : Second most efficient method

Table 4 :

 4 Kling-Gupta Efficiency values for 30% of missing data

	Stations	Methods	Tmax	Tmin	Tmoy	Rainfall
		imputeTS	0.9228503	0.9548006*	0.9726009*	0.8390918*
		knn	0.9322443**	0.9465732**	0.9629172	0.6078904
	Dakar	mice	0.8944558	0.9154454	0.9424125	0.8169906
		missForest	0.9324997*	0.9446708	0.9642721**	0.6432893
		ppca	0.7496204	0.7365367	0.7438693	0.829298**
		imputeTS	0.8891576**	0.8949092	0.8797889	0.6624315*
		knn	0.8865375	0.9202124*	0.9442642**	0.5724693
	Diourbel	mice	0.8321536	0.862689	0.9007464	0.6081533
		missForest	0.8966499*	0.917145**	0.9463423*	0.6184351
		ppca	0.7407879	0.7390653	0.7476448	0.6408472**
		imputeTS	0.8973604**	0.8838226**	0.8694179	0.7724092*
		knn	0.892247	0.8650311	0.8893751	0.6584461
	Kaolack	mice	0.8482048	0.879266	0.9084778**	0.7218114
		missForest	0.9011723*	0.9179111*	0.9336725*	0.6920713
		ppca	0.7428712	0.7414575	0.7455324	0.7528001**
		imputeTS	0.9087337	0.8620114	0.8829047	0.8104502*
		knn	0.9243209**	0.9332529**	0.9504369**	0.7193065
	Kedougou	mice	0.8702912	0.9028594	0.9233666	0.7002237
		missForest	0.9301424*	0.9424834*	0.95978*	0.7706161
		ppca	0.7531945	0.7391302	0.750186	0.7896617**
		imputeTS	0.9051777	0.8870052	0.8833441	0.8097123*
		knn	0.9065135**	0.9279021**	0.9403336**	0.6908563
	Kolda	mice	0.8444811	0.887341	0.9154191	0.776868
		missForest	0.9127943*	0.9499927*	0.9562014*	0.6711384
		ppca	0.7450693	0.7397894	0.7467644	0.7888154**
		imputeTS	0.9041598	0.8808717	0.8887467	0.7752905*
		knn	0.9109898**	0.886127	0.9176081	0.5320912
	Linguere	mice	0.8838301	0.8952219**	0.9298969**	0.7081255
		missForest	0.9199163*	0.9383304*	0.9590979*	0.7500961
		ppca	0.7378156	0.7450558	0.7458015	0.7634604**
		imputeTS	0.9150636	0.9048661	0.9206437	0.7273582*
		knn	0.9218779**	0.9073571	0.9311018	0.6689536
	Matam	mice	0.8951845	0.9088126**	0.9375384**	0.7132893
		missForest	0.9304248*	0.942168*	0.9616126*	0.671495
		ppca	0.7478506	0.7406403	0.7493904	0.7213824**
		imputeTS	0.9076218	0.8844121	0.9083581	0.7362469*
		knn	0.9288364**	0.9194323**	0.9420349**	0.4870299
	Podor	mice	0.9035971	0.9169244	0.9411934	0.6643826
		missForest	0.9350338*	0.9411229*	0.9607898*	0.702136
		ppca	0.8798987	0.8720476	0.9086679	0.7333422**
		imputeTS	0.8738798**	0.9379593*	0.936904*	0.6485662*
		knn	0.867566	0.9106304	0.9234925	0.493455
	Saint-louis	mice	0.8191599	0.8738503	0.8978311	0.5981901
		missForest	0.8775714*	0.911398**	0.9349973**	0.5046398
		ppca	0.7506856	0.7403021	0.745371	0.6441117**
		imputeTS	0.935663*	0.8885364**	0.9365367**	0.724632*
		knn	0.9214956	0.8367745	0.9340487	0.6289067
	Tamba	mice	0.8752414	0.8643785	0.9117643	0.7047774
		missForest	0.9241495**	0.903018*	0.94743*	0.6782567
		ppca	0.7498903	0.7379066	0.7497728	0.7091748**
		imputeTS	0.9005876*	0.9076701*	0.8903064	0.6932359*
		knn	0.8747723	0.8883877	0.8984108**	0.6498602
	Ziguinchor	mice	0.8131378	0.8411145	0.8569924	0.6275325
		missForest	0.8829549**	0.8929018**	0.9025141*	0.6219546
		ppca	0.7527299	0.7387578	0.7506435	0.6509218**
	* : Most efficient method
	** : Second most efficient method

Table 5 :

 5 Kling-Gupta Efficiency values for 40% of missing data

	Stations	Methods	Tmax	Tmin	Tmoy	Rainfall
		imputeTS	0.8856484*	0.9399577*	0.9605577*	0.7660358**
		knn	0.8407157	0.820518	0.8532427	0.3682654
	Dakar	mice	0.8217149	0.8550309	0.8889577	0.6291245
		missForest	0.8827918**	0.9018561**	0.9291653**	0.4484993
		ppca	0.6309381	0.614952	0.626679	0.7687713*
		imputeTS	0.8526575*	0.8516765	0.833122	0.5191244**
		knn	0.8242182	0.8603592**	0.8999033*	0.3831761
	Diourbel	mice	0.7595402	0.7746986	0.8437466	0.5553183*
		missForest	0.833535**	0.8703098*	0.895995**	0.5069784
		ppca	0.6361637	0.6220728	0.6328867	0.4885774
		imputeTS	0.8621449*	0.8274867**	0.8197836	0.6049044**
		knn	0.8302686	0.7436004	0.789068	0.5022114
	Kaolack	mice	0.7617875	0.7980448	0.8426663**	0.6236818*
		missForest	0.8469074**	0.841653*	0.8824627*	0.4045348
		ppca	0.6415341	0.6105613	0.6322836	0.5821172
		imputeTS	0.8805808	0.8104596	0.8301805	0.7455153*
		knn	0.8858372**	0.8745086**	0.9030213**	0.5769327
	Kedougou	mice	0.7831516	0.8264797	0.8712785	0.6951563
		missForest	0.8893422*	0.9190936*	0.9303633*	0.6506133
		ppca	0.655487	0.6256248	0.6330058	0.7157075**
		imputeTS	0.8705011*	0.8412936	0.836422	0.729596*
		knn	0.8423884	0.8585757**	0.8807279**	0.581001
	Kolda	mice	0.7629503	0.8029311	0.8589139	0.6496704
		missForest	0.857861**	0.9032349*	0.919949*	0.6639466
		ppca	0.6329923	0.6239965	0.6331516	0.7033501**
		imputeTS	0.8722742*	0.8248804**	0.8363292	0.7180198*
		knn	0.8452929	0.7809121	0.827412	0.5507138
	Linguere	mice	0.8098703	0.8204197	0.8719036**	0.6677765
		missForest	0.8694665**	0.8712147*	0.9158729*	0.6862811
		ppca	0.6317723	0.6217724	0.6197068	0.7059145**
		imputeTS	0.8829898*	0.8660074**	0.8852338**	0.6411213*
		knn	0.8645121	0.8145704	0.8551335	0.3899555
	Matam	mice	0.8235815	0.8371875	0.8784854	0.6105589
		missForest	0.8801762**	0.8870121*	0.9183331*	0.5017798
		ppca	0.6368766	0.6178205	0.6306486	0.6333343**
		imputeTS	0.8750853	0.8322942	0.8626633	0.4518334
		knn	0.8753888**	0.8381627	0.8820695	0.1650486
	Podor	mice	0.8329619	0.8416616**	0.8892205**	0.4808118*
		missForest	0.8806573*	0.8875104*	0.915009*	0.132102
		ppca	0.8236627	0.8064937	0.8452624	0.4550327**
		imputeTS	0.8205976*	0.9161803*	0.9071439*	0.7049484*
		knn	0.7821344	0.8412565	0.8656093	0.438566
	Saint-louis	mice	0.73688	0.7930482	0.8288616	0.6061936
		missForest	0.8062597**	0.8533662**	0.8821286**	0.3223085
		ppca	0.6446107	0.6228857	0.6302942	0.6955473**
		ImputeTS	0.916623*	0.8418315**	0.9101121*	0.709508*
		knn	0.8655534	0.7004157	0.8829522	0.4775905
	Tamba	mice	0.7973542	0.7748208	0.8498525	0.6633525
		missForest	0.8749291**	0.8442031*	0.9002511**	0.6572301
		ppca	0.6475122	0.6169807	0.6366385	0.6843639**
		imputeTS	0.8650835*	0.8690561*	0.8462582*	0.6888916*
		knn	0.7895692	0.7058349	0.815677	0.5566903
	Ziguinchor	mice	0.7294888	0.7518505	0.7778171	0.591241
		missForest	0.8124636**	0.8151284**	0.8420022**	0.5320508
		ppca	0.6480634	0.6136565	0.639697	0.6318854**
	* : Most efficient method
	** : Second most efficient method

R script for the 5 imputa- tion methods

  Representation of the imputeTS method on the Taylor diagram library(plotrix) taylor.diagram(as.numeric(Obs$Tmax),as.numeric(imp_ImputeTS$Tmax),

	imp_knn <-kNN(iris_NA)
	# mice
	library(mice)
	imp_mice <-complete(mice(iris_NA,m=10,method="pmm",seed=101))
	# missForest
	library(missForest)
	iris.imputed <-missForest(iris_NA)
	imp_missForest <-iris.imputed$ximp
	# ppca
	library(missMDA)
	library(FactoMineR)
	ncomp <-estim_ncpPCA(iris_NA)
	res.imp <-imputePCA(iris_NA,ncp = ncomp$ncp)
	imp_ppca <-res.imp$completeObs
	C R script for evaluating the
	performance of imputation
	methods
	a.) Taylor diagramm
	#
	# imputeTS
	library(imputeTS)
	imp_ImputeTS=na_kalman(iris_NA)
	# k-nn
	library(VIM)