1 Introduction
Among various environments where longitudinal data is gathered, the environment covered in this study is a multistream and realtime environment. Recent progress in sensor and data storage technologies has facilitated data collection from multiple sensors in realtime as well as the accumulation of historical signals from multiple similar units during their operational lifetime. This data structure where multiple signals across different units are collected is referred to as multistream longitudinal data. Examples include: vital health signals from patients collected through wearable devices Caldara et al. (2014); Magno et al. (2016), battery degradation signals from cars on the road Meeker & Hong (2014); Salamati et al. (2018) and energy usage patterns from different smart home appliances Hsu et al. (2017).
In this article, we propose an efficient approach to extrapolate multistream data for an inservice unit through borrowing strength from other historical units. An illustrative example is provided in Figure 1. In this figure, there are historical units and an inservice unit whose index is denoted by . Each unit has identical sensors from which each respective signal forms a stream. Multistream data from the inservice unit is partially observed up to the current time instance . Our goal is to extrapolate stream data from the inservice unit over a future period where is the time domain of interest.
In mathematical notation, let and be the respective index sets for all available units including the inservice unit and the units in our historical dataset. For each unit , we have streams of data where . For the th unit, the history of observed data for a specific stream is denoted as , where represents the observation time points and represents the number of observations for signal of unit . The underlying principle of our model is borrowing strength from a sample of curves to predict individual trajectories over a future time period . Without loss of generality, throughout the article we focus on predicting stream , which we refer to as the target stream. Note that the target stream in Figure 1 is stream . To achieve this goal we exploit functional principal component analysis (FPCA), which is a nonparametric tool for functional data analysis Ramsay & Silverman (2005). Indeed, FPCA has recently drawn increased attention due to its flexibility, uncertainty quantification capabilities and the ability to handle sparse and irregular data Peng & Paul (2009); Di et al. (2009); Wang et al. (2016); Xiao et al. (2018). However, advances in FPCA fall short of handling multistream data and realtime predictions.
Our overall framework is summarized at the bottom of Figure 1. Specifically, historical signals , , from the target stream are decomposed into a linear combination of orthonormal eigenfunctions that form their functional space. The coefficients of the linear combination are called functional principal component (FPC) scores. With the assumption that the target signal of the inservice unit
lies in the same functional space, a proper estimation of FPC scores associated with
is required. Here, we propose to establish a prior on these FPC scores using information from streams . Specifically, a Gaussian process (GP) prior for FPC scores of is built using a functional semimetric that measures similarities of streams between historical units and the inservice unit . The underlying principle is that unit will exhibit more commonalities with historical units that exhibit similar trends in streams . For example, if stream denotes degradation trajectories and denotes external factors such as temperature. Then will share more commonalities with a subset of historical signals , , degrading under similar external factors (similar temperatures). This approach allows us to address heterogeneity in the data. Lastly, an empirical Bayesian updating strategy is derived to update the established prior using realtime stream data obtained from the inservice unit.2 Literature Review
There has been extensive literature on the extrapolation of longitudinal signals under a single stream setting. However, literature has mainly focused on parametric models due to their computational efficiency and ease of implementation
Gebraeel et al. (2005); Gebraeel & Pan (2008); Si et al. (2012, 2013); Kontar et al. (2017). Such models have been applied in healthcare, manufacturing and mobility applications specifically to understand the remaining useful life of operational units. Unfortunately, in realworld applications, parametric modeling is vulnerable to model misspecifications, and if the specified form is far from the truth, predictive results will be misleading. For instance, parametric representations are specifically challenging when data is sparse or when the underlying physical and chemical theories guiding the process are unknown.To address this issue, recent attempts at nonparametric approaches have been based on FPCA Zhou et al. (2011, 2012); Fang et al. (2015) or multivariate Gaussian processes Álvarez & Lawrence (2011); Saul et al. (2016); Kontar et al. (2018, 2019). These studies show that such nonparametric approaches outperform parametric models in case where functional forms are complex and exhibit heterogeneity. Nevertheless, the foregoing works have dealt with only single stream cases.
On the other hand, the few literature that addressed multistream settings have focused on data fusion approaches. Data fusion in this case refers to aggregating all streams into a single stream using fusion mechanisms. In health related applications, this fused stream is coined as a healthindex which is often derived through a weighted combination of the data streams Liu et al. (2013); Song & Liu (2018). Such methods require regularly sampled observations and enforce strong parametric assumptions. An alternative data fusion approach includes multivariate FPCA Fang et al. (2017a, b). However, since data fusion methods are operated by aggregating multistreams into a single or a smaller group of streams, they are not capable of predicting individual stream trajectories and thus have limited applications.
Compared to current literature, our contribution can be summarized as follows. We propose an FPCAbased model that provides individualized predictions in a multistream environment. Our model is able to automatically account for heterogeneity in the data and screen the sharing of information between the inservice unit and units in our historical dataset. We then derive a computationally efficient Bayesian updating strategy to update predictions when data is collected in realtime. We demonstrate the advantageous features of our approach compared to stateoftheart methods using both synthetic and realworld data.
The rest of this paper is structured as follows. In section 3, we briefly revisit the FPCA. In section 4, we discuss our proposed model. Numerical experiments using synthetic data and realworld data are provided in section 5. Finally, section 6 discusses the computational complexity of our model. Technical proofs, a detailed code and additional numerical results are available in the supplementary materials.
3 Brief Review of FPCA
From an FPCA perspective, longitudinal signals observed in a given time domain can be decomposed into a linear combination of orthonormal basis functions with corresponding FPC scores as coefficients. Therefore, FPCA can be regarded as a dimensionality reduction method in which a signal corresponds to a vector in a functional space defined by the basis functions. The basis functions are referred to as eigenfunctions. Let us assume that the longitudinal signals, over a given time domain
, are generated from a squareintegrable stochastic process with its mean and covariance defined by a positive semidefinite kernel for . Using Mercer’s theorem on , we havewhere presents the th eigenfunction of the linear HilbertSchmidt operator
ordered by the corresponding eigenvalues
, . The eigenfunctions form a set of orthonormal basis in the Hilbert space . Following the KarhunenLoéve decomposition, the centered stochastic process can then be expressed aswhere presents FPC scores associated with
. The scores are uncorrelated normal random variables with zeromean and variance
; that is, and where denotes the Kronecker delta. Also, is additive Gaussian noise.This idea of projecting signals onto a functional space spanned by eigenfunctions was first introduced by Rao (1958) for growth curves in particular. Basic principles Castro et al. (1986); Besse & Ramsay (1986) and theoretical characteristics Silverman (1996); Boente & Fraiman (2000); Kneip & Utikal (2001) were then developed. These ideas were expanded to longitudinal data settings in the seminal work of Yao et al. (2005). After that, the FPCA was applied and extended to a wide variety of applications, where multiple works tackled fast and efficient estimation of the underlying covariance surface Huang et al. (2008); Di et al. (2009); Peng & Paul (2009); Goldsmith et al. (2013); Xiao et al. (2016).
4 Extrapolation of Multivariate Longitudinal Data
4.1 FPCA for Signal Approximation
Now we discuss our proposed nonparametric approach for extrapolation of multistream longitudinal data. Hereon, unless there is ambiguity, we suppress subscripting the target stream with . Using historical signals , , we decompose the target stream as
(1) 
where represents random effects characterizing stochastic deviations across different historical signals in stream and denotes additive noise. We assume and are independent. Through an FPCA decomposition, we have that . This decomposition is an infinite sum, however, only a small number of eigenvalues are commonly significantly nonzero. For these values the corresponding scores will also be approximately zero. Therefore, we approximate this decomposition as , where is the number of significantly nonzero eigenvalues.
(2) 
Here we follow the standard estimation procedures in Di et al. (2009) and Goldsmith et al. (2018) to estimate the model parameters where is obtained by local linear smoothers Fan & Gijbels (1996), while is selected to minimize the modified Akaike criterion. Now given that the inservice unit lies in the same functional space spanned by , our task is to find the individual distribution of using the partially observed multistream data from unit . Specifically, we aim to find .
4.2 Estimation for Prior Distribution of FPC scores via GP
Next, we estimate the prior distribution of based on the key premise that will behave more similarly to for some units whose signals for are similar to the corresponding signals of the inservice unit . To this end, for , we model a functional relationship between and for as
(3) 
where for , , and .
The idea here, is to model as a GP with a covariance function defined by a similarity measure between the observed signals, i.e., a functional similarity measure. Specifically, for any , the vector of FPC scores
will follow a multivariate Gaussian distribution
(4) 
where is constructed such that its th element is for , , , and denotes a covariance function defined as
in which is a semimetric providing as similarity measure across functions, and and
are hyperparameters for streams
. For notational simplicity, we introduce .To show the validity of the GP (4), we provide the following lemma.
Lemma 1.
The matrix corresponding to the covariance function a valid covariance matrix.
Proof.
See Section A.1 in supplementary document. ∎
One possible semimetric that represents the similarity between two signals can be derived based on FPCA. Let denote the time domain for observations up to . Note that we define since the signals of the inservice unit are available only for . For , , and , the semimetric based on FPCA for two signals and can be represented as
(5) 
where is th eigenfunction derived by the FPCA on for and , and is the number of eigenfunctions. We would like to point out that is the difference between the FPC scores of and associated with , which implies that this metric measures the Euclidean distance between two vectors composed of the corresponding FPC scores.
In order to optimize the hyperparameter for the multivariate Gaussian distribution (4), we maximize the marginal loglikelihood function of given . Let denote the observations of signals for units , that is where . Also, let denote the true underlying latent values corresponding to the FPC scores and let . Then the marginal likelihood is given as
where and . The second equality follows from the fact that the error is an additive Gaussian noise. Thus, and the loglikelihood of is
where and . As a consequence, the optimized hyperparameters denoted by are found by maximizing the marginal loglikelihood. More formally, we have
Following multivariate normal theory, the posterior predictive distribution of
, given (4) and , is derived as(6) 
with
Here we note that for each , we can derive and using an independent GP as the FPC scores from different orthonormal basis functions are uncorrelated. This facilitates scalability of computation as for different we can derive and in parallel. As shown in the computational complexity derivations in section 6, this aspect is important specifically in a realtime environment where predictions need to be continuously updated.
Here note that , and are model parameters corresponding to the estimated FPCA model in (2), where denotes the estimated variance of . Here we recall that as the index is dropped for the target stream.
4.3 Empirical Bayesian Updating with Online Data
In the previous section, we derive a prior for and using data observed from streams . Here, we develop an empirical Bayesian approach to update and given the target stream () observations from the inservice unit . Specifically, given the prior distributions for each and given the observations at , the posterior is given in Proposition 2.
Proposition 2.
Given that , where the prior distribution of is , and the FPC scores are pairwise independent. Then, the posterior distribution of the FPC scores, such that , is given as
where
with
Proof.
See Section A.2 in supplementary document. ∎
Based on the updated FPC scores for inservice unir , the posterior predicted mean , of for any future time point where is given as
Similarly, the posterior variance can be computed as
where indicates the th element of the covariance matrix .
Despite our focus on the target stream we note that our framework can predict every individual stream for the inservice unit . This ability to provide individualized predictions is a key feature of the proposed methodology compared the data fusion literature that predicts a single aggregated signal. Further, one differentiating factor is that we allow irregularly sampled data from each stream where time points of each signal do not need to be identical or regularly spaced across streams. Indeed, such situations are quite common in practice because most multistream data is gathered from different types of sensors. Therefore, the proposed approach is applicable to a wide array of practical situations.
5 Numerical Case Study
5.1 General Settings
In this section, we discuss the general settings used to assess the proposed model, denoted as FPCAGP. We evaluate the model by performing experiments with both synthetic and realworld data. We report the prediction accuracy at varying time points for the partially observed unit . Specifically, for the time domain , we assume that the online signals from the inservice unit are partially observed in the range of , referred as observation. We set , and for every case study. In the extrapolation interval , we use the mean absolute error (MAE) between the true signal value and its predicted value at evenly spaced test points (denoted as for ) as the criterion to evaluate our prediction accuracy.
(8) 
We report the distribution of the errors across repetitions using a group of boxplots representing the MAE for the testing unit at diffrent observation percentiles. Further, we benchmark our method with two other reference methods for comparison: (i) The FPCA approach for single stream settings denoted as FPCAB. In this method we only consider the target stream Zhou et al. (2011); Kontar et al. (2018). Note that we incorporate our Bayesain updating procedure to update predictions as new data is observed. (ii) The Bayesian mixed effect model with a general polynomial function whose degree is determined through an Akaike information criteria (AIC) Rizopoulos (2011); Son et al. (2013); Kontar et al. (2017). We denote this methods as ME. The ME model intrinsically applies a Bayesian updating scheme as more data is obtained from the inservice unit. Detailed codes for both reference methods are included in our supplementary materials.
5.2 Numerical Study with Synthetic Data
First, we show the numerical results of the proposed model performed on synthetic data. For this experiment, we assume that two streams of data are observed from two different sensors embedded in each unit. The target stream of interest is . To generate signals possessing heterogeneity, we suppose there are two separating environments, denoted by environment I and II. We generate signals for each unit using different underlying functions depending on which environment the unit is in. This is illustrated in Figure 2. As shown in the figure, the underlying trend of the target stream () will vary under different profiles of stream . To relate this setting with realworld application, consider as the degradation level and as the temperature profile. Then from Figure 2, we have that units operating under different temperature profiles will exhibit different trends.
We generate a training set of units and one testing unit whose signals are partially observed. Also we repeat the experiment times. Historical units are operated in either environment I or II whereas the inservice unit is operated in environment II. The population of historical units is created according to three levels of heterogeneity: (i) 0% heterogeneity where all units in the historical database are operated under environment II (similar to that of the testing unit) (ii) 50% heterogeneity where 25 units are distributed to each environment (iii) 90% heterogeneity where only 5 units are assigned to environment II. Conducting the experiments across a homogeneous setting and a heterogeneous setting, where the inservice unit belongs to the minority group with only 10% ratio, will allow us to investigate the robustness of our approach.
For units in environment I, the signals from respective streams are generated according to and , where and , where
denotes the uniform distribution. For units in environment II, we generate the signals as
and where . Measurement error is assumed similar across both streams.Figure 2 illustrates training signals in the case of 90% heterogeneity. It is crucial to note that at early stages (ex: ), it is hard to distinguish between the two different trends in stream . We model that on purpose to check if our model can leverage information from stream to uncover the underlying heterogeneity at early stages. This in fact is a common feature in many health related applications, as many diseases remain dormant at early stages and it is only through measuring other factors we can predict there evolution early on.
The results are illustrated in both Figure 3 and 4. Based on the figures we can obtain some important insights. First, the FPCAGP clearly outperforms the FPCAB. This is specifically obvious at early stages () and when the data exhibits heterogeneity (90% and 50% heterogeneity). This confirms the ability of our model to borrow strength from information across different streams to discern the heterogeneity and enhance predictive accuracy at early stages. This result is very motivating specifically since at data from inservice unit is sparse and all signals in stream have similar behaviour which makes it hard to uncover future heterogeneity. It further implies that the FPC scores of the testing unit are appropriately estimated by the proposed approach, as shown in the first column of Figure 3. From the figure, we observe that the estimated prior mean from the FPCAGP appropriately follows the signals in environment II, whereas the prior mean from the FPCAB follows the signals in environment I, which is the majority. Second, as expected, prediction errors significantly decrease as the percentiles increase. Thus, our Bayesian updating framework is able to efficiently utilize new collected data and provide more accurate predictions as
increases. Third, the results show that ME behaved the worst and its predictions accuracy merely decreases at later stages. This result illustrates the vulnerabilities of parametric modeling and demonstrates the ability of our nonparametric modeling to avoid model misspecifications. Fourth, the results confirm that even in the case where other streams have no effect on the target stream (
0% heterogeneity) the FPCAGP is competitive compared to FPCAB. This highlights the robustness of the FPCAGP.5.3 Numerical Study with Realworld Data
In this section, we discuss the numerical study using realworld data provided by the National Aeronautics and Space Administration (NASA). The dataset contains degradation signals collected from multiple sensors on an aircraft turbofan engine. This dataset was generated from a simulation model, developed in Matlab Simulink, called commercial modular aeropropulsion system simulation (CMAPSS). This system simulates degradation signals from multiplesensors, installed in several components of an aero turbofan engine, under a variety of environmental conditions. The list of the components includes Fan, LPC, HPC, and LPT, and are illustrated in Figure 5. Refer to Saxena & Simon (2008) for more details about turbofan engine data. The dataset is available at Saxena & Goebel (2008). The dataset is composed of 21 sensor streams from 100 training and 100 testing units. Following the analysis of Liu et al. (2013), we select the 11 most crucial streams. Some signals from these streams are shown in Figure 6. We provide the detailed list and description of sensors in the supplementary materials. In our analysis we truncate the time range and predict the testing signal over the time range .
Table 1 demonstrates that the MAE results of stream 4 and 15. Note that these two streams have shown to have the largest impact on failure Fang et al. (2017b)
and therefore, due to space limitation, we only focus on their predictive results. Results for other signals are provided in the supplementary materials. Note that we include the standard deviation of MAE across the testing units.
The results clearly show that our approach is far more superior than benchmarks for the realworld data. For all provided cases, the mean of MAE for the FPCAGP is less than that of the FPCA. Once again this highlights the importance of leveraging information from all streams of the data. Another important insight from this study is that our model was able to outperform the ME even though the curves from Figure 6 seem to exhibit a clear parametric trend. This further highlights the robustness of our method and its ability to safeguard against parametric misspecifications.
Sensor 4  Sensor 15 ()  

Model  25%  50%  75%  25%  50%  75% 
FGP  3.26  3.21  3.19  1.62  1.62  1.57 
(std.)  (0.36)  (0.42)  (0.48)  (0.18)  (0.19)  (0.33) 
FB  3.49  3.37  3.31  1.76  1.75  1.63 
(std.)  (0.45)  (0.56)  (0.54)  (0.28)  (0.31)  (0.36) 
ME  3.51  3.38  3.34  1.79  1.77  1.65 
(std.)  (0.45)  (0.59)  (0.55)  (0.28)  (0.32)  (0.37) 
6 Discussion
In this study, we developed a nonparametric statistical model that can extrapolate individual signals in a multistream data setting. Using both synthetic and realworld data, we demonstrate our models ability to borrow strength across all streams of data, predict individual streams, account for heterogeneity and provide accurate realtime predictions where an empirical Bayesian approach updates our predictor as new data is observed in realtime. Since we work in the regime of streaming data, the frequency with which we receive data is very high. Due to this, our model needs to be efficient in terms of the time taken to make each update. With the assumption that all signals from streams have observations, the complexity of multivariate FPCA for multistream data is Fang et al. (2017b). In our model, the computationally expensive steps are the FPCA for the target stream (Section 4.1) and the implementation of GP for estimating the FPC scores (Section 4.2). Following Xiao et al. (2016), the complexity of the former is . While complexity of a GP with an covariance matrix is Rasmussen & Williams (2005). Given that we implement independent GP models the complexity of estimating the FPC scores is . Combing the above observations, we conclude that the complexity of our procedure is . Typically, we have that , also, in realtime is increasing. Thus, our model is clearly more efficient than multivariate the FPCA and applicable in practice in a realtime streaming environment.
7 Software and Data
Technical proofs, the used dataset, a detailed code and additional numerical results are available in the supplementary materials.
References

Álvarez & Lawrence (2011)
Álvarez, M. A. and Lawrence, L. D.
Computationally efficient convolved multiple output gaussian
processes.
Journal of Machine Learning Research
, 12(May):1459–1500, 2011.  Besse & Ramsay (1986) Besse, P. and Ramsay, J. Principal components analysis of sampled functions. Psychometrika, 51(2):285–311, 1986.

Boente & Fraiman (2000)
Boente, G. and Fraiman, R.
Kernelbased functional principal components.
Statistics & probability letters
, 48(4):335–345, 2000.  Caldara et al. (2014) Caldara, M., Colleoni, C., Guido, E., Re, V., Rosace, G., and Vitali, A. A wearable sweat pH and body temperature sensor platform for health, fitness, and wellness applications. In Sensors and Microsystems, volume 268, pp. 431–434. Springer, 2014.
 Castro et al. (1986) Castro, P. E., Lawton, W. H., and Sylvestre, E. A. Principal modes of variation for processe with continuous sample curves. Technometrics, 28(4):329–337, 1986.
 Di et al. (2009) Di, C.Z., Crainiceanu, C. M., Caffo, B., and Punjabi, N. M. Multilevel functional principal component analysis. The Annals of Applied Statistics, 3(1):458–488, 2009.
 Fan & Gijbels (1996) Fan, J. and Gijbels, I. Local Polynomial Modelling and Its Applications. Chapman and Hall, London, 1996.
 Fang et al. (2015) Fang, X., Zhou, R., and Gebraeel, N. Z. An adaptive functional regressionbased prognostic model for applications with missing data. Reliability Engineering & System Safety, 133:266–274, 2015.
 Fang et al. (2017a) Fang, X., Gebraeel, N. Z., and Paynabar, K. Scalable prognostic models for largescale condition monitoring applications. IISE Transactions, 49(7):698–710, 2017a.
 Fang et al. (2017b) Fang, X., Paynabar, K., and Gebraeel, N. Z. Multistream sensor fusionbased prognostics model for systems with single failure modes. Reliability Engineering & System Safety, 159:322–331, 2017b.
 Gebraeel & Pan (2008) Gebraeel, N. Z. and Pan, J. Prognostic degradation models for computing and updating residual life distributions in a timevarying environment. IEEE Transactions on Reliability, 57(4):539–550, 2008.
 Gebraeel et al. (2005) Gebraeel, N. Z., Lawley, M. A., Li, R., and Ryan, J. K. Residuallife distributions from component degradation signals: A bayesian approach. IIE Transactions, 37(6):543–557, 2005.
 Goldsmith et al. (2013) Goldsmith, J., Greven, S., and Crainiceanu, C. Corrected confidence bands for functional data using principal components. Biometrics, 69(1):41–51, 2013.
 Goldsmith et al. (2018) Goldsmith, J., Scheipl, F., Huang, L., Wrobel, J., Gellar, J., Harezlak, J., McLean, M. W., Swihart, B., Xiao, L., Crainiceanu, C., and Reiss, P. T. refund: Regression with Functional Data, 2018. URL https://CRAN.Rproject.org/package=refund. R package version 0.117.
 Hsu et al. (2017) Hsu, Y.L., Chou, P.H., Chang, H.C., Lin, S.L., Yang, S.C., Su, H.Y., Chang, C.C., Cheng, Y.S., and Kuo, Y.C. Design and implementation of a smart home system using multisensor data fusion technology. Sensors, 17(7):1631, 2017.
 Huang et al. (2008) Huang, J., Shen, H., and Buja, A. Functional principal components analysis via penalized rank one approximation. Electronic Journal of Statistics, 2:678–695, 2008.
 Kneip & Utikal (2001) Kneip, A. and Utikal, K. Inference for density families using functional principal component analysis. Journal of the American Statistical Association, 96(454):519–532, 2001.
 Kontar et al. (2017) Kontar, R., Son, J., Zhou, S., Sankavaram, C., Zhang, Y., and Du, X. Remaining useful life prediction based on the mixed effects model with mixture prior distribution. IISE Transactions, 49(7):682–697, 2017.
 Kontar et al. (2018) Kontar, R., Zhou, S., Sankavaram, C., Du, X., and Zhang, Y. Nonparametric modeling and prognosis of condition monitoring signals using multivariate gaussian convolution processes. Technometrics, 60(4):484–496, 2018.
 Kontar et al. (2019) Kontar, R., Raskutti, G., and Zhou, S. Minimizing negative transfer of knowledge in multivariate gaussian processes: A scalable and regularized approach, 2019.
 Liu et al. (2013) Liu, K., Gebraeel, N. Z., and Shi, J. A datalevel fusion model for developing composite health indices for degradation modeling and prognostic analysis. IEEE Transactions on Automation Science and Engineering, 10(3):652–664, 2013.
 Liu et al. (2012) Liu, Y., Frederick, D. K., DeCastro, J. A., Litt, J. S., and Chan, W. W. User’s guide for the commercial modular aeropropulsion system simulation (CMAPSS). Technical report, National Aeronautics and Space Administration (NASA), Cleveland, OH, 2012.
 Magno et al. (2016) Magno, M., Brunelli, D., Sigrist, L., Andri, R., Cavigelli, L., Gomez, A., and Benini, L. Infinitime: Multisensor wearable bracelet with human body harvesting. Sustainable Computing: Informatics and Systems, 11:38–49, 2016.
 Meeker & Hong (2014) Meeker, W. Q. and Hong, Y. Reliability meets big data: opportunities and challenges. Quality Engineering, 26(1):102–116, 2014.
 Peng & Paul (2009) Peng, J. and Paul, D. A geometric approach to maximum likelihood estimation of the functional principal components from sparse longitudinal data. Journal of Computational and Graphical Statistics, 18(4):995–1015, 2009.
 Ramsay & Silverman (2005) Ramsay, J. O. and Silverman, B. W. Functional Data Analysis. Springer, NY, 2nd edition, 2005.
 Rao (1958) Rao, C. R. Some statistical methods for comparison of growth curves. Biometrics, 14:1–17, 1958.
 Rasmussen & Williams (2005) Rasmussen, C. E. and Williams, C. K. I. Gaussian Processes for Machine Learning. The MIT Press, MA, 2005.
 Rizopoulos (2011) Rizopoulos, D. Dynamic predictions and prospective accuracy in joint models for longitudinal and timetoevent data. Biometrics, 67(3):819–829, 2011.
 Salamati et al. (2018) Salamati, S. M., Huang, C. S., Balagopal, B., and Chow, M.Y. Experimental battery monitoring system design for electric vehicle applications. In 2018 IEEE International Conference on Industrial Electronics for Sustainable Energy Systems (IESES), pp. 38–43. IEEE, 2018.
 Saul et al. (2016) Saul, A. D., Hensman, J., Vehtari, A., and Lawrence, N. D. Chained gaussian processes. In Artificial Intelligence and Statistics, pp. 1431–1440, 2016.
 Saxena & Goebel (2008) Saxena, A. and Goebel, K. PHM08 challenge data set, 2008. URL https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognosticdatarepository/.
 Saxena & Simon (2008) Saxena, A. and Simon, D. Damage propagation modeling for aircraft engine runtofailure simulation. In International Conference on Prognostics and Health Management, pp. 1–9, Denvor, CO, 2008.
 Si et al. (2012) Si, X.S., Wang, W., Hu, C.H., Zhou, D.H., and Pecht, M. G. Remaining useful life estimation based on a nonlinear diffusion degradation process. IEEE Transactions on Reliability, 61(1):50–67, 2012.
 Si et al. (2013) Si, X.S., Wang, W., Hu, C.H., Chen, M.Y., and Pecht, M. G. A wienerprocessbased degradation model with a recursive filter algorithm for remaining useful life estimation. Mechanical Systems and Signal Processing, 35(1):219–237, 2013.
 Silverman (1996) Silverman, B. Smoothed functional principal components analysis by choice of norm. The Annals of Statistics, 24(1):1–24, 1996.
 Son et al. (2013) Son, J., Zhou, Q., Zhou, S., Mao, X., and Salman, M. Evaluation and comparison of mixed effects model based prognosis for hard failure. IEEE Transactions on Reliability, 62(2):379–394, 2013.
 Song & Liu (2018) Song, C. and Liu, K. Statistical degradation modeling and prognostics of multiple sensor signals via data fusion: A composite health index approach. IISE Transactions, 50(10):853–867, 2018.
 Wang et al. (2016) Wang, J.L., Chiou, J. M., and Müller, H.G. Functional data analysis. Annual Review of Statistics and Its Application, 3:257–295, 2016.
 Xiao et al. (2016) Xiao, L., Zipunnikov, V., Ruppert, D., and Crainiceanu, C. Fast covariance estimation for highdimensional functional data. Statistics and Computing, 26(12):409–421, 2016.
 Xiao et al. (2018) Xiao, L., Cai, L., Checkley, W., and Crainiceanu, C. Fast covariance estimation for sparse functional data. Statistics and Computing, 28(3):511–522, 2018.
 Yao et al. (2005) Yao, F., Müller, H.G., and Wang, J.L. Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association, 100(470):577–590, 2005.
 Zhou et al. (2012) Zhou, R., Gebraeel, N., and Serban, N. Degradation modeling and monitoring of truncated degradation signals. IIE Transactions, 44(9):793–803, 2012.
 Zhou et al. (2011) Zhou, R. R., Serban, N., and Gebraeel, N. Degradation modeling applied to residual lifetime prediction using functional data analysis. The Annals of Applied Statistics, 5(2B):1586–1610, 2011.
Comments
There are no comments yet.