Missing Values In Time Series Research Articles

Linearly interpolating missing values in time series helps little for land cover classification using recurrent or attention networks

Satellite time series data, widely used for land cover classification, often contain missing values due to cloud contamination, which can negatively affect classification. Numerous strategies have been developed to reconstruct the missing values to produce regular time series for machine learning classifiers, among which the compositing followed by the linear interpolation is most widely used. However, the classification improvement of linear interpolation for land cover classification has not been examined. Recently developed deep learning models such as long short term memory (LSTM) and Transformer allow such examination as they can classify time series with missing values. In this study, we compared the time series composites with missing values (without linear interpolation) and the linearly interpolated time series composites (without missing values) for land cover classification. About 18 thousand Harmonized Landsat Sentinel-2 (HLS) images acquired over Amur River Basin of China (890,308 km2) in 2021 were composited to 14 16-day periods. Two time series composites were classified, i.e., (i) the 16-day composites without interpolation that have on average 15.35% 16-day periods with missing values and (ii) the linearly interpolated 16-day composites with no missing values. The classifications showed that (1) between classifications with and without linear interpolation there was < 0.2% overall accuracy differences for the bidirectional LSTM (Bi-LSTM) and < 0.5% for the Transformer both of which were smaller than model training randomness; and (2) the computation time can be saved using composites without linear interpolation. The findings suggested that it is unnecessary to use the time-consuming linear interpolation in Bi-LSTM and Transformer-based land cover classifications. The findings were confirmed by experiments for sensitivity to the number of cloud-free composites and to different classification legends using crop type classifications. It implied the linear interpolation algorithm cannot reconstruct reliable time series for land cover classifications and historical use of such method is more about mitigating the inability of traditional classifiers to handle missing values rather than improving classifications. Linear interpolation is not necessary for LSTM and Transformer with capability to handle missing values. The training datasets and developed codes in this study are made publicly available.

A gap-filling method for satellite-derived chlorophyll-a time series based on neighborhood spatiotemporal information

Satellite-derived Chlorophyll-a concentration (Chla) time series products are essential for large-scale marine environmental monitoring. However, the plenty of missing pixels in current satellite Chla products severely hinder their applications for marine research, due to cloud contamination, solar glint, and unfavorable observation conditions. This study proposed a Chla time series gap-filling method for MODIS 8-day composite Chla product by integrating spatiotemporal information (STGF). This method employed spatially neighboring pixels with similar temporal variation to fill the missing values in time series, without involving training or auxiliary data. The performance of the STGF is assessed quantitatively and qualitatively. The correlation coefficients (CC) between gap-filled data and actual observations for years of 2004, 2010, 2016, and 2022 across the entire study area are all greater than 0.97. The mean absolute percentage error (MAPE) and root mean square error (RMSE) are less than 16.1 % and 0.233 mg/m3, respectively. The proposed STGF outperformed linear interpolation and the DINEOF algorithm from both spatial and temporal perspectives, suggesting the effectiveness of STGF in handling continuous data gaps and capturing detailed Chla variation patterns, especially in regions with significant variability. The findings suggest that the proposed STGF method offers a viable alternative for filling missing values in Chla time series data. This supports the demand for long-term, large-scale, and high-coverage ocean color remote sensing data in marine environmental studies.

A time series continuous missing values imputation method based on generative adversarial networks

Generative adversarial networks (GANs) have been widely utilized in time series analysis and modeling, wherein generators and discriminators interact to generate realistic data. However, when addressing the challenge of imputing continuous missing values in time series, the generator struggles to learn meaningful features due to the loss of local information, and the discriminator’s loss function exhibits significant deviations in determining the probability, making it difficult to effectively update model parameters during training. In response to these issues, this study presents a novel imputation model called cue Wasserstein generative adversarial network with gradient penalty (CWGAIN-GP). CWGAIN-GP incorporates the contextual cue information matrixs into the generator, enabling obtain and capture the potential data evolution features that hidden beyond the missing positions. This approach constrains the generator’s generation results to be closer to the true values probability distribution. Meanwhile, it optimizes the generator and discriminator neural network structure and loss function computation strategy of the original GAN model, which not only improves the accuracy of continuous missing data interpolation, but also improves the training stability of the model. Finally, We used three imputation accuracies on two real-world datasets, including experiments on imputation accuracy for eight comparison models, ablation study, and experiments on the effect of consecutive missing lengths. The experimental results show that CWGAIN-GP achieves impressive performance in imputing continuous missing time series.

Imputation of missing values in environmental time series by D-vine copulas

Missing values in environmental time series are common and must be imputed before carrying out an analysis requiring complete data. We propose an imputation method for the time series of a target station using information of neighboring stations measuring the same variable. The method allows these neighboring stations to have missing values themselves. The multivariate dataset comprising the time series of the target station and its neighboring stations is jointly modeled by a vine copula and parametric margins. Multiple imputation takes into account the uncertainty of missing data by generating several plausible values for each missing value in the time series of the target station. This is done in a Bayesian framework by sampling the posterior distribution of a missing value, which is conditional on the observed stations for the date. The method is suitable for extremes because the vine copula can model the eventual tail dependence between stations. The application to a skew surge time series is presented, with cross-validated results and a focus on the performance for the upper extremes.

Attention-Based Sequence-to-Sequence Model for Time Series Imputation.

Time series data are usually characterized by having missing values, high dimensionality, and large data volume. To solve the problem of high-dimensional time series with missing values, this paper proposes an attention-based sequence-to-sequence model to imputation missing values in time series (ASSM), which is a sequence-to-sequence model based on the combination of feature learning and data computation. The model consists of two parts, encoder and decoder. The encoder part is a BIGRU recurrent neural network and incorporates a self-attentive mechanism to make the model more capable of handling long-range time series; The decoder part is a GRU recurrent neural network and incorporates a cross-attentive mechanism into associate with the encoder part. The relationship weights between the generated sequences in the decoder part and the known sequences in the encoder part are calculated to achieve the purpose of focusing on the sequences with a high degree of correlation. In this paper, we conduct comparison experiments with four evaluation metrics and six models on four real datasets. The experimental results show that the model proposed in this paper outperforms the six comparative missing value interpolation algorithms.

Comparative characteristics of gap filling methods in high-frequency data of micrometeorological measurements

This article is devoted to one of the most complex technical aspects of the primary processing of micrometeorological data. The article contains a description of basic algorithms for restoring missing values in time series of data. The gap filling methods implemented by the authors on the basis of the thus described algorithms are tested using an example of filling-in generated gaps in series of temperature and vertical wind speed. The series of high-frequency measurements of an acoustic anemometer installed on the micrometeorological tower of the meteorological observatory (MO) of Moscow State University (MSU) are used as test data. As a result of this review of gap-filling methods, the effectiveness of various methods for filling-in missing values is assessed.

Data-Driven Copy-Paste Imputation for Energy Time Series

A cornerstone of the worldwide transition to smart grids are smart meters. Smart meters typically collect and provide energy time series that are vital for various applications, such as grid simulations, fault-detection, load forecasting, load analysis, and load management. Unfortunately, these time series are often characterized by missing values that must be handled before the data can be used. A common approach to handle missing values in time series is imputation. However, existing imputation methods are designed for power time series and do not take into account the total energy of gaps, resulting in jumps or constant shifts when imputing energy time series. In order to overcome these issues, the present paper introduces the new Copy-Paste Imputation (CPI) method for energy time series. The CPI method copies data blocks with similar properties and pastes them into gaps of the time series while preserving the total energy of each gap. The new method is evaluated on a real-world dataset that contains six shares of artificially inserted missing values between 1 and 30%. It outperforms by far the three benchmark imputation methods selected for comparison. The comparison furthermore shows that the CPI method uses matching patterns and preserves the total energy of each gap while requiring only a moderate run-time.

Kalman Filter Algorithm versus Other Methods of Estimating Missing Values: Time Series Evidence

Ideally, we think data are carefully collected and have regular patterns with no missing values, but in reality, this does not always happen. This study examines four (4) methods—mean imputation (MI), median imputation (MDI), linear imputation (LI) and Kalman filter algorithm (KAL)—of estimating missing values in time series. The study utilized pairs of nine (9) simulated series; each pair constitutes “actual series” and “12% missingness series”. The three (3) sample sizes i.e. small (50), medium (200) and large (1000) were varied over the additive models linear, quadratic and exponential forms of trend. The 12% missingness series were estimated using MI, MDI, LI and KAL. The performances of the method were checked using the root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE), while the overall performances of the estimating methods were accessed using the average of the accuracy measures (RMSE, MAE and MAPE). The results of the average-accuracy measures show that KAL outperformed other methods (MI, MDI and LI) at the three sample sizes when the trend was linear; also, MDI outperformed other methods at the three (3) sample sizes when the trend was exponential. Furthermore, MI outperformed others at small and large sample sizes when the trend was quadratic. However, the Kalman filter algorithm proved better when the sample size was medium. Hence, KAL, MI and MDI methods are recommended to estimate missing data in time series when the trend is linear, quadratic and exponential respectively, until further study proves otherwise.

Optimization methods for the imputation of missing values in Educational Institutions Data

The imputation of missing values in the detail data of Educational Institutions is a difficult task. These data contain multivariate time series, which cannot be satisfactory imputed by many existing imputation techniques. Moreover, almost all the data of an Institution are interconnected: the number of graduates is not independent from the number of students, the expenditure is not independent from the staff, etc. In other words, each imputed value has an impact on the whole set of data of the institution. Therefore, imputation techniques for this specific case should be designed very carefully. We describe here the methods and the codes of the imputation methodology developed to impute the various patterns of missing values which appear in similar interconnected data. In particular, a first part of the proposed methodology, called ``trend smoothing imputation'', is designed to impute missing values in time series by respecting the trend and the other features of an Institution. The second part of the proposed methodology, called ``donor imputation'', is designed to impute larger chunks of missing data by using values taken form similar Institutions in order to respect again their size and trend.•Trend smoothing imputation can handle missing subsequences in time series, and is given by a weighted combination of: (a) weighed average of the other available values of the sequence, and (b) linear regression.•Donor imputation can handle full sequence missing in time series. It imputes the Recipient Institution using the values taken from a similar institution, called Donor, selected using optimization criteria.•The values imputed by our techniques should respect the trend, the size and the ratios of each Institution.

A comparison of the Bayesian classifier and the Estimates-based algorithm for crop identification by Terra/MODIS 250- m data

We studied and compared the Bayesian classifier (BC) and the Estimates-based algorithm (EBA) in this research for crop identification by a set of Terra/MODIS 250-m data. Methods are based on time-series calculated by satellite images. EBA has an undoubted advantage. It uses time series even with missing values whereas BC requires reconstructing missing values of time series a priori. Experiments are based on data obtained for the Samara region in 2014-2016 years. The numerical results are given for the different length of time series to compare two methods throughout the growing season.

A Spatiotemporal Prediction Framework for Air Pollution Based on Deep RNN

Abstract. Time series data in practical applications always contain missing values due to sensor malfunction, network failure, outliers etc. In order to handle missing values in time series, as well as the lack of considering temporal properties in machine learning models, we propose a spatiotemporal prediction framework based on missing value processing algorithms and deep recurrent neural network (DRNN). By using missing tag and missing interval to represent time series patterns, we implement three different missing value fixing algorithms, which are further incorporated into deep neural network that consists of LSTM (Long Short-term Memory) layers and fully connected layers. Real-world air quality and meteorological datasets (Jingjinji area, China) are used for model training and testing. Deep feed forward neural networks (DFNN) and gradient boosting decision trees (GBDT) are trained as baseline models against the proposed DRNN. Performances of three missing value fixing algorithms, as well as different machine learning models are evaluated and analysed. Experiments show that the proposed DRNN framework outperforms both DFNN and GBDT, therefore validating the capacity of the proposed framework. Our results also provides useful insights for better understanding of different strategies that handle missing values.

Cuckoo search-designated fractal interpolation functions with winner combination for estimating missing values in time series

Reliable data have always played a vital role in time series analysis and research. Nevertheless, missing data, which can bias the original properties of the time series if the pattern of missed data is systematic, is also a very common phenomenon in observed processes. Thus, how to address missing values is a very important challenge, especially in the upcoming big data era. This paper proposes the cuckoo search-designated fractal interpolation functions (CS-DFIFs) method and the CS-DFIFs-winners combining (CS-DFIFs-WC) method for estimating missing values. The former method skillfully transforms Fractal Interpolation Functions (FIFs) to make it possible to calculate a specified point's missing value, which is difficult to obtain with the traditional approach. Then, to optimize the parameters that transforming process generates, the cuckoo search algorithm (CS) and DFIFs are synthesized into a novel model, CS-DFIFs. Considering classical interpolation methods, such as Linear, Cubic Spline and Piecewise Cubic Hermite Interpolation Polynomial (PCHIP), having some born advantages, the inspiration of winners combining is sparked. CS algorithm is used to obtain the best weights of winners in this combined model, CS-DFIFs-WC. Two databases, electricity demands and prices, are chosen to be the numerical testing object at 7 missing levels in this paper. The results show that CS-DFIFs-WC overcomes the deficiency of FIFs that cannot calculate a specified point's missing value, and outperforms benchmarks at almost every level by four criteria.

연속적 결측이 존재하는 기온 자료에 대한 결측복원 기법의 비교

Abstract Consecutive missing values are likely to occur in long climate data due to system error or defective equipment.Furthermore, it is difficult to impute missing values. However, these complicated problems can be overcameby imputing missing values with reference time series. Reference time series must be composed of similartime series to time series that include missing values. We performed a simulation to compare three missingimputation methods (the adjusted normal ratio method, the regression method and the IDW method) tocomplete the missing values of time series. A comparison of the three missing imputation methods for thedaily mean temperatures at 14 climatological stations indicated that the IDW method was better thanxothers at south seaside stations. We also found the regression method was better than others at moststations (except south seaside stations).Keywords: consecutive missing value, missing value imputation, adjusted normal ratio methods, regressionmethod, IDW method

Application of structured low-rank approximation methods for imputing missing values in time series

Application of structured low-rank approximation methods for imputing missing values in time series

A Note on Likelihood Estimation of Missing Values in Time Series

Missing values in time series can be treated as unknown parameters and estimated by maximum likelihood, or as random variables and predicted by the expectation of the unknown values given the data. The difference between these two procedures is illustrated by an example. It is argued that the second procedure is, in general, more relevant for estimating missing values in time series.

A Note on Likelihood Estimation of Missing Values in Time Series

Abstract Missing values in time series can be treated as unknown parameters and estimated by maximum likelihood or as random variables and predicted by the expectation of the unknown values given the data. The difference between these two procedures is illustrated by an example. It is argued that the second procedure is, in general, more relevant for estimating missing values in time series.

ESTIMATION AND INTERPOLATION OF MISSING VALUES OF A STATIONARY TIME SERIES

Abstract. A complete solution of the important problem of estimating (interpolating) the missing values of a stationary time series is obtained by decomposing it into a prediction plus regression problem. This makes it possible to estimate the missing values by finding the multistep‐ahead predictors and using the existing computer packages for time series analysis. Such a solution is vital for the E step of the EM algorithm, and it is shown how this algorithm can be used to develop a simultaneous procedure for estimating the parameters and missing values of a time series.

Gap Repair In Water Level Measurement DataUsing Neural Networks

The paper presents a method for completing missing values in time series by applying neural network computation. Conventional methods to complete fragmentary time series like linear interpolation are chosen for small amounts of missing values; problems occur when there are larger intervals of missing values. So far such gaps are repaired by estimating techniques that use parameter estimation of mathematical models. However such an approach fails when there is not enough information for calibrating the model or when the model is too simplified for reliably completing the data series. Neural networks with their generalisation and memory properties are predestined for this category of problems. In the proposed method a two-layer feed-forward network is used to build spatial patterns out of time series intervals with some variations on one hand and on the other hand directly model processing is used for producing time series information as a function of additional and external information. A representative values approach combines characteristics of both the spatial and the direct approach. Parameters in the approach are the number of representativ e values in each input pattern and the sampling distance of representativ e values in the original time series. The discussed example refers to water level data that contains a basic tidal oscillation (which is computable by astronomical components) and additional set-up which is mainly a function of local meteorological conditions like wind speed and wind direction.