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Bridging the Gap between
Applications and Tools: Modeling
Multivariate Time Series
X Liu, S Swift & A Tucker
Department of Computer Science
Birkbeck College
University of London
MTS Applications at Birkbeck
 Screening
 Forecasting
 Explanation
Forecasting
Predicting Visual Field Deterioration of
Glaucoma Patients
 Function Prediction for Novel Proteins from
Multiple Sequence/Structure Data

Explanation
Input (observations):
t-0
t-3
: Tail Gas Flow in_state 0
: Reboiler Temperature in_state 1
Output (explanation):
t - 7 : Top Temperature in_state 0 with probability=0.92
t - 54 : Feed Rate in_state 1 with probability=0.71
t - 75 : Reactor Temperature in_state 0 with probability=0.65
The Gaps

Screening
 Automatic
/ Semi- Automatic Analysis of
Outliers

Forecasting
 Analysing

Short Multivariate Time Series
Explanation
 Coping
with Huge Search Spaces
The Problem - What/Why/How
Short-Term Forecasting of Visual Field Progression
 Using a Statistical MTS Model
 The Vector Auto-Regressive Process - VAR(P)
 There Could be Problems if the MTS is Short
 A Modified Genetic Algorithm (GA) can be Used
 VARGA

The Prediction of Visual Field Deterioration Plays an
Important Role in the Management of the Condition
Background - The Dataset
The interval between tests
is about 6 months
76
75
18
19
74
73
72
15
16
17
71
70
69
68
11
12
13
14
67
66
65
64
63
6
7
8
9
10
62
61
60
59
58
1
2
3
4
5
43
42
41
40
39
20
21
22
23
24
48
47
46
45
44
25
26
27
28
29
52
51
50
49
30
31
32
33
55
54
53
34
35
36
57
56
37
38
Typically, 76 points
are measured
Values Range Between
60 =very good, 0 = blind
The number of tests can
range between 10 and 44
x
x
Points used in this
paper (Right Eye)
Usual Position of
Blind Spot (Right Eye)
Background - The VAR Process
Vector Auto-Regressive Process of Order P: VAR(P)
x(t) VF Test for Data Points at Time t (K1)
Ai Parameter Matrix at Lag i (KK)
x(t-i) VF Test for Data Points at lag i from t (K1)
 (t) Observational Noise at time t (K1)
The Genetic Algorithm
“A Search/Optimisation method that solves a problem
through maintaining and improving a population of
suitable candidate solutions using biological metaphors”
Generate a Population of random Chromosomes (Solutions)
Repeat for a number of Generations
Cross Over the current Population
Mutate the current Population
Select the Fittest for the next Population
Loop
The best solution to the problem is the Chromosome in
the last generation which has the highest Fitness
GAs - Chromosome Example
X
Y
0-127
0-31
0000000-1111111
00000-11111
0000000.00000-1111111.11111
GAs - Mutation
Each Bit (gene) of a Chromosome is Given
a Chance MP of inverting
 A ‘1’ becomes a ‘0’, and a ‘0’ becomes a 1’

01101101
These Ones!
00101111
GAs - Crossover (2)
A
B
01011101
11101010
X=4
11101101
01011010
C
D
VARGA - Representation
Chromosome
a111 …
…
…
…
a1ij …
…
a1KK
A1
a211 …
…
…
…
a2ij …
…
A2
a1KK
...
am11 …
…
…
…
amij …
…
amKK
Am
...
ap11 …
…
…
…
apij …
…
Ap
apKK
VARGA - The Genetic Algorithm
GA With Extra Mutation
 Order Mutation After Gene Mutation
 Parents and Children Mutate (Both)
 Genes are Bound Natural Numbers
 Fitness is -ve Forecast Error
 Minimisation Problem - Roulette Wheel
 Run for EACH Patient

Evaluation - Methods for
Comparison
SPlus: Yule Walker Equations, AIC and
Whittles Recursion, NK(P+1), Standard Package
 Holt-Winters Univariate Forecasting Method,
Is the Data Univariate? (GA Solution)
 Pure Noise Model, VAR(0), Worst Case
Forecast, (Non-Differenced = 0)
 54 out of the Possible 82 Patients VF Records
 Could not be Used : SPlus Implementation

Results - Graph Comparison
Scores for Cases 0 to 6
2000
1500
HW
S-Plus
VARGA
Noise
Score 1000
500
0
0
1
2
3
4
5
6
Case Number
The Lower the Score - the Better
 Score is the One Step Ahead Forecast Error

Results - Table Summary
Method
VARGA
S-Plus
HW
Noise
Order
Average
(number of order) Score
26 of 1, 2 of 2
12 of 0, 14 of 1, 1 of 2, 1 of 3
N/A
28 of 0
559.82
616.12
683.79
816.53
Average = The Average One Step Forecast Error
For the 28 Patients (Both GA’s Fitness)
(The Lower - The Better)
Conclusion - Results
VARGA Has a Better Performance
 VARGA Can Model Short MTS
 The Visual Field Data is Definitely Multivariate
 Data Has a High Proportion of Noise

Conclusion - Remarks
Non-Linear Methods and Transformations
 Performance Enhancements for the GA
 Improve Crossover
 Irregularly Spaced Methods
 Space-Time Series Methods
 Time Dependant Relationships Between Variables

Generating Explanations in MTS





Useful to know probable explanations for a
given set of observations within a time series
E.g. Oil Refinery: ‘Why a temperature has
become high whilst a pressure has fallen below
a certain value?’
Possible paradigm which facilitates Explanation
is the Bayesian Network
Evolutionary Methods to learn BNs
Extend work to Dynamic Bayesian Networks
Dynamic Bayesian Networks



Static BNs repeated over t time slices
Contemporaneous / Non-Contemporaneous Links
Used for Prediction / Diagnosis within dynamic
systems
n
P( X 1... X n )   P( X i |  i )
i 1
Assumptions - 1




Assume all variables take at least one time slice to
impose an effect on another.
The more frequently a system generates data, the
more likely this will be true.
Contemporaneous Links can be excluded from the
DBN
Each variable at time, t, will be considered
independent of one another
Representation


P pairs of the form (ParentVar, TimeLag)
Each pair represents a link from a node at a previous time
slice to the node in question at time t.
Examples :
Variable 1: { (1,1); (2,2); (0,3)}
Variable 4: { (4,1); (2,5)}
Search Space

Given the first assumption and proposed
representation the Search Space for each
variable will be:
2
NMaxLag
Algorithm
Structure Search : Evolutionary
Algorithms, Hill Climbing etc.
Multivariate
Time Series
Parameter Calculation
given structure
Dynamic Bayesian Network Library
for Different Operating States
User
Explanation Algorithm
(e.g. using Stochastic
Simulation)
Generating Synthetic Data
(1)
(2)
Oil Refinery Data






Data recorded every minute
Hundreds of variables
Selected 11 interrelated variables
Discretised each variable into k states
Large Time Lags (up to 120 minutes between
some variables)
Different Operating States
Results
SOT
FF
TGF
TT
RinT
Explanations - using Stochastic
Simulation
Explanations - using Stochastic
Simulation
1
0.9
0.8
0.7
SOF-SP
SOT
TT
BPF-SP
BPF
P(y=1)
0.6
0.5
0.4
0.3
0.2
0.1
0
1
7
13 19 25 31 37 43 49 55 61 67 73 79 85 91 97
time-x
Explanation
Input (observations):
t-0
t-3
: Tail Gas Flow in_state 0
: Reboiler Temperature in_state 1
Output (explanation):
t - 7 : Top Temperature in_state 0 with probability=0.92
t - 54 : Feed Rate in_state 1 with probability=0.71
t - 75 : Reactor Temperature in_state 0 with probability=0.65
Future Work




Exploring the use of different searches and metrics
Improving accuracy
(e.g. different discretisation policies, continuous
DBNs)
Using the library of DBNs in order to quickly
classify the current state of a system
Automatically Detecting Changing Dependency
Structure
Acknowledgements
BBSRC
BP-AMOCO
British Council for Prevention of Blindness
EPSRC
Honeywell Hi-Spec Solutions
Honeywell Technology Center
Institute of Opthalmology
Moorfields Eye Hospital
MRC
Intelligent Data Analysis
X Liu
Department of Computer Science
Birkbeck College
University of London
Intelligent Data Analysis
An interdisciplinary study concerned with
effective analysis of data
 Intelligent application of data analytic
tools
 Application of “intelligent” data analytic
tools

IDA Requires
Careful thinking at every stage of an
analysis process (strategic aspects)
 Intelligent application of relevant domain
knowledge
 Assessment and selection of appropriate
analysis methods

IDA Conferences
 IDA-95,
Baden-Baden
 IDA-97, London
 IDA-99, Amsterdam
 IDA-2001, Lisbon
IDA in Medicine and Pharmacology
IDAMAP-96, Budapest
 IDAMAP-97, Nagoya
 IDAMAP-98, Brighton
 IDAMAP-99, Washington DC
 IDAMAP-2000, Berlin

Other IDA Activities
IDA Journal (Elsevier 1997)
 Journal Special Issues (1997 -)
 Introductory Books (Springer 1999)
 The Dagstuhl Seminar (Germany 2000)
 European Summer School (Italy 2000)
 Special Sessions at Conferences

Concluding Remarks
Strategies for data analysis and mining
 Strategies for human-computer
collaboration in IDA
 Principles for exploring and analysing “big
data”
 Benchmarking interesting real-world datasets as well as computational methods
 A long term interdisciplinary effort

The Screening Architecture
Results from a GP Clinic