Missing data is a recurring problem in machine learning and in particular for clinical datasets, where it is common that numerous feature values are not present for reasons including incomplete data collection and discrepancies in data formats and data corruption (Luo, 2022; Emmanuel et al., 2021; Pedersen et al., 2017; Myers, 2000). Machine learning is routinely used in life science and clinical research for prediction tasks, such as diagnostics (Qin et al., 2019) and prognostics (Booth et al., 2021), as well as estimation tasks, such as biomarker proxies (Wang et al., 2017) and digital biomarkers (Rendleman et al., 2019). Beyond the research setting, machine learning is becoming more commonplace as regulated Software as a Medical Device, where machine learning models are influencing – or making – clinical decisions that affect patient care.

Machine learning algorithms typically require complete datasets and missing values can significantly affect the quality of the machine learning models trained on such data. This is in large part due to the fact that there can be different underlying reasons for the missingness: for example, feature values can be missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR), each one with their own characteristics.

Despite its importance for clinical trials, there is no canonical approach for dealing with missingness and finding appropriate, effective and reproducible methods remains a challenge. A common way to deal with missing clinical data is to exclude subjects that do not have the complete set of feature values present. A drawback of this approach is that excluding subjects can in fact introduce significant biases in the final model. For example, it can result in the model being trained to be more effective for the type of subjects that are likely to have complete data than for those that do not. Moreover, the effectiveness and reliability of clinical trials are reduced when subjects with missing feature values are excluded from the clinical trial.

Data imputation is an alternative to the complete dataset approach, where subjects with missing feature values are not excluded. Instead, missing values are imputed to create a complete dataset that is then used for a classification task as shown in Figure 1. There are different ways to achieve this, including ‘filling’ missing values with zeros, or with the mean value of the feature across all subjects that have such a value present. These methods provide consistent imputation results, but there are important caveats for using such simple methods since they ignore possible correlations between features and can make the dataset appear more homogeneous than it really is. More advanced data imputation methods have been proposed in the literature: iterative methods include the multivariate imputation by chained equations (MICE) (Groothuis-Oudshoorn, 2011) and MissForest (Stekhoven and Bühlmann, 2012) algorithms, and deep learning methods include GAIN (generative adversarial imputation nets) Yoon and Jordon, 2018 and MIWAE (missing data importance-weighted autoencoder) (Mattei and Frellsen, 2019). Recent results Shadbahr et al., 2022 have shown that for clinical data two iterative imputation methods, MiceRanger, which uses predictive mean matching, and MissForest, which uses Random Forests to predict the missing values of each feature using the other features, provide the best results and have been used here as a baseline.

Figure 1

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Several metrics are routinely used to quantify the quality of data imputation: point-wise discrepancy measures include root mean square error, mean absolute error, and coefficient of determination ($R^2$). Feature-wise discrepancy measures include Kullback–Leibler divergence, two-sample Kolmogorov–Smirnov statistic or (2-)Wasserstein distance. Ultimately, the quality and reliability of imputations can be measured by the performance of a downstream predictor, which is usually the area under the receiver operating curve (AUC) for a classification task. In practical terms, the performance of the downstream classifier is usually of highest importance for clinical datasets: for example, in one of our datasets, the classifier denominates a binary outcome of a critical care unit stay (e.g., survival) for each patient. Accordingly, we have used AUC for the classification task here on different holdout sets (see Figure 2) to assess the performance of our novel methods.

Figure 2

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In order to increase the resulting AUC, we combine the MissForest and MiceRanger imputation methods with determinantal sampling, based on determinantal point processes (DPP) (Dereziński and Mahoney, 2021; Kulezsa and Taskar, 2011), which favors samples that are diverse and thus reduces the variance of the training of each decision tree that in turn provides more accurate models. In essence, determinantal sampling picks subsets of data according to a distribution that gives more weight to subsets of data that contain diverse data points. More precisely, each subset of data points is picked according to the volume encapsulated by these data points. The determinantal distribution increases the attention given to uncommon or out-of-the-ordinary data points rather than biasing the learning process towards the more commonly found data, which can improve the overall prediction accuracy in particular for unbalanced datasets as is often the case for clinical data (Derezińsk and Mahoney). Determinantal sampling for regression and classification tasks with full data has been proposed previously for linear regressors (Dereziński et al., 2018) and for Random Forest training for a financial data classification use case where it outperformed the standard Random Forest model (Thakkar et al., 2023). However, an inherent feature of standard Random Forest and determinantal sampling algorithms is randomness that produces data imputations that vary from one run of the algorithm to the next. This is often undesirable since the downstream classification performance can also be affected, which motivated us to apply a deterministic version of determinantal sampling (Schreurs and Suykens, 2021) within the Random Forests of the imputation methods to provide more robust and reliable imputations.

Through deterministic determinantal sampling, we address two challenges in data imputation: first, we provide improved data imputation methods that can increase the performance of the downstream classifier; and second, we remove the variance of the common stochastic and multiple imputation methods, thus ensuring reproducibility, easier integration in machine learning workflows, and compliance with healthcare regulations. While these improvements are of particular relevance for clinical data, our algorithms can also be advantageous for other imputation tasks where improving downstream classification and removing variance is of importance.

In order to demonstrate this improvement, we apply our methods to two classification datasets: a synthetic dataset and a public clinical dataset where the predicted outcome is the survival of the patient.

In addition, we explore the potential of quantum computing to speed up these novel imputation methods: we provide a quantum circuit implementation of the determinantal sampling algorithm that offers a computational advantage compared to its classical counterpart. The best classical algorithms for determinantal sampling take in practice cubic time in the number of features to provide a sample (Dereziński and Mahoney, 2021). In contrast, the quantum algorithm we present here, based on theoretical analysis in Kerenidis and Prakash, 2022, has running time that scales linearly with the number of features. We measure running time as the depth of the necessary quantum circuits, given that the quantum processing units that are being developed currently offer the possibility of performing parallel operations on disjoint qubits.

This suggests that with the advent of next-generation quantum computers with more and better qubits, one could also expect a computational speedup in performing determinantal sampling using a quantum computer. Here, we demonstrate competitive results with up to 10 qubits for small-scale imputation tasks on a state-of-the-art IBM quantum processor.

This work combines classical (Dereziński and Mahoney, 2021) and quantum (Kerenidis and Prakash, 2022) DPP algorithms with widely used data imputation methods, resulting in novel data imputation algorithms that can improve performance on classical computers while also having the potential of a quantum speedup in the future.

In ‘Methods’, we provide a detailed description of our four imputation methods, DPP-MICE, DPP-MissForest, detDPP-MICE, and detDPP-MissForest. All of them are based on iterative imputation methods that use the observed values of every column to predict the missing values. The model used to fill missing values in each column is the Random Forest classifier. Our imputation methods replace the standard Random Forest used by the original miceRanger and MissForest imputers by the DPP-Random Forest model, for our first two imputers, and the detDPP-Random Forest for the latter two. The DPP-Random Forest model subsamples the data for each decision tree using determinantal sampling instead of uniform sampling, while the detDPP-Random Forest model deterministically picks for each decision tree the subset of data that has the maximum probability according to the determinantal distribution. We also demonstrate a computationally advantageous way to perform the determinantal sampling on quantum computers.

In order to benchmark the different imputation methods, we used two types of datasets with a categorical outcome variable. First, a synthetic dataset, created using the scikit-learn method make_classification. It consists of 2000 rows with 25 informative features. This is useful to study the imputation quality where features have equal importance. Second, the MIMIC-III dataset (Johnson et al., 2016): the Medical Information Mart for Intensive Care (MIMIC) dataset, which is a freely available clinical database. It is comprised of data for patients who stayed in critical care units at the Beth Israel Deaconess Medical Center between 2001 and 2012. It contains the data of 7214 patients with 14 features.

We also applied two types of missingness on these datasets: MCAR, where the missingness distribution is independent of any observed or unobserved variable, and MNAR, where the missingness distribution depends on the outcome variable. We expect similar results to hold for the MAR case as well, but it was not considered in this work.

We present the numerical results in terms of the AUC of the downstream classification task in Table 1 and provide graphs of the results in Tables 2 and 3. Each experiment was run 10 times with different random seeds to get the variance of the results.

Table 2

	MCAR	MNAR
SYNTH
MIMIC

1. AUC = area under the receiver operating curve; MCAR = missing completely at random; MNAR = missing not at random.

Table 3

	MCAR	MNAR
SYNTH
MIMIC

1. AUC = area under the receiver operating curve; MCAR = missing completely at random; MNAR = missing not at random.

Overall, DPP-MICE and DPP-MissForest provide improved results compared to their classical baseline MICE and MissForest. This is the case for both the synthetic and the MIMIC datasets and for both MCAR and MNAR missingness. Even more interestingly, the detDPP-MICE and detDPP-MissForest collapse the variance of the imputed data to 0 and moreover lead in most cases to even higher AUC than the expectation of the previous methods.

Quantum hardware implementation of DPP-MissForest results in competitive downstream classification

As we describe in ‘Methods’, quantum computers can in principle be used to offer a computational advantage in determinantal sampling. In order to better understand the state of the art of current quantum hardware, we used a currently available quantum computer to perform determinantal sampling within a DPP-MissForest imputation method for scaled-down versions of the synthetic and MIMIC datasets.

• Reduced synthetic dataset: 100 points and three features, created using the sklearn method make_classification.

• Reduced MIMIC dataset: 200 points and three features. The three features were chosen from the original dataset features based on low degree of missingness and their predictiveness of the downstream classifier, and they were ‘Oxygen saturation std’, ‘Oxygen saturation mean’, and ‘Diastolic blood pressure mean’.

For the purposes of our experiments, we used the ‘ibm_hanoi’ 27-qubit quantum processor shown in Figure 3. We implemented quantum circuits with up to 10 qubits. We also performed quantum simulations using the qiskit noiseless simulator. The decision trees of the DPP-Random Forests used by the imputation models are trained using batches of decreasing sizes (see Table 4). For example, for the algorithm with batch size equal to 10, the algorithm first samples 2 out of the 10 data points to use for the first decision tree, then from the remaining 8 data points it picks another 2 for the second tree, then 2 from the remaining 6, and last 2 from the remaining 4. In other words, we train four different trees, and each time we use a quantum circuit with number of qubits equal to 10, 8, 6, and 4, to perform the respective determinantal sampling.

Figure 3

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Table 4

Batch size	Tree 1	Tree 2	Tree 3	Tree 4
7	(7,2)	(5,2)	-	-
8	(8,2)	(6,2)	(4,2)	-
10	(10,2)	(8,2)	(6,2)	(4,2)

In the figures of Table 5 and in Table 6, we provide for the different dataset experiments the AUC for MissForest, the simulated results of the quantum version of DPP-MissForest, and the actual hardware experimental results of running the quantum version of DPP-MissForest. Even for these very small datasets, when simulating the quantum version of DPP-MissForest, we demonstrate an increase in the AUC compared to the MissForest algorithm. This further highlights the potential advantages of determinantal sampling within imputation methods. Of note, running our algorithms on current hardware introduces variance in the downstream classifier. Importantly, this variance is due to noise in the hardware rather than inherent to the algorithm.

Table 5

	Batch size: 7Number of trees: 2	Batch size: 8Number of trees: 3	Batch size: 10Number of trees: 4
MCAR SYNTH
MCAR MIMIC
MNAR SYNTH
MNAR MIMIC

1. AUC = area under the receiver operating curve; MCAR = missing completely at random; MNAR = missing not at random.

Table 6

Dataset	Missingness	Batch size	Trees	MissForest	detDPP-MissForest (simulator)	detDPP-MissForest (hardware)
SYNTH	MCAR	7	2	0.868 ± 0.0302	0.9026	0.8598 ± 0.021
		8	3	0.8667 ± 0.0342	0.9256	0.8923 ± 0.027
		10	4	0.8725 ± 0.0275	0.9028	0.8902 ± 0.024
	MNAR	7	2	0.7122 ± 0.0264	0.78	0.7149 ± 0.02
		8	3	0.7153 ± 0.022	0.729	0.7036 ± 0.0167
		10	4	0.7258 ± 0.0157	0.7868	0.7082 ± 0.036
MIMIC	MCAR	7	2	0.7127 ± 0.038	0.7522	0.7117 ± 0.0315
		8	3	0.7136 ± 0.03	0.7728	0.7448 ± 0.0258
		10	4	0.6968 ± 0.03	0.7327	0.7262 ± 0.0299
	MNAR	7	2	0.7697 ± 0.0133	0.7794	0.7742 ± 0.0108
		8	3	0.7713 ± 0.0112	0.7943	0.767 ± 0.0125
		10	4	0.7712 ± 0.0116	0.7922	0.7675 ± .01545

1. AUC = area under the receiver operating curve; MCAR = missing completely at random; MNAR = missing not at random.

Our quantum hardware results are competitive with standard methods and in many cases close to the values expected from the simulation. In some cases, we observed a clear deterioration of the AUC due to the noise and errors in the quantum hardware. The results are closer to the simulations when using MCAR missingness with larger batch sizes that use more trees both for synthetic and the MIMIC datasets. As explained above, even though the algorithm with batch size 10 means using a quantum circuit with 10 qubits, the fact that we use four trees overall with a decreasing number of data points each time, and thus a decreasing number of qubits (namely, 10, 8, 6, and 4), results in an overall more reliable imputation.

Missing data is a critical issue for machine learning practitioners as complete datasets are usually required for training machine learning algorithms. To achieve complete datasets, missing values are usually imputed. In the case of clinical data, missing values and imputation can be a potential source of bias and can considerably influence the robustness and interpretability of results. Nevertheless, there is no canonical way to deal with missing data, which makes improvements in data imputation methods an attractive and impactful approach to increase the effectiveness and reliability of clinical trials. In this proof-of-concept study, we assessed the downstream consequences of implementing such improvements focussing on MCAR and MNAR to assess the usefulness of our approach. MNAR and MCAR represent two extreme cases of missingness with importance for clinical data imputation applications.

Determinantal point processing methods increase the diversity of the data picked to train the models, showcasing also that data gathering and preprocessing are important to remove biases related to over-representation of particular data types. This is more important when dealing with unbalanced datasets, as is the case often with clinical data. Determinantal sampling is an important tool not only for Random Forest models, but also for linear regression, where data diversity results in more robust and fair models (Dereziński et al., 2018). Moreover, such sampling methods based on DPP are computationally intensive and quantum computers are expected to be useful in this case: quantum computers offer an asymptotic speedup for performing this sampling, and it is expected that next-generation quantum computers will provide a speedup in practice.

We show that, as expected, the quantum version of detDPP-MissForest does not introduce any variance in the downstream classifier when simulated in the absence of hardware noise. While the AUC improvements achieved in our experiments may seem modest, it is the consistency of improvements we observed in our simulation results coupled with removal of variance that makes our approach attractive for clinical data applications where these characteristics are extremely desirable. When implemented on quantum hardware, we observed variance that is caused by the noise in the hardware itself. More precisely, the output of the quantum circuit is not a sample from the precise determinantal distribution but from a noisy version of it, and this noise depends on the particular quantum circuit implemented and the quality of the hardware. Thus when attempting to compute the highest probability element using samples from the quantum circuit on current hardware, the result is not deterministic. Importantly, unlike for standard MissForest, this variance is not inherent in the algorithm and is expected to reduce considerably with the advent of better quality quantum computers. The quantum circuits needed to efficiently perform determinantal sampling require a number of qubits equal to the batch size used for each decision tree within the Random Forest training and the depth of the quantum circuit is roughly proportional to the number of features. As an example, if we would like to perform the quantum version of the determinantal imputation methods we used for MIMIC-III, then we would need a quantum computer with 150 qubits (the batch size) that can be reliably used to perform a quantum circuit of depth around 400 (the depth is given by $4d\log n$, where $n=150$ is the batch size and $d=14$ is the number of features; Kerenidis and Prakash, 2022). While quantum hardware with a few hundred qubits that can perform computations of a few hundred steps are not available right now, it seems quite possible that they will be available in the not so far future. In the meantime, further optimization could also help reduce the quantum resources needed for such imputation methods.

While our DPP-based imputation methods can run classically on small datasets such as MIMIC-III, they are computationally intensive and are hard to parallelize due to the sequential nature of the algorithm. This results in less and less efficient imputation for larger datasets where DPP sampling is applied to bigger batches. For example, when a DPP-MICE imputation is run on a dataset of 200 features and batches of size 400, then the training is expected to take multiple hours on a single GPU. The quantum DPP algorithm therefore could provide a way to speed up the hardest part of the imputer using a next-generation quantum computer. For instance, if $d=200$, and batch size is 400, the number of qubits will be 400 and the depth of the quantum circuit would be ≈ 6400, whereas it would take $\sim 8*{10}^6$ classical steps for DPP sampling. These are of course simply illustrative calculations and will require more detailed analysis as these machines become available and will need to include parameters such as clock speeds and error correction overheads. Only then can it be experimentally proven that this theoretical asymptotic speedup can translate to a practical speedup for this particular algorithm.

In summary, here we propose novel data imputation methods that, first, improve the widely used iterative imputation methods – MiceRanger and MissForest – as measured by the AUC of a downstream classifier; second, remove the variance of the imputation methods, thus ensuring reproducibility and simpler integration into machine learning workflows; and third, become even more efficient when run on quantum computers. Based on our results, we anticipate an impact of our algorithms on the reliability of models in high-precision value settings, including in pharmaceutical drug trials where they can provide higher confidence in the predictions made by eradicating the stochastic variance due to multiple imputations. In addition, tasks that are currently overwhelmed by the challenges of missingness become more tractable through the approaches introduced here, which is a common problem with real-world-evidence investigations, where detDPP-MICE and detDPP-MissForest can yield improved performance in the face of missingness.

The synthetic dataset can be generated using the make classification method from scikit-learn. The MIMIC-III dataset is publicly available at https://mimic.mit.edu/.

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Author details

1. Skander Kazdaghli

   QC Ware, Paris, France

   Contribution

   Conceptualization, Data curation, Software, Visualization, Methodology, Writing – original draft, Writing – review and editing

   For correspondence

   skander.kazdaghli@gmail.com

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0009-0005-9044-0919

2. Iordanis Kerenidis

   1. QC Ware, Paris, France
   2. Universite de Paris, CNRS, IRIF, Paris, France

   Contribution

   Conceptualization, Supervision, Methodology, Writing – original draft, Project administration, Writing – review and editing

   Competing interests

   No competing interests declared

3. Jens Kieckbusch

   Emerging Innovations Unit, Discovery Sciences, BioPharmaceuticals R&D, AstraZeneca, Cambridge, United Kingdom

   Contribution

   Supervision, Validation, Writing – original draft, Project administration, Writing – review and editing

   Competing interests

   are employees of AstraZeneca. The authors declares that no other competing interests exist

4. Philip Teare

   Centre for AI, Data Science & AI, BioPharmaceuticals R&D, AstraZeneca, Cambridge, United Kingdom

   Contribution

   Validation, Visualization, Methodology, Writing – original draft, Project administration, Writing – review and editing

   Competing interests

   are employees of AstraZeneca. The authors declares that no other competing interests exist

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