Handling Missing Data in the Time-Series Data from Wearables

Abstract

Wearable technology is being used for tracking continuous events in various sectors of our lives. Wearables contain different types of sensors which can acquire movement data, blood pressure, blood sugar, temperature, and other physiological parameters. These parameters are recorded in the form of seamless univariate or multivariate time-series data. Very often, however, the data contains missing datum which disrupts the continuity of the data making it difficult to analyze the data. The missing part of the data needs to be imputed to make the remaining available data applicable. Choosing the proper imputation method is crucial for fruitful analysis and extracting underlined features from the data. In this context, this chapter discusses sensors associated with wearable technology which generate the time-series data, missing data in the wearables’ time-series data, and various imputation methods being used for imputing the missing data.

1. Introduction

Wearable technology refers to the electronic devices which users can wear. Wearables are available in different manifestations like watches, eyeglasses, tattoos, strips, gloves, belts, footwears, clothes, and can be associated with different parts of the body like wrist, eyes, skin, chest, waist, and foot [1, 2]. These wearables are often called smart devices as they are remotely connected by different wireless communication technologies like the internet and Bluetooth. The data is also stored on cloud platforms and consequently shared. It helps to synchronize wearables with mobile phones and laptops. This chapter is focused on healthcare applications. The major purpose of wearing these smart devices is to track real-time physiological information from the body. The information gathered by wearables is further analyzed by different algorithms for extracting features relevant to the users’ health. Finally, it is shared with users, and users’ healthcare providers. This information is utilized for the purpose of self-management, monitoring health conditions, clinical decision-making, and preventive measures for different health conditions [3, 4].

The popularity of wearables is astonishing with a forecast of 20% annual growth rate for its market [5]. The major reason for the popularity of wearables is that it collects data in a non-invasive or a minimal invasive manner. For many devices, no intervention of clinicians is required. This advantage has turned wearable devices into smart consumer devices. Among the diverse wearable devices and gadgets, smartwatches are the most popular. Interested users can buy it from the market and use it for self-monitoring. The advancement of technology has turned it into a helpful supplementary in healthcare and disease monitoring for clinicians [6]. Smartwatches can primarily track heart rate, measure calories burnt, count steps, and monitor sleep. Moreover, the use of smartwatches has extended beyond self-monitoring. Smartwatches are being used for routine remote monitoring, diagnostics purposes, predicting disease risk factors and even for treatments. Smartwatch-derived data is being used for diagnosing cardiovascular diseases [7, 8], neurological disorders [9], fatty liver diseases [10], Parkinson’s disease [11], and metabolic disorders including diabetes [12].

Data collection has become feasible due to technological advancement in developing various types of sensors. Wearable devices contain inertial sensors, optical sensors, temperature sensors, barometric sensors, and others. This is a rapidly growing area of research as improving the sensitivity of sensors, incorporating multiple sensors in a limited space, energy efficiency of the sensors, connectivity of the sensors, and lighter weight materials are the associated challenges. Sensors are also being specially designed for specific purposes [13].

Due to the presence of different types of sensors and their continuous operations, wearable devices generate a huge amount of multivariate data often considered as “big data” which is personalized also [14, 15]. Data consists of values of different parameters recorded continuously maintaining a regular interval of time. The time-series data from smartwatches contain heart rate, step counts, elevation, sleep duration, and sleep quality. Wearables also provide blood pressure, blood oxygen level, electrocardiogram (ECG), and body temperature. The cardiac data from wearables are being used for diagnosing and monitoring several disease conditions like coronary artery diseases [16], atrial fibrillation [17, 18, 19], and atrial arrhythmias [20]. The gyroscopes used in the wearables provide time series data of rotation along three mutually perpendicular directions. This data can further be analyzed to identify different activities and body postures like walking, running, sitting, moving up or down along the staircase, and standing [21]. It also detects tremors, and bradykinesia in case of users having Parkinson’s disease [22].

However, very often due to several reasons time series data generated by wearables contain missing data. The reasons may include but not limited to that the device was not used for a certain interval of time, the device was used but data was not transmitted, or the recorded data was noisy and hence removed [23]. The presence of missing data or missing values of any recorded parameter downgrades the data quality. Wearable data are very often analyzed by artificial intelligence-based machine learning algorithms, for which continuous data is preferred [24]. Ignoring the timestamps of missing data is not a good choice as it is a complete loss of data. Also, multivariate data may have missing values only for a particular parameter. Imputing the missing data is a necessity for making the entire dataset useful. Hence, missing values are substituted with predicted values. Moreover, there are different aspects of missing data that need to be considered during imputation. Data may or may not be missing at random fashion. Data may be missing continuously or in a discrete manner [25].

Researchers have concentrated on developing effective ways of imputing missing values in time-series data. This chapter is focused on wearable technology and frequently used imputation methods in imputing missing values of time-series data in the context of wearables.

2. Wearable technology

Wearable technology comprising portable sensors is being used to collect various types of information in a continuous seamless fashion. It has appeared as a huge source of time-series data. Data is collected from biofluids like sweat, tears, saliva, and interstitial fluid.

Wearable devices are loaded with different types of sensors which include accelerometers, gyroscopes, magnetometers, heart rate sensors, oximetry sensors, barometric pressure sensors, ambient temperature sensors, skin conductance sensors, etc.

The most often used sensors are inertial sensors like accelerometers and gyroscopes. Such sensors can detect the motion of body parts like hands and limbs. The acceleration and angular velocity of the associated organ are measured in three mutually perpendicular directions using combinations of three accelerometers and gyroscopes. The acceleration is measured by measuring the displacement of a test mass attached to a suspension system [26]. This information has been analyzed to successfully detect different daily activities like walking, sitting, running, and moving up/down through steps [21]. It can even predict the risk of possible falls [27] and the user and even the caregivers can be alerted by remote monitoring. Such sensors are particularly useful for patients having gait-related disorders. Patients suffering from Parkinson’s disease are also benefitted [28]. Magnetometers are also associated with motion sensors. It measures the earth’s magnetic field to detect the direction of the motion.

Wrist-worn wearable devices, smartwatches, use optical sensors to measure heart rate by the photoplethysmography (PPG) method. They use green LED light which is absorbed by blood as red-colored blood absorbs green light most. The photosensors detect light reflected from blood and measure absorbance which is then quantified to the volume of the blood. The synchronous cardiac beats change blood volume within the vessel in each beat. Heart rate is extracted from the volume change measured from the changes in absorbance [29]. Apart from measuring heart rates, smartwatches like KickLL, are also designed to measure respiratory rates using the PPG method [30].

Smartwatches also measure peripheral blood oxygen saturation levels by measuring light absorbed by oxygenated and non-oxygenated hemoglobin in blood vessels. The oxygenated and non-oxygenated hemoglobin molecules absorb light of different wavelengths (940 nm and 660 nm, respectively) [31]. Smartwatches use red and infrared lights and have inbuilt sensors for measuring blood oxygen levels only when the user is not physically active. Blood with higher oxygen levels will absorb more red light compared to infrared light [32, 33].

Wearables can provide an estimate of body temperature. It is based on the epidermal measurement and hence does not provide the core body temperature of the user. However, long-term, continuous, and accurate measurement is possible [34]. Apart from monitoring regular health conditions, the recorded temperature plays an important role in detecting diseases like epilepsy [9] and COVID-19 [35].

Bioimpedance (BIA) is the response to the externally applied electric signal. Smartwatches are equipped with bioimpedance sensors. These sensors are used to measure body fat, composition, blood pressure, and body glucose levels [36]. A recent study, involving 75 participants reports that the measurements of smartwatch BIAs is comparable with laboratory measures for measuring body fat [37]. BIA sensors are also being used for blood pressure measurements [38]. The advantage of wearable sensors is that they can continuously measure blood pressure with a comfortable cuffless setup. Kireev et al. have designed self-adhesive grapheme-made electronic tattoos for measuring blood pressure [39]. BIA sensors measure pulse transit time and further machine learning based analysis estimates both diastolic and systolic blood pressures [40].

The wearables are often coupled with biosensors. A typical biosensor has two parts receptor and transducer. The receptor may consist of a specific enzyme, antibody, DNA or cell which identifies target analytes by bio-reaction. The transducer translates the bio-reaction into a measurable electrical signal. Depending on the bio-reaction, the transducer can be electrochemical, optical, thermal, or piezoelectric type [1].

3. Missing data

Time series data in biology is generated when information on a physiological parameter or a phenomenon is recorded with time. The presence of missing values can lead to the loss of information and statistical power [41], making typical data analysis techniques ineffective or challenging to use and can inject bias into estimates obtained from a statistical model [42, 43]. Missing data typically falls into one of the following three categories according to Little et al. [44]:

• Missing completely at random (MCAR): In cases where missing data is said to be missing completely at random (MCAR), the missing values are independent of both the observed and unobserved data. This means that there is no systematic relationship between the existing data and the missing values [44, 45].

• Missing at random (MAR): In situations where missing data is said to be missing at random (MAR), the missing values are systematically related to the observed data, but not the unobserved data. In other words, although they cannot be predicted, the missing values have some relationship with other variables in the dataset that can be used to explain why they are missing [44, 45].

• Not missing at random (NMAR): Missing data is considered missing not at random (MNAR) when the missing values are systematically related to the unobserved data, which cannot be explained by other variables in the dataset. This means that there is a pattern to the missing data that is not accounted for by the available information [46].

High data quality criteria are essential for machine learning (ML) applications to provide reliable prediction performance and appropriate use of automated decision-making. Eliminating the missing data is one approach to solving this issue. However, if we just delete the data, we run the danger of losing important information. To impute the missing data would be a preferable course of action. In other words, we must extrapolate those missing numbers from the data that is already available. This chapter goes into further detail on the substantial work that the statistics community has done on the imputation of missing data. Techniques for handling missing data have been divided into three groups by Song et al. [47]:

1. Delete missing data: With this approach, the row with the missing data is simply deleted. However, due to its ease of use, this strategy is popular. Due to the possibility of leaving out important details, it can only be utilized when the missing data are modest in number as compared to the available data.

2. Tolerate missing data: In contrast to the missing data deletion approach, this method uses the whole dataset for the analysis and replaces the missing data points with the special value NULL. This approach cannot be used when the goal is to forecast the data point.

3. Impute missing data: The process of imputing missing data entails replacing estimated values for missing data points and then evaluating the entire dataset as if the imputed data were original. This approach is considered to be better when we have to deal with limited data, which is usually the case in most practical scenarios. Let us discuss some techniques used to impute missing data in the following section.

4. Imputation methods

Missing data imputation refers to replacing missing values with approximated ones. The current portion of this book chapter will go through various missing data imputation techniques. Statistical procedures, such as substituting missing values with the mean, median, or mode of the column containing missing values, are one of the most used approaches for dealing with missing data. While it is simple and quick, it alters the statistical character of the data set. This not only skews our histograms, but it also understates the variation in our data because we are making several values the same whereas, in reality, they evidently would not be the same [48]. Time series data can be seasonal or cyclical in nature, and they can follow a pattern. The standard statistical data imputation techniques compromise these components of time series data. Therefore, while this is a popular practice, it should be avoided while handling time-series data.

When mean imputation is not the best fit for the missing values, interpolation is used. Interpolation is a technique for estimating missing values between two known positions. An equally-spaced continuous time-series data can be represented as $y=\left\{y_t|t=t_0,t_0+a,\dots ,t_0+\mathit{Na}\right\}$

$$y_{t_0},y_{t_0+a},y_{t_0+2a},\dots ,y_{t_0+\mathit{Na}}$$E1

where the series starts at time $t_0$, $y_t$ is the measured value of the variable at time $t$, the series continues for N timestamps, and a is the interval between two consecutive observations, also called “sampling interval”. If data for n continuous timestamps are missing, the series looks like

$$y_{t_0},y_{t_0+a},y_{t_0+2a},\dots ,y_{t_0+\left(i-1\right)a},\mathit{nan},\mathit{nan},\dots ,\mathit{nan},\mathit{nan},y_{t_0+\left(i+n+1\right)a},\dots ,y_{t_0+\mathit{Na}}$$E2

Here we consider data is missing continuously from time $t_0+\mathit{ia}$ to time $t_0+\left(i+n\right)a$, for n number of timestamps. The missing values are represented as nan (not a number).

The following are the most often used algorithms for imputing missing values in time series.

4.1 Linear interpolation

Linear interpolation is the simplest imputation method. In this method, missing data points are estimated considering the linearity between the data points available before and after missing data points. The missing period is replaced by [49, 50]:

$$y_t=y_{t_0+\left(i-1\right)a}+m\left(t-\left(t_0+\left(i-1\right)a\right)\right)$$E3

Where, $m=\left(y_{t_0+\left(i-1\right)a}-y_{t_0+\left(i+n+1\right)a}\right)/\left(n+2\right)a$ is the slope of the straight line. This method is simple and fast; however, it is not useful if the data is seasonal. It misses the time-dependent variation in the data.

4.2 Exponential weighted moving average (EWMA)

The Exponential Weighted Moving Average (EWMA) method estimates the missing values as a weighted average of the historical data points, where the weighting factors decrease exponentially for the older timestamps. The timestamps in the near past have more weightage as compared to the timestamps in the distant past [51]. EWMA relates the predicted value and error in prediction.

$$\mathit{\text{EWMA}}=y_{t+1}^{\ast }=y_t^{\ast }+\lambda \left(y_t-y_t^{\ast }\right)$$E4

Where, $y_t^{\ast }$ is the estimation of y at time t, $\lambda \left(0<\lambda <1\right)$ is the weight of the data at the time $t$ and $y_t$ is the data point at time t. It is a recursive process where the weighting constant λ determines the memory of the process [51].

Intuitively, having a weightage of $\lambda$ implies how much “older” data is considered. λ = 1 means only the most recent data is considered. A higher value of λ means recent data is given weightage, whereas a lower value of λ imposes more weightage on older data. An optimum choice of λ is necessary. J. Stuart Hunter recommends a choice of λ between 0.1 and 0.3 [51].

According to Wijesekara et al. [52], EWMA outperformed mean imputation and interpolation in data imputation. EWMA resulted in an MSE of 0.297 when imputing 5% of missing data. Some advantages of using EWMA for data imputation include its simplicity and flexibility, making it suitable for a wide range of datasets with different patterns of missing data. Additionally, EWMA is known to preserve trend and seasonality in time-series data. However, there are some limitations to EWMA as an imputation method. For example, EWMA is also sensitive to parameter selection, which can significantly affect the imputed values.

4.3 k-nearest neighbor (kNN)

Another approach to imputing missing values in data is the k-nearest neighbor (kNN) method. This method is used when we have limited knowledge about the distribution of the data [53]. This technique is derived from the nearest neighbor (NN) approach, which involves using the value of the closest data point to fill in a missing value. However, the NN approach can sometimes be affected by outliers and result in overfitting. The kNN method addresses this issue by using the average or median of the target values from the k nearest neighbors to make a prediction, rather than just relying on a single nearest neighbor. This can help to reduce the risk of overfitting and improve the generalizability of the model [54, 55]. Using the kNN technique categorical missing values can be imputed using the majority rule among k neighbors, whereas missing numerical values can be handled by imputing the average value of k – closest neighbors. This is known as the majority/mean rule. kNN first finds the k nearest neighbors and imputes the missing points with the weighted average of these neighbor points. The number of nearest neighbors k to be considered appears as a hyperparameter and is to be tuned in the experiments. For a set of k nearest neighbors, $D_k=\left\{\left(t_j,y_j,1\right)\right\}$ where $j=1,2,\dots ,k$, the kNN estimator predicts [55].

$$y=\mathit{arg}{\mathit{max}}_v\left\{\sum _{\left(t_j,y_j,1\right)\in D_k}\mathbf{1}\left(y_j=v\right)\right\}$$E5

when y is categorical. Here v is a value in the domain of target feature y. $\mathbf{1}\left(y_j=v\right)$ is an indicator function that returns 1 when $v\in y_j$ and returns 0 otherwise. For numerical outcomes, the kNN estimator predicts [55]

$$y=\frac{1}{k}\sum _{j=1}^k{y}_j$$E6

The advantages of the kNN imputation method include simplicity and model-free approach. However, there are challenges associated with the proper choice of k and the subsequent selection of k nearest neighbors. Lall et al. recommend $k=\sqrt{N}$ for $N>100$, N being total number of observations or timestamps [56]. Minkowski distance is a commonly used approach to define the k nearest neighbors. Generally, this method gives good accuracy but it requires a lot of calculation and memory to predict the missing values [55]. kNN works well for randomly missing values; however, wearables data are often correlated over time. Lagged kNN method combined with Fourier transform imputation has shown better imputation for biomedical time series data [57]. Several modified kNN methods have been applied to impute ECG signal and hence can be applied to wearables data [58].

Kenyhercz et al. [59] conducted a study on data imputation using various methods such as KNN, mean imputation, hot deck imputation, and iterative robust model, with 25% and 50% data missing. The study used craniometric data from 352 individuals representing four population groups, which were publicly available in Howell’s dataset. The results showed that KNN performed better than mean imputation and hot deck imputation, and iterative robust model.

4.4 Expectation maximization (EM)

The EM method is an iterative process that comprises two steps i.e., estimation step (E-step) and maximization step (M-step). The E-step calculates expected values with the help of complete data points. In M-step, the parameters are optimized for best estimates [60]. The process repeats itself until the changes between expected values in each iteration become negligible or reaches convergence.

If we consider our time series data set as $y=\left\{y_{\mathit{obs}},y_{\mathit{mis}}\right\}$ where $y_{\mathit{obs}}$ and $y_{\mathit{mis}}$ represent observed and missing data set, respectively. The data set can be described by probability or density function $p\left(y|\theta \right)$ governed by a set of parameters. Here $p\left(y|\theta \right)$ is a function of y for a given $\theta$. The likelihood function can be defined from the density function $L\left(\theta |y\right)=p\left(y|\theta \right)$ where$L\left(\theta |y\right)$ is a function of parameter $\theta$ for data y. The initial estimate parameter is ${\theta }^0$. The E-step finds the objective function $Q\left(\theta |{\theta }^t\right)$ which provides the expected value of the observed data log-likelihood ($l\left(\theta |y\right)$) for the given observed data and current parameters

$$Q\left(\theta |{\theta }^t\right)=E\left[l\left(\theta |y\right)|y^0,{\theta }^t\right]$$E7

The M-step determines parameter vector ${\theta }^{t+1}$ which maximizes the log likelihood of the imputed data. At each iteration, the observed data likelihood increases but due to imposed upper bound, convergence is achieved. For all $\theta$, the objective function satisfies,

$$Q\left({\theta }^{t+1}|{\theta }^t\right)\ge Q\left(\theta |{\theta }^t\right)$$E8

EM is deemed to be superior then many substitution algorithms such as mean, median, and mode. EM is assumed to produce an unbiased estimation of values MCAR and little biased estimation for values MAR. The limitation of the algorithm is that, while it delivers reliable and correct estimation for missing data, the standard error is low, resulting in inaccuracy in several statistical tests (e.g., t-test). Thus, this method can only be used when the standard error is not crucial like in the case of factor analysis, which does not have the p-values [61].

EM algorithm has been applied to impute missing data of fetal heart rate data [62] as well as to ischemic heart rate data [63]. Aljuaid et al. [64] used the EM (Expectation-Maximization) algorithm to impute missing data in datasets obtained from the UCI machine learning repository. The study compared the performance of the EM algorithm with other imputation methods such as mean imputation, hot deck, kNN, and C5.0. The results showed that the EM algorithm produced the lowest RMSE (Root Mean Square Error) value, with an approximate value of 0.36. These findings suggest that the EM algorithm is a highly effective method for imputing missing data, and can outperform other popular imputation methods in terms of accuracy.

4.5 Kalman prediction

In 1960 R. E. Kalman proposed a method of predicting a signal from its past observations [65]. It is a linear model for discrete data and is widely used for predicting time-series data [66]. It estimates observations based on a set of measurements taken over time that contains noise. This algorithm consists of two processes: a prediction stage and a correction stage. The state and error covariance of the next data point is predicted with the help of the current state and error covariance in the prediction stage. The Kalman filter can be formally expressed as:

1. Step 1: Prediction process

It predicts a priori state estimate ${\widehat{y}}_{k+1}^{-}$and a posteriori state estimate ${\widehat{y}}_k$ by measurement equation,

$${\widehat{y}}_{\left(k+1\right)}^{-}=A_k{\widehat{y}}_k$$E9

Here A is a $n\times n$ matrix that relates state ${\widehat{y}}_k$ at time step k with the state of the next time step (k + 1). It further estimates the next state a posteriori estimate error covariance $P_{k+1}^{-}$ based on the current state covariance $P_k$ considering the noise covariance $Q$ by transition equation

$$P_{k+1}^{-}=A{P}_k{A}^T+Q$$E10

1. Step 2: Correction process

The prediction is further corrected by Kalman gain K (a $n\times m$matrix) which reflects confidence between the predicted result and actual measurement $z_k$.

$$K_{k+1}=P_{k+1}^{-}{H}^T{\left(H{P}_{k+1}^{-}{H}^T+R\right)}^{-1}$$E11

Here R is the measurement noise covariance matrix. The matrix H ($m\times n$ matrix) relates with the state of the measurement $z_k$. The gain matrix updates the estimation by

$${\widehat{y}}_{k+1}={\widehat{y}}_{k+1}^{-}+K_k\left(z_k-H{\widehat{y}}_{k+1}^{-}\right)$$E12

The covariance is also updated,

$$P_{k+1}=\left(I-K_{k+1}H\right)P_{k+1}^{-}$$E13

We observe that there is a feedback relation between a priori estimate and a posteriori estimate. In Kalman gain, noise in measurement determines the weightage on the predicted values [67]. Being a recursive relation, the Kalman filter is used for the imputation of missing values by measurement and transition equations [68].

Kalman prediction is used for various types of physiological data like heart rate variability and body weight variability [69, 70, 71] as well as for other time series data [72]. Luis Alfonso et al. [73] used data imputation techniques on air quality data and evaluated the results using RMSE values. The study compared the performance of the Kalman smoothing algorithm with other imputation methods such as kNN and RF (Random Forest). The results showed that the Kalman smoothing algorithm performed well and outperformed the other methods in terms of accuracy, as evidenced by the lower RMSE values obtained.

The Kalman filter is a popular method for data imputation in time series data. It can incorporate temporal relationships between observations for more accurate imputations. However, it can be computationally intensive and require knowledge of system dynamics and parameters, which can be challenging to estimate. Despite these limitations, the Kalman filter remains effective for time series data imputation.

5. Conclusion

We observe that the wearables generate a massive amount of personalized multivariate time-series data. This data is generated by different types of sensors which include gyroscopes, accelerometers, magnetometers, light sensors, and temperature sensors. The technological advances have improved the performance of different types of sensors which are also energy efficient, and occupy lesser space. The advancement of computational efficiency has made the analysis of these data feasible. Data is processed in real-time and further analyzed by software which is often powered by artificial intelligence. Apart from real-time monitoring of patients, data extracted from wearable devices have successfully been used for monitoring and diagnosing different diseases. The diseases include cardiological disorders, metabolic disorders, neurological disorders, sleep quality, and others. The robust analysis of this data is a requisite to make the data insightful and beneficiary to the users. The analysis demands the availability of complete data without any missing parts. However, in reality, data contains missing data, compensation for which requires imputation. There are various types of missing data, depending on the randomness of missing. The choice of imputation method is critical as subsequent analysis depends on the imputed data. Several imputation methods are being used whose success depends on the type, trend, and amount of available data as well as the missing data. We have observed that instead of using entire days’ data, use of data binned around the missing period generates better imputed data [74]. Finally, we can conclude that the data which is missing is cardinal along with the available time-series data and hence should be imputed in a worthwhile way.

Conflict of interest

The authors declare no conflict of interest.