Hierarchical Time-series Modelling for Haemodialysis. Tingting Zhu 24 th October 2016

Transcription

1 Hierarchical Time-series Modelling for Haemodialysis Tingting Zhu 24 th October 2016

2 Content Background on haemodialysis Methods o Gaussian Process Regression (GPR) o Hierarchical Gaussian Process Regression (HGPR) Application of GPR to the haemodialysis dataset o Pre-processing and Fitting Time-series using GPR o Univariate GPR Forecasting - Prediction of Deterioration o HGPR Forecasting for Multiple Time-series o Clustering of Time-series Trajectories

3 Background - Haemodialysis Background: - Prevalence of end-stage renal failure is 861 per million population in the UK; - 50% of them requires renal replacement therapy haemodialysis. The process: (1) Remove excess fluid from the body; (2) Restore electrolyte (such as sodium, potassium, and phosphate) balance; (3) Clear the waste products (such as urine) in blood; (4) 3 sessions per week, each lasts upto 4 hours. Image taken from

4 Background - Haemodialysis The problem(s): The rapid changes in blood fluid levels that occur during the treatment and causing: - cramps, nausea, and vomiting; - Intra-dialytic hypotension (IDH) sudden fall in blood pressure [reduction in blood volume resulted from an imbalance between rapid fluid withdraw and vascular refilling from the interstitial space to the blood stream]. The challenges: - Discrepancy in definition of IDH : - haemodynamic (fluid dynamic of blood flow) status varies between sessions; - Inaccurate measurement of non-invasive systolic blood pressure (derived from mean arterial pressure). - Real-time detection/early warning of IDH. Definition of IDH from Kidney Disease Outcomes and Quality Initiative: A decrease in systolic blood pressure by 20 mmhg or a decrease in mean arterial pressure by 10 mmhg associated with symptoms that include: abdominal discomfort; yawning; sighing; nausea; vomiting; muscle cramps; restlessness; dizziness or fainting; and anxiety.

5 Gaussian Process Regression (GPR) Gaussian Process for Regression: y can be considered as related to an underlying function f x through a Gaussian noise model: y = f x + N(0, σ n 2 ) Assuming f x has a Gaussian Process (GP) prior: or y~n(f(x), σ n 2 ) f x ~ GP(m x, k x, x ) where m x is the mean function of the GP and k x, x is a covariance function which describes the relationship among the y values that is determined according to the distance of the x values. Hence, we can define y as: y ~ GP (m x, k x, x + σ n 2 I) Noting that there are different covariance functions available to be considered, a common one is called the squaredexponential covariance function: k x, x = αexp { γ(x x ) 2 }) where the amplitude (α) and relative length-scale (γ) are hyperparameters.

6 Gaussian Process Regression (GPR) GPR Prediction: Condition on the training set {x, y}, the distribution of an unknown output y at x is defined as: y x, x, y ~N(E y, Var y ) where E y = k(x, x)k(x, x) 1 y Var y = k x, x k(x, x)k(x, x) 1 k(x, x) T Log Marginal Likelihood: To infer the hyperparameters (denoted as θ) in the covariance function, we can compute the probability of the data given the hyperparameters (i.e., marginal likelihood): p y x, θ = p y f, x p f x, θ df where we have marginalised over the function values f, and by taking the log of the likelihood, we have: L = log p y x, θ = 1 2 log k x, x + σ n 2 I 1 2 y μ T [k x, x + σ n 2 I] 1 y μ N 2 log(2π) Complexity Penalty Data-fit Measure Constant

7 Hierarchical Gaussian Process Regression (HGPR) Assuming we have N groups of time-series (such as physiological measurements over a time vector t n ), and they are similar to each other. The observed data for N groups of timeseries can be defined as Y = y nr r=1 taken at times T = t N nr r=1 N. Subject 1 N=3 Under the model assumption, there is a latent GP function which governs all the time series, denoted as g n t. Given a draw for g n, each group of data is then drawn from a GP as: Session 1 Session 2 Session 3 f nr t ~GP g n t, k f t, t The hierarchical structure of GPs can be computed as: g n t ~GP 0, k g t, t, f nr t ~GP g n t, k f t, t. Note two points on f nr t are jointly Gaussian distributed with zero mean and covariance k g t, t + k f t, t. But two points in different time-series are jointly distributed with covariance k g t, t. Figure taken from Hensman et al. 2013

8 Deeper HGPR Session 1 Subject ID=1 Session K Cluster 1 Subject ID=2 Subject ID=3 Session 1 Session M Session 1 What does a cluster mean: - Normal vs abnormal clusters of subjects - Different clusters of an abnormal population - Different clusters of a normal population h i t ~GP 0, k h t, t, g n t ~GP h i t, k g t, t, f nr t ~GP g n t, k f t, t. Cluster G Session 1 Subject ID=N Session M

9 Haemodialysis Dataset 60 recruited patients for the haemodialysis study, however, only 35 subjects had continuous blood pressure measurements. 4 vital signs (HR, SBP, MAP, and SpO2) were considered, and each vital sign is extracted from different sensors. Data taken from Clare R. MacEwen PhD Thesis 2016 Definition of Intradialytic hypotension (IDH) event: o MAP < 60mmHg (cerebral ischemia) AND SBP < 80%SBP_0 (i.e., SBP at baseline) o OR MAP < 60mmHg when there is no baseline SBP.

10 Pre-processing and Fitting Time-series using GPR

11 Raw session data Downsample GPR outlier removal GPR fitted time-series

12 Normalised LML Log SBP (mmhg) Raw session data Downsample GPR outlier removal GPR fitted time-series Time (min) Time (min) GPR Outlier Removal: - Estimate the normalised LMLs with respect to the userdefined window (such as ±5min window with 10min overlap); - Identify and remove outliers using a threshold; - Iteratively fitting GPR after removal of outliers; - Stop when over 5% data removed.

13 Raw session data Downsample GPR outlier removal GPR fitted time-series GPR Outlier Removal: - Estimate the normalised LMLs with respect to the userdefined window (such as ±5min window with 10min overlap); - Identify and remove outliers using a threshold; - Iteratively fitting GPR after removal of outliers; - Stop when over 5% data removed.

14 GPR fitting for an individual session: intervention SBP<80%SBP_baseline +MAP<60mmHg

15 Univariate GPR Forecasting - Prediction of Deterioration

16 Prediction of Deterioration using the Log Marginal Likelihood (LML) Adaptive Training and Forecasting: Training mins Training window Forecast window LML

17 Mean Predictive Log-Likelihood VS 160 Training Forecasting mins Original Signal IDH LML For each training set: - estimate the mean of LMLs in a 3-min forecasted window Time (mins) Abnormal blood pressure (MAP < 60mmHg) occurred at 684 mins; Nurse intervention occurred at 687 mins.

18 HGPR Forecasting for Multiple Time-series

19 LML GPR Forecasting for Multiple Vital Signs HR MAP SBP spo Time(mins) LML_HR LML_MAP LML_SBP Fused/Latent LML LML_SPO2

20 Data Fusion on Forecasted LMLs using BCLA-MAP Assumptions: the LML values are independent, and each vital sign LML is conditionally independent.

21 Data Fusion on Forecasted LMLs using HGPR Raw LML values HR MAP SBP SPO2 Normalised LML values (zero mean unit variance) HR MAP SBP SPO2

22 Data Fusion on GPR mean function of the Forecasted LMLs using HGPR HR MAP SBP SPO2 GPR mean function of the LML values LML_HR LML_MAP LML_SBP LML_SPO2 Mean function of LML_HR Mean function of LML_MAP Mean function of LML_SBP Mean function of LML_SPO2 Fused/Latent LML

23 Raw LML values HR MAP SBP SPO2 GPR mean function of the LML values HR MAP SBP SPO2

24 Clustering of Time-series Trajectories

25 Derivation of Latent Trajectories using HGPR (1) Derive latent GPR mean function from session-wise GPR mean functions Abnormal Sessions Normal Sessions

26 Hierarchical Clustering of Latent Trajectories (2) Clustering of latent mean functions using hierarchical clustering Abnormal Population (N=29) Normal Population (N=24) Abnormal Population + Normal Population (N=53)

27 Session 1 Subject ID=1 Deeper HGPR of Latent Trajectories Cluster 1 Session K Session 1 Subject ID=10 Session M

28 Future Works Normalise LML trajectory of each vital sign to better infer a latent LML trajectory for a session; Time-series modelling across sessions; Focus on windowed HGPR (2hrs) prior to an event; Deeper windowed HGPR; Non-parametric clustering: o using Mixture of hierarchical GPRs using Dirichlet distribution/process; Overlapping mixtures of GPRs. Cluster 1 Subject ID=1 Session 1 Session K Session 1 Subject ID=2 Session M

29 Thank You Acknowledgements: o o o o Kate Niehaus and Glen Colopy; Prof David Clifton and Prof Chris Pugh; The CHI lab; Funding bodies: NIHR and the EPSRC. References: o Ebden, M.: Gaussian processes: A quick introduction, arxiv: v2. o Rasmussen, C. E.: "Gaussian processes for machine learning." (2006). o MacEwen, C.R.: Can data fusion techniques predict adverse physiological events during haemodialysis?. PhD thesis o Colopy, G.W., Pimentel, M.A.F., Roberts, S.J., and Clifton, D.A.: Bayesian Gaussian Processes for Identifying the Deteriorating Patient. IEEE Engineering in Medicine & Biology Conference, Orlando, Florida, USA, 2016, pp o Zhu, T.T., Dunkley, N., Behar, J., Clifton, D.A., and Clifford, G.D.: Fusing Continuous-Valued Medical Labels Using a Bayesian Model. Annals of Biomedical Engineering 43(12), 2015, pp o Hensman J, Lawrence ND, Rattray M.: Hierarchical Bayesian modelling of gene expression time series across irregularly sampled replicates and clusters. BMC bioinformatics Aug 20;14(1):252. o Hensman, J., Rattray, M. and Lawrence, N.D.: Fast nonparametric clustering of structured time-series. IEEE transactions on pattern analysis and machine intelligence, 37(2), 2015, pp o Lázaro-Gredilla, M., Van Vaerenbergh, S. and Lawrence, N.D.: Overlapping mixtures of Gaussian processes for the data association problem. Pattern Recognition, 45(4), 2012, pp