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MBDiff: Multi-view Behavior-aware Diffusion Model for Probabilistic Utility Data Imputation
Rongchao Xu, Lin Jiang, Dahai Yu, Ximiao Li, Guang Wang
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
The paper introduces MBDiff, a multi-view behavior-aware diffusion model designed for probabilistic imputation of missing utility data (electricity, water, gas). It addresses challenges in learning user behavior from incomplete, long-term time series by employing a Multi-view User Behavior Extraction (MUBE) module that captures global, local, and instance-level patterns. These patterns guide a Behavior-aware Conditional Diffusion Model (BCDiff) with a reference selection module and an attentional denoising network to efficiently impute missing values. The model was evaluated on real-world data from a Florida utility provider, showing significant improvements over state-of-the-art baselines.
Entities (8)
Relation Signals (8)
MBDiff → appliedto → Utility Data
confidence 95% · MBDiff: Multi-view Behavior-aware Diffusion Model for Probabilistic Utility Data Imputation
MBDiff → contains → BCDiff
confidence 95% · (ii) a behavior-aware conditional diffusion model... to impute utility data
MBDiff → contains → MUBE
confidence 95% · MBDiff incorporates two key technical components: (i) a multi-view User Behavior Extraction module
MBDiff → improves → 7.04%
confidence 95% · it improves 7.04% and 29.1% on the electricity and water usage datasets
MBDiff → improves → 29.1%
confidence 95% · it improves 7.04% and 29.1% on the electricity and water usage datasets
BCDiff → uses → BCADN
confidence 92% · a behavior-aware conditional diffusion model consisting of... a conditional attentional denoising network
MBDiff → outperforms → State-of-the-art baselines
confidence 90% · Experimental results demonstrate our proposed MBDiff effectively outperforms state-of-the-art baselines
MUBE → uses → TVAE
confidence 90% · We then design a Transformer-based Variational Autoencoder (TVAE... to learn a unified latent representation
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Abstract
Abstract:Utility data (e.g., electricity, water, and gas consumption), collected by ubiquitous sensors and embedded devices, often contains substantial missing values due to various factors such as device failures and data transmission issues. The data missingness can severely impact utility billing accuracy, hinder demand forecasting, and disrupt efficient utility supply management. As a result, utility data imputation has attracted much interest from both industry and academia. While many studies have attempted to address this issue, most of them rely on aggregated datasets for training, overlooking rich user behavior information, which could provide valuable insights for more accurate imputation. However, learning comprehensive user behavior from long-term, diverse, and incomplete utility data remains a significant challenge. Moreover, leveraging user behavior information to guide imputation is nontrivial due to the indirect nature of the correlations. To address these challenges, we propose MBDiff, a Multi-view Behavior-aware Diffusion Model for Probabilistic Utility Data Imputation. MBDiff incorporates two key technical components: (i) a multi-view User Behavior Extraction module that learns comprehensive user behavior from multiple perspectives, including global, local, and instance-level views; and (ii) a behavior-aware conditional diffusion model consisting of a reference selection module and a conditional attentional denoising network to impute utility data in a computationally efficient manner. We implement and evaluate MBDiff by collaborating with one of the largest municipal utility providers in Florida. Experimental results demonstrate our proposed MBDiff effectively outperforms state-of-the-art baselines, e.g., it improves 7.04% and 29.1% on the electricity and water usage datasets for block missingness imputation, respectively.
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- Source: https://arxiv.org/abs/2607.29177v1
- Canonical: https://arxiv.org/abs/2607.29177v1
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MBDiff: Multi-view Behavior-aware Diffusion Model for Probabilistic Utility Data Imputation Rongchao Xu * Lin Jiang * Dahai Yu * Ximiao Li * Guang Wang * † Abstract. Utility data (e.g., electricity, water, and gas consumption), collected by ubiquitous sensors and embed- ded devices, often contains substantial missing values due to various factors such as device failures and data transmis- sion issues. The data missingness can severely impact utility billing accuracy, hinder demand forecasting, and disrupt ef- ficient utility supply management. As a result, utility data imputation has attracted much interest from both industry and academia. While many studies have attempted to ad- dress this issue, most of them rely on aggregated datasets for training, overlooking rich user behavior information, which could provide valuable insights for more accurate imputa- tion. However, learning comprehensive user behavior from long-term, diverse, and incomplete utility data remains a sig- nificant challenge. Moreover, leveraging user behavior in- formation to guide imputation is nontrivial due to the indi- rect nature of the correlations. To address these challenges, we propose MBDiff, a Multi-view Behavior-aware Diffusion Model for Probabilistic Utility Data Imputation. MBDiff in- corporates two key technical components: (i) a multi-view User Behavior Extraction module that learns comprehensive user behavior from multiple perspectives, including global, local, and instance-level views; and (i) a behavior-aware conditional diffusion model consisting of a reference selec- tion module and a conditional attentional denoising network to impute utility data in a computationally efficient manner. We implement and evaluate MBDiff by collaborating with one of the largest municipal utility providers in Florida. Ex- perimental results demonstrate our proposed MBDiff effec- tively outperforms state-of-the-art baselines, e.g., it improves 7.04% and 29.1% on the electricity and water usage datasets for block missingness imputation, respectively. 1 Introduction Missing values are common in utility data (e.g., electricity load, water usage, and gas consump- tion) due to various factors such as sensor malfunctions, extreme weather events (e.g., hurricanes) Li et al. (2025); Shen et al. (2025), and irregular sampling, which can signif- icantly affect real-world utility services in different aspects * Department of Computer Science, Florida State University, Tal- lahassee, FL, USA (rx21a@fsu.edu, lj23d@fsu.edu, dahai.yu@fsu.edu, xl24g@fsu.edu, guang@cs.fsu.edu). † Corresponding author. including billing, resource allocation, and demand-supply re- balancing. According to our collaboration with one of the largest utility providers in Florida, missing utility records during residential transitions present considerable challenges for both providers and customers. For instance, inaccurate or incomplete data may lead to unfair billing, where outgo- ing residents are either undercharged or overcharged, and in- coming residents may be billed for consumption they did not cause. These issues can erode customer trust, prompt dis- putes, and potentially lead to regulatory non-compliance. Due to its importance, utility data imputation has at- tracted much interest from both academia and industry, and various approaches have been proposed to address this task. Early statistical and machine learning methods, such as in- terpolation, k-nearest neighbors (KNN) (Altman, 1992), and matrix factorization (Fang and Wang, 2020), were widely used but often struggled with long and complex sequences exhibiting intrinsic dependencies. Recently, deep learning methods have advanced utility data imputation (e.g., RNNs Yoon, Zame, and van der Schaar (2018); Cao et al. (2018), CNNs Wu et al. (2023), VAEs (Fortuin et al., 2020), GANs (Brophy et al., 2023), diffusion models (Tashiro et al., 2021; Liu et al., 2023)), which aim to learn underlying distributions for accurate imputation. However, most existing works typi- cally rely on aggregated data for model training, often over- looking a critical factor, i.e., user-specific behaviors. These behaviors, characterized by unique trends and habitual pat- terns, can offer valuable information to improve imputation accuracy. More detailed motivation and data analysis are pre- sented in Appendix A.1, which further illustrate diverse user behaviors across different utility types and time periods. In this paper, we focus on user-specific utility data im- putation by considering unique user behaviors. However, there are two major challenges to achieving this objective: (1) How to effectively learn comprehensive user behavior from long-term, dynamic, and diverse historical utility data? (2) How to accurately impute the data with extracted user behavior in a computationally efficient way? First, behav- ioral patterns can vary significantly not only across users but also for the same user over time, making it difficult to extract comprehensive and meaningful patterns. Further- more, adopting machine learning for data imputation may cause the user-specific data to be temporally discontinuous Copyright © 2026 by SIAM Unauthorized reproduction of this article is prohibited arXiv:2607.29177v1 [cs.LG] 31 Jul 2026 due to training and testing data splitting. For instance, data may only be available for certain days (e.g., days 1, 4, 5, 7, and 14) in the training data, resulting in large temporal gaps that further complicate behavior modeling. A straight- forward way is to represent user behavior by generating a global user embedding or a chronologically sorted latent rep- resentation. However, such representations often fail to pro- vide adequate and contextually relevant references for imput- ing target records on different dates. Secondly, leveraging the learned user behavior to efficiently guide the imputation of target data records remains a challenge due to implicit cor- relations and high dimensionality. In particular, it is difficult to establish dependencies between user behavior represen- tations and the attributes of target records (e.g., timestamps or values), as such relationships are often complex, indirect, and not explicitly observable. To address these challenges, we propose MBDiff, a Multi-view Behavior-aware Diffusion Model for Probabilis- tic Utility Data Imputation, which consists of two key com- ponents: a Multi-view User Behavior Extraction module (MUBE) and a Behavior-aware Conditional Diffusion model (BCDiff). MUBE comprehensively extracts users’ behavior patterns from different aspects (e.g., global, local, and in- stance view) based on long-term, diverse, and discontinuous historical time series data. In BCDiff, a reference selection module is firstly introduced to identify the most informa- tive behavior patterns, thereby reducing computational cost. Subsequently, a novel behavior-aware conditional attentional denoising network is designed to accurately impute missing values by leveraging the selected reference behaviors. In summary, the key contributions are as follows: • Motivated by real-world significance, in this paper, we address the utility data imputation problem by consider- ing complicated user behaviors, which can benefit both utility providers and customers through accurate utility billing and resource allocation for societal impact. • We propose a behavior-aware utility data imputation framework called MBDiff, which includes a multi-view user behavior extraction module that learns comprehen- sive user behaviors from various aspects (e.g., global, local, and instance view), and a computationally effi- cient behavior-aware conditional diffusion model with an attentional denoising network that leverages the in- formative behavior to impute missing values. • We implement and comprehensively evaluate our pro- posed MBDiff on three types of real-world utility data in collaboration with a large municipal utility provider. Extensive results show that MBDiff significantly out- performs state-of-the-art baselines, e.g., it improves 7.04% and 29.1% on the electricity and water usage datasets for block missingness imputation, demonstrat- ing the effectiveness of our design. More importantly, our MBDiff has helped our collaborator impute over 20 million missing values in their utility data to improve their utility services, indicating its social impact. 2 Preliminary 2.1 ProblemDefinitionDefinition1:(Utility Data) For a utility dataset collected from a group of users U= u 1 ,u 2 ,· ,u N , where Nde- notes the total number of users, we define D u = (X u 1 ,d u 1 ,m u 1 ), (X u 2 ,d u 2 ,m u 2 ),· , (X u N u ,d u N u ,m u N u ) as the historical utility data of user u. Here, X u j represents the j-th data record (time series) of user u, and N i is the total number of data records of user u. The timestamp d u j = (d u j ,md u j ,yd u j ) consists of the day of a week (d u j ∈ [1, 7]), month (md u j ∈ [1, 12]), and year yd u j of X u j . The variable m u j ∈ 0, 1 is a binary observation mask to indicate whether a value (data point) is observed or missing in a data record, where m l = 1 represents the data point is observed and m l = 0 represents the data point is missing. Definition 2: (Utility Data Imputation) Given a historical dataset D u i of user u i , deterministic/probabilistic utility data imputation aims to estimate the values/distributions of the missing values in each utility data record ̃ X at the date ̃ d = ( ̃ d, ̃ md, ̃ yd), where ̃ d, ̃ md, and ̃ yd represent the day, month, and year, respectively. 2.2 Denoising Diffusion Probabilistic Models De- noising Diffusion Probabilistic Models (DDPMs) (Ho, Jain, and Abbeel, 2020) are a class of generative models designed to learn complex data distributions through a sequence of noise-adding and denoising steps. It consists of a forward diffusion process and a reverse generative process. The for- ward process gradually corrupts the data x 0 ∈ R d by adding Gaussian noise over T discrete time steps: (2.1) q(x t |x t−1 ) =N (x t ; √ α t x t−1 , (1− α t )I), (2.2)q(x 1:T |x 0 ) = T Y t=1 q(x t |x t−1 ), where α t ∈ (0, 1) controls the variance schedule. After T steps, the data becomes nearly isotropic Gaussian noise. The reverse process aims to iteratively reconstruct x 0 from x T . This is achieved by learning a parameterized model p θ (x t−1 |x t ), which is also Gaussian: (2.3)p θ (x t−1 |x t ) =N (x t−1 ;μ θ (x t ,t), Σ θ (t)). The mean μ θ (x t ,t) is predicted by a neural network trained to approximate the noise added during the forward process. The model is trained by minimizing a simplified variational lower bound on the negative log-likelihood of the data. DDPMs have demonstrated state-of-the-art performance in generating high-quality samples across various domains, including images, videos, and time-series data. Copyright © 2026 by SIAM Unauthorized reproduction of this article is prohibited dv ... 퐷 !,# $ ... ... ... 푋 ! L 푑 ! 푔푦 ! 푙푝 ! ... ... ... TF DM M YM 3 * 푑 Concatenate SW 푧̃ STF TF TF mv yv TM STF L 2 L 3 푁 0 L L Instance-view Pattern Selection ... ... ... BCADN Diffusion Framework Local-view Pattern Selection Global-view Pattern Selection SW 퐷 !,% $ 퐷 !,# & 퐷 !,#' & 퐷 !,# ( 퐷 !,) ! ( 푔푚 ! 푔푑 ! 푁 " 푁 ! 푑 * 푋 ! 푑푖 # 푑푖 % 푚푖 # 푚푖 #' 푦푖 # 푦푖 ) ! TVAE #$% & TVAE #$% ' TVAE #$% ' - 푋 ! 푋 ! ℎ ' 8 푖푝 ! 8 푙푝 ! 8 푔푝 ! ℎ # 푁 ! ; 푋 ! 8 푔푑 ! < 푔푚 ! 8 푔푦 ! = 푋 = > 푋 ! 푋 ! 푋 +,# ! 푋 + ! 푋 ) Figure 3.1: The pipeline of MBDiff. The shapes outlined by the red and pink lines represent the entire and selected user behavior patterns, respectively. The black dashed box and blue dashed box represent the historical user-specific utility data and the input of the imputation stage. TF refers to data filtering based on time (day, month, year). SW represents the splitting of subsequences using sliding windows. STF between two tensors A→ B denotes the process of selecting entries from A based on the top values in B. The detailed architectures of TVAE and BCADN can be found in Figures 3.2 and 3.3, respectively. Notations in grey denote the dimensional information about the data. 3 Methodology The overall framework of our pro- posed MBDiff is illustrated in Figure 3.1. MBDiff comprises two core modules. The first, Multi-view User Behavior Ex- traction (MUBE), captures global, local, and instance-level behaviors to build a comprehensive user profile. The sec- ond, Behavior-aware Conditional Diffusion Model (BCDiff), identifies the most informative features from the profile and leverages them to impute missing data in a computationally efficient manner. 3.1 Multi-view User Behavior Extraction In this part, we design a multi-view user behavior extraction mod- ule (MUBE) to learn comprehensive user behavior from different aspects (e.g., global, local, and instance view) based on long-term and diverse historical utility data records, which will provide rich information for effective data im- putation. Specifically, for a utility time series D u = (X u 1 ,d u 1 ,m u 1 ),· , (X u N u ,d u N u ,m u N u ) collected from user u, we target to construct a user profile P = (GP,LP,IP ), where GP,LP,IP captures the global, local, and instance- view behavior patterns. 3.1.1 Global-view Behavior Pattern Extraction Global-view behavior patterns provide prior knowledge for user u, offering coarse-grained bounds for each data point in the utility time series and thereby alleviating the incom- pleteness of historical data records. We define three types of global-view patterns for each feature i based on aggregation over the same day of the week, month, and year. For the daily pattern gd u d , the mean μ d,i and variance σ 2 d,i are calculated over all data points where the day of the week is d∈ [1, 7]: (3.1)gd u =gd u 1 ,· ,gd u 7 , (3.2)gd u d =(μ u d,i ,σ u d,i 2 ),· , (μ u d,L ,σ u d,L 2 ), (3.3)μ u d,i = 1 |D u,d | X (X u k ,d u k )∈D u,d X u k,i , (3.4) σ u d,i 2 = 1 |D u,d | X (X i k ,d i k )∈D u,d (X u k,i − μ u d,i ) 2 , where D u,d is a subset of user u’s historical data D u ; the day of the week d i k is equal to d and k ∈ [1,|D u,d |]; X u k,i denotes the i-th value in data record X u k and L represents the length of the utility data. The monthly patterns gm u m and yearly gy u y patterns are computed in a similar way, where m ∈ [1, 12] and y belongs to the year set Y of collected data. These global-view behavior patterns GP = (gp 1 ,gp 2 ,· ,gp |U| ) represent the typical user behavior across different time scales and form a statistical foundation for predicting or imputing values of data points, where gp u = [gd u ,gm u ,gy u ]. Copyright © 2026 by SIAM Unauthorized reproduction of this article is prohibited Positional Encoding Masked Multi-Head Attention Add & Norm Multi-Head Attention Add & Norm Latent Representation z Positional Encoding Multi-Head Attention Add & Norm Feed Forward Add & Norm Input Series Feed Forward Add & Norm Linear LinearLinear TVAE !"# TVAE $%# Reconstructed Series Minimize Distance Figure 3.2: The architecture of the designed TVAE for utility time series embedding and subsequence embedding. 3.1.2 Local-view Behavior Pattern Extraction Al- though global-view behavior patterns offer a possible range for each data point, they fail to capture the temporal depen- dencies between data points in the data record (i.e., a time series), which are important for accurate missing value impu- tation. To address this, we introduce fine-grained local-view behavior patterns, extracted from consecutive data points in D u , that reflect the correlations between these points. Ini- tially, due to the potential incompleteness of data points, we utilize the global-view patterns g u for missing value X u i,j fill- ing in each X u i , where m u i,j = 0. Specifically, we obtain pseudo-imputed utility data record ˆ X u i from original data (X u i ,d u i ,m u i ) through: (3.5) ˆ X u i,j = gd u d ′ ,j or gm u m ′ ,j or gy u y ′ ,j if m u i,j = 0 X u i,j if m u i,j = 1 where j is the index in the data record, and d ′ ,m ′ ,y ′ are the day, month, and year indices of d u i . Next, we apply a sliding window of size l w and step s to extract subsequences from each pseudo-imputed utility record ˆ X u i , represented as: (3.6) w u i =w u i,j N w j=1 , w u i,j = ˆ X u i,s j ,· , ˆ X u i,e j where w u i denotes all subsequences of user u’s i-th historical utility data record, and w u i,j represents the j-th subsequence. The starting and ending indices are calculated as j start = s × (j−1), j end = j start + l w − 1, and the number of subsequences is given by N w = T−l w s + 1. By splitting a full record into multiple subsequences, we extract behavioral segments that capture local dependencies across different positions and offer more fine-grained local references for imputation. We then design a Transformer- based Variational Autoencoder (TVAE, whose architecture is shown in Figure 3.2) to learn a unified latent representation z u i,j for each subsequence w u i,j : (3.7)z u i,j = TVAE d encoder (w u i,j ), z u i,j ∈ R d z , where d z is the dimension of the latent representation, and TVAE d encoder is the encoder of TVAE for dynamic patterns. We train TVAE d on subsequences W =w 1 ,w 2 ,· ,w |U| from all users. Finally, we represent dynamic behavioral patterns LP = (lp 1 ,lp 2 ,· ,lp |U| ) by aggregating over subsequence positions in each data record: (3.8) lp u =lp u j N w j=1 , lp u j =(w u i,j ,z u i,j ) N u i=1 , 3.1.3 Instance-view Behavior Pattern Extraction We design an instance-view behavior pattern extraction mod- ule to provide record-level information for imputation, offer- ing a broader context than local-view patterns. Specifically, we employ another Transformer-based Variational Autoen- coder, denoted as TVAE i , to encode the i-th pseudo-imputed utility record of user u, ˆ X u i , into a shared latent space: v u i = TVAE i encoder ( ˆ X u i ). The model TVAE i is trained on all users’ pseudo-imputed records, ˆ X = ˆ X 1 , ˆ X 2 ,· , ˆ X |U| . To capture instance-view behavior patterns with respect to various time points ̃ d in the target record, we design a multi- dimensional temporal matching mechanism based on three matching matrices: ip u = (DM u ,M u ,Y M u ), where DM u ∈ R 7×7 , M u ∈ R 12×12 , and Y M u ∈ R N y ×N y correspond to day, month, and year dimensions, respec- tively. We first compute the Euclidean distance matrix D u ∈ R N u ×N u , where each element is given by D u i,j = ∥v u i − v u j ∥ 2 . For each time pair (p,q), we then aggregate the rele- vant distances to form the three temporal matching matrices DM u , M u , and Y M u , as defined below: (3.9) DM u p,q = 1 |di p |×|di q | |di p | X a=1 |di q | X b=1 Dist u di p a ,di q b , (3.10) M u p,q = 1 |mi p |×|mi q | |mi p | X a=1 |mi q | X b=1 Dist u mi p a ,mi q b , (3.11) Y M u p,q = 1 |yi p |×|yi q | |yi p | X a=1 |yi q | X b=1 Dist u yi p a ,yi q b , where di p , mi p , yi p denotes the index set where each index di p j , mi p j , yi p j satisfies d u di p j = p, md u mi p j = p, yd u yi p j = p, respectively. Finally, each matrix is normalized through the row, indicating the overall historical dependency between each time p and q in three temporal dimensions. Copyright © 2026 by SIAM Unauthorized reproduction of this article is prohibited 3.2 Behavior-aware Conditional Diffusion Model In this part, we design a Behavior-aware Conditional Diffu- sion Model called BCDiff to impute missing values in utility data ̃ X . Due to the high-dimensional nature of the construc- ted user profile P , directly applying diffusion models would incur considerable computational cost. To address this, we first introduce a reference selection module that identifies the most informative behavioral patterns from the extracted user profile. We then develop a probabilistic diffusion model equipped with a novel behavior-aware conditional attentional denoising network, which is guided by the selected reference behaviors to impute missing values. 3.2.1 Reference Selection First, we select reference global-view patterns ̃ gp u = [ ̃ gd u , ̃ gm u , ̃ gy u ] ∈ R 3×L×2 fromgp u by indexing with the three input temporal elements. Specifically, ̃ gd u = gd u ̃ d , ̃ gm u = gm u ̃ md , and ̃ gy u = gy u ̃ yd denote the selected reference global day, month, and year patterns, respectively. Each element ̃ gp u i,j,0 and ̃ gp u i,j,1 represents the mean and variance, respectively. Next, to obtain reference local-view behavior pat- terns, we first generate the pseudo-imputed utility data record ˆ ̃ X and extract unified latent representation pairs ( ̃w 1 , ̃z 1 ),· , ( ̃w N w , ̃z N w ) from ̃ X , following Equa- tions 3.5, 3.6, and 3.7. We compute the Euclidean distance between each latent representation ̃z i and each historical la- tent representation in lp u i illustrated in Equation 3.8 and ob- tain a local dynamic set of size h 2 ̃w i = ̃w i,1 ,· , ̃w i,h 2 for each location of subsequences, where i denotes the in- dex of subsequence in the data record. The entire selec- tive reference local-view behavior patterns are represented as ̃ lp u = ̃w 1 ,· , ̃w N w . In terms of the selection of instance-view behavior patterns, we initially obtain three weighted vectors dv ∈ R 7 , mv ∈ R 7 , yv ∈ R N y from corresponding rows, where dv = DM u ̃ d,: , mv = M u ̃ d,: , yv = Y M u ̃ yd,: . A temporal weight matrix TM is computed through the outer product of dv, mv, yv as follows: (3.12) TM i,j,k = dv i · mv j · yv k , TM∈ R 7×12×N y , We then select h 2 samples ̃ ip u = n ̃ X u i , ̃ d u i , ̃m u i o h 2 i=1 , whose dates correspond to the top h 2 weights in the temporal matching matrix TM , from the user’s historical utility data D u as reference instance-view behavior patterns. In summary, three types of selected behavior patterns ̃ gp u ∈ R 3×L×2 , ̃ lp u ∈ R N w ×h 2 ×l w , ̃ ip u ∈ R h 2 ×L serve as the references that steer the imputation process of the target utility data records. 3.2.2 Conditional Probabilistic Diffusion Imputa- tion We propose a steered conditional probabilistic diffu- sion model armed with all sorts of guidance specifically cho- sen for ̃ X to impute missing values. Within the diffusion IF Conv MLP N !푥 ! m IF LF GF 퐿 · ( ) · softmax 퐿 ! · () ·softmax L 퐿 ! 푁 " "푥 " $ 푖푝 # ! 푙푝 ! $푔푝 ! 3 MLP AH 1 m L L 푍푆 LF L N GF L L L L 3 휀’ n 2 푁 ! *푤 퐿 ! L Input Input Local-view Attention Instance-view Attention L ℎ $ ℎ % Split MLP Figure 3.3: The architecture of the proposed behavior-aware conditional attentional denoising network (BCADN). model, we design a novel behavior-aware conditional atten- tional denoising network (BCADN) that fully extracts the multi-types of correlations among missing values, observed values, and chosen references. The architecture of BCADN is shown in Figure 3.3. Given the noisy utility data record ̃ X n , at diffusion step n, observation mask ̃m and selected patterns ̃ gp u , ̃ lp u , ̃ ip u , the noise prediction network ε θ pre- dicts the added noise ε ′ = ε θ ( ̃ gp u , ̃ lp u , ̃ ip u ,n, ̃m, ̃ X n ). We compute Z-scores ZS using selected global patterns ̃ gp u as the reference distribution to measure the deviation of ob- served values. Based on these scores, we derive an intermedi- ate global pattern-steered feature GF that captures the over- all deviation across the three temporal dimensions, which are formalized as: (3.13) ZS i,j = X j − ̃ gp u i,j,0 ̃ gp u i,j,1 , GF = MLP(Norm(ZS)), where ZS ∈ R 3×L , GF ∈ R L , MLP denotes the multi- layer perceptron, and Norm represents the normalization op- eration along the row dimension. To capture the dependency between subsequences of ̃ X or the entire ̃ X and either local- view or instance-view patterns, we design Attention layers that capture the correlation between adjacent or distant data points as follows: H l = Attn( ̃w j , ̃ lp u j , ̃ lp u j ) N w j=1 , (3.14) H i = Attn( ̃ X, ̃ ip u , ̃ ip u ), H l ∈ R N w ×l w , H i ∈ R L (3.15) where Attn(Q,K,V ) = softmax( Q·K √ l ) · V represents the attention layer and l is equal to l w and L for H l and H i respectively. Considering the different locations of row vectors in H l representing the subsequences in the time Copyright © 2026 by SIAM Unauthorized reproduction of this article is prohibited series, we align them at corresponding positions as follows: (3.16) AH i,j = H d i,j−i start if 0≤ j− i start < l w 0otherwise where AH d ∈ R N w ×L and i start = s × (i − 1) repre- sents the starting index of i-th subsequence. The interme- diate local-view and instance-view pattern-steered feature LF = MLP(AH T ) ∈ R L and IF = MLP(H i ) ∈ R L are derived through MLP, which naturally incorporates the positional information without the need to introduce po- sitional embedding to lower the computational complex- ity. Finally, we fuse three intermediate features, diffusion step N = Pos(n) ∈ L embedded with positional encod- ing and observation mask ̃m with noisy data record ̃ X n from previous step and obtain the predicted noise ε ′ = 1DConv(Concat( ̃ X n ,N, ̃m,GF,LF,IF ))∈ R L . We train ε θ by minimizing the divergence between the added ground- truth noise ε and predicted noise ε ′ . 4 Evaluation 4.1 Evaluation Settings 4.1.1 Dataset We evaluate our MBDiff using three different types of utility datasets (gas, water, electricity), which are collected from over 120K users. The water and gas datasets are sparser (containing more zeros) compared to the electricity data. Each data record represents a user’s utility usage time series for a day. We preprocess the three datasets by filtering out users with fewer than 200 records. Utility data records for each user are randomly split into training, validation, and test sets with a 7:1:2 ratio, ensuring a balanced distribution across users. A data statistics of the three datasets is summarized in Table 4.1. Table 4.1: Statistics of the Three Utility Datasets. ElectricityWaterGas Length of Each Series482424 Sampling Interval30 mins60 mins60 mins Number of Users60,90790,70431,683 Mean # Series per User701317709 Data Duration2018-201920192019-2020 4.1.2 Baselines We compare our MBDiff with ten representative baseline methods covering statistical, deter- ministic, and generative approaches: • G-MEAN (Amiri and Jensen, 2016): Imputes missing values using the global historical average across all users. • U-MEAN (Amiri and Jensen, 2016): Imputes missing values using the target user’s historical average. • LINEAR (Huang, 2021): Performs linear interpolation to fill missing values. • KF (Bakibillah et al., 2022): Uses Kalman Filtering to estimate and impute missing entries. • SAITS (Du, Côté, and Liu, 2023): A deterministic imputation model based on self-attention mechanisms for time series. • GP-VAE (Fortuin et al., 2020): A probabilistic model combining variational autoencoders and Gaussian Pro- cesses to capture temporal dependencies. • CSDI (Tashiro et al., 2021): A conditional diffusion model utilizing Transformer-based denoising for time series imputation. • PriSTI (Liu et al., 2023): Enhances CSDI by incorpo- rating spatiotemporal attention to better model spatial and temporal correlations. • ImputeFormer (Nie et al., 2024): A low-rank Trans- former that balances model expressiveness and induc- tive bias for spatiotemporal imputation tasks. • TimeMixer++ (Wang et al., 2025): A general-purpose time series modeling architecture designed for tasks such as forecasting, classification, and imputation. 4.1.3 Metrics We use three metrics to evaluate the overall imputation performance: Mean Absolute Er- ror (MAE), Mean Squared Error (MSE), and Continuous Ranked Probability Score (CRPS). MAE quantifies the aver- age magnitude of the errors between the imputed values and the ground truth data, defined as MAE = 1 n P n i=1 |ˆx i − x i |, where x i is the true value, ˆx i is the imputed value, and n is the total number of instances. MSE evaluates the aver- age squared difference between the imputed and true val- ues, MSE = 1 n P n i=1 (ˆx i − x i ) 2 ; unlike MAE, it penalizes larger deviations more heavily, making it sensitive to out- liers. CRPS measures the probabilistic accuracy of imputed values when the imputation method provides an estimated probability distribution rather than a single point estimate, thus capturing uncertainty as well. Given a cumulative dis- tribution function (CDF) F and true value x, it is defined as (4.1)CRPS(F,x) = Z ∞ −∞ [F (y)− Iy ≥ x] 2 dy, where I(y ≥ x) is an indicator function and a lower value indicates better calibration and sharpness of the predicted distribution. Among them, MAE and MSE are applied to all methods, while CRPS i s only used for probabilistic models such as GP-VAE, CSDI, PriSTI, and our MBDiff. Copyright © 2026 by SIAM Unauthorized reproduction of this article is prohibited Table 4.2: Evaluation results on three different datasets with a missing rate of 20%. ElectricityGasWater MethodPoint MissingBlock MissingPoint MissingBlock MissingPoint MissingBlock Missing MAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSE G-Mean0.50210.45300.50620.47810.99631.60340.98121.42331.04101.11961.04761.1347 U-Mean0.37730.30450.38130.31760.59465.07420.53504.31120.60140.60760.60400.6080 LINEAR0.21300.12550.27060.21150.12230.26100.13490.28740.03160.49200.03170.0286 KF0.32860.50990.50200.78790.18991.18880.22500.84210.04050.03510.04600.0523 SAITS0.21700.01410.20190.04380.12950.18710.12670.19700.30410.09410.17400.0320 GP-VAE0.03820.00310.03830.00340.01870.00250.02630.00200.02710.00750.04320.0070 CSDI0.02930.00250.03010.00310.02670.00970.02140.00780.01960.00710.02850.0112 ImputeFormer0.13470.09520.33280.12520.14650.05490.25370.08930.04720.00860.23230.1514 PriSTI0.02880.00310.03260.00340.0193 0.00360.02130.00890.02040.00690.03010.0091 TimeMixer++0.01990.00310.15610.13680.11340.01870.12820.06460.07510.01040.09230.0532 MBDiff0.02760.00240.02930.00280.02130.00850.01980.00690.02510.00540.02020.0065 4.1.4 Implementation Details We implemented the MBDiff framework in PyTorch and conducted all experi- ments on an NVIDIA A100 GPU. The model is trained us- ing a two-stage process. In the first stage, we train the TVAE models for behavior pattern extraction. The utility consump- tion data is structured into tensors of shape R 144×48 , repre- senting three days of readings taken at 30-minute intervals. Local patterns are subsequently extracted using a sliding window with a length of 24 and a stride of 12. These TVAE models, configured with a latent dimension of d z = 64, 4 attention heads, and 2 Transformer layers, are trained for 100 epochs using the Adam optimizer with a learning rate of α = 0.001 and a weight decay of λ = 10 −6 . In the sec- ond stage, we train the diffusion model. This stage runs for 200 epochs using T = 500 diffusion timesteps and a linear noise schedule with β 1 starting at 10 −4 and β T ending at 0.5. The learning rate is decayed at the 75% and 90% train- ing milestones. For all training procedures, we used a batch size of 16 and performed validation every 20 epochs. For the baselines, parameters were configured to en- sure a robust comparison. The statistical methods G-MEAN (Amiri and Jensen, 2016), U-MEAN (Amiri and Jensen, 2016), and LINEAR (Huang, 2021) are hyperparameter-free. For the KF (Bakibillah et al., 2022) model, the process and measurement noise covariance matrices were initialized as identity matrices. The deep learning models were all trained using the Adam optimizer with a learning rate of 1× 10 −3 and a batch size of 32. Specifically, SAITS (Du, Côté, and Liu, 2023) was configured with 2 attention layers, 4 heads, and a hidden dimension of 128. GP-VAE (Fortuin et al., 2020) used a latent dimension of 64 and a Radial Basis Func- tion (RBF) kernel for its Gaussian Process component. The diffusion models, CSDI (Tashiro et al., 2021) and PriSTI (Liu et al., 2023), were implemented with T = 100 diffu- sion steps and a 4-layer Transformer backbone. Lastly, both Table 4.3: The results of CRPS on three different datasets. MethodElectricityGasWater MissingPointBlockPointBlockPointBlock GP-VAE0.0292 0.0414 0.0188 0.0192 0.0312 0.0331 CSDI0.0233 0.03770.0169 0.0174 0.0298 0.0317 PriSTI0.0225 0.0389 0.01350.0147 0.02570.0263 ImputeFormer 0.1347 0.3328 0.1465 0.2537 0.472 0.2323 TimeMixer++ 0.0199 0.1561 0.1134 0.1282 0.0751 0.0923 MBDiff0.0168 0.0359 0.0113 0.01680.0239 0.0281 ImputeFormer (Nie et al., 2024) and TimeMixer++ (Wang et al., 2025) were structured with 4 layers, a model dimen- sion of 128, and 4 attention heads. 4.2 Overall Performance We first present the impu- tation results of different methods on the three datasets with 20% missing data (Tables 4.2 and 4.3). To reflect real-world scenarios, we consider two missing patterns: point missing, where individual values are randomly removed, and block missing, where several values are removed in sequence, start- ing from a random point. Overall, traditional statistical methods do not perform well on any of the datasets, even with user-specific informa- tion. This suggests that user behavior in utility usage is often irregular and hard to predict, especially in the gas and water datasets, which are sparser. In contrast, neural network-based methods perform better by capturing patterns in the data. Among all models, those that generate data rather than pre- dict fixed values tend to produce more stable and accurate re- sults. On the electricity dataset, our method MBDiff achieves the best performance in both point and block missing cases. For example, it achieves an MAE of 0.0293 and an MSE of 0.0024, which are 2.32% and 4.0% better than the next-best Copyright © 2026 by SIAM Unauthorized reproduction of this article is prohibited model in the block and point missing setting, respectively. It also improves CRPS by 15.57%, 7.71% over the second-best one in the same settings. Electricity usage records are relatively dense and con- sistent, which helps MUBE identify and use meaningful pat- terns for filling in missing values. On the other hand, gas and water data are sparser and harder to learn from. Even so, MUBE still ranks in the top two across all settings. For gas data, GP-VAE works better on point missing cases, while MBDiff performs better on block missing. On water data, MBDiff shows strong advantages. For instance, it reduces block missing MAE by 29.1% compared to CSDI, and im- proves point missing CRPS by 7% over PriSTI, showing its ability to handle more irregular and incomplete data. Table 4.4: Sensitivity analysis under different missing ratios (point missing). Method 20%35%50% MAEMSEMAEMSEMAEMSE GP-VAE0.0382 0.0031 0.0523 0.0049 0.0522 0.0068 CSDI0.0293 0.00250.0514 0.0055 0.0519 0.0059 PriSTI0.0288 0.0031 0.04020.0043 0.04720.0052 ImputeFormer 0.1347 0.0952 0.1480 0.1025 0.1567 0.1236 TimeMixer++ 0.0199 0.0031 0.0983 0.0565 0.1804 0.1583 MBDiff0.02760.0024 0.0387 0.00490.0396 0.0050 4.3 Sensitivity Analysis To address the diverse types and rates of missing data in real-world scenarios, we evaluate the robustness of each method under varying missing pro- portions. Due to the sparsity of the water and gas datasets, we focus our analysis on the electricity dataset. As shown in Table 4.4, we report MAE and MSE under point miss- ing settings at different rates (20%, 35%, 50%). Our method, MBDiff, consistently ranks first or second across all condi- tions, outperforming other baselines overall. For example, with a low missing rate of 20%, aided by the user pro- file extracted by MUBE, BCDiff selects more informative local- and instance-view references to guide accurate impu- tations. Even at a 50% missing rate, MBDiff remains robust, achieving an MAE of 0.0396, which is better than PriSTI (0.0472) and CSDI (0.0519). It also achieves the lowest MSE of 0.0050. These results confirm MBDiff’s robustness under severe missing conditions. To address various missing data types and rates encoun- tered in real-world scenarios, we evaluate the robustness of different methods by assessing their performance across dif- ferent missing types and imputation rates. Due to the spar- sity of the water and gas datasets, we focus on the electricity dataset in this part. As shown in Tab. A.2 and Tab. A.1, at low missing rates (20%), MBDiff achieves superior perfor- mance across metrics, with the best MAE and MSE in both point and block missing scenarios, improving 4.2% in point missing MAE compared to PriSTI. To handle a small portion of data points missing (20%), given the extracted user pro- file by MUBE, BCDiff can select more instructive local-view and instance-view reference that steers the imputation. With the missing rate increases, the selection of local-view ref- erences may be impacted, as the matching subsequences in the data records to be imputed are filled with coarse-grained priors. At 35% missing rate, MBDiff maintains strong per- formance with the best MAE in both point missing (0.0387, 3.7% better than PriSTI) and block missing (0.0346, 10.8% better than PriSTI). Even at a 50% missing rate, MBDiff demonstrates remarkable resilience, achieving the best point missing data imputation performance (MAE: 0.0396, im- proving 16.1% over PriSTI) and remaining competitive in block missing scenarios. Despite this, with the aid of global- view and instance-view behavior selected solely based on temporal information, MBDiff still outperforms other base- lines in most cases. Similarly, Tab. A.1 shows the compar- ison of CRPS across different missing rates. From this ta- ble, we found that MBDiff demonstrates consistent strong performance. At a 20% missing rate, MBDiff achieves the best point missing imputation CRPS of 0.0210 (6.7% im- provement over PriSTI). With the missing rate increasing to 35%, MBDiff maintains its advantage with the best CRPS in both point (0.0374) and block (0.0386) scenarios. Even at a 50% missing rate, MBDiff still shows strong performance with competitive CRPS values, particularly in block miss- ing (0.0479), which is 7.7% better than PriSTI. These results further validate MBDiff’s robust performance across differ- ent evaluation metrics and missing scenarios. Table 4.5: Ablation study of different user behavior views. The abbreviations ’glb’, ’loc’, and ’ins’ refer to the global- view, local-view, and instance-view behavior patterns, re- spectively. Method20%35%50% MAEMSEMAEMSEMAEMSE glb + loc 0.0280 0.0027 0.0390 0.0055 0.0416 0.0054 glb + ins 0.0291 0.0035 0.0401 0.0072 0.0449 0.0060 loc + ins 0.0293 0.0032 0.0395 0.0067 0.0421 0.0053 MBDiff0.0276 0.0024 0.0387 0.0049 0.0396 0.0050 4.4 Ablation Study To further assess the effective- ness of incorporating three different views of user behav- iors in our proposed MBDiff, we conduct ablation studies on the electricity dataset, comparing the full model against variants that leverage only two types of extracted behav- iors. As shown in Table 4.5, our full MBDiff model con- sistently achieves the best performance across all missing rates, demonstrating the strong complementarity of the three views. Notably, removing the local-view behavior (the glb + Copyright © 2026 by SIAM Unauthorized reproduction of this article is prohibited ins variant) leads to the most significant performance drop, particularly at the severe 50% missing rate, which indi- cates that fine-grained, short-term temporal dependencies are crucial for guiding imputation when data becomes highly incomplete. Conversely, the glb + loc variant exhibits the most stable performance among the ablated pairs, suggest- ing that combining short-term local dependencies with long- term global statistical boundaries provides a highly robust foundation. Ultimately, while eliminating any single behav- ior pattern leads to a performance decline compared to the full model, these ablated versions still maintain competitive performance against most standard baselines, validating that a comprehensive, multi-view approach is essential for accu- rate utility data imputation. 5 Related Work 5.1 Time Series Data Imputation Time series data imputation fills in missing values in sequential data to en- sure completeness for analysis and applications, and it has been widely studied in various fields such as finance, health- care, and transportation. In the early years, traditional meth- ods such as linear interpolation (LinITP)(Huang, 2021) and Kalman filtering (KF)(Bakibillah et al., 2022) relied heavily on statistical prediction, but struggled to capture the com- plex temporal patterns inherent in real-world data. Recent advances in deep learning have led to rapid progress in time series imputation, which can be broadly divided into predic- tive and generative approaches. Predictive methods estimate missing values by leveraging temporal context from both past and future observations, typically using reconstruction- based models such as RNNs Cao et al. (2018), CNNs Wu et al. (2023), GNNs Cini, Marisca, and Alippi (2022), or at- tention mechanisms Shan, Li, and Oliva (2023). While effec- tive in capturing local patterns, these models often struggle with scalability and modeling global structures in long se- quences. In this work, we focus on generative methods, as they can produce diverse outputs for missing observations and are better suited to capturing overall data distributions. Among these, VAE-based Kim et al. (2023) and GAN-based Liu et al. (2019); Miao et al. (2021) methods rely on direct sam- pling from the latent space, which limits their generaliza- tion. Recently, diffusion models have demonstrated strong capabilities in modeling complex data through multi-step noise injection. CSDI Tashiro et al. (2021) first adopts condi- tional training using partial observations, SSSD Alcaraz and Strodthoff (2022) introduces structured state space models, and PriSTI Liu et al. (2023) incorporates spatiotemporal de- pendencies. However, these methods do not explicitly model user-specific behaviors. In our work, we address this limita- tion by incorporating fine-grained user characteristics from historical data as prior knowledge. This approach improves both model performance and real-world applicability. 5.2 Utility Data Mining High-quality utility time se- ries data is essential for real-world applications such as billing, resource allocation, and system optimization. Many data mining tasks, including demand forecasting and anom- aly detection, rely on these data to uncover meaningful pat- terns. In the water systems, DTW-based clustering reveals correlations between consumption behavior and socioeco- nomic status Steffelbauer et al. (2021). Smart meter data in gas systems enables pipeline topology inference and leakage detection Matalkah and Coyle (2020). In electricity systems, fine-grained records support load profiling, demand predic- tion, and operational planning Wang et al. (2018), and re- cent studies further advance accurate and reliable user-level energy usage prediction Yu et al. (2026b,a). Utility mining techniques have also been applied in other domains, such as wastewater system analysis Cipolla and Maglionico (2014) and heating system optimization Calikus et al. (2019). These diverse downstream applications fundamentally rely on real- istic and high-quality utility data. Therefore, our work lays a vital foundation for a broad range of utility data mining tasks. 6 Conclusion In this paper, we propose MBDiff, a novel behavior-aware utility data imputation framework that effectively incorporates comprehensive user behavior to steer accurate missing value imputation. MBDiff consists of two core modules: (i) a multi-view user behavior extraction module to learn comprehensive user behavior from differ- ent aspects (e.g., global, local, and instance view), and (i) a behavior-aware conditional diffusion model consisting of a reference selection module and a conditional attentional denoising network to impute utility data in a computation- ally efficient way. We implement and evaluate MBDiff by collaborating with a utility provider, and extensive experi- mental results demonstrate that MBDiff outperforms state- of-the-art baselines across three datasets, e.g., it improves 7.04% and 29.1% on the electricity and water usage datasets for block missingness imputation, showcasing the effective- ness of utilizing user behavior as guidance for imputation and highlighting its ability to address real-world challenges of missingness in utility data. It has great potential to impute missing values in utility data to improve utility services and business intelligence. Beyond utility time series imputation, behavioral patterns extracted from existing data hold signif- icant potential to guide imputation across various types of temporal or spatio-temporal time series, offering a promis- ing avenue for future exploration. Acknowledgment We thank all the reviewers for their insightful feedback and comments, which helped us improve this paper. 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A Appendix A.1 Motivation and Data Analysis We present some analysis and visualization of the datasets in this part. As shown in Fig. A.1, daily usage patterns of different utilities are distinct and each utility usage also varies significantly across different months. For electricity consumption, March and November show almost the same pattern, but July shows notably higher consumption, especially between 10 AM and 10 PM, which is likely due to increasing air conditioner us- age in the hot summer in Florida. The usage patterns of gas and electricity consumption are opposite. Gas consumption in July is much less than that in other months. Water con- sumption shows consistent patterns across all three months and there is always a morning peak and an evening peak. This regular pattern stays stable from March to November, though July has a slightly higher daytime usage. Fig. A.2 illustrates individual-level electricity consump- tion patterns for three randomly chosen users in March, July, and November. It indicates that different users exhibit distinct energy consumption behaviors. During these three months, user C consistently shows higher electricity usage compared to user A and user B, with broader fluctuations throughout the day, while user A and user B show more steady consumption patterns. Even for the same user, the electricity consumption varies across different months. User A maintains relatively low electricity usage throughout these three months with minimum fluctuations throughout the day. User B consumes similar electricity to user A in March and July, but the consumption is much higher in November com- pared to user A. Overall, this analysis demonstrates the im- pact of both user-specific behaviors and ambient factors like seasonal changes on electricity consumption patterns. Fig. A.3 shows the electricity consumption pattern of three randomly selected users across different days of the week, revealing significant variations in daily usage behav- iors. During weekdays, user A consistently consumes the most electricity compared to user B and user C, with sev- eral peaks during the day, especially around 8 AM to 10 AM and 6 PM to 9 PM, which aligns with our previous monthly analysis. Unlike user A, user B keeps their usage low and steady, and user C shows moderate usage with some ups and downs. There are the highest peaks on Monday, especially for user A, while there are longer periods of high electric- ity consumption on Thursday and Friday. On Saturday, all users show similar electricity usage patterns with two main peaks: one around 1 PM, which is likely during lunchtime, and another one around 8 PM, which is likely for evening activities. This similar pattern indicates that people’s activity is more similar on weekends, unlike weekdays, where indi- vidual schedules create more distinct usage patterns. Copyright © 2026 by SIAM Unauthorized reproduction of this article is prohibited 6 AM12 PM6 PM12 AM Time of Day 0.5 1.0 1.5 2.0 Value Mar Jul Nov (a) Electricity 6 AM12 PM6 PM12 AM Time of Day 0.02 0.03 0.04 0.05 0.06 0.07 Value Mar Jul Nov (b) Gas 6 AM12 PM6 PM12 AM Time of Day 0.08 0.10 0.12 0.14 Value Mar Jul Nov (c) Water Figure A.1: Aggregated statistics of each whole utility dataset on March, July, and November. The solid line represents the global mean of each data point in the time series, while the shaded area indicates the range between the 25th and 75th percentiles for the electricity time series. Due to the sparsity of data points in the gas and water datasets, we omit the quantile plots for these datasets. 6 AM12 PM6 PM12 AM Time of Day 1 2 3 Value User A User B User C (a) 2018/03 6 AM12 PM6 PM12 AM Time of Day 0.5 1.0 1.5 2.0 2.5 Value User A User B User C (b) 2018/07 6 AM12 PM6 PM12 AM Time of Day 0 1 2 3 Value User A User B User C (c) 2018/11 Figure A.2: Aggregated statistics of each user in different months on electricity dataset. 6 AM12 PM6 PM12 AM Time of Day 0 1 2 3 Value User A User B User C (a) Monday 6 AM12 PM6 PM12 AM Time of Day 0 1 2 3 4 Value User A User B User C (b) Tuesday 6 AM12 PM6 PM12 AM Time of Day 0 1 2 3 Value User A User B User C (c) Wednesday 6 AM12 PM6 PM12 AM Time of Day 0 1 2 3 Value User A User B User C (d) Thursday 6 AM12 PM6 PM12 AM Time of Day 0 1 2 3 4 Value User A User B User C (e) Friday 6 AM12 PM6 PM12 AM Time of Day 0 1 2 3 Value User A User B User C (f) Saturday Figure A.3: Aggregated statistics of each user on different days in a week on the electricity dataset. Copyright © 2026 by SIAM Unauthorized reproduction of this article is prohibited A.2 Extended Sensitivity Analysis We report the complete sensitivity analysis results of all methods on the electricity dataset under point and block missingness at 20%, 35%, and 50% missing rates. Table A.1 and Table A.2 sum- marize the CRPS and the MAE/MSE results, respectively. Table A.1: Sensitivity Analysis: The results of CRPS for electricity data imputation under different missing ratios. Method20%35%50% MissingPointBlockPointBlockPointBlock GP-VAE0.0292 0.0300 0.0420 0.0363 0.0415 0.0611 CSDI0.0233 0.0377 0.0411 0.03940.0419 0.0528 PriSTI0.0225 0.0389 0.0387 0.0399 0.0396 0.0519 ImputeFormer 0.1347 0.3328 0.1480 0.3595 0.1567 0.3851 TimeMixer++ 0.0199 0.1561 0.0983 0.1521 0.1804 0.1924 MBDiff0.02100.03590.0374 0.0386 0.04030.0479 Table A.2: Sensitivity Analysis: The results of MAE and MSE for electricity data imputation under different missing ratios. 20%35%50% MethodPoint MissingBlock MissingPoint MissingBlock MissingPoint MissingBlock Missing MAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSE G-Mean0.50210.45300.50620.47810.50690.46950.51260.47440.50640.47080.49910.4584 U-Mean0.33730.30450.38130.31760.37500.30500.38220.31400.37880.30440.37320.3021 LINEAR0.21300.12550.27060.21150.26590.20080.32900.29500.36390.35680.38410.4281 KF0.32860.50990.50200.78790.50120.87650.80002.53860.89683.21821.45449.7296 SAITS0.21700.01410.20190.04380.15870.05980.29460.09170.16860.03680.15320.0367 GP-VAE0.03820.00310.03830.00340.05230.00490.04590.00520.05220.00680.08180.0141 CSDI0.02930.00250.03010.00310.05140.00550.04230.00380.05190.00590.06590.0153 PriSTI0.02880.00310.03260.00340.0402 0.00430.03880.00390.04720.00520.05140.0129 TimeMixer++0.01990.00310.15610.13680.09830.05650.15210.14270.18040.15830.19240.1621 ImputeFormer0.13470.09520.33280.12520.14800.10250.35950.22350.15670.12360.38510.2568 MBDiff0.02760.00240.02930.00280.03870.00490.03460.00390.03960.00500.05390.0149 Copyright © 2026 by SIAM Unauthorized reproduction of this article is prohibited