Original: T ABULAR D ATA I MPUTATION C HOOSE KNN OVER D EEP L EARNING
Revised: T ABULAR D ATA I MPUTATION C HOOSE KNN OVER D EEP L EARNING

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Recent infatuation for Artiﬁcial Neural Networks have led to the development of complex and powerful algorithms for data imputation.
Revised: Recent infatuation for Artiﬁcial Neural Networks have led to the development of complex and powerful algorithms for data imputation

Original: This study is  the ﬁrst to compare state-of-the-art deep-learning models with the well-established KNN algorithm (1951).
Revised: This study is compares the best state-of-the-art deep -learning models with the well-established KNN algorithm (1951).

Original: By using real-world and generated datasets in various missing data scenarios, we show that the good old KNN algorithm is still competitive (nay better) than powerful deep-learning algorithms for tabular data imputation.
Revised: By using real- world and generated datasets in various missing data scenarios, we show that the good old KNN algorithm is still competitive (nay better) than powerful deep-learning algorithms for tabular data imputation.

Original: This work advocates foran appropriate and reasonable use of machine learning.
Revised: This work advocates for an appropriate and reasonable use of machine learning

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Missing values constitute a serious issue in data science.
Revised: Missing values constitute a serious issue in data science.

Original: Incomplete datasets result from uncollected, lost or voluntarily deleted observations.
Revised: Incomplete datasets result from uncollected, lost or voluntarily deleted observations.

Original: Most machine learning algorithms cannot handle datasets with missing values, and meticulous data preprocessing is therefore needed.
Revised: Most machine learning algorithms cannot handle datasets with missing values, and meticulous data preprocessing is therefore needed.

Original: The most common preparatory processing consists of Complete Case Analysis, where only complete observations are kept for further data analysis.
Revised: The most common preparatory processing consists of Listwise deletion, where only complete observations are kept for further data analysis.

Original: This solution, although straightforward to implement, has two main disadvantages: itcan signiﬁcantly reduce the size of the dataset and induce a bias.
Revised: This solution, although straightforward to implement, has two main disadvantages: it signiﬁcantly reduces the size of the dataset and induces a bias, which may lead to false positive research claims (Lall (2016)).

Original: Data imputation is an alternative preprocessing method which involves estimation and replacement of the missing values.
Revised: Data imputation is an alternative preprocessing method which involves estimation and replacement of the missing values.

Original: It allows to preserve the whole dataset for subsequent analysis but requires careful handling as it can also introduce a bias in the imputed dataset.
Revised: It allows to preserve the whole dataset for subsequent analysis but requires careful handling as it can also introduce a bias in the imputed dataset (Gelman & Hill (2006)).

Original: In particular, only ”wellbehaved” values can be imputed and missing values of outlier observations will get overlooked.
Revised: In particular, only ”well-behaved” values can be imputed and missing values of outlier observations will get overlooked (Binder (1996)).

Original: Several data imputation algorithms have been proposed over the last 70 years: Hot-deck, Colddeck, Mean-value substitution (Kalton & Kasprzyk (1986)) , C4.5 algorithm
Revised: Several data imputation algorithms have been proposed over the last 70 years: Hot-deck, Colddeck, Mean-value substitution (Kalton & Kasprzyk (1986)) , C4.5 algorithm

Original: (Salzberg (1994)), CN induction algorithm
Revised: (Salzberg (1994)), CN induction algorithm

Original: (Clark & Niblett (1989)), KNN (originally developed in 1951 - Fix & Hodges (1989) - and later expanded by Altman (1992)).
Revised: (Clark & Niblett (1989)), KNN (originally developed in 1951 - Fix & Hodges (1989) - and later expanded by Altman (1992)).

Original: These traditional machine learning algorithms have been shown useful in the past.
Revised: These traditional machine learning algorithms have been shown useful in the past.

Original: Notably, KNN has shown best imputation quality amongst these various data imputation algorithms (Batista & Monard (2002)).
Revised: Notably, KNN has shown best imputation quality amongst these various data imputation algorithms (Batista & Monard (2002), Bertsimas et al. Poulos &

Revised: Valle (2018), Jadhav et al.

Original: Over the last decade, Artiﬁcial Neural Networks (ANN) have transformed the ﬁelds of statistics, computer science and data analysis.
Revised: Over the last decade, Artiﬁcial Neural Networks (ANN) have transformed the ﬁelds of statistics, computer science and data analysis.

Original: More precisely, Generative Adversarial Networks (GANs) have shown spectacular results at generating new observations from a given data distribution.
Revised: More precisely, Generative Adversarial Networks (GANs) have shown spectacular results at generating new observations from a given data distribution.

Original: As such,
Revised: As such,

Original: GAN models have been developed to tackle the problem of missing values in data science.
Revised: GAN models have been developed to tackle the problem of missing values in data science.

Original: GAIN (Yoon et al.
Revised: GAIN (Yoon et al.

Original: and MisGAN (Li et al.
Revised: and MisGAN (Li et al.

Original: (2019)) claim state-of-the-art data imputation results.
Revised: (2019)) claim state-of-the-art data imputation results.

Original: While GAIN has been speciﬁcally developed for tabular data imputation, MisGAN is primarily designed to impute degraded images and can be adapted to work with tabular data.
Revised: While GAIN has been speciﬁcally developed for tabular data imputation, MisGAN is primarily designed to impute degraded images and can be adapted to work with tabular data.

Original: This work intends to compare data imputation performances of state-of-the-art GAN models (GAIN and MisGAN) with the standard already established K-nearest neighbours algorithm (KNN).
Revised: This work intends to compare data imputation performances of state-of-the-art GAN models (GAIN and MisGAN) with the standard already established K-nearest neighbours algorithm (KNN).

Original: We evaluate the imputation quality in MCAR, MAR and MNAR scenarios (see Section 2).
Revised: We evaluate the imputation quality in MCAR, MAR and MNAR scenarios (see Section 2).

Original: The present study is restricted to tabular data, that is numerical data we can arrange into rows and columns in the form of a table with cells.
Revised: The present study is restricted to tabular data, that is numerical data we can arrange into rows and columns in the form of a table with cells.

Original: Section 2 explains the various missing data scenarios and their properties.
Revised: Section 2 explains the various missing data scenarios and their properties.

Original: Section 3 presents the algorithms used in this work.
Revised: Section 3 presents the algorithms used in this work.

Original: Section 4 introduces the datasets.
Revised: Section 4 introduces the datasets.

Original: Section 5 details the method of this study.
Revised: Section 5 details the method of this study.

Original: Section 6 shows the data imputation results.
Revised: Section 6 shows the data imputation results.

Original: Section 7 includes a summary, a discussion and ﬁnal recommendations.
Revised: Section 7 includes a summary, a discussion and ﬁnal recommendations.

Original:  M ISSING D ATA S ETTINGS
Revised:  M ISSING D ATA S ETTINGS

Original: Let x denote the complete vector of a given observation and m its missing value binary mask, so that the actual available data is the element-wise product (cid:101) x
Revised: Let x denote the complete vector of a given observation and m its missing value binary mask, so that the actual available data is the element-wise product (cid:101) x

Original: = x (cid:12) m .
Revised: = x (cid:12) m .

Original: Data imputation methods seek to estimate the missing values of (cid:101) x by taking advantage of patterns in the dataset and using observed properties of (cid:101) x .
Revised: Data imputation methods seek to estimate the missing values of (cid:101) x by taking advantage of patterns in the dataset and using observed properties of (cid:101) x .

Original: The probability distribution of m is referred to as the missing data mechanism .
Revised: The probability distribution of m is referred to as the missing data mechanism .

Original: Following the classiﬁcation established by Little & Rubin (2014), missing data scenarios belong to one of the following three settings.
Revised: Following the classiﬁcation established by Little & Rubin (2014), missing data scenarios belong to one of the following three settings.

Original: [Equation] [Equation]
Revised: 2. M ISSING C OMPLETELY A T R ANDOM (MCAR)

Original: In MCAR scenarios, the missing data mechanism is assumed to be independent of the intrinsic probability distribution of x , and occurs completely at random, such that p ( m | x )
Revised: In MCAR scenarios, the missing data mechanism is assumed to be independent of the intrinsic probability distribution of x , and occurs completely at random, such that p ( m | x )

Original: = p ( m ) .
Revised: = p ( m ) .

Original: We can generate MCAR missing values in a complete dataset by hiding each cell with a given probability, for example 10%.
Revised: We can generate MCAR missing values in a complete dataset by hiding each cell with a given probability, for example 10%.

Original: There exist statistical tests to assess whether the MCAR assumption holds or not (Little (1988)).
Revised: There exist statistical tests to assess whether the MCAR assumption holds or not (Little (1988)).

Original: Apart from the obvious information loss, MCAR scenarios are convenient assumptions.
Revised: Apart from the obvious information loss, MCAR scenarios are convenient assumptions.

Original: Indeed, the subset of complete observations is free of bias, and we may therefore safely proceed with further data analysis techniques.
Revised: Indeed, the subset of complete observations is free of bias, and we may therefore safely proceed with further data analysis techniques.

Original: However, MCAR is often an unrealistically strong assumption in practice.
Revised: However, MCAR is often an unrealistically strong assumption in practice.

Original: 2. M ISSING A T R ANDOM (MAR)
Revised: 2. M ISSING A T R ANDOM (MAR)

Original: MAR scenarios occur when the missing data mechanism can be fully explained by the observed data.
Revised: MAR scenarios occur when the missing data mechanism can be fully explained by the observed data.

Original: We may write p ( m | x ) = p ( m | (cid:101) x ) .
Revised: We may write p ( m | x ) = p ( m | (cid:101) x ) .

Original: For example, males are less likely to complete a depression survey, regardless of their depression level.
Revised: For example, males are less likely to complete a depression survey, regardless of their depression level.

Original: As long as the gender of every participant is known, the dataset can be treated as MAR by assuming that missingness only depends on maleness.
Revised: As long as the gender of every participant is known, the dataset can be treated as MAR by assuming that missingness only depends on maleness.

Original: MAR assumption is more general than MCAR, but cannot be statistically tested and one must rely on its fairness.
Revised: MAR assumption is more general than MCAR, but cannot be statistically tested and one must rely on its fairness.

Original: Subsequent statistical analyses may include biases which can theoretically be corrected under the MAR assumption.
Revised: Subsequent statistical analyses may include biases which can theoretically be corrected under the MAR assumption.

Original: [Equation]
Revised: 2. M ISSING N OT A T R ANDOM (MNAR)

Original: Finally, MNAR scenarios encompass every other scenario: the reason why data is missing depends on unobserved variables.
Revised: Finally, MNAR scenarios encompass every other scenario: the reason why data is missing depends on unobserved variables.

Original: In the previous example, this is the case if respondents do not ﬁll the depression survey because of their actual depression level, or because of another unrecorded variable like their income for instance.
Revised: In the previous example, this is the case if respondents do not ﬁll the depression survey because of their actual depression level, or because of another unrecorded variable like their income for instance.

Original: We argue that all real-world datasets have missing values distributed in MNAR settings; MCAR and MAR are convenient assumptions to facilitate the treatment of incomplete datasets.
Revised: As MCAR and MAR are convenient assumptions to facilitate the treatment of incomplete datasets, most missing data mechanisms are actually distributed in a MNAR setting.

Original: While being truthful, MNAR scenarios have no systematic ways to handle missing values.
Revised: While being truthful, MNAR scenarios have no systematic ways to handle missing values.

Original: One strategy is to accumulate more data in the hope of understanding the missing data mechanism.
Revised: One strategy is to accumulate more data in the hope of understanding the missing data mechanism.

Original:  D ATA I MPUTATION A LGORITHMS
Revised:  D ATA I MPUTATION A LGORITHMS

Original: We select, analyse and compare the performances of the best algorithms to date.
Revised: We select, analyse and compare the performances of the best algorithms to date.

Original: While some tabular data imputation methods learn a probability distribution on a complete dataset before performing data imputation, we focus on algorithms which use the entire dataset (including missing values) to perform in-place imputations.
Revised: While some tabular data imputation methods learn a probability distribution on a complete dataset before performing data imputation, we focus on algorithms which use the entire dataset (including missing values) to perform in-place imputations.

Original: On the one hand, GAIN (Yoon et al.
Revised: On the one hand, GAIN (Yoon et al.

Original: and MisGAN (Li et al.
Revised: and MisGAN (Li et al.

Original: (2019)) are new state-of-the-art ANN generative models tailored for data imputation.
Revised: (2019)) are new state-of-the-art ANN generative models tailored for data imputation.

Original: On the other hand, the KNN algorithm is a non-parametric method which can be used for data imputation.
Revised: On the other hand, the KNN algorithm is a non-parametric method which can be used for data imputation.

Original: 3. P RESENTATION OF GAIN
Revised: 3. P RESENTATION OF GAIN

Original: Generative Adversarial Imputation Nets (GAIN) have been proposed in 2018 as a generative model speciﬁcally designed for tabular data imputation problems.
Revised: Generative Adversarial Imputation Nets (GAIN) have been proposed in 2018 as a generative model speciﬁcally designed for tabular data imputation problems.

Original: GAIN generalises the well-established architecture of GAN models by looking at individual cells rather than complete rows.
Revised: GAIN generalises the well-established architecture of GAN models by looking at individual cells rather than complete rows.

Original: The authors claim state-of-the art imputation quality performances (Yoon et al.
Revised: The authors claim state-of-the art imputation quality performances (Yoon et al.

Original: Given an incomplete observation, GAIN works as follows:
Revised: Given an incomplete observation, GAIN works as follows:

Original: Cells with missing values are ﬁlled by random noise while observed values are kept unchanged.
Revised: Cells with missing values are ﬁlled by random noise while observed values are kept unchanged.

Original: A binary mask telling whether each cell is genuine or comes from the random noise is concatenated to the observation, therefore doubling its length.
Revised: A binary mask telling whether each cell is genuine or comes from the random noise is concatenated to the observation, therefore doubling its length.

Original: This vector is provided to the GAIN generator which outputs a generated vector.
Revised: This vector is provided to the GAIN generator which outputs a generated vector.

Original: The original data vector is imputed by conserving genuine cells and replacing missing cells by their corresponding values in the generated vector.
Revised: The original data vector is imputed by conserving genuine cells and replacing missing cells by their corresponding values in the generated vector.

Original: The imputed vector is presented to the GAIN discriminator which tells apart genuine values from imputed values.
Revised: The imputed vector is presented to the GAIN discriminator which tells apart genuine values from imputed values.

Original: The GAIN discriminator works on a cell-by-cell basis instead of considering the row as a whole.
Revised: The GAIN discriminator works on a cell-by-cell basis instead of considering the row as a whole.

Original: Both GAIN generator’s and discriminator’s parameters are trainedtrained following the standard adversarial process used by GAN models.GAIN’s parameters are
Revised: Both GAIN generator’s and discriminator’s parameters are trained following the standard adversarial process used by GAN models.

Original:  updated to minimise the binary cross-entropy loss function for 
Revised: GAIN’s parameters are updated to minimise the binary cross-entropy loss function for the discriminator given by

Original: the

Original: discriminator

Original: given by

Original: [Equation]
Revised: [Equation]

Original: where m ij is 0 if cell ( i, j ) is missing or 1 otherwise, b ij = 0 indicates cells that have not been hinted to the discriminator, and (cid:99)
Revised: where m ij is 0 if cell ( i, j ) is missing or 1 otherwise, b ij = 0 indicates cells that have not been hinted to the discriminator, and (cid:99)

Original: y ij is the discriminator prediction for cell ( i, j ) ; and a combination of binary cross-entropy (for the generated cells) and RMSE for the discriminator given by
Revised: y ij is the discriminator prediction for cell ( i, j ) ; and a combination of binary cross-entropy (for the generated cells) and RMSE for the discriminator given by

Original: [Equation]
Revised: [Equation]

Original: with
Revised: with

Original: [Equation]
Revised: [Equation]

Revised: −

Revised: m ij ) log( (cid:99) y ij )

Original: and
Revised: and

Revised: L G

Revised: 2 ( i )

Revised: = − (cid:88)

Original: [Equation]
Revised: [Equation]

Original: where ˜ x ij is the generator output and x ij the actual observation for cell ( i, j ) .
Revised: where ˜ x ij is the generator output and x ij the actual observation for cell ( i, j ) .

Original: GAIN has two hyperparameters to manually tune: the hint fraction h of true labels to provide the discriminator to stabilise learning, and α the relative weight between cross-entropy and RMSE for the generator loss.
Revised: GAIN has two hyperparameters to manually tune: the hint fraction h of true labels to provide the discriminator to stabilise learning, and α the relative weight between cross-entropy and RMSE for the generator loss.

Original: After several trails, we decide to use the recommended default values of Yoon et al.
Revised: After several trails, we decide to use the recommended default values of Yoon et al.

Original: Besides, the most relevant hyperparameter for GAIN is the number of training epochs, which we discuss in Section 5.
Revised: Besides, the most relevant hyperparameter for GAIN is the number of training epochs, which we discuss in Section 5.

Original: 3. P RESENTATION OF M IS GAN
Revised: 3. P RESENTATION OF M IS GAN

Original: MisGAN has been introduced in 2019 as another GAN model framework capable of handling complex datasets with missing values.
Revised: MisGAN has been introduced in 2019 as another GAN model framework capable of handling complex datasets with missing values.

Original: Primarily developed for image completion, it can be adapted to handle tabular data.
Revised: Primarily developed for image completion, it can be adapted to handle tabular data.

Original: MisGAN claims state-of-the-art imputation quality for images (Li et al.
Revised: MisGAN claims state-of-the-art imputation quality for images (Li et al.

Original: In its original design, MisGAN is composed of three pairs of generator and discriminator.
Revised: In its original design, MisGAN is composed of three pairs of generator and discriminator.

Original: A ﬁrst pair ( G x , D x ) attempts to model the probability distribution of the data, while another pair ( G m , D m ) tries to model the missing data mechanism.
Revised: A ﬁrst pair ( G x , D x ) attempts to model the probability distribution of the data, while another pair ( G m , D m ) tries to model the missing data mechanism.

Original: Both generators G x and G m are used in parallel to produce fake deteriorated observations.
Revised: Both generators G x and G m are used in parallel to produce fake deteriorated observations.

Original: Finally, a third pair ( G i , D i ) is used to perform the imputation of missing values.
Revised: Finally, a third pair ( G i , D i ) is used to perform the imputation of missing values.

Original: When adapted to tabular data, the missing data mechanism has a simple distribution and does not require complex neural networks to be modelled.
Revised: When adapted to tabular data, the missing data mechanism has a simple distribution and does not require complex neural networks to be modelled.

Original: Instead, we choose to draw missing masks directly from the observations in the dataset.
Revised: Instead, we choose to draw missing masks directly from the observations in the dataset.

Original: Like for GAIN, the performances of MisGAN mostly depends on the number of training epochs.
Revised: Like for GAIN, the performances of MisGAN mostly depends on the number of training epochs.

Original: Section 5 explains how we ﬁx the number of training epochs for MisGAN.
Revised: Section 5 explains how we ﬁx the number of training epochs for MisGAN.

Original: The artiﬁcial neural network architecture of GAIN and MisGAN are both presented in Appendix A.
Revised: The artiﬁcial neural network architecture of GAIN and MisGAN are both presented in Appendix A.

Original: 3. P RESENTATION OF KNN
Revised: 3. P RESENTATION OF KNN

Original: The K-Nearest Neighbours algorithm is a non-parametric method, originally developed for classiﬁcation tasks in 1951 (Altman (1992)).
Revised: The K-Nearest Neighbours algorithm is a non-parametric method, originally developed for classiﬁcation tasks in 1951 (Altman (1992)).

Original: For data imputation tasks, the KNN algorithm selects the
Revised: For data imputation tasks, the KNN algorithm selects the

Original: K nearest neighbours of a given incomplete observation, and uses available data from the selected neighbours to estimate missing values.
Revised: K nearest neighbours of a given incomplete observation, and uses available data from the selected neighbours to estimate missing values.

Original: Despite its simplicity and its age, the KNN algorithm has been shown to outperform traditional data imputation algorithms (Batista & Monard (2002)).
Revised: Despite its simplicity and its age, the KNN algorithm has been shown to outperform traditional data imputation algorithms (Batista & Monard (2002)).

Original: KNN imputes missing values using a weighted average of the selected neighbours.
Revised: KNN imputes missing values using a weighted average of the selected neighbours.

Original: The most two common weighting systems are uniform and distance , respectively deﬁned by 1 K (where K is the number of neighbours) and 1 d ij (where d ij is the distance between observations i and j .
Revised: The most two common weighting systems are uniform and distance , respectively deﬁned by 1 K (where K is the number of neighbours) and 1 d ij (where d ij is the distance between observations i and j .

Original: We respectively refer to these algorithms with the names KNN-uniform and KNN-distance.
Revised: We respectively refer to these algorithms with the names KNN-uniform and KNN-distance.

Original:  D ATASETS
Revised:  D ATASETS

Original: Real-world and well-behaved simulated datasets are both used in this comparative study.
Revised: Real-world and well-behaved simulated datasets are both used in this comparative study.

Original: The realworld datasets are taken from the open access UCI Machine Learning Repository (Dua & Graff (2019)).
Revised: The realworld datasets are taken from the open access UCI Machine Learning Repository (Dua & Graff (2019)).

Original: For the sake of reproducibility, we make use of the same datasets used by authors of GAIN (Yoon et al.
Revised: For the sake of reproducibility, we make use of the same datasets used by authors of GAIN (Yoon et al.

Original: And for the sake variety, two additional datasets have been selected: white wine and red wine datasets.
Revised: And for the sake variety, two additional datasets have been selected: white wine and red wine datasets.

Original: Table 1 shows the size of the nine datasets.
Revised: Table 1 shows the size of the nine datasets.

Original: We insist that a comprehensive understanding of the data at play is the most important preliminary step before any data analysis.
Revised: We insist that a comprehensive understanding of the data at play is the most important preliminary step before any data analysis.

Original: Accordingly, Appendix B provides all essential details about the data used in this work.
Revised: Accordingly, Appendix B provides all essential details about the data used in this work.

Original: It is worth noting that all following datasets are originally complete, i.e. they do not have missing values in the ﬁrst place.
Revised: It is worth noting that all following datasets are originally complete, i.e. they do not have missing values in the ﬁrst place.

Original:  M ETHODS
Revised:  M ETHODS

Original: This section describes the methodology of this work.
Revised: This section describes the methodology of this work.

Original: We begin with ﬁnding the appropriate hyperparameters of the algorithms used.
Revised: We begin with ﬁnding the appropriate hyperparameters of the algorithms used.

Original: Then, we conduct several experiments with varying missing rates, missing data scenarios and datasets.
Revised: Then, we conduct several experiments with varying missing rates, missing data scenarios and datasets.

Original: Throughout the following experiments, we ﬁrst inject missing values and immediately scale the datasets in the interval [0, 1].
Revised: Throughout the following experiments, we ﬁrst inject missing values and immediately scale the datasets in the interval [0, 1].

Original: After imputation, the performances are computed using the RMSE between ground truth and imputed missing values directly on the scaled variables.
Revised: After imputation, the performances are computed using the RMSE between ground truth and imputed missing values directly on the scaled variables.

Original: Each experiment is repeated 20 times to allow for the computation of the mean and the standard deviation of the RMSE reported in each subsequent plot.
Revised: Each experiment is repeated 20 times to allow for the computation of the mean and the standard deviation of the RMSE reported in each subsequent plot.

Original: 5. T UNING THE HYPERPARAMETERS
Revised: 5. T UNING THE HYPERPARAMETERS

Original: We use the generated mixture of three Gaussians dataset (c.f. mydata2 dataset, Appendix B.7) in MCAR scenario with 20% missing rate to select appropriate hyperparameters for the various algorithms of this comparative study. 
Revised: We use the generated mixture of three Gaussians dataset (c.f. mydata2 dataset, Appendix B.7) in MCAR scenario with 20% missing rate to select appropriate hyperparameters for the various algorithms of this comparative study. For the following hyperparameter search, we ﬁrst introduce missing values and then use the algorithms with their pre-selected hyperparameters values and repeat this experiemtn 20 times. We report the mean and standard deviation of the imputation RMSE in Appendix C to select the best hyperparameter values.

Original: GAIN and MisGAN need the number of training epochs to be speciﬁed.
Revised: GAIN and MisGAN need the number of training epochs to be speciﬁed.

Original: We train from 1,000 to 20,000 epochs by steps of 1,000 epochs and choose the number of training epochs which minimises the imputation RMSE (see Appendix C.1): 20,000 epochs for GAIN and 5,000 epochs for MisGAN.
Revised: We train from 1,000 to 20,000 epochs by steps of 1,000 epochs and choose the number of training epochs which minimises the imputation RMSE (see Appendix C.1): 20,000 epochs for GAIN and 5,000 epochs for MisGAN.

Original: MisGAN becomes unstable after 7,000 epochs.
Revised: MisGAN becomes unstable after 7,000 epochs.

Original: As GAIN performances keep decreasing, we train with more epochs (see Appendix C.2) and decide to stick with 20,000 epochs.
Revised: As GAIN performances keep decreasing, we train with more epochs (see Appendix C.2) and decide to stick with 20,000 epochs.

Original: For KNN, the hyperparameter is the number of neighbours K .
Revised: For KNN, the hyperparameter is the number of neighbours K .

Original: We use from 2 to 300 neighbours by steps of 2, and select K = 50 neighbours to minimise the imputation RMSE for both weight systems (see Appendix C.3).
Revised: We use from 2 to 300 neighbours by steps of 2, and select K = 50 neighbours to minimise the imputation RMSE for both weight systems (see Appendix C.3).

Original: Note that these hyperparameters may strongly depend on the dataset size.
Revised: Note that these hyperparameters may strongly depend on the dataset size.

Original: The dataset used for hyperparameters tuning has 1,000 rows.
Revised: The dataset used for hyperparameters tuning has 1,000 rows.

Original: For following experiments, we compute a multiplicative factor to preserve similar proportions f
Revised: For following experiments, we compute a multiplicative factor to preserve similar proportions f

Original: = n 1000 , where n is the number of rows in the current dataset.
Revised: = n 1000 , where n is the number of rows in the current dataset.

Original: The number of training epochs accordingly becomes 5000 f for MisGAN and 20000 f for GAIN, and the number of neighbours is 50 f for both KNN algorithms.
Revised: The number of training epochs accordingly becomes 5000 f for MisGAN and 20000 f for GAIN, and the number of neighbours is 50 f for both KNN algorithms.

Original: As an example, a dataset two times larger will use twice as many neighbours and half the number of training epochs.
Revised: As an example, a dataset two times larger will use twice as many neighbours and half the number of training epochs.

Revised: We experimented this approach on two datasets: wine white and breast .

Revised: The former has about ﬁve times more entries than mydata2 dataset, while the later has about half.

Revised: Appendix C.4 discuss the reliability of the hyperparameter selection tuning process.

Original: 5. E XPERIMENTS
Revised: 5. E XPERIMENTS

Original: The following experiments are conducted 20 times to compute the mean and standard deviation of the imputation RMSE.
Revised: The following experiments are conducted 20 times to compute the mean and standard deviation of the imputation RMSE.

Original: Results and interpretation are provided in the next section.
Revised: Results and interpretation are provided in the next section.

Original: Varying the missing rate in MCAR setting from 10% to 80% by steps of 10%, with the generated mixture of three Gaussians dataset. 2.
Revised: Varying the missing rate in MCAR setting from 10% to 80% by steps of 10%, with the generated mixture of three Gaussians dataset. 2.

Original: Performances on real-world datasets in MCAR setting with 20% missing rate.
Revised: Performances on real-world datasets in MCAR setting with 20% missing rate.

Original: Performances over all datasets in MAR setting, with overall 20% and 45% missing rates.
Revised: Performances over all datasets in MAR setting, with overall 20% and 45% missing rates.

Original: Performances over all datasets in MNAR setting, with overall 20% and 45% missing rates.
Revised: Performances over all datasets in MNAR setting, with overall 20% and 45% missing rates.

Original: To generate MAR missing values, we select a column that we keep untouched throughout the full
Revised: To generate MAR missing values, we select a column that we keep untouched throughout the full

Original: MAR experiment.
Revised: MAR experiment.

Original: We compute the quantiles of the selected variable, scale the quantiles between 0 and 2 µ (where µ is the overall missing rate) and interpret these values as common missing rates for the remaining columns.
Revised: We compute the quantiles of the selected column, scale the quantiles betweenand 2 µ (where µ is the overall missing rate) and interpret these values as common missing rates for all other columns.

Original: For example, in MAR setting with an overall 20% missing rate, the quantiles of the selected column are scaled in the range [0% , 40%] .
Revised: For example, in MAR setting with an overall 20% missing rate, the quantiles of the selected column are scaled in the range [0% , 40%] .

Original: The selected variable for each dataset along with its meaning are reported in Table 1.[Table] [Table] [Table]
Revised: The selected variable for each dataset along with its meaning are reported in Table 1.

Original: credit (30000, 14)

Original: [Table]

Original: letter (20000, 16)

Original: [Table]

Original: news (39644, 44)

Original: [Table]

Original: spam (4601, 57)

Original: [Table]

Original: wine red (1599, 12)

Original: [Table]
Revised: [Table]

Original: wine white (4898, 12)

Original: [Table]

Original: mydata (1000, 5)

Original: [Table] mydata (1000, 5) [Table] To generate MNAR missing values, we compute the quantiles for every individual column (in the same way as MAR setting) which we scale in the interval [0 , 2 µ ] .
Revised: To generate MNAR missing values, we compute the quantiles for every individual column (in the same way as MAR setting) which we scale in the interval [0 , 2 µ ] .

Original: We interpret these values as the missing probability for every corresponding cell.
Revised: We interpret these values as the missing probability for every corresponding cell.

Original: In MNAR settings, the higher the value of a cell relatively to its column, the more likely it will be missing.
Revised: In MNAR settings, the higher the value of a cell relatively to its column, the more likely it will be missing.

Original: Looking at the ﬁrst column of the mixture of three Gaussian dataset for example, data from the orange cluster is more likely to be missing than data from the blue cluster (see Fig. 8 in Appendix B).
Revised: Looking at the ﬁrst column of the mixture of three Gaussian dataset for example, data from the orange cluster is more likely to be missing than data from the blue cluster (see Fig. 5 in Appendix B).

Original:  R ESULTS AND I NTERPRETATION 6. V ARYING THE MISSING RATE IN MCAR SETTING
Revised:  R ESULTS AND I NTERPRETATION 6. V ARYING THE MISSING RATE IN MCAR SETTING

Original: The ﬁrst experiment consists in changing the missing rate from 10% to 80% (by steps of 10%) on the generated mixture of three Gaussians dataset in MCAR setting.
Revised: The ﬁrst experiment consists in changing the missing rate from 10% to 80% (by steps of 10%) on the generated mixture of three Gaussians dataset in MCAR setting ( mydata2 ).

Original: The imputation results are shown in 
Revised: The imputation results are shown in Fig.

Original: Fig.
Revised: 1 (a).

Original: As the RMSE for MisGAN is comparatively large, we only display the results of GAIN and KNN for clarity (for complete results and discussion on MisGAN, see Fig.
Revised: As the RMSE for MisGAN is comparatively large, we only display the results of GAIN and KNN for clarity (for complete results and discussion on MisGAN, see Fig.

Original: 12 in Appendix D.1).
Revised: 11 in Appendix D.1).

Original: We decide not to show the results of MisGAN from now on (provided in the supplementary materials).
Revised: We decide not to show the results of MisGAN from now on (they are provided in the supplementary materials, in Appendix D.1). For numerical values, see Tables in Appendix G. [Figure] e a

Original: Not surprisingly, the performances of GAIN and KNN deteriorate with increasing missing rates.
Revised: [Figure] c r e d i t [Figure] e tt e r

Revised: [Figure]

Revised: p a m

Revised: [Figure]

Revised: e

Revised: [Figure]

Revised: n

Revised: [Figure]

Revised: w

Revised: [Figure]

Revised: e

Revised: [Figure]

Revised: Not surprisingly, the performances of GAIN and KNN deteriorate with increasing missing rates.

Original: GAIN remains competitive with both KNN algorithms.
Revised: GAIN remains competitive with both KNN algorithms.

Original: It is worth noting that KNN-uniform performs better at low missing rates, while KNN-distance performs better at high missing rates.
Revised: It is worth noting that KNN-uniform performs better at low missing rates, while KNN-distance performs better at high missing rates.

Original: Intuitively, the weight system provides KNN-distance with better resilience to bad neighbours (which are more common with high missing rates).
Revised: Intuitively, the weight system provides KNN-distance with better resilience to bad neighbours (which are more common with high missing rates).

Original: 6. R EAL - WORLD DATASETS IN MCAR SETTING
Revised: 6. R EAL - WORLD DATASETS IN MCAR SETTING

Original: The next experiment uses real-world datasets in MCAR setting with 20% missing rate.
Revised: The next experiment uses real-world datasets in MCAR setting with 20% missing rate.

Original: Fig.
Revised: Fig.

Original: 2 shows the imputation quality results, where we can see that both KNN algorithms perform better than
Revised: 1 (b) shows the imputation quality results, where we can see that both KNN algorithms perform better than

Original: GAIN over all available datasets.
Revised: GAIN over all available datasets.

Original: From now, we also decide not to report the RMSE for the news dataset since the performances are chaotic regardless of the imputation method (for explanation, discussion and the complete results, see Fig.13 in Appendix D.2). [Figure] three Gaussian dataset
Revised: From now, we also decide not to report the RMSE for the news dataset since the performances are chaotic regardless of the imputation method (for explanation, discussion and the complete results, see Fig.

Original: [Figure] 1000x [Figure] [Figure]
Revised: 12 in Appendix D.2).

Original: We could not reproduce GAIN results.
Revised: We could not reproduce GAIN results.

Original: Intriguingly, the imputation of GAIN in this work outperforms the imputation from the original authors on the credit dataset: RMSE of 0.063 here [Figure] and 0.186 in Yoon et al.
Revised: Intriguingly, the imputation of GAIN in this work outperforms the imputation from the original authors on the credit dataset: RMSE of 0.063 here, and 0.186 in Yoon et al.

Original: For the other datasets, the performances of GAIN in this work are slightly lower than the original authors’ reported RMSE.
Revised: For the other datasets, the performances of GAIN in this work are slightly lower than the original authors’ reported RMSE.

Revised: Exhaustive comparison is available in Appendix E.

Original: 6. MAR EXPERIMENTS
Revised: 6. MAR EXPERIMENTS

Original:  With a missing rate of 20% in MAR setting (see Fig. 3), KNN algorithms perform slightly better than GAIN over the datasets provided.
Revised: This section refers to Fig. With a missing rate of 20% in MAR setting (left hand panel), both KNN algorithms perform slightly better than GAIN over all datasets.

Original: When we increase the missing rate to 45% (see Fig. 4) [Figure] we see that the imputation quality overall decays (note the y-axis scale).
Revised: When we increase the missing rate to 45% (right hand panel), we see that the imputation quality overall decays.

Original: The performances of GAIN, KNN-uniform and KNN-distance are now similar, with sometimes one method signiﬁcantly performing better.
Revised: The performances of GAIN, KNN-uniform and KNN-distance are now comparable, with sometimes one method signiﬁcantly performing better.

Revised: We notice that the trends and orders of magnitude are being preserved across datasets when the missing rate increases.

Revised: Look at the breast dataset for instance: it has a high RMSE with GAIN both with 20% or 45% missing rate data.

Original: 6. MNAR EXPERIMENTS
Revised: 6. MNAR EXPERIMENTS

Original:  In MNAR setting with a missing rate of 20% (see Fig. 5), there is no clear best imputation method.
Revised: This paragraph refers to Fig. In MNAR setting with a missing rate of 20% (left hand panel), there is no clear best imputation method.

Revised: All imputation RMSEs remain below 0.15 on average, and are very close to the imputation RMSEs obtained in the MAR setting (cf.

Original:  With a missing rate of 45% (see Fig. 6), we again note that the RMSE becomes  larger(even more than in the MAR experiments).
Revised: Fig. 2 (a)). With a missing rate of 45% (right hand panel), the RMSE becomes even larger than with the MAR experiments.

Original: We can argue that MNAR missing values are harder to recover than MAR missing values, since the missing data mechanism hides itself in the process of generating missing values.
Revised: We can argue that MNAR missing values are harder to recover than MAR missing values, since the missing data mechanism hides itself in the process of generating missing values.

Original: GAIN performs  better imputations than KNN in MNAR setting with 45% missing rate.
Revised: GAIN performs [Figure] better imputations than KNN in MNAR setting with 45% missing rate.

Original: For the large variance in the spam dataset RMSE, see Appendix D.2.
Revised: For the large variance in the spam dataset RMSE, see Appendix D.2.

Original: [Figure] [Figure] It is worth noting that KNN algorithms perform poorly against GAIN with the generated Gaussians mixture dataset, on which we benchmarked the hyperparameters.
Revised: It is worth noting that KNN algorithms perform poorly against GAIN with the generated Gaussians mixture dataset, on which we benchmarked the hyperparameters.

Original: The mixture of three Gaussians dataset indeed suffers a lot from the strong case of MNAR mechanism using quantiles
Revised: The mixture of three Gaussians dataset indeed suffers a lot from the strong case of MNAR mechanism using quantiles

Original: [Figure]
Revised: [Figure]

Original: as clusters with large value variables will undergo higher missing rates than clusters high low value variables
Revised: as clusters with large value variables will undergo higher missing rates than clusters high low value variables.

Original: [Figure]

Original: Consequently
Revised: Consequently

Original: [Figure]
Revised: [Figure]

Original: some clusters concentrate most of the missing data and become challenging to impute with KNN as they have few neighbours with complete observations
Revised: some clusters concentrate most of the missing data and become challenging to impute with KNN as they have few neighbours with complete observations

Original: [Figure]
Revised: [Figure]

Original: performances, MNAR

Original: [Table]

Original: [Figure]

Original: r e a s t c r e d i t l e tt e r s p a m w i n e _

Original: r e d w i n e _ w h

Original: i t

Original: [-e m y d a t a m y d a t a 
-]
Revised: [-e m y d a t a m y d a t a 
-]

Original: [Figure]

Original:  S UMMARY , D 
Revised:  S UMMARY , D ISCUSSION AND M ESSAGE 7. S UMMING UP

Original: AND M 7. S UMMING UP

Original: MisGAN was initially designed to impute missing rectangular pixel blocs in images.
Revised: MisGAN was initially designed to impute missing rectangular pixel blocs in images.

Original: While MisGAN claims state-of-the-art results for image imputation, this framework does not perform well for tabular data imputation, with large conﬁdence intervals in its predictions.
Revised: While MisGAN claims state-of-the-art results for image imputation, this framework does not perform well for tabular data imputation, with large conﬁdence intervals in its predictions.

Original: That being said, MisGAN can remain competitive on well behaved datasets with low missing rates.
Revised: That being said, MisGAN can remain competitive on well behaved datasets with low missing rates.

Original: GAIN has been developed for tabular data imputation and shows good imputation quality, especially in MNAR settings with high missing rates.
Revised: GAIN has been developed for tabular data imputation and shows good imputation quality, especially in MNAR settings with high missing rates.

Original: Like most ANN models, tuning the hyperparameters of MisGAN and GAIN is key to avoid over-training or under-training.
Revised: Like most ANN models, tuning the hyperparameters of MisGAN and GAIN is key to avoid over-training or under-training.

Original: Unlike lazy learning algorithms like KNN, Generative
Revised: Unlike lazy learning algorithms like KNN, Generative

Original: Adversarial Network architectures can learn complex relationships between variables.
Revised: Adversarial Network architectures can learn complex relationships between variables.

Original: This feature can explain the performances of GAIN in MAR and MNAR settings, where the network could ﬁnd meaningful patterns in the missingness.
Revised: This feature can explain the performances of GAIN in MAR and MNAR settings, where the network could ﬁnd meaningful patterns in the missingness.

Original: MCAR is an assumption that can be tested (as opposed to MAR assumption).
Revised: MCAR is an assumption that can be tested (as opposed to MAR assumption).

Original: Little’s Test of Missing
Revised: Little’s Test of Missing

Original: Completely At Random (Little (1988)) can be used to assess whether the MCAR hypothesis holds.
Revised: Completely At Random (Little (1988)) can be used to assess whether the MCAR hypothesis holds.

Original: The MAR and MNAR missing data scenarios used in this study are strong cases of dependence between the data distribution and the missing data mechanism.
Revised: The MAR and MNAR missing data scenarios used in this study are strong cases of dependence between the data distribution and the missing data mechanism.

Original: Lighter dependencies exist (see supplementary materials for GAIN, Yoon et al.
Revised: Lighter dependencies exist (see supplementary materials for GAIN, Yoon et al.

Original: Choosing a missing data mechanism is subjective (except for MCAR) and obviously affects the subsequent data imputation quality.
Revised: Choosing a missing data mechanism is subjective (except for MCAR) and obviously affects the subsequent data imputation quality.

Revised: This study shows that the use of massive deep-learning algorithms are not justiﬁed for tabular data imputation, and reinforces previous results: Bertsimas et al.

Revised: (2018), Jadhav et al.

Original:  Lastly, it appears that the imputation quality of an algorithm rather depends on the dataset itself than on the missing data mechanism per se: see the breast dataset where KNN systematically performs better than GAIN, and mydata2 (mixture of three Gaussian dataset) where GAIN performs better than KNN most of the time.
Revised: Poulos & Valle (2018), J¨ager et al. (2021). Tabular data imputation problems do not require complex machine learning algorithms as it was already established by Batista & Monard (2002). Lastly, it appears that the imputation quality of an algorithm rather depends on the dataset itself than on the missing data mechanism per se: see the breast dataset where KNN systematically performs better than GAIN, and mydata2 (mixture of three Gaussian dataset) where GAIN performs better than KNN most of the time.

Original: For this reason, we recommend to start any data processing with a thorough investigation of the dataset at play, like we show in Appendix B.
Revised: For this reason, we recommend to start any data processing with a thorough investigation of the dataset at play, like we show in Appendix B.

Original: 7. D ISCUSSION ON M ACHINE L EARNING
Revised: 7. D ISCUSSION ON M ACHINE L EARNING

Original: The development of ANN along with ”Deep Learning” is unquestionably an important milestone in Data Science.
Revised: The development of ANN along with ”Deep Learning” is unquestionably an important milestone in Data Science.

Original: For example, CNN have shown outstanding results in image processing where they can outperform most other traditional methods when properly used.
Revised: For example, Convolutional Neural Networks (CNNs) have shown outstanding results in image processing where they can outperform most other traditional methods when properly used.

Original: But as for tabular data imputation, traditional and well-established statistical methods are not yet obsolete.
Revised: But as for tabular data imputation, traditional and well-established statistical methods are not yet obsolete.

Original: There is also an inherent limitation in data imputation problems: we are left with what the (observed) data is.
Revised: There is also an inherent limitation in data imputation problems: we are left with what the (observed) data is.

Original: Consequently, it is particularly challenging to recover missing outliers.
Revised: Consequently, it is particularly challenging to recover missing outliers.

Original: On one hand, KNN algorithms require a distance matrix between each observation in the dataset to be computed.
Revised: On one hand, KNN algorithms require a distance matrix between each observation in the dataset to be computed.

Original: This step can take a couple of minutes if the dataset is large (see Appendix E).
Revised: This step can take a couple of minutes if the dataset is large (see Appendix E).

Original: Once computed, we can instantaneously try any desired number of neighbours for the imputation, which make KNN easily to ﬁne tune.
Revised: Once computed, we can instantaneously try any desired number of neighbours for the imputation, which make KNN easily to ﬁne tune.

Original: Also, KNN is a deterministic algorithm that can be straightforwardly reproduced and its results are easy to interpret.
Revised: Also, KNN is a deterministic algorithm that can be straightforwardly reproduced and its results are easy to interpret.

Original: On the other hand, it takes several minutes to train GAIN or MisGAN (for 20,000 and 5,000 epochs respectively, on a dataset with 1,000 observations).
Revised: On the other hand, it takes several minutes to train GAIN or MisGAN (for 20,000 and 5,000 epochs respectively, on a dataset with 1,000 observations).

Original: Fitting massive ANN models can become time and resource consuming.
Revised: Fitting massive ANN models can become time and resource consuming.

Original: In addition, the ANN ”black-box” architecture is often criticised.
Revised: In addition, the ANN ”black-box” architecture is often criticised.

Original: Stochastic imputations can be useful to derive probability distributions, but prevent from unambiguous replicability.
Revised: Stochastic imputations can be useful to derive probability distributions, but prevent from unambiguous replicability.

Original: Table 2 gives a summary of this comparison.
Revised: Table 2 gives a summary of this comparison.

Original: For more details on the computational training time, see Appendix E.
Revised: For more details on the computational training time, see Appendix E.

Original: [Table]
Revised: [Table]

Original: Advances in Machine Learning tend to showcase several ”state-of-the-art” algorithms, although only one may exist by deﬁnition.
Revised: Advances in Machine Learning tend to showcase several ”state-of-the-art” algorithms, although only one may exist by deﬁnition.

Original: Exhaustive comparative studies allow to share common benchmarks and fairly assess the performances of each method.
Revised: Exhaustive comparative studies allow to share common benchmarks and fairly assess the performances of each method.

Original: Finally, Machine Learning comes alongside with the improvement of computing power, smaller processors and a digital era with huge amount of data.
Revised: Finally, Machine Learning comes alongside with the improvement of computing power, smaller processors and a digital era with huge amount of data.

Original: But this era is also associated with overconsumption, resources problems and alarming ecological consequences.
Revised: But this era is also associated with overconsumption, resources problems and alarming ecological consequences.

Original: Digital sobriety advocates for using the Internet and technologies in a mindful and responsible way.
Revised: Digital sobriety advocates for using the Internet and technologies in a mindful and responsible way.

Original: In light of it, we urge researchers in Machine Learning to develop and promote appropriate and reasonable tools, adapted to the data and problem that we aim to address.
Revised: In light of it, we urge researchers in Machine Learning to develop and promote appropriate and reasonable tools, adapted to the data and problem that we aim to address.

Original: 7. T ABULAR DATA ANALYSIS RECIPE
Revised: 7. T ABULAR DATA ANALYSIS RECIPE

Original: Acknowledging and handling missing values is of major importance in data analysis.
Revised: Acknowledging and handling missing values is of major importance in data analysis.

Original: Since most
Revised: Since most

Original: Data Science algorithms cannot handle missing values, a straightforward solution consists in deleting incomplete observations.
Revised: Data Science algorithms cannot handle missing values, a straightforward solution consists in deleting incomplete observations.

Original: We strongly discourage from doing so: the reason why data is missing may tell a lot about an observation, and a missing value could even provide more information than an actual value, if utilised properly.
Revised: We strongly discourage from doing so: the reason why data is missing may tell a lot about an observation, and a missing value could even provide more information than an actual value, if utilised properly.

Original: It goes without saying that a great understanding of the dataset is the most important factor for a successful data analysis.
Revised: It goes without saying that a great understanding of the dataset is the most important factor for a successful data analysis.

Original: We suggest to follow these steps before any tabular data analysis with missing values:
Revised: We suggest to follow these steps before any tabular data analysis with missing values:

Original: Do not discard incomplete observations.
Revised: Do not discard incomplete observations.

Original: If possible, handle individual missing values by hand.
Revised: If possible, handle individual missing values by hand.

Original: Look for pattern in the missingness, and understand the missing data mechanism.
Revised: Look for pattern in the missingness, and understand the missing data mechanism.

Original: Test whether the MCAR assumption holds.
Revised: Test whether the MCAR assumption holds.

Original: If not, assess (empirically) if the MAR assumption seems valid.
Revised: If not, assess (empirically) if the MAR assumption seems valid.

Original: Perform the imputation with a suitable algorithm.
Revised: Perform the imputation with a suitable algorithm.

Original: KNN imputation is almost instantaneous while deep-learning models take time and energy to be trained.
Revised: KNN imputation is almost instantaneous while deep-learning models take time and energy to be trained.

Original: For its simplicity, reproducibility and intuitive interpretation, we encourage to use KNN for data imputation when possible.
Revised: For its simplicity, reproducibility and intuitive interpretation, we encourage to use KNN for data imputation when possible.

Original: E THICS STATEMENT
Revised: E THICS STATEMENT

Original: We assure that this review is the authors’ own original work and reﬂect the authors’ own research and analysis in a truthful and complete manner.
Revised: We assure that this review is the authors’ own original work and reﬂect the authors’ own research and analysis in a truthful and complete manner.

Original: The developers of the algorithms used in this study are appropriately credited.
Revised: The developers of the algorithms used in this study are appropriately credited.

Original: We also declare no conﬂict of interest or discrimination bias in this study.
Revised: We also declare no conﬂict of interest or discrimination bias in this study.

Original: This work reﬂects fairness concerns and responsible research practices.
Revised: This work reﬂects fairness concerns and responsible research practices.

Original: We are grateful to the developers of GAIN for their valuable input regarding the hyperparameters’ tuning of their algorithm.
Revised: We are grateful to the developers of GAIN for their valuable input regarding the hyperparameters’ tuning of their algorithm.

Original: R EPRODUCIBILITY STATEMENT
Revised: R EPRODUCIBILITY STATEMENT

Original: The code and the data used in this work are provided as supplementary materials in a zip folder.
Revised: The code and the data used in this work are provided as supplementary materials in a zip folder.

Original: We have used a pseudorandom number generator seed so that interested readers can faithfully reproduce all the plots of this work.
Revised: We have used a pseudorandom number generator seed, so that interested readers can faithfully reproduce all the plots of this work.

Original: We also welcome readers to change the seed to assess the statistical consistency of our results.
Revised: We also welcome readers to change the seed to assess the statistical consistency of our results.

Original: – Supplementary materials– A A PPENDIX : GAIN AND MI IS GAN ARCHITECTURES
Revised: Supplementary materials

Revised: A A PPENDIX : GAIN AND MI IS GAN ARCHITECTURES

Original: Table 3 and Table 4 present the architecture of both the generator and the discriminator of GAIN and
Revised: Table 3 and Table 4 present the architecture of both the generator and the discriminator of GAIN and

Original: MisGAN models respectively.
Revised: MisGAN models respectively.

Original: Following the architecture of GAIN (Yoon et al.
Revised: Following the architecture of GAIN (Yoon et al.

Original: , we adapted the input and output layers to the number c of columns in the dataset.
Revised: , we adapted the input and output layers to the number c of columns in the dataset.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: B A PPENDIX : D ATASETS DESCRIPTION
Revised: B A PPENDIX : D ATASETS DESCRIPTION

Original: Real-world and well-behaved simulated datasets are both used in this comparative study.
Revised: Real-world and well-behaved simulated datasets are both used in this comparative study.

Original: Seven real-world datasets are taken from the open access UCI Machine Learning Repository (Dua & Graff (2019)).
Revised: Seven real-world datasets are taken from the open access UCI Machine Learning Repository (Dua & Graff (2019)).

Original: Because an exhaustive comprehension of the data at play is the most important preliminary step before any data analysis, this appendix provides essential details about the data used in this work.
Revised: Because an exhaustive comprehension of the data at play is the most important preliminary step before any data analysis, this appendix provides essential details about the data used in this work.

Original: Note that all following datasets are complete: they do not show missing values in the ﬁrst place.
Revised: Note that all following datasets are complete: they do not show missing values in the ﬁrst place.

Original: The ﬁrst two columns of Table 1 shows the dataset names and sizes.
Revised: The ﬁrst two columns of Table 1 shows the dataset names and sizes.

Original: B. 
Revised: B.

Original: [Equation]
Revised: [Equation]

Original: Ten features are computed from a digitised image of a ﬁne-needle aspiration of a breast mass.
Revised: Ten features are computed from a digitised image of a ﬁne-needle aspiration of a breast mass.

Original: They describe characteristics of the cell nuclei shown in the image (Bennett & Mangasarian (1992)).
Revised: They describe characteristics of the cell nuclei shown in the image (Bennett & Mangasarian (1992)).

Original: Fineneedle aspiration is a diagnostic procedure which consists in sampling and examining cells under microscope (biopsy), in the search for lumps or tumours.
Revised: Fineneedle aspiration is a diagnostic procedure which consists in sampling and examining cells under microscope (biopsy), in the search for lumps or tumours.

Original: The dataset contains 596 observations and 32 columns.
Revised: The dataset contains 596 observations and 32 columns.

Original: The ﬁrst two columns are the ID number and the diagnosis of the cell (malignant or benign).
Revised: The ﬁrst two columns are the ID number and the diagnosis of the cell (malignant or benign).

Revised: The last 30 columns are comprised of the mean, the standard error and the mean of the largest three values of the following ten cell features: radius, texture, perimeter, area, smoothness, compactness, concavity, concave points, symmetry and fractal dimension.

Original: The last 30 columns are comprised of the
Revised: These last 30 columns are kept for data imputation comparison.

Original: B. D EFAULT OF CREDIT CARD CLIENTS D ATA S ET - CREDIT
Revised: B. D EFAULT OF CREDIT CARD CLIENTS D ATA S ET - CREDIT

Original: Default payments of 30000 customers in Taiwan have been recorded over six months.
Revised: Default payments of 30000 customers in Taiwan have been recorded over six months.

Original: The dataset comprises the following explanatory variables: total amount of given credit, gender, education, marital status, age, past payment records, amount of bill statement and amount of previous statements (Yeh & hui Lien (2009)).
Revised: The dataset comprises the following explanatory variables: total amount of given credit, gender, education, marital status, age, past payment records, amount of bill statement and amount of previous statements (Yeh & hui Lien (2009)).

Original: Regarding data imputation purposes, the categorical socio-demographic variables (gender, education and marital status) and the past payment records are removed.
Revised: Regarding data imputation purposes, the categorical socio-demographic variables (gender, education and marital status) and the past payment records are removed.

Original: In the end, the dataset includesrows and 14 columns.
Revised: In the end, the dataset includesrows and 14 columns.

Original: B. L ETTER R ECOGNITION D ATA S ET - LETTER
Revised: B. L ETTER R ECOGNITION D ATA S ET - LETTER

Original: Black and white images of the 26 uppercase letters from the roman alphabet are produced usingdifferent font styles and random distortions.
Revised: Black and white images of the 26 uppercase letters from the roman alphabet are produced usingdifferent font styles and random distortions.

Original: Sixteen features have been computed on the original images such that the resulting dataset comprises 20000 rows and 17 columns, where the ﬁrst column tells the corresponding letter and is therefore discarded for this study.
Revised: Sixteen features have been computed on the original images such that the resulting dataset comprises 20000 rows and 17 columns, where the ﬁrst column tells the corresponding letter and is therefore discarded for this study.

Original: The following 16 columns contain integer values ranging from 0 to 15, that we treat as continuous.
Revised: The following 16 columns contain integer values ranging from 0 to 15, that we treat as continuous.

Original: These 16 columns include features like the height and length of the image in pixels, the number of black pixels in the image, the horizontal and vertical average of black pixels or higher statistical moments (Frey & Slate (1991)).
Revised: These 16 columns include features like the height and length of the image in pixels, the number of black pixels in the image, the horizontal and vertical average of black pixels or higher statistical moments (Frey & Slate (1991)).

Original: B. O NLINE N EWS P OPULARITY D ATA S ET - NEWS
Revised: B. O NLINE N EWS P OPULARITY D ATA S ET - NEWS

Original: A large number of online news articles published by Mashable have been collected over the course of two years (Fernandes et al. (2015)).
Revised: A large number of online news articles published by Mashable have been collected over the course of two years (Fernandes et al. (2015)).

Original: The actual articles can be seen been accessing their corresponding URL link provided in the dataset.
Revised: The actual articles can be seen been accessing their corresponding URL link provided in the dataset.

Original: A series of 58 features have been computed on the news articles.
Revised: A series of 58 features have been computed on the news articles.

Original: They include 44 continuous variables - like number of words in the title - and 14 binary variables - indicating whether the article has been published a Monday, for example.
Revised: They include 44 continuous variables - like number of words in the title - and 14 binary variables - indicating whether the article has been published a Monday, for example.

Original: The dataset has 39644 rows and 61 columns.
Revised: The dataset has 39644 rows and 61 columns.

Original: Three additional columns have been discarded for data imputation purposes.
Revised: Three additional columns have been discarded for data imputation purposes.

Original: They include the URL link of the article and the time delay between the news article publication and the data acquisition (both of which are non-informative) as well as the number of shares of the news article (which is the original target variable).
Revised: They include the URL link of the article and the time delay between the news article publication and the data acquisition (both of which are non-informative) as well as the number of shares of the news article (which is the original target variable).

Original: B. S PAMBASE D ATA S ET - SPAM
Revised: B. S PAMBASE D ATA S ET - SPAM

Original: Features are extracted from both spam and non-spam e-mails.
Revised: Features are extracted from both spam and non-spam e-mails.

Original: The dataset comprises 4601 observations and 58 columns corresponding to the following attributes.
Revised: The dataset comprises 4601 observations and 58 columns corresponding to the following attributes.

Original: The ﬁrst 48 attributes are the frequency (in percent of words in the e-mail) of preselected words like ”will”, ”you” or ”free”.
Revised: The ﬁrst 48 attributes are the frequency (in percent of words in the e-mail) of preselected words like ”will”, ”you” or ”free”.

Original: The next six columns are the frequency (in percent) of the following preselected characters: ’;’, ’(’, ’[’, ’!’
Revised: The next six columns are the frequency (in percent) of the following preselected characters: ’;’, ’(’, ’[’, ’!’

Original: , ’$’, ’#’.
Revised: , ’$’, ’#’.

Original: The next three columns show the average length of uninterrupted sequence of uppercase letters, the length of the longest uninterrupted sequence of uppercase letters, and the total number of uppercase letters in the e-mail respectively.
Revised: The next three columns show the average length of uninterrupted sequence of uppercase letters, the length of the longest uninterrupted sequence of uppercase letters, and the total number of uppercase letters in the e-mail respectively.

Original: Finally, the last column indicates whether the corresponding e-mail was considered spam or not and has been discarded for data imputation purposes.
Revised: Finally, the last column indicates whether the corresponding e-mail was considered spam or not and has been discarded for data imputation purposes.

Original: B. W INE Q UALITY D ATA S ET - WINE RED AND WINE WHITE
Revised: B. W INE Q UALITY D ATA S ET - WINE RED AND WINE WHITE

Original: Various variables have been collected for red and white variants of the Portuguese Vinho Verde wines (Cortez et al.
Revised: Various variables have been collected for red and white variants of the Portuguese Vinho Verde wines (Cortez et al.

Original: The red wines dataset contains 1599 observations while the white wines one has 4989 observations.
Revised: The red wines dataset contains 1599 observations while the white wines one has 4989 observations.

Original: Both datasets have 12 columns corresponding to the following attributes: ﬁxed acidity, volatile acidity, citric acid, residual sugar, chlorides, free sulfur dioxide, total sulfur dioxide, density, pH, sulphates, alcohol and a ﬁnal score between 0 and 10 for the wine quality.
Revised: Both datasets have 12 columns corresponding to the following attributes: ﬁxed acidity, volatile acidity, citric acid, residual sugar, chlorides, free sulfur dioxide, total sulfur dioxide, density, pH, sulphates, alcohol and a ﬁnal score between 0 and 10 for the wine quality.

Original: B. G AUSSIAN DATASETS - MYDATA 1 AND MYDATA
Revised: B. G AUSSIAN DATASETS - MYDATA 1 AND MYDATA

Original: Two Gaussian datasets have been generated for comparison.
Revised: Two Gaussian datasets have been generated for comparison.

Original: We referred them as ”well behaved” datasets, in the sense that they do not show outliers as the other seven real-world datasets.
Revised: We referred them as ”well behaved” datasets, in the sense that they do not show outliers as the other seven real-world datasets.

Original: Also, their intrinsic probability distributions are perfectly determined.
Revised: Also, their intrinsic probability distributions are perfectly determined.

Original: Both Gaussian datasets have 1000 rows and 5 columns.
Revised: Both Gaussian datasets have 1000 rows and 5 columns.

Original: Correlation matrices with high coefﬁcients have been obtained by using random factors (see Appendix F).
Revised: Correlation matrices with high coefﬁcients have been obtained by using random factors (see Appendix F).

Original: The ﬁrst Gaussian dataset comprises a single multivariate Gaussian distribution (see pairplot, Fig.
Revised: The ﬁrst Gaussian dataset comprises a single multivariate Gaussian distribution (see pairplot, Fig.

Original: The second Gaussian dataset has been generated with a mixture of 3 Gaussians from independent correlation matrices obtained with with random factors (see pairplot, Fig. 
Revised: The second Gaussian dataset has been generated with a mixture of 3 Gaussians from independent correlation matrices obtained with with random factors (see pairplot, Fig. 5).

Original: The ﬁrst class (in blue) has 150 observations, the second class (in orange) has 300 observations and the third class (in green) has 550 observations.
Revised: The ﬁrst class (in blue) has 150 observations, the second class (in orange) has 300 observations and the third class (in green) has 550 observations.

Original: LabelClass0Class1Class
Revised: LabelClass0Class1Class

Original: C
Revised: C

Original: A PPENDIX : HYPERPARAMETERS SELECTION
Revised: A PPENDIX : HYPERPARAMETERS SELECTION

Original: C.
Revised: C.

Original: C HOOSING THE NUMBER OF EPOCHS FOR GAIN AND M IS GAN
Revised: C HOOSING THE NUMBER OF EPOCHS FOR GAIN AND M IS GAN

Original: [Figure]
Revised: [Figure]

Original: Mixture of three Gaussian dataset [1000x

Original: [Figure]

Original: See Fig.
Revised: See Fig.

Original: While the imputation performances for GAIN are consistently decreasing with a low variance when we increase the number of training epochs, MisGAN shows poor performances with a high variability after 7,000 training epochs (notice the large standard deviations).
Revised: While the imputation performances for GAIN are consistently decreasing with a low variance when we increase the number of training epochs, MisGAN shows poor performances with a high variability after 7,000 training epochs (notice the large standard deviations).

Original: Consequently, we respectively select 20,000 epochs and 5,000 epochs for GAIN and MisGAN hyperparameters.
Revised: Consequently, we respectively select 20,000 epochs and 5,000 epochs for GAIN and MisGAN hyperparameters.

Original: As the RMSE steadily decreases from 1,000 to 20,000 epochs in the case of GAIN, we decide to investigate more training epochs.
Revised: As the RMSE steadily decreases from 1,000 to 20,000 epochs in the case of GAIN, we decide to investigate more training epochs.

Original: See Appendix C.2 below.
Revised: See Appendix C.2 below.

Original: C. A DDITIONAL NUMBER OF EPOCHS FOR GAIN
Revised: C. A DDITIONAL NUMBER OF EPOCHS FOR GAIN

Original: See Fig.
Revised: See Fig.

Original: If there is a clear improvement from 10,000 to 20,000 epochs, there is no signiﬁcant change between 20,000 and 100,000 training epochs for GAIN imputation performances.
Revised: If there is a clear improvement from 10,000 to 20,000 epochs, there is no signiﬁcant change between 20,000 and 100,000 training epochs for GAIN imputation performances.

Original: We decide to stick with 20,000 training epochs (per 1,000 observations) for GAIN so that the computational cost does not become prohibitive.
Revised: We decide to stick with 20,000 training epochs (per 1,000 observations) for GAIN so that the computational cost does not become prohibitive.

Original: C. C HOOSING THE NUMBER OF NEIGHBOURS FOR KNN
Revised: C. C HOOSING THE NUMBER OF NEIGHBOURS FOR KNN

Original: See Fig. 
Revised: See Fig. 8.

Original: Both KNN algorithms have better imputation quality when we increase the number of neighbours from 2 to about 50.
Revised: Both KNN algorithms have better imputation quality when we increase the number of neighbours from 2 to about 50.

Original: KNN-uniform shows better imputation quality than KNN-distance for the generated mixture of Gaussians dataset with MCAR setting at 20% missing rate.
Revised: KNN-uniform shows better imputation quality than KNN-distance for the generated mixture of Gaussians dataset with MCAR setting at 20% missing rate.

Original: KNNuniform becomes worse than KNN-distance when the number of neighbours becomes to large.
Revised: KNNuniform becomes worse than KNN-distance when the number of neighbours becomes to large.

Original: Indeed, with the distance weight system, KNN-distance is less sensitive to newly included poor predictors when the number of neighbours becomes unnecessarily large.
Revised: Indeed, with the distance weight system, KNN-distance is less sensitive to newly included poor predictors when the number of neighbours becomes unnecessarily large.

Original: Despite the global minimum being around 70 neighbours for KNN-uniform and 120 for KNNdistance, we decide to select K = 50 neighbours (per 1,000 observations) as the hyperparameter for both KNN algorithms.
Revised: Despite the global minimum being around 70 neighbours for KNN-uniform and 120 for KNNdistance, we decide to select K = 50 neighbours (per 1,000 observations) as the hyperparameter for both KNN algorithms.

Original: The reason behind this choice is that the RMSE plateaus on a large number of neighbours and its standard deviation is too large to justify meticulous ﬁne-tuning on the number of neighbours.
Revised: The reason behind this choice is that the RMSE plateaus on a large number of neighbours and its standard deviation is too large to justify meticulous ﬁne-tuning on the number of neighbours.

Original: [Figure]
Revised: [Figure]

Original: three Gaussian
Revised: [Figure] C. V ERIFICATION OF THE HYPERPARAMETER SCALING PROCESS We see in Fig. 9 and 10 the imputation RMSE for GAIN with respect to the training epochs on the left panel, and for both KNN algorithms with respect to the number of neighbours on the right panel.

Original: [Figure]
Revised: Looking at Fig.

Revised: 9 corresponding to the wine white dataset ﬁrst, the scaling process introduced in Section 5.1 recommends to use 50004898 × 1000 ≈ 1000 epochs, and 50 × 48981000 ≈ 250 neighbours.

Original: [Figure]
Revised: We see that these approximations do not corresponds to the optimal hyperparameters. The optimal hyperparameters correspond to 5000 epochs for GAIN and approximately 25 neighbours for KNN. Now looking at Fig. 10 which uses the breast dataset, the recommend values obtained by the scaling process correspond to the optimal values for the algorithms hyperparameters.

Revised: Indeed, for

Revised: GAIN the number of epochs obtained with the scaling process is frac 5000569 × 1000 ≈ 9000 and for KNN the computed number of epochs is 50 × 5691000 ≈ 28 .

Revised: Looking at the RMSE, the optimal number of epochs for GAIN is 10,000 epochs, and the optimal number of neighbours for the KNN algorithms is 6 (which is lower than our estimate of 28).

Revised: Even if we see that the scaling process does not always yield optimal results, we decide to perform the whole analysis of this work following this scaling method.

Revised: The reason behind this choice is that an extensive hyperparameter search is not only extremely time and energy consuming, but it also impedes unambiguous reproducibility.

Original: D A PPENDIX : S KIPPED OVER PERFORMANCES D. P OOR IMPUTATION QUALITY WITH M IS GAN
Revised: D A PPENDIX : S KIPPED OVER PERFORMANCES D. P OOR IMPUTATION QUALITY WITH M IS GAN

Original: Fig.
Revised: Fig.

Original: 12 shows the imputation quality of the four algorithms presented in this study (including MisGAN), on the mixture of three Gaussians dataset with varying missing rates in MCAR setting.
Revised: 11 shows the imputation quality of the four algorithms presented in this study (including MisGAN), on the mixture of three Gaussians dataset with varying missing rates in MCAR setting.

Original: The imputation quality of MisGAN rapidly deteriorates when the missing rate increases.
Revised: The imputation quality of MisGAN rapidly deteriorates when the missing rate increases.

Original: Given the poor performances of MisGAN, we decide not to include any of its results in the study for clarity.
Revised: Given the poor performances of MisGAN, we decide not to include any of its results in the study for clarity.

Original: This can be explained because MisGAN has been developed as a deep-learning architecture tailored for deteriorated images restoration, while this work focuses on tabular data.
Revised: This can be explained because MisGAN has been developed as a deep-learning architecture tailored for deteriorated images restoration, while this work focuses on tabular data.

Original: We refer the interested reader to Li et al.
Revised: We refer the interested reader to Li et al.

Original: for further details on MisGAN.
Revised: for further details on MisGAN.

Original: Imputations with MisGAN have been conducted with all experiments of this work (see Fig. 13), but the results are not plotted for clarity.
Revised: Imputations with MisGAN have been conducted with all experiments of this work (see Fig. 12), but the results are not plotted for clarity.

Original: [Figure]
Revised: [Figure]

Original: of three Gaussian dataset
Revised: of three Gaussian dataset

Original: [Figure]
Revised: [Figure]

Original: 1000x
Revised: 1000x

Original: [Figure]

Original: D

Original: [Figure]
Revised: [Figure]

Original:  C HAOTIC IMPUTATION PERFORMANCES FOR THE NEWS DATASET
Revised: D. C HAOTIC IMPUTATION PERFORMANCES FOR THE NEWS DATASET

Original: Fig.
Revised: Fig.

Original: 13 presents the imputation performances of the four algorithms in this study for the seven real-world datasets in MCAR setting with 20% missing rate.
Revised: 12 presents the imputation performances of the four algorithms in this study for the seven real-world datasets in MCAR setting with 20% missing rate.

Original: We see that MisGAN systematically performs worst, and we therefore decide not to include its results for other experiments.
Revised: We see that MisGAN systematically performs worst, and we therefore decide not to include its results for other experiments

Original: [Figure]
Revised: [Figure]

Original: We note the huge variability in the imputation performances for the four algorithms over the news dataset.
Revised: We note the huge variability in the imputation performances for the four algorithms over the news dataset.

Original: The reason is that the news dataset has few extreme outliers, whose prediction will be particularly bad when randomly hidden before scaling the remaining data between [0 , 1] .
Revised: The reason is that the news dataset has few extreme outliers, whose prediction will be particularly bad when randomly hidden before scaling the remaining data between [0 , 1] .

Original: The same phenomenon happens on the spam dataset in the MNAR setting with an average missing rate of 45%: extreme outliers are consequently more likely to become missing values in the hiding process (because of the chosen MNAR setting using quantiles).
Revised: The same phenomenon happens on the spam dataset in the MNAR setting with an average missing rate of 45%: extreme outliers are consequently more likely to become missing values in the hiding process (because of the chosen MNAR setting using quantiles).

Original: We would like to mention that the news and spam datasets are not really suited for data imputation problems.
Revised: We would like to mention that the news and spam datasets are not really suited for data imputation problems.

Original: Using descriptive statistics (not presented in this work), we see that these two datasets are sparse with a lot of zeroes, and have extreme outliers.
Revised: Using descriptive statistics (not presented in this work), we see that these two datasets are sparse with a lot of zeroes, and have extreme outliers.

Original: This is not a problem per se, but we think that a manual treatment of missing values in such datasets is more appropriate.
Revised: This is not a problem per se, but we think that a manual treatment of missing values in such datasets is more appropriate.

Original: Unfortunately, they have been extensively used to benchmark Machine Learning algorithms.
Revised: Unfortunately, they have been extensively used to benchmark Machine Learning algorithms.

Original: We decided to include them in this comparative study since they have been used by Yoon et al.
Revised: We decided to include them in this comparative study since they have been used by Yoon et al.

Original: E A PPENDIX : Q UANTITATIVE ANALYSIS ON THE TRAINING TIME
Revised: E A PPENDIX : Q UANTITATIVE ANALYSIS ON THE TRAINING TIME

Original: Recall that the training strategy consists of using 20,000 and 5,000 training epochs per 1,000 observations for GAIN and MisGAN respectively, and using 50 neighbours per 1,000 observations for both KNN algorithms.
Revised: Recall that the training strategy consists of using 20,000 and 5,000 training epochs per 1,000 observations for GAIN and MisGAN respectively, and using 50 neighbours per 1,000 observations for both KNN algorithms.

Original: Table 5 shows the average training time for the algorithms used in this study.
Revised: Table 5 shows the average training time for the algorithms used in this study.

Original: KNN-uniform and KNN-distance having really similar training procedures, with grouped them into a common column.
Revised: KNN-uniform and KNN-distance having really similar training procedures, with grouped them into a common column.

Original: [Table]
Revised: [Table]

Original: The training time for GAIN and MisGAN remains nearly constant across datasets.
Revised: The training time for GAIN and MisGAN remains nearly constant across datasets.

Original: This is because of the multiplicative factor (introduced in Section 5.1) used to keep similar proportion when training the
Revised: This is because of the multiplicative factor (introduced in Section 5.1) used to keep similar proportion when training the

Original: ANN models.
Revised: ANN models.

Original: To trian GAIN for example, we use 20,000 training epochs for the dataset mydata which has 1,000 observations, but we use only 1,000 training for the letter dataset which shows 20,000 observations.
Revised: To trian GAIN for example, we use 20,000 training epochs for the dataset mydata which has 1,000 observations [Table] but we use only 1,000 training for the letter dataset which shows 20,000 observations.

Original: However, the training time for the KNN algorithms is strongly dependent on the dataset size.
Revised: However, the training time for the KNN algorithms is strongly dependent on the dataset size.

Original: Indeed, computing the distance matrix between each pair of rows can become computationally expensive for large datasets.
Revised: Indeed, computing the distance matrix between each pair of rows can become computationally expensive for large datasets.

Original: For the largest dataset, news , it takes about 3 minutes and 30 seconds to compute the distance matrix.
Revised: For the largest dataset, news , it takes about 3 minutes and 30 seconds to compute the distance matrix.

Original: That being said, there is no need to compute the distance matrix again when tuning the hyperparameter K of the number of neighbours used for imputation, while we need to retrain
Revised: That being said, there is no need to compute the distance matrix again when tuning the hyperparameter K of the number of neighbours used for imputation, while we need to retrain

Original: GAIN or MisGAN from scratch if we change the number of training epochs, their hyperparameter in this case.
Revised: GAIN or MisGAN from scratch if we change the number of training epochs, their hyperparameter in this case.

Original: F A PPENDIX : R ANDOM FACTOR METHOD FOR CORRELATION MATRICES
Revised: F A PPENDIX : R ANDOM FACTOR METHOD FOR CORRELATION MATRICES

Original: Generating random correlation matrices can be easily done by transforming a random square matrix
Revised: Generating random correlation matrices can be easily done by transforming a random square matrix

Original: A into a symmetric positive deﬁnitive matrix.
Revised: A into a symmetric positive deﬁnitive matrix.

Original: In our case, we draw the entries of A from a standard
Revised: In our case, we draw the entries of A from a standard

Original: Gaussian distribution, and we compute A T
Revised: Gaussian distribution, and we compute A T

Original: A symmetric positive deﬁnite matrix.
Revised: A symmetric positive deﬁnite matrix.

Original: However, this method yields matrices with non-diagonal elements close to zeros.
Revised: However, this method yields matrices with non-diagonal elements close to zeros.

Original: To obtain a symmetric deﬁnite positive matrix of dimension n with higher correlation coefﬁcients outside the diagonal, we start by choosing a number of factors 0 < k < n .
Revised: To obtain a symmetric deﬁnite positive matrix of dimension n with higher correlation coefﬁcients outside the diagonal, we start by choosing a number of factors 0 < k < n .

Original: We draw a rectangular matrix A of size ( k, n ) whose entries are independently sampled from a standard Gaussian distribution.
Revised: We draw a rectangular matrix A of size ( k, n ) whose entries are independently sampled from a standard Gaussian distribution.

Original: We compute the matrix A T A which is symmetric of size ( n, n ) , but not full rank.
Revised: We compute the matrix A T A which is symmetric of size ( n, n ) , but not full rank.

Original: We draw a diagonal matrix D whose entries have random positive noise, such that W =
Revised: We draw a diagonal matrix D whose entries have random positive noise, such that W =

Original: A T A
Revised: A T A

Original: + D is now invertible.
Revised: + D is now invertible.

Original: The code to generate the Gaussian data used is this study is provided as supplementary material.
Revised: The code to generate the Gaussian data used is this study is provided as supplementary material.

Revised: G A PPENDIX : N UMERICAL IMPUTATION RMSE RESULTS

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: 44) mydata

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: 82) mydata

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]