Probabilistic Missing Value Imputation for Mixed Categorical and Ordered Data
Authors: Yuxuan Zhao
Alex Townsend
Madeleine Udell
Venue: NeurIPS 2022 Accept (2022)
Keywords: Categorical data
missing value imputation
mixed data

Abstract: Many real-world datasets contain missing entries and mixed data types including categorical and ordered (e.g. continuous and ordinal) variables. Imputing the missing entries is necessary, since many data analysis pipelines require complete data, but challenging especially for mixed data. This paper proposes a probabilistic imputation method using an extended Gaussian copula model that supports both single and multiple imputation. The method models mixed categorical and ordered data using a latent Gaussian distribution. The unordered characteristics of categorical variables is explicitly modeled using the argmax operator. The method makes no assumptions on the data marginals nor does it require tuning any hyperparameters. Experimental results on synthetic and real datasets show that imputation with the extended Gaussian copula outperforms the current state-of-the-art for both categorical and ordered variables in mixed data.

Probabilistic Missing Value Imputation for Mixed Categorical and Ordered Data
Yuxuan Zhao Cornell University y222950cornel1.eed
Alex Townsend Comell University tornsend@cornell.c edu
Madeleine Udell Stanford University udell@stanf ord. edu
Abstract
Many real-world datasets contain missing entries and mixed data types including categorical and ordered (e.g. continuous and ordinal) variables. Imputing the missing entries necessary, since many data analysis pipelines require complete data, but this is challenging especially for mixed data. This paper proposes proba- bilistic imputation method using an extended Gaussian copula model that supports both single and multiple imputation. The method models mixed categorical and ordered data using a latent Gaussian distribution. The unordered characteristics of categorical variables is explicitly modeled using the argmax operator. The method makes no assumptions on the data marginals nor does it require tuning any hyperpa- rameters. Experimental results synthetic and real datasets show that imputation with the extended Gaussian copula outperforms the current state-of-the- for both categorical and ordered variables in mixed data.
Introduction
Modern datasets from healthcare, the sciences, and the social sciences often contain missing entries and mixed data types such as continuous, ordinal, and categorical. Social survey datasets, for example, typically mixed because they include variables like age (continuous), demographic group (categorical), and Likert scales (ordinal) measuring how strongly a respondent agrees with certain stated opinions. Continuous variables are encoded real numbers and sometimes called numeric. We refer to variables that admit a total order (e.g. continuous and ordinal) as ordered variables. In contrast, categorical variable, also called nominal, take one of fixed number of unordered values such as "A", "B", "AB", or "O" for blood type.
Most data analysis techniques require complete dataset, missing data imputation is an essential preprocessing step. also often of interest to propagate imputation uncertainty into subsequent analyses through multiple imputation, which generates several potentially different imputed datasets (25). An imputation method should ideally use all the collected data regardegssdsess of data type- to impute any missing data entries. However, most imputation approaches, whether explicitly or implicitly, assume that each variable admits total order and, as a result, cannot impute categorical variables without proper preprocessing.
There is no satisfying, successful, and widely adopted method for imputing categorical variables, especially in mixed datasets. It is tempting to reduce categorical imputation to ordinal imputation using an integer encoding in which each category is assigned a number; however, this encoding requires choosing an arbitrary ordering for the categories that may affect the results of downstream (e.g.. predictive) models. (This problem most severe for linear and neural network models, whereas tree-based models are less sensitive represent a categorical variable using a binary vector with one entry for each category. With this encoding, imputation methods developed for continuous or binary data, such as the popular low rank matrix completion methods (LRMC) 24
(31].) Instead, it is more common to use one-hot encoding to
can be jerry-rigged for categorical imputation.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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However, LRMC is only proper when the encoded matrix (both ordered and encoded categorical variables) is approximately low rank. It has been observed that LRMC performs poorly on long skinny (many samples n and few features p) datasets because the low rank assumption fails p 35
Iterative imputation methods including MICE and missForest 22 directly operate on the unordered categories. MICE learns the conditional distribution of each variable (whether ordered or unordered) using all other variables via linear and logistic regression. However, the learned conditional models may be incompatible in the sense that the joint distribution cannot exist [ MissForest uses random forest to predict each variable and yields more accurate single imputations 127. but cannot provide multiple imputation. Both methods converge slowly (or sometimes diverge) for large datasets because they train many models on (possibly) ill ill-conditioned data.
This paper expands on the idea of modeling categorical data and multivariate data interaction using latent continuous space [11 (). This approach can explicitly model the categorical distribution, which is critical for providing multiple imputation. This paper models the latent space as a Gaussian distribution and models each categorical variable as the argmax of latent Gaussian vector. This
choice is inspired by the Gaussian copula model [14 8 10] which yields high quality imputations
for ordered data [35 See (3) for a concise review and [2] for comprehensive methodology about Gaussian copula imputation. However, it cannot be used when the data contains categorical variables. This paper proposes the extended Gaussian copula to overcome this limitation by explicitly modeling each data type. In contrast, other imputation methods typically require bespoke preprocessing for the input data perform well. For mixed categorical and ordered data, the extended Gaussian copula model generates each categorical variable using a latent Gaussian vector and each ordered variable using a latent Gaussian scalar. The latent Gaussian correlations capture the dependence structure.
Contributions. This paper makes three main contributions to the literature:
1. A probabilistic model, the extended Gaussian copula, for mixed data including continuous, ordinal, and categorical variables. The model is free of hyperparameters and makes assumptions on the marginal distribution of each data type.
2. A single imputation method that empirically provides state-of-the-art accuracy for mixed data and multiple imputation method that can quantify imputation uncertainty by measures such as the category probability for categorical variables and confidence intervals for continuous variables.
3. A robust and efficient parameter fitting algorithm for the extended Gaussian copula that supports further acceleration through parallelization, mini-batch training, and low rank structure.
Related work. Modeling a categorical variable as the argmax of a latent continuous vector is a classical technique used in the multinomial probit model B [2 in the context of supervised learning. Instead, our model more similar to 6). which proposed one way to accommodate categorical variables in the Gaussian copula. However, the model in [6] has many redundant parameters and thus unidentifiable, while the extended Gaussian copula is identifiable and can match any categorical marginal distribution.
Methodology
Notation, For J: R - IR, we define the (possibly set-valued) pre-image of J as (y) = fx f()==)), We define argmax(z) of a vector z = (Z1,..., xp) as the index of the maximum entry so that (urgmax(x) max 24. We use Nu, Σ)) to denote the multivariate normal distribution with mean μ and covariance Σ. We use to denote the all-zero vector and 1 for the all-one vector, where the context determines the length of the vector.
We introduce our model in Section 2.1 then derive the model-based imputation methods in Section 2.2 and finally present model estimation algorithms in Section 2.3 Throughout the paper, we assume the missing complete at random (MCAR) mechanism: missing values are uniform and independent of any data. Nevertheless, we show our method performs reasonably well under missing at random (MAR) and missing not at random (MNAR) assumptions through experiments. We also discuss how violating the MCAR assumption may affect the assumptions of the proposed model in Section 4

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2.1 Extended Gaussian copula with categorical variables
We first show how to model categorical variable by transforming a latent Gaussian vector. Then we extend this model to generate categorical vectors and mixed categorical and ordered vectors. To ease the notation, we assume that all categorical variables have K categories encoded {"1", ........ "K"). It is straightforward to allow categorical variables with different numbers of categories.
2.1.1 Univariate categorical variable
We model univariate categorical variable x with K categories as the argmax of K-dim latent Gaussian z = (21,-..; 2x) with some mean (M11--..: :M and identity covariance. That is,
I == argmax(z + p), z~ ~0,,I0,
(1)
We call the distribution Eq. the Gaussian-Max distribution. Without loss of generality, we assume |41 0, as the argmax is invariant under translations, i.e., / μ + al for a E TR. While dense covariance matrix sometimes used for z [6, we prefer the model of Eq. ( as is identifiable. Theorem D states that any categorical distribution corresponds to unique choice of under Eq. Q All proofs are in the supplement. Section 2.3.I we describe an algorithm to estimate μ for a given categorical distribution.
Theorem 1 (Existence and Uniqueness). For any categorical distribution P(x = "k"] = Pk > 0 for k ..... K such that Σ Pk= 1, there is a unique μ R with μr 0 such that
P Px~ Jargmax(z + p4) = k] Pk. k = #==..... K.
(2)
2.1.2 Multivariate categorical vector
We model a categorical vector x = (Z1....,, mp)) by supposing each of its entries æj follows the Gaussian-Man distribution. That is,
argmax(z() + eد for some nهبر and isotropic Gaussian 2 Additionally, the latent Gaussian variables z(4) corresponding to different categorical variables may be correlated. We call this model the categorical latent Gaussian (CLG) model (see Definition]] [ use [j] denote the indices of z(5) in so [1z] z for j .....,P. P..
For = (z()) (P)).
Definition 1 (Categorical latent Gaussian). For a categorical vector x (I....., Ip). we say follows the categorical latent Gaussian x ~ CLG(E, (). if there exists a correlation matrix E and such that (1) for z NO, (). Tj = argmax(Zis) + Mig)): (2) E1 for every l.....P
In the CLG model, the value of [d]] suffices determine the marginal distribution of each categorical Ty,, the marginal distribution of z is (0, and independent of Σ. The correlation matrix Σ introduces dependencies between different categorical variables in x.
Consider two categorical variables T.j, and Fi2 whose joint distribution is described by P E R K where P&i = P((r, = k J,, = 2). The CLG model captures the dependence between 2j, and Tjz by the correlation submatrix Eisl- Note that has only (K- - 1)² free parameters as the rows and columns must sum to the associated marginal probabilities. This means that 1.b1) in the CLG model is not identifiable. We develop an identifiable variant of the CLG model in the supplement that uses only (K 1)² parameters in [i1l-11)) Both models share similar imputation performance, but the identifiable model is not invariant under permutation of the categorical labels while the model in Definition 1 is invariant. For the rest of this paper,
the permutation-invariat CLG model.
2.1.3 Mixed categorical and ordered vector
We model mixed data that contains both categorical and ordered variables by combining the CLG model for categorical variables with the Gaussian copula model for ordered variables [10 35
To model x with ordered variables, the Gaussian copula assumes GC(E, f) is generated as where each f3 is monotonic increasing function and z ~ (o, E). Denoting the cumulative distribution function (CDF) of æt as F,. the transformation ʃj uniquely determined as ʃz = F Φ where & denotes the standard Gaussian CDF and F7 (v) := inf (z E IR Fz( 2 y}. Specifically, ordinals result from thresholding the latent Gaussian variable. For ordinal Iy, the transformation fz( has the
an elementwise transformed Gaussian, i.e., f(z) =
3

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form ELEs 12 > s) where 1(2 > is if > and O otherwise. Here, the set S is determined by F.. The CLG model has the same form as x f(z) by writing
2j ( argmax(z + Mb))
(3)
Hence, we propose the extended Gaussian copula (EGC) model.
Definition 2 (Extended Gaussian copula). Write mixed data vector as = (Xeat Xord) where Xeat collects all categorical variables and Xord collects all ordered variables. We say x follows the extended Gaussian copula X ~ EGC(E, (1) if there exists a correlation matrix E, an elementwise monotone fool, and a μt such that Xea CLG((icat, cats (1) and Xead GC(2 fund).
Eeat, Eeat, ord Eorl, cat Eardood
where E =
We can also write x = f(z) (f (Zeati (e), furi (Zer)) for 2 (Zeats Zord) ~ (0 where is defined by all transformation parameters, i.e., fond and μ.. Unlike the original Gaussian copula, the dimension of the latent does not match that of the data x; instead, is larger as subvectors correspond to categorical entries of x. The correlation matrix E must also obey constraints the submatrix Eear, according to Definition
2.2 Missing data imputation
Now we show how to impute missing values given an EGC model with known parameters. For model parameter estimation, see Section 2.3
Note each categorical zy corresponds to K consecutive entries in z: the subvector 2 that generates Zz. We use [4] to denote the set of indices in z that corresponds to a set indices from x. For example, [j] follows this notation with I = {ii.. Two sets of indices are particularly important: the observed entries O and missing entries M for a given mixed data vector x. Thus [O] and [M] are the latent dimensions in that generate xo and XM,, respectively.
Our imputation strategy follows [35] First, the algorithm identifies region of the latent space % constrained by the observed entries. Then, it imputes the latent dimensions by exploiting the Gaussian distribution of Z[M] conditional on [o] Finally, the algorithm uses the marginal transformation to produce imputations in the data space X,M Pictorially, can visualize this process as
X l 2 IMMi
fM, X.-
Multiple imputation. Multiple imputation creates several imputed datasets by sampling from the distribution of missing entries conditional the observations. For the EGC model, Algorithm T samples from the distribution of xM using two facts: (1) given Xo, the variable Z(o: follows
(xz)): Z(M] [[o is Gaussian distributed. Section 2.3.3 shows how sample from the truncated Gaussian random variable given by | Using the empirical distribution of drawn samples, we can estimate the category probability for a missing categorical variable and build confidence intervals for missing continuous variable.
truncated Gaussian distribution truncated to f (xx) = Ih ) and (2) the random variable
Algorithm 1 Multiple imputation via the extended Gaussian Copula
1: Input: of imputations m, data vector observed O, model parameters f and Σ.
2: for 1,2,.... do
3: Sample [ Foo||xo
NoO, (1o1,1o) truncated to (xx(x)
4: Sample a, [MM] ~ Z [o]
N (Imm0oo138)) IMMWil - - of
5: Compute (()) = (((
6: end for
7: Output: (x(e) 1 m..,m).

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Single imputation. To provide an accurate single imputation under the EGC model, we impute based on the conditional mean of the latent Gaussian, i.e..
MM fm((EZ xo: I, f]) Ei
I,
(4)
To compute E [(o Xo; 2, fl. see Section 2.3.3 With a known mean [i 2, ff and marginal distribution fMM. computing the imputation X.M is straightforward. Specifically, for categori- cal missing 2j, we first compute the K K-dim [ : |xz Σ, and then impute as ž,
argmax (2 (b))
Online imputation. Online imputation is the task to impute missing entries in data streams arriving at different times. We conduct online imputation using the EGC model following [3]: upon observing an incomplete data batch, impute the missing entries with a current saved model, and then update the model using new observation. See Appendix A how to update the model.
2.3 Parameter estimation
Parameter estimation for the EGC model consists two steps: (I) we form an estimate f of the marginal transformation f. and (2) we estimate the copula correlation Σ given The marginal transformation differs for ordered and categorical variables. For an ordered zy, since fj F;-1 $ we can simply estimate F using the empirical quantile function on the observed entries of I as in 20|35. For categorical variable zj. we show how to estimate ( (b)) in Section 2.3. and how to estimate E in Eq. by extending estimation algorithms from 35
2.3.1 Marginal estimation for categorical variables
Here we estimate the mean μi for which the EGC model, EGC(E, food pt), matches the observed categorical marginal distribution. Each subvector Hbj corresponding to categorical zj may be estimated independently of all the others, so we drop the variable subscript j to ease the notation.
We must find μ in Eq. (2) so that the resulting category frequency py matches the observed frequency, subject to μ = O: this setup gives K nonlinear equations with K unknowns. Unfortunately, there is closed form solution. Thus resort to iterative root finding algorithms, which use as oracles the value and derivative of P/argmax(z + μ4) k] in Eq. 2). We use two approximation techniques to evaluate these oracles efficiently.
First we approximate the probability of argmax the expectation of softmax by noticing
P (P>>((O..)) |argmax(z 14) kt]
lim (g) (softmax(((2= + pk): B(z + #))) 8
where softmax zn; z) exp(=x) Σ
exp(21). This approximation is accurate for large 3. Second we approximate the expectation using Monte Carlo samples with the reparameterization trick [19]
i-1
E [softmax(((2z + pk): B(z p))
softmax(3(22 px):B(z p)),
(5)
M
where z s.d.d. Ix). With these approximations, we solve for subject to = 0 in Eq.
6
softmax(3(E) + (a)::(2' + #)) pk:
M
for l = 2, 2.... K. The modified Powell method implemented in SciPy computes accurate solution.
Optimization hyperparameter selection. Both M and / are easy to select as larger values always improve approximation accuracy: larger M leads to better MC approximation and larger 3 leads to better argmax probability approximation. We use M 5000 and B 1000 as the default in our experiments, and find this setting works well across all our experiments. Theoretically, a large M needed when the number of categories K is large for accurate MC approximation; conversely, when
5

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K' is small 3 must be large for accurate argmax probability approximation. However, in practice, it rare have K larger than 50. Also, for very rare categories, e.g., those with probability < 10 the limited samples may not suffice to learn correlations; one solution is to drop rare categories or merge rare categories into larger groups. Our default setting achieves satisfying accuracy in synthetic experiment variants even with K = 50 and tiny category probabilities (as low as 10 10-1).
Gaussian-max versus Gumbel-softmax. Both the Gaussian-max and the Gumbel-softmax distri- argmax(z + (1) for some
butions models categorical variable taking value in K categories μ E Rx with independent identically distributed entries of z. The difference lies the underlying distribution of the latent continuous z: the Gumbel-softmax uses the Gumbel distribution, while we use the Gaussian distribution. Different choices serve for different goals. Using the Gaussian distribution, it is easy to model and estimate the joint distribution for categorical variables well as their mix with ordered variables. Using the Gumbel distribution, we can write down the closed form expression of ş4 in Eq. (2): P(argmax(z + 44) k) softmax( 11) and pa log Pk. Thus it is simple to compute the gradients of the argmax probability F(argmax(z (z + 11) = k) w.r.t. its parameter, making it widely used in neural networks [16]
2.3.2 Copula correlation estimation
Now we show how to find the maximum likelihood estimator (MLE) of copula correlation Σ, which models the multivariate dependence structure. Given i.i.d samples x² jid EGC(E, f 44) observed at O2 and missing at M, for i = 1, ...... n. We maximize the observed likelihood by integrating over the latent space that maps to the observation x6,, as in Eq. (7)
Lata (2;
olz o O, oo, dzjo.
1=1 (6,
The form of this likelihood matches that of the Gaussian copula for ordered variables 35 except that the latent integration dimensions [O:] instead of O, Hence the Expectation Maximization (EM) algorithm in (35) also works in our case by integrating over the appropriate latent indices. Concretely, for the E-step at iteration t + 1, we first compute the expectation of the joint likelihood of (z' x over the distribution of z conditional on the observation x6, and the previous correlation estimate {()) using the Gaussianity of z denoted QE,; E(0):
QEE; (00) cc
1 log det())
Σ Σ
Blz' Blz'(z) X 2
where e is a constant. Then we compute argmaxy QEE I()) in the M-step, which is the expected "empirical covariance matrix" of zz. We update ((++)) to its corresponding correlation matrix to satisfy the copula parameter constraint similar to 12 35
((++1) <
('E
(1 n
Σ
n
Elz' Elz'(z') xxo. 2001
where Poor(E) (Σ) returns the correlation matrix corresponding to a covariance matrix Σ. The EM algorithm is guaranteed to strictly increase the likelihood in Eq. 7 and converges to a stationary point [23 Chapter 3J. The main computation is the expected sample covariance of z Dropping the row index, evaluating E zz |xo, [0) requires computing Bizio] xo, [ and Cov[zioj X ()) We give details in the supplement and show how to evaluate the reduced quantities in Section 2.3.3
Recall the correlation Σ must satisfy the constraint in Definition π the submatrix Eg1bi for each categorical index j must be the identity. The correlation computed by EM generally does not satisfy this constraint. Instead, we use the function Pean (E) to correct the correlation Σ:
Σ- Peae Σ)
1/2 Σ (raal Pua): AEAT T where A diag (>
(10)
by recognizing that Cov Cov/E 1/2 Xord has Pord entries.
(1) [b] = Ix for each categorical index j. when Xea has Peat entries and

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Algorithm 2 Estimation algorithm for the extended Gaussian Copula
1: Input: Partial observation xx J correlation initialization , M 5000, 8 1000.
2: for each variable index j do
> Marginal estimation starts
3: Collect all observed data: X, = {at "..............
4: if x, is categorical then
5:
Solve Eq. (6) for ji with input M, 3 and pk equals to the frequency of "k" in Xj.
6:
Compute f: (b)) in Eq. (3) with Mby as the solved solution ft.
7: else
8:
Compute f f o where F/= is the empirical quantile function of Xj.
9:
end if
10:
end for
> Marginal estimation ends D EM algorithm starts
11: for = 0, -.. until convergence do
Compute (z') |xo, E0)), f] for i ,....
13: Update (1++)) as in Eq. (9)
14:
Correct ( + Part (+))) as Eq. (0)
15:
end for
> EM algorithm ends
16: Output: marginal estimation f, correlation estimation ==(++))
Computational cost. Algorithm 2 summarizes the complete estimation algorithm for the EGC model. Marginal estimation only takes small fraction of total runtime (less than 4% in all experi- ments of this paper). We observe that the EM algorithm mostly converges in less than 50 iterations. Each iteration has time complexity ((and)) where a denotes the observed entry ratio, denotes the number of samples and d Pord + Pea K denotes the latent dimension. Line 12 of Algorithm P takes the vast majority of the computation time. There are many ways to accelerate the computation. For large n., we can parallelize the computation in line 12 of Algorithm 2 or conduct minibatch EM training as in [33]. The minibatch EM performs model updates using randomly sampled data batch and achieves sigmificant speedup without sacrificing accuracy. For large d. we can reduce the cubic time complexity in terms of linear by imposing a low rank structure on following [34, that = WWT T + o I where W E R with l < d and o² E (0, 1). See Appendix B for details.
2.3.3 Truncated Gaussian with categorical variables
Finally, show how to estimate moments and sample from Z(o] xo, these operations required for imputation and correlation estimation. Note M|o| |xo follows a truncated Gaussian distribution truncated into the region r (xxo). For each ordered variable with index j E O, (x;) is an interval. (For continuous Tjj. this interval is a single point.) Thus without observed categoricals, Gaussian, fo (xx) is Cartesian product of intervals. We call this distribution the interval-truncatl
for which accurate sampling 2 and effective moment estimation (35) methods have been developed.
The truncation region is no longer a Cartesian product of intervals with observed categoricals. Fortunately, for any X can find an involution Ax (i.e., A I) so that AxZ(o) (xo follows an interval-truncated Gaussian distribution. Consequently,
E [ = A₂ A,E[AAZ(o)
(11)
tid Similarly, to sample 2; ~ 2/0 |xo, we can first sample 2; ~ Ax xxo and then use 2; = Az To
iid
derive Ax, first consider that when categorical variable x takes value k by Eq. (3) we have:
(x:p)= :rgmax(z+ p1) k} {z I
12 24 + uk, for j # k), (12)
Define = AgZ and ja = Asp where A IK entry and 0 elsewhere). We can rewrite Eq. 12) to { ž of intervals. We can find a matrix A, as shown above for every categorical variable Ty. Let Ay be (Ang(AA,.... Apas I. Then Az(o) follows interval truncated Gaussian and Ag is an involution.
Ex + Eus (matrix Eij has at (i. -th ر 0, for j + k). Cartesian product
7

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Table 1: Mean(sd) of runtime in seconds and imputation error for each variable type (cat for categorical, cont for continuous, ord for ordinal) over 10 repetitions. See Figure I for the error metric.
π = 2000
Runtime
Cat Error
Cont Error
Ord Error
EGC our
(2(
0.64(0.01)
1.81(0.06) 2.06(0.07)
0.50(0.02)
missForest
(11)
0.68(0.01)
0.59(0.02)
n = 10000
Runtime
Cat Error
Cont Error
Ord Error
EGC _ur
147(4)
0. 0.64
1.81(0.04)
0.45(0.01) 0.52(..02)
missForest
1006(70)
0.66(0.01)
2.04()
π = 20000
Runtime
Cat Error
Cont Error
Ord Error
EGC_our
202(9)
1.81(0.06) 0.64(..01) 0.66(0.01) 2.04(0.04)
0.42(0.01)
missForest 3714(267)
0.48(0.01)
3 Experiments
In the experiments, we compare the accuracy of single imputations using our method EGC and several competitors. It is more challenging compare the accuracy for multiple imputation (MI) as MI seeks to recover the correct distribution which is mostly unknown in practice. Nevertheless, we designed synthetic experiment in Appendix C As shown in Algorithm 2 EGC does not require any model hyperparameter but only optimization hyperparamters M and 3, for which we use fixed values M 5000 and B 1000 across all our experiments. We show in Section 333 that this option already accurately estimates the categorical marginals. All codes are provided in a Github repo
Competitors. We implement several imputation methods for mixed categorical and ordered data. Among iterative imputation algorithms, we implement miss missForest (27 26 and MICE
4 30
Among LRMC algorithms, we implement imputeFAMD [ [8] and soft softImptte 21 methods one-hot encodes categorical variables. ImputeFAMD includes special weighting strategy incorporating different variables types, while softImpute simply treats the data as numerical. We also implement a baseline imputation: majority vote for categorical variables and median imputation for ordered variables. We only need to choose hyperparameters for imputeFAMD and soft Impute have satisfying performance, for which further mask 20% of observed training entries as a validation set. Through all experiments in this paper, softImpute is the fastest method and EGC comes second overall. We include all implementation and runtime details in the supplement.
13] Both
Evaluation. To evaluate an imputation method, we compute miselassification error (ME) for categorical variables and mean absolute error (MAE) for ordered variables at masked entries.
3.1 Synthetic experiments
We generate synthetic datasets with 2000 samples and {11, 13, 15] variables from the EGC model in Definition 2 with randomly drawn parameters. The p variables include 5 exponentially distributed continuous variables, 5 ordinal variables with five ordinal levels, and Peat E {1.3,5) categorical variables with six categories (K = 6). We randomly remove 30% entries of X for imputation evaluation and generate 10 masked datasets using differrntt seeds.
Fig. 2 shows our method EGC performs the best for all variable types in all scenarios. For categorical variables, EGC yields larger improvement with fewer categorical variables. For both continuous and ordinal variables, EGC substantially outperforms all other methods. Several methods that impute categorical variables quite well struggle to outperform simple baseline on ordered variables. The supplement includes more experiments that differ by the number of categories for categorical variables
(K 3, 9), the missing ratio (20% and 40%) and missing mechanisms (MAR and MNAR). In general, these experiments show EGC performs well for both categorical and ordered variables in mixed data when the model is well-specified. We also ran experiments with much larger sample size and found EGC is much faster compared to missForest, as shown in Table
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Imputation of categorical
Imputation of continuous
Imputation of ordinal
0.80
2.2
0.65
0.75
0.60
0.70
2.0
MAE
MAE
0.65
0.55
1.8
0.60
0.55
1.6
0.45
Number of categoricals
Number of categoricals
Number of categoricals
EGC_our
mizsForest
imputeFAMD
softimpute
Figure Imputation error on synthetic mixed data with 2000 samples. There are 5 continuous variables, ordinal variables and 1/3/5 categorical variables with six categories, reported over 10 repetitions (error bars indicate standard deviation).
Imputation error of categorical
Imputation error of ordered
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0.95
0.8
SME
0.90
SMAE
0.85
0.80
0.75
0.4
0.2
0.65
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MICE
imputeFAMD
missForest
softimpute
Figure 2: Imputation error of categorical variables and of ordered variables, i.e., ordinal and continu- ous, on UCI datasets at 20% missingness. Results shown as mean standard deviation.
3.2 Real data experiments
We employ six real-world datasets from the UCI machine learning repository here [7]. Each dataset contains both categorical and ordered variables (details in the supplement). We randomly remove 20% of observed entries as test set to evaluate imputation accuracy and generate 10 masked datasets using different seeds. Since variables in real data often have very different scales, e.g., number of categories for categorical variables and standard deviation for continuous variables, we use an evaluation metric that compensates for the imputation difficulty across variables. For an imputation method, we normalize its imputation error for each variable by the imputation error of basel inn and then compute the average normalized error for each categorical variable and ordered variable, similar to [35]. We denote the scaled imputation error for categorical variables and ordered variables as SME and SMAE, respectively.
Fig. 2 shows that our method EGC is one of the best method in almost all evaluation metrics. missForest comes second but its total runtime four times longer than that of EGC. Interest- ingly. softImpute imputes quite accurately for categorical data but poorly for ordered data. The supplement includes more experiments that differ by the missing ratio and missing mechanism, demonstrating that EGC consistently outperforms other methods.
3.3 Categorical marginal estimation accuracy
Here we verify that EGC can learn arbitrary categorical distribution, i.e. the algorithm in Section 2.3. finds the root of Eq. ( accurately. Concretely, we compare two sides of Eq. (2). where the right side the empirical category probability and the left side approximated at the estimated μ using 10000 random samples. Across all used datasets in Section 3.1 and Section 3.2 Fig. shows the

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Categorical probability estimation
Error
0.015
Dim
True probability
Absolute value
Dim
Dim
0.010
Dim 14
0.005
0.2
0.0
0.000
Estimated probability
Dim: number of categories
Figure 3: Performance of categorical distribution fitting. The results are collected from 70 datasets: each of datasets (1 synthetic and real data) is randomly masked times. Dim the number of categories for each categorical variable. Left plots the true probability vs the estimated probability of each categorical value, colored by dim, and Right plots the absolute values of the estimation error as boxplot, grouped by dim.
estimated probability indeed matches the true probability well and the performance is robust to the number of categories. Using larger M in Eq. (6) can further reduce the estimation error, but we find it does not significantly improve the imputation performance.
4 Discussion and conclusion
We have presented probabilistic model, the EGC model, for mixed categorical and ordered data. We developed an imputation method based on the EGC model that empirically demonstrates state-of- the-art performance. Here we discuss two limitations of this paper. First, the estimation algorithm for the EGC model admits guarantees under the MCAR mechanism. Concretely, the marginal estimation requires the MCAR mechanism to accurately match the empirical marginal distribution. Supposing the marginals have been estimated accurately, estimating the correlations consistently possible under the less restrictive MAR mechanism. While the proposed algorithm this paper works reasonably well empirically under other missing mechanisms, it may be possible to improve performance by estimating and exploiting the missing mechanism for model estimation. Second, the EGC model uses Gaussian dependence structure after marginal transformation. This paper demonstrates the benefits modeling the marginals explicitly. Learning a more complex dependence structure, e.g., with neural networks, may yield further improvements.
Acknowledgement
MU and YZ gratefully acknowledge support from NSF Award IIS-194313,, the ONR Young Investiga- tor Program, the Alfred Sloan Foundation. AT is supported by NSF Grants No. DMS-195211 and No. DMS-2045646 as well as Simons Fellowship in Mathematics.
10

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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you descrihe the limitations of your work? [Yes] See Section f
(c) Did you discuss any potential negative societal impacts of your work? [N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes]
(b) Did you include complete proofs of all theoretical results? [Yes] See the supplement
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)? [Yes] Provided the supplemental material
(b) Did you specify all the training details (e.g.. data splits, hyperparameters, how they were chosen)? [Yes] See the supplement
(c) Did you report error bars (e.g.. with respect the random seed after running experi- ments multiple times)? [Yes]
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes] See the supplement
4. If you using existing assets (e.g., code, data, models) or curating/releusing
assets...
(a) If your work uses existing assets, did you cite the creators? [Yes]
(b) Did you mention the license of the assets? [No] No license provided
(c) Did you include any new assets either in the supplemental material
URL.? [No]
(d) Did you discuss whether and how consent was obtained from people whose data you't rou're
using/curating? [No] All used data is publicly available
(c) Did you discuss whether the data you are using/curating contains personally identifi- able information or offensive content? [No] The used data does not contain any ID information offensive content
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A] Research does not involve human subjects
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] Research does not involve human subjects
(c) Did you include the estimated hourly wage paid participants and the total amount spent on participant compensation? [N/A] Research does not involve human subjects
A Online imputation and minibatch training
Here we introduce how update the parameter of the extended Gaussian copula on a new incomplete data batch, following 333 For the purpose of online imputation, we update both the marginal and copula correlation at newly observed data. For the purpose of offline minibatch training, we use all available data to estimate the marginal and only update the copula correlation at each data batch.
To update the marginal, we first update the stored observation window, the recent (a hyperpa- rameter) observations for each variable, as [33]. then re-compute the empirical category probability, and re-solve the nonlinear equations in Eq. (2) with previous solution the initialization.
To update the copula correlation (()) we first conduct a normal EM iteration on the new data batch and denote the estimation as S. Then we perform an incremental update: {(++) 7 (1) where "Y a learning rate in (0, 1). The updating rule a special case on the online EM algorithm in [5). See 33 for more details.
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Method
Cross Entropy Loss
Runtime
EGC (our)
0.278 (0.039)
14(1)
MICE
0.340 (0.004)
(1)
Table 2: Distribution estimation error of 100 multiple imputation samples the missing categorical. The error is measured using cross entropy loss (smaller is better), averaged over all missing entries. The runtime is recorded in seconds. Numbers stand for mean (sd) over 10 repetitions.
B Low rank Gaussian copula for categorical and ordered mixed data
We introduce a special case of the extended Gaussian copula here following the low rank Gaussian copula [3]. Shortly, we assume the latent Gaussian vector R which generates the data follows the probabilistic principal component analysis model [28]:
z Wt + e, (O,, ((0, LI))
where t and e are independent and W E RRdxr with d > r. Consequently, copula correlation Σ WWT + o contains only dr + free parameters. An EM algorithm has been developed in 34 to solve W and o². Similar to the results Section 2.3.2 the EM algorithm for the low rank Gaussian copula also works here by integrating over the appropriate latent indices. The adjustment is to make sure that for a general estimate W and o², the formed correlation WW + o²La satisfies a that the submatrix Eb1 Wi W I at each categorical index j must be identity. This requirement implies constraint in the rank it must be smaller than K. In other words, the used rank must be no smaller than the largest number of categories among all categorical variables, otherwise Wi) W will have rank r and cannot be (1 x²) L We can do correction on W only (o² remains unchanged) similar to Eq. (10). That is, at each categorical index j. with the singular value decomposition of Wi [)) U,D,VJ we correct WG) + U,vI- - o-, VJ. where I E Rxr has in its diagonal.
b]
C An experiment on multiple imputation
Multiple imputation (MI) is often used in specific case studies (11 e.g.) where no ground truth is present. There is no generally accepted metric compare MI methods. Nevertheless, we designed synthetic experiment scenario to showcase that our MI provides more accurate distribution estimation of the missing entries than MICE using much less time, and MICE the only other imputation method implemented in this paper that supports MI. For the designed synthetic experiment, the true distribution of missing entries can be accurately approximated. Thus we can compare the estimated distribution provided from MI to the true distribution.
synthetic dataset with 5 exponentially distributed continuous variables and categorical variable with categories, 2000 samples and 30% missing ratio, generated similar to that in Section 3.1 For straightforward evaluation, we focus the categorical variable, for which we use its conditional distribution given all continuous observations as the true distribution. We evaluate the distribution estimation error through the cross entropy loss (smaller is better) between true and estimated category probability. averaged across all missing categorical entries. For both our EGC method and MICE, we take 100 multiple imputation samples and compute the empirical probability for each category as an estimate. The results are reported in Table 2 Our EGC MI achieves significantly smaller loss and smaller runtime than MICE.
Specifically,
14


Reviews:
[{"review_id": "QEz_M8UcDAx", "reviewer": "Reviewer_5kUW", "summary": "This manuscript proposes an extension of the Gaussian copula model for imputing missing values on mixed non-ordinal (e.g., categorical) and ordered (e.g., continuous) data. The authors show the superiority of the proposed method on 1 synthetic and 6 real datasets in comparison to several competing approaches.", "questions": "1) in lines 93-94, why the assumption of $\\mu_1=0$ is necessary for the model to work?\n2) in lines 102-103, why the Gaussian-Max distribution assumption is necessary for $x_j$, and how this assumption affects the generality of the proposed model? Does it always hold? What happens if it does not hold? How can we check in practice whether this assumption holds?\n3) how does the proposed method perform when the input data has a very large rank?\n", "limitations": "The authors must include a section to discuss the limitations of the proposed method.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n1) The paper addresses an important and well-defined problem in the machine learning context.\n2) The text is clear and the technical contributions are well presented.\n3) The proposed method shows a substantial improvement over the competing methods.\n\n# Weaknesses\n1) The main contribution of this manuscript overlaps substantially with a previously published work \"Missing Value Imputation for Mixed Data via Gaussian Copula\" (https://dl.acm.org/doi/abs/10.1145/3394486.3403106, which is not even mentioned in the related works), which affects negatively the significance of this work. The authors must i) discuss the added value of their new method and ii) compare its performance with the existing one. \n2) Even though some results for the MAR and MNAR scenarios are presented in the supplementary material since all the contributions in the manuscript are made based on the MCAR assumption, the author must discuss thoroughly how violating the MCAR assumption may affect the assumptions of the proposed model.\n3) The computational complexity of the method is cubic with respect to the dimension of the latent which is restricting in practice. The authors propose to use a low-rank approximation of $\\Sigma$, but the effect of such an approximation is not evaluated in the experimental results. \n4) While promised as a method for multiple imputation (lines 63-65), the method is not evaluated for multiple imputation in the experiments (stating that the correct distribution is mostly unknown). Maybe this could be evaluated on synthetic data at least.\n5)  While the proposed method has two hyperparameters (referred to as optimization hyperparameters $M$ and $\\beta$), the authors claim that their method has no hyperparameters. Further, it is not very clear how the values for these hyperparameters are specified.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Gm_Cubo-GLu", "reviewer": "Reviewer_JaBb", "summary": "The authors propose a single and multiple missing value imputation method for mixed data under the MCAR (missing completely at random) assumption. The method is based on using a latent Gaussian distribution in the form of an ordinary Gaussian copula model for ordered data (ordinal and continuous) and an *extended* Gaussian copula model proposed by the authors for categorical variables. The author showcase impressive results for their approach, which outperforms standard imputation methods in both synthetic and real-world experiments.", "questions": "Would it make sense to relax the constraint for the covariance of **z** in (1), for instance if there are variables that are partially ordered?", "limitations": "There are no potential negative societal impacts for this work that I can imagine.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This work represents a novel and sound method for handling single and multiple imputation for mixed data. The extended Gaussian copula probabilistic model using the argmax transformation is, to the best of my knowledge, novel. The results showcased are impressive and suggest that the work may have significant impact. Finally, the paper is almost flawlessly written and clearly structured.\n\nMinor comments:\n- p1 (page 1), l4 (line 4): Insert 'this is' before \"challenging\".\n- p1, l19: \"we\" needs to be capitalized.\n- p5, l189: \"expection\" $\\to$ 'expectation'", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "AhGUlieRiir", "reviewer": "Reviewer_ZBKz", "summary": "The authors through this manuscript propose a probabilistic model to impute missing values of mixed data including continuous, categorical and ordinal variables that supports both single and multiple imputation. The proposed model is based on extended Gaussian copula, is free of hyperparameters, and makes no assumptions on the marginal distribution of the data types. Authors run a series of experiments to compare the accuracy of the imputations of the proposed model with several competitors. ", "questions": "1) The dataset used in Synthetic experiment has 2000 samples, and in Real data experiments a maximum of just 4177 samples. Did the authors test the convergence of the EGC model on a much larger dataset? If yes, what were the drawbacks you noticed with the performance?\n2) Also, noticed a minor typographical error at Line 49, \"approach can explicitly modeling the categorical distribution\"", "limitations": "Other than the scalability of the proposed EGC model, authors did address the limitations of the study.", "rating": 5, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Authors did a good job in articulating the importance of their study and it's contributions to the literature including the shortcomings of the existing imputation methods and the proposed algorithm methodology. The proposed methodology is evaluated through a series of experiments to compare the accuracy with other competing methods. I agree with the authors that several existing methods such as MICE, MissForest etc. are good but converge slowly especially for large datasets, however, I strongly believe as part of their experiments authors should have evaluated real world applicability by testing the performance of the proposed EGC model on a large dataset and test it's scalability.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept