1. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Longitudinal and time-to-event models
Anthony Ebert
Supervisor: Dr Edward Cripps
Co-supervisor: Professor Melinda Hodkiewicz
September 29, 2015

2. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Bearings
Figure : Left: Diagram of angular contact bearing (NBC Bearings, 2015).
Right: Disassembled bearing (Karacay and Akturk, 2009)

3. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Feature plot
0 0.2 0.4 0.6 0.8 1
AgeZ
P1V_Par5
●
●
● ●
●
●
●
●
●
●
●
●
P1V_Par5
0 0.2 0.4 0.6 0.8 1
−2−101234
●
●
● ●
●
●
●
●
●
●
●
●
P1V_Gs
Range standardised time
Normalisedfeature
Figure : Longitudinal data from two features

4. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Life line plot
0.0 0.2 0.4 0.6 0.8 1.0
q
q
q
q
q
q
q
q
q
q
q
q
Range standardised observation time
Life line plot of failure time data

5. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Thesis aims
Aims:
Model timetoevent and vibration data jointly
Determine the best predictive model for timetoevent, and
Compare results to Sundin et al. (2007)
Dataset: Pump failures and vibration readings from Irving Pulp and
Paper Mill in New Brunswick, Canada. Reported in series of journal
articles (Banjevic and Jardine, 2006; Lin et al., 2006) and a
conference paper (Sundin et al., 2007).

6. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Denitions
yi: vector of feature measurements for item i,
Yi(t): feature history up to time t for item i,
hi(t): risk at time t,
Ti: failure time, and
mi(t): tted value of feature at time t.

7. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Exogeneous and Endogenous covariates
Endogeneity:
Figure : Relationship between variables
Exogeneity:
∀s, t such that 0 s ≤ t
P(Yi(t)|Yi(s), Ti ≥ s) = P(Yi(t)|Yi(s), Ti = s).

8. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Joint model
hi(t) = h0(t) exp (αmi(t))
yi ∼ N(mi(ti), σ2
e),
Mixed eects model
yi = Xi[β − bi] +
∼ N(0, σ2
e)
bi ∼ N (0, D)

9. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
B-Spline functions
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Range standardised time
B−Splines
Figure : B-Spline functions. The red lines indicate the knot positions.

10. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Spline ts
AgeZ
P1V_Par4
0 0.2 0.4 0.6 0.8 1
AgeZ
P1V_Par5
●●
●
●
●
●
●
●
●
●
●
P1V_Par5
0 0.2 0.4 0.6 0.8 1
−2−10123
Range standardised time
Normalisedfeature
●
●
●
●
●
●
●
●
●
●
●
●
P1V_Gs
Figure : Fitted values from longitudinal model

11. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Log likelihood table
Feature Two-stage Joint model
P1H_Gs 12.6477 -224.53
P1H_Overall -5.5770 -292.06
P1H_Par1 -5.6857 -587.79
P1H_Par2 -5.5953 -220.82
P1H_Par3 -5.4412 -319.63
P1H_Par4 -3.3896 -467.61
P1H_Par5 15.0031 -283.94
P1V_Gs 14.5775 -213.02
P1V_Overall -0.8569 -296.67
P1V_Par1 -3.6330 -634.75
P1V_Par2 -5.5561 -351.27
P1V_Par3 -4.6813 -375.93
P1V_Par4 0.5258 -384.74
P1V_Par5 16.1976 -227.93
Table : Log likelihood values

12. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Vibration features
Figure : Bearing fault patterns (STI Field Application Note, 2012)

13. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
ROC curve
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
P1V_Gs
P1V_Par5
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
1 - Specificity
Sensitivity

14. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Conclusions and Future Work
Conclusions:
The highest frequency band P1V_Par5 and acceleration P1V_Gs
are the best t features for diagnosis,
The best t features are not necessarily the most useful for
diagnosis and not necessarily the most useful for prognosis,
Our results dier from Sundin et al. (2007), and
Joint models are more useful where we have fewer longitudinal
datapoints
Future Work:
Change point model
Prediction of longitudinal outcome: Spline extrapolation and
Markov Chain

15. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
References
Banjevic, D. and Jardine, A. (2006). Calculation of reliability function and
remaining useful life for a Markov failure time process. IMA Journal of
Management Mathematics, 17(2):115130.
Karacay, T. and Akturk, N. (2009). Experimental diagnostics of ball bearings
using statistical and spectral methods. Tribology International,
42(6):836843.
Lin, D., Banjevic, D., and Jardine, A. (2006). Using principal components in a
proportional hazards model with applications in condition-based maintenance.
Journal of the Operational Research Society, 57(8):910919.
NBC Bearings (2015). www.nbcbearings.com/product/single-row-deep-groove.
Online; accessed 7-August-2015.
STI Field Application Note (2012). Rolling element bearings. Technical report,
REB, League City TX.
Sundin, P., Montgomery, N., and Jardine, A. (2007). Pulp mill on-site
implementation of CBM decision support software. In Proceedings of
International Conference of Maintenance Societies, Melbourne, Australia.

16. Longitudinal
and
time-to-event
models
Anthony
Ebert
Introduction
Exploratory
data analysis
Joint model
Results and
discussion
Conclusions
and future
work
References
Joint model likelihood
li(θ|(qi, δi), yi) = log(p((qi, δi), yi|θ))
= log p((qi, δi), yi, bi|θ)dbi
= log


 p((qi, δi)|bi, θPH)
prop hazards
f (yi, bi|θlme)
longitudinal
dbi


