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Learning Qualitative Models from Physiological Signals by
David TakWai Hau S.B., Massachusetts Institute of Technology (1992) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering and Computer Science
at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 1994
(
David TakWai Hau, MCMXCIV. All rights reserved.
The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part, and to grant others the right to do so.
Author
. ...........................
.
.
Department of Electrical Engineering and Computer Science May 17, 1994
Certified by ..................................................
( Eico Project Manager, HewlettP ,~7o~ Certified by ....................
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W. Coiera
ard Laboratories Thesis Supervisor
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Ror G. Mark Grover Hermann Professor in Health Sciences and Technology
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LIBRARSES
Learning Qualitative Models from Physiological Signals by David TakWai Hau Submitted to the Department of Electrical Engineering and Computer Science on May 17, 1994, in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering and Computer Science
Abstract Physiological models represent a useful form of knowledge, but are both difficult and time
consuming to generate by hand. Further, most physiological systems are incompletely understood. This thesis addresses these two issues with a system that learns qualitative models from physiological signals. The qualitative representation of models allows incom
plete knowledgeto be encapsulated, and is based on Kuipers' approach used in his QSIM algorithm. The learning algorithm allows automatic generation of such models, and is based on Coiera's GENMODEL algorithm. We first show that QSIM models are efficiently PAC learnable from positive examples only, and that GENMODEL is an algorithm for efficiently constructing a QSIM model consistent with a given set of examples, if one exists. We then describe the learning system in detail, including the frontend processing and segmenting
stages that transform a signal into a set of qualitative states, and GENMODEL that uses these qualitative states as positive examples to learn a QSIM model. Next we report re
sults of experiments using the learning system on data segments obtained from six patients during cardiac bypass surgery. Useful model constraints were obtained, representing both general physiological knowledge and knowledge particular to the patient being monitored.
Model variation across time and across different levels of temporal abstraction and fault tolerance is examined. Thesis Supervisor: Enrico W. Coiera
Title: Project Manager, HewlettPackard Laboratories Thesis Supervisor: Roger G. Mark Title: Grover Hermann Professor in Health Sciences and Technology
Acknowledgements First I would like to thank Enrico Coiera, my supervisor at HP Labs, for offering me an extremely interesting project which combines my interests in artificial intelligence, signal processing and medicine, and for being a really great supervisor. I especially thank him for his superb guidance throughout the project, and for his encouragement and patience in the writeup process.. It has been my pleasure to have him as a mentor and a friend. I would also like to thank Dave Reynolds at HP Labs for offering me a lot of help and advice in the project, and for reading part of a draft of the thesis. The overall frontend processing scheme and the idea of using Gaussian filters came from him. I also appreciate his excellent sense of humor. Every discussion with him has been a lot of fun. I am grateful to Professor Roger Mark for taking time out of his busy schedule to supervise my thesis. I thank him for his patience and his comments in the writeup process. I really enjoyed the discussions with him in which I learned a lot about cardiovascular iphysiology.
My friend David Maw kindly let me store my enormous number of data files in his server at LCS. I thank him deeply for his help. Most of all, I thank my family for their neverending love and encouragement. I dedicate this thesis to my mother.
This thesis describes work done while the author was a VIA student at HewlettPackard Laboratories, Bristol, U.K. in Spring and Summer 1993. 3
To my mother
4
Contents 1 Introduction
14
2 Qualitative Reasoning
17
2.1 Qualitative Model Constraints
........
17
2.2
Qualitative System Behavior.
19
2.3
Qualitative Simulation: QSIM.........
21
2.4
Learning Qualitative Models: GENMODEL .
21
2.5 An Example: The Utube ...........
22
2.5.1 Qualitative Simulation of the Utube .
24
2.5.2 Learning the Utube Model ......
24
3 Learning Qualitative Models
28
...... .... . .28
3.1 Probably Approximately Correct Learning ........ 3.1.1
Definitions.
3.1.2
PAC Learnability ..................
3.1.3
Proving PAC Learnability .............
3.1.4
An Occam Algorithm for Learning Conjunctions
. ...
3.2 The GENMODEL Algorithm ............... 3.3
3.4
QSIM Models are PAC Learnable ............. 3.3.1
The Class of QSIM Models is PolynomialSized .
3.3.2
QSIM Models are PolynomialTime Identifiable
GENMODEL ........................ 3.4.1
Dimensional Analysis.
3.4.2
Performance on Learning the UTube Model
3.4.3
Version Space .................... 5
.......... . .28 .......... . .29 .......... . .32 .......... . .33 .......... . .35 .......... . .35 .......... . .35 .......... . .37 .......... . .37 .......... . .38 .......... . .38
. .
...
...
.
.30
3.5
3.4.4
Fault Tolerance ...........................
39
3.4.5
Comparison of GENMODEL with Other Learning Approaches
40
Applicability of PAC Learning
......................
4 Physiological Signals and Mode 4.1
4.2
4.1.1
Primary Measurements
4.1.2
Derived Values ....
A Qualitative
Cardiovascular
5 System Architecture 5.1
Overvievw .
5.2 FrontEnd System ...... 5.2.1 Artifact Filter..... 5.2.2
Median Filter.
5.2.3
Temporal Abstraction
5.2.4 Gaussian Filter ....
5.3
...................... ...................... ...................... ......................
Hemodynamic Monitoring
5.2.5
Differentiator.
5.2.6
An Example.
Segmenter ...........
I
.......................... .......................... .......................... .......................... .......................... .......................... .......................... .......................... ..........................
6 Results and Interpretation 6.1
6.2
InterPatient
Model Comparison:
42
44 44 44 46 49
54 54 54 56 57 57 60 66 68 68
72 Patients
73
1
.5 . . . . . . . . . . . . . . . ..
6.1.1
Patient 1 .
. . . . .
73
6.1.2
Patient 2 .........
. . . . .
80
6.1.3
Patient 3 .........
. . . . .
87
6.1.4
Patient 4 .........
. . . . .
94
6.1.5
Patient 5 .........
. . . . .
IntraPatient
Model Comparison: Patient
,Sement 16..........
6,
101
108
6.2.1 Segment i .........
. . . . . Se me t . 16.. . . . . . . . . . .
108
6.2.2
Segment 2 .........
. . . . . . . . . . . . . . . . . . . . . .
115
6.2.3
Segment 3 .........
. . . . .
122
6.2.4
Segment 4 .........
. . . . .
131
6
6.2.5
Segment 5 .................................
138
6.2.6
Segment 6 .................................
145
7 Discussion 7.1
7.2
152
Validity of Models Learned
. . . . . . . . . . . . . . . .
.
........
7.1.1
Model Variation Across Time ......................
153
7.1.2
Model Variation Across Different Levels of Temporal Abstraction . .
153
7.1.3
Model Variation Across Different Levels of Fault Tolerance
154
7.1.4
Sources of Error
.....
.............................
Applicability in Diagnostic Patient Monitoring
155 ................
156
7.2.1
A LearningBased Approach to Diagnostic Patient Monitoring
7.2.2
Generating Diagnoses Based on Models Learned
.
..........
8 Conclusion and Future Work 8.1
152
Future Work 8.1.1
.
156 157
159
...................................
160
IJsing Corresponding Values in Noisy Learning Data .........
.Bibliography
160
165
7
List of Figures 21
Qualitative reasoning is an abstraction of mathematical reasoning with differential equations and continuously differentiable functions
22
..........
Qualitative states following addition of an increment of fluid to one arm of a
Utube.................................... 23
24
18
...
22
Prolog representation for the Utube model, initialized for an increment of
fluid added to arm a ................................
23
Qualitative model of a Utube.
24
..........................
25 Output of the system behavior generated by running QSIM on the Utube
model ................... 26
.....................
25
Qualitative plots for the qualitative states of a Utube following addition of
an increment of fluid to one arm until equilibrium is reached. ........ 27
26
Output of the Utube model generated by GENMODEL on the given behavior. Note that the model learned is the same as the original model shown previously ......................................
31 GENMODEL algorithm
27
..............................
34
32 Using the upper and lower boundary sets to represent the version space...
39
33 GENMODEL algorithm with fault tolerance.
40
.................
41 Deriving the systolic, diastolic and mean pressures from the arterial blood pressure waveform .................................
45
42 Deriving the stroke volume from the arterial blood pressure waveform....
47
43 Cardiac output curves for the normal heart and for hypo and hypereffective
hearts................................
.........
51 Overall architecture of the learning system. 8
..................
50 54
52 Architecture used for frontend processing of physiological signals.
.....
55
53 An artifact found in blood pressure signals, caused by flushing the blood pressure line with high pressure saline solution 54
.................
56
An artifact found in blood pressure signals, caused by blood sampling from
the blood pressure catheter ........................... 55
56
A general artifact caused by the transducer being left open to air.
.....
57
56 The median filter removes features with durations less than half of its window length, but retains sharp edges of remaining features .............. 57
The effect of aliasing in the segmentation of a signal with the temporal ab
straction parameter set to
T.
g(t) is aliased into h(t) in the qualitative be
havior produced ........... 58
58
.......................
Plot of the Hanning window wn].
61
........................
64
59 Plots of impulse responses h[n] of Gaussian filters for a = 20,40, 60,80, 100.
64
510 Plots of frequency responses H(Q) of Gaussian filters for a = 20, 40, 60, 80, 100. 65 511 Impulse response of an FIR bandlimited differentiator
.............
512 Frequency response of an FIR bandlimited differentiator ............
66 67
513 Equivalent frequency responses of a Gaussian filter in cascade with a bandlimited differentiator for a = 20, 40, 60, 80, 100 .................
67
514 A segment of the mean arterial blood pressure (ABPM) signal. Note the artifacts at t = 600, 1000, 3400 seconds.
.....................
68
515 The ABPM data segment processed by the artifact filter. Note that the artifacts have been filtered but the signal contains many impulsive features.
68
516 The ABPM data segment processed by the median filter. Note that the impulsive features have been filtered
.......................
69
517 The ABPM data segment processed by the Gaussian filter. Note that the signal has been smoothed
............................
518 The ABPM data segment processed by the differentiator.
69 ..........
69
519 Overall scheme of segmentation to produce a qualitative behavior from processed signals and derivatives .......................... 61 Patient
70
: Original Signals. Note the relatively constant heart rate due to
the effect of betablockers
............................
9
73
62 Patient 1: Filtered Signals (L = 61)
74
63 Patient 1: Filtered Signals (L = 121) . . . . . . . . . . . . . . . . . . . . . .
75
64 Patient 1: Filtered Signals (L = 241)
. . . . . . . . . . . ..
76
65 Patient 1: Filtered Signals (L = 361)
. . . . . . . . . . . . . . . . . . . . .
77
66 Patient 1: Filtered Signals (L = 481)
. . . . . . . . . . . . . . . . . . . . .
78
67 Patient 1: Filtered Signals (L = 601)
..............
79
...... . . . . .
. . .....
. ...
68 Patient 2: Original Signals. Note the relatively constant heart rate due to artificial pacing ..............
. . . . . . . . . . . . . . . .
80
69 Patient 2: Filtered Signals (L = 61)
. . . . . . . . . . . . . . . .
81
610 Patient 2: Filtered Signals (L = 121)
. . . . . . . . . . . . . . . .
82
611 Patient 2: Filtered Signals (L = 241)
. . . . . . . . . . . . . . . .
83
612 Patient 2: Filtered Signals (L = 361)
. . . . . . . . . . . . . . . .
613 Patient 2: Filtered Signals (L = 481)
. . . . . . . . . . . . . . . .
85
614 Patient 2: Filtered Signals (L = 601) .
. . . . . . . . . . . . . . . .
86
615 Patient :3: Original Signals .......
. . . . . . . . . . . . . . . .
616 Patient 3: Filtered Signals (L = 61)
. . . . . . . . . . . . . . . .
88
617 Patient :3: Filtered Signals (L = 121)
. . . . . . . . . . . . . . . .
89
618 Patient 3: Filtered Signals (L = 241)
. . . . . . . . . . . . . . . .
90
619 Patient 3: Filtered Signals (L = 361)
. . . . . . . . . . . . . . . .
91
620 Patient 3: Filtered Signals (L = 481)
. . . . . . . . . . . . . . . .
92
. . . . . . . . . . . . . . . .
93
621 Patient :3: Filtered Signals (L = 601) .
.
.
.
.
.
.84
.87
622 Patient 4: Original Signals. Note the relatively constant heart rate due to the effect of betablockers
......
94
623 Patient 4: Filtered Signals (L = 61). Note that the trends of the relatively
constant heart rate are amplified. ..
. . . . . . . . . . . . . .
95
624 Patient 4: Filtered Signals (L = 121)
. . . . . . . . . . . . . .
96
625 Patient 4: Filtered Signals (L = 241)
. . . . . . . . . . . . . .
97
626 Patient 4: Filtered Signals (L = 361)
. . . . . . . . . . . . . .
98
627 Patient 4: Filtered Signals (L = 481)
. . . . . . . . . . . . . .
99
628 Patient 4: Filtered Signals (L = 601)
. . . . . . . . . . . . . . . 100
629 Patient 5: Original Signals. Note the relatively constant heart rate due to the effect of betablockers ......................
10
101
630 Patient 5: Filtered Signals (L = 61). Note that the trends of the relatively
constant heart rate are amplified .
. . . . . . . . . . . . . . . . 102
631 Patient
5:
Filtered Signals (L = 121)
. . . . . . . . . . . . . . . . 103
632 Patient
5:
Filtered Signals (L = 241)
. . . . . . . . . . . . . . . .104
633 Patient
5: Filtered Signals (L = 361)
. . . . . . . . . . . . . . . .105
634 Patient
5: Filtered Signals (L = 481)
. . . . . . . . . . . . . . . . 106
635 Patient
5:
Filtered Signals (L = 601)
. . . . . . . . . . . . . . . ..107
636 Patient 6, Segment 1: Original Signals
.
. . . . . . . . . . . . . . . . 108
1: Filtered Signals (1 = 61)
. . . . . . . . . . . . . . . . 109
638 Patient 6, Segment
1: Filtered Signals (1L = 121)
. . . . . . . . . . . . . . . ..110
639 Patient 6, Segment
1: Filtered Signals (1L = 241)
. . . . . . . . . . . . . . .
640 Patient
6, Segment
1: Filtered Signals (1L = 361)
. . . . . . . . . . . . . . . . 112
641 Patient 6, Segment
1: Filtered Signals (1L = 481)
. . . . . . . . . . . . . . . . 113
642 Patient 6, Segment 1: Filtered Signals (1L = 601)
. . . . . . . . . . . . . . . . 114
643 Patient 6, Segment 2: Original Signals .
. . . . . . . . . . . . . . . . 115
644 Patient 6, Segment
2: Filtered Signals (L = 61)
. . . . . . . . . . . . . . . . 116
645 Patient 6, Segment
2:
Filtered Signals (L = 121) . . . . . . . . . . . . . . . ..117
646 Patient 6, Segment
2:
Filtered Signals (L = 241) . . . . . . . . . . . . . . . . 118
647 Patient 6, Segment
2:
Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 119
648 Patient 6, Segment
2:
Filtered Signals (L = 481) . . . . . . . . . . . . . . . ..120
649 Patient 6, Segment
2: Filtered Signals (L = 601)
. . . . . . . . . . . . . . . . 121
650 Patient 6, Segment
3: Original Signals ......
. . . . . . . . . . . . . . . . 122
651 Patient 6, Segment
3:
637 Patient
6, Segment
Filtered Signals (L = 61)
l...111
. . . . . . . . . . . . . . . . 123
6, Segment 3:
Filtered Signals (L = 121) . . . . . . . . . . . . . . . ..124
653 Patient 6, Segment 3:
Filtered Signals (L = 241) . . . . . . . . . . . . . . . . 125
654 Patient
6, Segment 3:
Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 126
655 Patient
6, Segment 3:
Filtered Signals (L = 481) . . . . . . . . . . . . . . . . 127
656 Patient
6, Segment 3: Filtered Signals (L = 601) . . . . . . . . . . . . . . .
657 Patient
6, Segment 4: Original Signals ......
. . . . . . . . . . . . . . . . 131
658 Patient
6, Segment 4: Filtered Signals (L = 61)
. . . . . . . . . . . . . . . . 132
659 Patient
6, Segment 4: Filtered Signals (L = 121) . . . . . . . . . . . . . . . . 133
652 Patient
660 Patient 6, Segment 4: Filtered Signals (L = 241)
11
...
.....
.129
134
661 Patient
6, Segment 4:
Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 135
662 Patient
6, Segment 4:
Filtered Signals (L = 481) . . . . . . . . . . . . . . . . 136
663 Patient
6, Segment 4:
Filtered Signals (L = 601) . . . . . . . . . . . . . . .
664 Patient
6, Segment 5: Original Signals ......
665 Patient
6, Segment 5:
Filtered Signals (L = 61)
666 Patient
6, Segment 5:
Filtered Signals (L = 121)
667 Patient
6, Segment 5:
Filtered Signals (L = 241)
668 Patient
6, Segment 5:
Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 142
669 Patient
6, Segment 5: Filtered Signals (L = 481)
. . . . . . . . . . . . . . .
.143
670 Patient
6, Segment 5: Filtered Signals (L = 601)
. . . . . . . . . . . . . . .
.144
671 Patient
6, Segment 6: Original Signals ......
. . . . . . . . . . . . . . . . 145
672 Patient
6, Segment 6:
Filtered Signals (L = 61)
673 Patient
6, Segment 6:
Filtered Signals (L = 121) . . . . . . . . . . . . . . .
674 Patient
6, Segment 6:
Filtered Signals (L = 241) . . . . . . . . . . . . . . . . 148
675 Patient
6, Segment 6: Filtered Signals (L = 361) . . . . . . . . . . . . . . . . 149
676 Patient
6, Segment 6: Filtered Signals (L = 481) . . . . . . . . . . . . . . .
.137
. . . . . . . . . . . . . . . . 138 . . . . . . . . . . . . . . .
.139
......... ...... ....
141
.................... 140
. . . . . . . . . . . . . . . . 146 .147
.150
677 Patient 6, Segment 6: Filtered Signals (L = 601) . . . . . . . . . . . . . . . . 151
71 Two approaches to diagnostic patient monitoring. In the learningbased approach, models are continually learned from the patient data. In the historybased approach, a hypothesizetestrefine cycle is used to generate models that best match the patient data.
based on the current model.
In each approach, diagnoses are made
...........................
157
81 The two functions a and b shown obey directionofchange consistency for
the qualitative constraint M+ (a, b), but not magnitude consistency......
162
82 Incorporating intervals into corresponding value sets is analogous to adding a bounding envelope to monotonic functions .................
12
164
List of Tables 5.1
Table showing the orders M and lengths L of the Gaussian filters corresponding to a = 10, 20, 40, 60, 80, 100. . . . . . . . . . . . . . . .
13
.
......
65
Chapter
1
Introduction Physiological models represent a useful form of knowledge because they encapsulate structural information of the system and allow deep forms of reasoning techniques to be applied. For example, such models are used in many prototype intelligent monitoring systems and medical expert systems. However, generating physiological models by hand is difficult and time consuming. Further, most physiological systems are incompletely understood. These factors have hindered the development of modelbased reasoning systems for clinical decision
support. Qualitative models permit useful representations of a system to be developed in the absence of extensive knowledge of the system. In the medical domain, such models have
been applied to: * diagnostic patient monitoring of acidbase balance [6]
* qualitative simulation of the iron metabolism mechanism [14] * qualitative simulation of urea extraction during dialysis [2] * qualitative simulation of the water balance mechanism and its disorders [19]
* qualitative simulation of the mechanism for regulation of blood pressure [19] Further, recent developments in the machine learning community have produced methods of automatically inducing qualitative models from system behaviors. Applying such techniques to learning physiological models should not only benefit knowledge acquisition, but also provide a useful tool for physiologists who need to process vast amounts of data and 14
induce useful theoretical models of the systems they study. The learning system could also serve as a tool for modelbased diagnosis. For example, it could be incorporated into a diagnostic patient monitoring system to perform adaptive model construction for diagnosis in a dynamic environment. In this thesis, we describe a system for learning qualitative models from physiological signals. The qualitative representation of physiological models used is based on Kuipers' approach, used in his qualitative simulation system QSIM [20]. The learning algorithm adopted is based on Coiera's GENMODEL system described in [5]. There has been much work in the machine learning community on learning qualitative models from system behaviors [4, 38, 28]. However, most of these efforts involve learning from readymade qualitative behaviors, or at best simulated quantitative data only. Few, if any, address the issues of learning qualitative models from real, noisy data. Yet such domains are precisely ones where incomplete information prevails, and where automatic generation of qualitative models is most useful. This thesis addresses such a need with a learning system that constructs qualitative models from physiological signals, which are often corrupted with artifacts and other kinds of noise. In our system, we use signals derived from hemodynamic measurements, including:
* heart rate (HR) * stroke volume (SV)
* cardiac output (CO) * mean arterial blood pressure (ABPM) * mean central venous pressure (CVPM)
* ventricular contractility (VC) * rate pressure product (RPP) * skintocore temperature gradient (AT) The thesis is organized into the following chapters: Chapter 2 provides an overview of qualitative reasoning, with focus on qualitative simulation, learning qualitative models, and the relationship between these two tasks. The classic example of the Utube model is discussed, showing the sequence of qualitative states obtained by running QSIM on the model, and the result of using GENMODEL to learn the original model from these states. 15
Chapter 3 analyzes the learnability of qualitative models. QSIM models are shown to be PAC learnable from positive examples only, and GENMODEL is shown to be an algorithm for efficiently constructing a QSIM model consistent with a given set of examples, if one exists. The GENMODEL algorithm and the newly added features of dimensional analysis and fault tolerance are discussed in detail. The chapter ends with a comparison of GENMODEL with other approaches of learning qualitative models, and a discussion of difficulties of applying the PAC learning algorithm to learning qualitative models from physiological signals. Chapter 4 describes the various signals from hemodynamic monitoring used for our learning experiments. A qualitative model of the human cardiovascular system using these signals is provided as a 'gold standard' model, representing the author's best estimate
of the target concepts learnable from the data. Chapter 5 discusses the architecture of the learning system in detail, with emphasis on the frontend processing stage and the segmenter, and how these two stages provide
temporal abstraction. Chapter 6 reports results of using the learning system on data segments obtained from six patients during cardiac bypass surgery. Chapter 7 discusses validity of the models learned, analyzes sources of error, and discusses applicability of the system in diagnostic patient monitoring. Model variation across time and across different levels of temporal abstraction and fault tolerance is also examined. Chapter 8 provides a summary of the main points from the thesis, and discusses promising directions for future work.
16
Chapter 2
Qualitative Reasoning In studying the behavior of dynamic systems, we often model the system with a set of differential equations.
The differential equations capture the structure of the system by
specifying the relationships that exist among the functions of the system. From the differ
ential equations and an initial state, we can derive a quantitative system behavior using analytical methods or numerical simulation. A qualitative abstraction of the above procedure allows us to work with an incomplete
specification of the system. A qualitative model can be represented by a set of qualitative differential equations, or qualitative constraints. From the constraints and an initial state, we can derive a qualitative system behavior using qualitative simulation. Figure 21
!illustratesthis abstraction. Different qualitative representations for models and their behavior have resulted from research in different problem domains [7]. In this chapter, we describe Kuipers' representation in [20], used in his qualitative simulation system QSIM.
'2.1 Qualitative Model Constraints QSIM represents a model of a system by a set of qualitative constraints applied to the functions of the system. These include arithmetic constraints, which correspond to the basic arithmetic and differential operators, and monotonic function constraints, which correspond to monotonically increasing and decreasing functions that exist between two functions.
17
Perturbation Dynamic System
System Behavior
Abstraction
Abstraction Analytical Methods/Numerical Simulation Quantitative System Behavior (Numerical Data)
Differential Equations Statistical Analysis (e.g. RegressionMethods) Abstraction
Abstraction Qualitative Simulation (e.g. QSIM)
Qualitative Constraints (Qualitative Differential Equations)
Qualitative System Behavior (Qualitative States) Inductive Learning (e.g.GENMODEL)
Figure 21: Qualitative reasoning is an abstraction of mathematical reasoning with differential equations and continuously differentiable functions.
Arithmetic Constraints Four arithmetic constraints are included in the QSIM representation:
1. add(f, g,h) 2. mult(f,g,h)
f(t) +g(t) = h(t) f(t) x g(t) = h(t)
3. minus(f,g)
f(t) =g(t)
4. deriv(f, g) a
f'(t) = g(t)
Monotonic Function Constraints When working with incomplete knowledge of a system, we may need to express a functional relationship that exists between two system functions, without specifying the functional relationship completely. In QSIM, the relationship can be described in terms of regions that are monotonically increasing or decreasing, and landmark values that the functions pass through. The QSIM representation includes two constraints for strictly monotonically
18
increasing and decreasing functional relationships. 1
f(t) = H(g(t)) where H(x) is a strictly monotonically increasing func
1. M+(f, g) tion 2. M(f,g)
=
f(t) = H(g(t)) where H(x) is a strictly monotonically decreasing
function Note that since H(x) is a strictly monotonic function in both cases, both M+(f,g) and M(f,g)
require values of f(t)
and g(t) to have a onetoone correspondence, i.e.
f(tl) = f(t 2 ) = g(tl) = g(t2). Also note that the two function constraints M + and M
map onto many different
functions including exponential, logarithmic, linear and other monotonically increasing or decreasing functions.
This manytoone mapping enables qualitative models to capture
incomplete knowledge of a system. Qualitative constraints can be considered as an abstraction of ordinary differential equations (ODE). Every suitable ODE can be decomposed into a corresponding set of qualitative constraints. For example, Hooke's Law which relates the force of a spring to its displacement with the Hooke's constant k [20]: d2 x dt 2
k m
=x
can be decomposed into the following qualitative constraints:
deriv(x, v)
v= d
a d
d2 x

dv
dt == deriv(v,a)
a =  x == M(a, x) m
2.2
Qualitative System Behavior
A dynamic system is characterized by a number of system functions which vary over time. The system behavior can be described in terms of these functions. In QSIM, system func1
In this thesis, M+ is also referred to as mplus and M as mminus.
19
tions must be reasonablefunctions f: [a, b] + R* where * = [oo, oo], which satisfy the following criteria: 1. f is continuous on [a, b] 2. f is continuously differentiable on (a, b) 3. f has only finitely many critical points in any bounded interval
4. limt,a+f'(t) and limtbf'(t)
exist in 2*. We define f'(a) and f'(b) to be equal to
these limits. Every system function f(t) has associated with it a finite and totally ordered set of landmark values. These include 0, the value of f(t) at each of its critical points, and the value of f(t) at each of the endpoints of its domain.
The set of landmark values for a
function form its quantity space which includes all the values of interest for the function. A value can be either at a landmark value, or in an interval between two landmark values. The system has associated with it a finite and totally ordered set of distinguished time points. These include all time points at which any of the system functions reaches a land
mark value. The set of distinguished time points form a time space. A qualitative time can be either a distinguished time point, or an interval between two adjacent distinguished
time points. The qualitative state of f at t is defined as a pair < qval, qdir >. qval is the value of the function at t, and is either a landmark value or an interval between two landmark values. qdir is the direction of change of the function at t, and is one of inc, std, or dec depending
on whether the function is increasing, steady or decreasing at t. Since f is a reasonable function, the qualitative state of f must be constant over intervals between adjacent distinguished time points. Therefore a function can be completely
characterized by its qualitative states at all its distinguished time points and at all intervals between adjacent distinguished time points. Such a temporal sequence of qualitative states
of f forms the qualitative state history or qualitative behaviorof f. Since a reasonable function f is continuously differentiable, the Intermediate Value
Theorem and the Mean Value Theorem restrict the possible transitions from one qualitative state to the next. [20] includes a table listing all possible transitions.
20
2.3
Qualitative Simulation: QSIM
QSIM takes a qualitative model and an initial state, and generates all possible behaviors
from the initial state consistent with the qualitative constraints in the model. Starting with the initial state, the QSIM algorithm works by repeatedly taking an active state and generating all possible nextstate transitions according to the table of possible transitions mentioned in the previous section. These transitions are then filtered according to restrictions posed by the constraints in the system model. Because a model may allow multiple behaviors, QSIM builds a tree of states representing all possible behaviors. Any path across the tree from the given initial state to a final state is a possible behavior of the system.
2.4
Learning Qualitative Models: GENMODEL
GENMODEL [5]goes in the opposite direction to QSIM. It takes a system behavior and di
mensional information about the system functions, and generates all qualitative constraints that permits the system behavior. The GENMODEL algorithm works by first generating all possible dimensionally correct qualitative constraints that may exist among the system functions, according to different permutations of the functions. Then it progresses along
the state history, successivelypruning all constraints that are inconsistent with each state transition. The set of qualitative constraints remaining at the end represent the most specific model that permits the given behavior.
Any subset of this model also permits the
given behavior, and therefore is also a possible model of the system.
2.5
An Example: The Utube
The QSIM representation can be illustrated by an example of a Utube [20]. Figure 22
shows a Utube (two partially filled tanks connected at the bottom by a tube). If an increment of water is added to one arm (A) of the Utube, the system will undergo three different states before reaching equilibrium again: Initial State The water level in A will rise to a new level with the water level in the other arm (B) unchanged initially. There is a net pressure difference between the two arms, resulting in a net flow of water from A to B.
21
A
B
I
I
I
I
(Normal)
t(O)
t(O)/t(l)
t(l) Time, t
Figure 22: Qualitative states following addition of an increment of fluid to one arm of a Utube. ITransitional State As time progresses, the level in A decreases and the level in B increases, while the pressure difference and the water flow diminishes.
Equilibrium State When the water in the two arms reach the same level, the pressure difference and the water flow becomes zero and a new equilibrium is reached. A simple model of the Utube consists of six qualitative constraints, as illustrated in Table 23 [6]:
1. The pressure in A increases with the level in A. 2. The pressure in B increases with the level in B. 3. The pressure difference between A and B is the difference in the two pressures in A and B. 4. The flow from A to B increases with the pressure difference between A and B. 5. The level in A is inversely proportional to the derivative of the flow from A to B. 6. The level in B is proportional to the derivative of the flow from A to B. These constraints are represented in a diagram in Figure 24.
2.5.1
Qualitative Simulation of the Utube
With this simple model and a partial specification of the initial state, QSIM deduces the remaining states according to the QSIM algorithm mentioned previously. The output of the system behavior generated by running QSIM 2 on the Utube model is shown in Figure 25. 2
We used an implementation
of QSIM in UNSW Prolog V4.2 on the HP9000/720 [6].
22
model(u_tube).
% function definitions fn(a). fn(b).
% level in arm a % level in arm b
fn(pa). fn(pb). fn(pdiff). fn(flow_ab).
% pressure in arm a % pressure in arm b % pressure difference % flow rate from arm a to arm b
% model constraints mplus(a,pa). mplus(b,pb). add(pb,pdiff,pa). mplus(pdiff,flowab) . inv_deriv(a,flow_ab). deriv(b,flow_ab). % initial conditions
of qualitative
state at t = t(O)
initialize(a,t(0),a(O)/inf,dec). initialize (b,t () ,b(O),_). % ordinal values for function landmarks
landmarks(a,[O,a(O),inf]). landmarks(b,[O,b(O),inf]). % definition of function values at normal equilibrium normal( [a/a(O), b/b(O), pa/pa(O),
pb/pb(O), pdiff/0, flow_ab/O]).
:Figure 23: Prolog representation for the Utube model, initialized for an increment of fluid added to arm a.
23
_1
IT
Level A
·_
Pressure A I\
I
I~ L1
Pressure Differenct
I
~zh M+nA s rluw
LauC.
'
Figure 24: Qualitative model of a Utube. The qualitative states generated can be presented in qualitative plots as shown in Figure
26. In these plots, landmark values are placed on the vertical axis, and distinguished time points and corresponding intervals are placed on the horizontal axis. The only positions
that can be taken at each time point are at or between landmark values.
2.5.2
Learning the Utube Model
The system behavior described previously together with dimensional information on the system functions can be given to GENMODEL 3 to generate a set of all qualitative constraints which are consistent with the behavior. Dimensional information on the system functions of the Utube are specified in Prolog as follows:
units(a,mass). units(b,mass). units(pa,pressure). units(pb, pressure). units(pdiff,pressure). units(flow_ab,mass/t).
The output of the Utube model generated by GENMODEL on the given behavior 3
GENMODEL is implemented in UNSW Prolog V4.2 on the HP9000/720 [5].
24
Simulating u_tube t(O)
t(O) / t(1) t(1)
simulation completed landmarks used in simulation: landmarks(a, landmarks(b,
[0, a(O), a(1), [0, b(O), b(1),
inf]). inf]).
landmarks(pa, [0, pa(O), pa(1), landmarks(pb, [0, pb(O), pb(1), landmarks(pdiff, [0, inf]). landmarks(flowab, [0, inf]).
inf]). inf]).
The simulation generated the following qualitative states: t(0) a b
a(O) / inf b(O)
dec inc
pa pb
pa(O) / inf pb(O)
dec inc
pdiff
0 / inf
dec
flowab
0 / inf
dec
a b
a(O) / inf b(O) / inf
dec inc
pa
pa(O) / inf
dec
pb
pb(O) / inf
inc
pdiff
0 / inf
dec
flowab
0 / inf
dec
a b
a(1) b(1)
std std
pa
pa(1)
std
pb pdiff flowab
pb(1) 0 0
std std std
t(O) / t(1)
t(1)
Figure 25: Output of the system behavior generated by running QSIM on the Utube model.
25
lnf
IlA(A)
inf
:(I)
vrl(B)
b(I)
b(O)
a(O t(0)
t(0)
t(1)
'n
t(1)
inf
inf.
prss..r(A)
inf
preun.(B)
pb(l)' pb(O)
1
p( ) P(O t(O)
t(l)
(t()
inf
'
t()
inf I
t(o)
l t (AB)
prrressru diffnce
inf
t(1)
inf I
I
Figure 26: Qualitative plots for the qualitative states of a Utube following addition of an increment of fluid to one arm until equilibrium is reached. is shown in Figure 27. In the case of the Utube, the generated model is equivalent to the original model after redundancy elimination by GENMODEL. GENMODEL will be described further in Chapter 3.
26
Generating possible constraints mplus(pdiff, flowab) mplus(pdiff, a) mplus(pdiff, pa) mplus(flowab, pdiff) mplus(flowab, a) mplus(flowab, pa) mplus(a, pdiff) invderiv(a, flowab) mplus(a, flow_ab) mplus(a, pa) deriv(b, flowab) mplus(b, pb) mplus(pa, pdiff) mplus(pa, flowab) mplus(pa, a) mplus(pb, b) add(pdiff, pb, pa) add(pa, pb, pdiff) add(pb, pdiff, pa) add(pb, pa, pdiff) filtering with state t(O) / t(1) filtering with state t(1) filtering : mplLus(pdiff, a) filtering : mp].us(pdiff, pa) filtering : mpllus(flowab, a) filtering : mp].us(flow_ab,pa) filtering : mp]lus(a, pdiff) filtering : mpl.us(a, flowab) filtering : mp]lus(pa, pdiff) filtering : mpllus(pa, flowab) filtering : addL(pa, pb, pdiff) filtering : addL(pb, pa, pdiff) Checking for redundancies filtering : mpllus(flowab, pdiff) simple redundancy filtering : mp].us(pa, a) simple redundancy filtering : mpllus(pb, b) simple redundancy filtering: add((pb, pdiff, pa) simple redundancy Model Constraints: deriv(b, flowab) invderiv (a, flowab) mplus(pdiff, flowab) mplus(a, pa) mplus(b, pb) add(pdiff, pb, pa)
Figure 27: Output of the Utube model generated by GENMODEL on the given behavior. Note that the model learned is the same as the original model shown previously.
27
Chapter 3
Learning Qualitative Models In this chapter, we examine the learnability of qualitative models employing the QSIM formalism [20]. We review the Probably Approximately Correct (PAC) model of learning [37], and show that QSIM models are efficiently PAC learnable from positive examples only. The proof is based on the algorithm GENMODEL [5]which can efficiently construct a QSIM model consistent with a given set of examples, if one exists. The chapter continues with a detailed coverage of GENMODEL, including the newly added features of dimensional analysis and fault tolerance, and a comparison of GENMODEL with other approaches of learning qualitative models. We end the chapter with a discussion of the difficulties of applying PAC results to our task of learning QSIM models from physiological signals.
3.1 Probably Approximately Correct Learning A common setting in machine learning is as follows: Given a set of examples, produce a concept consistent with the examples that is likely to correctly classify future instances. We
are interested in algorithms that can perform this task efficiently. The Probably Approximately Correct (PAC) model of learning introduced by Valiant [37] is an attempt to make precise the notion of "learnable from examples" in such a setting [17, 30].
3.1.1
Definitions
Let X be the set of encodings of all instances.
X is called the instance space. In our
learning task, X is the set of all qualitative states within a given set of landmark values and distinguished time points. 28
A concept c is a subset of the instance space, i.e. c C X. In our learning task, c is a QSIM model which can be seen as a concept specifying a subset of X (the set of all qualitative states) as legal states. We can equivalently define a concept c to be a boolean mapping applied to X, i.e. c : X+ (0, 1}. An example is considered to be a positive example if c(x) = 1, and a negative example if c(x) = 0. A concept class C over X is a collection of concepts over X. In our case, C is the set of all QSIM models with a given number of system functions. The goal of the learning algorithm is to determine which concept in C (the target concept c) is actually being used in classifying the examples seen so far.
We define n to be the size parameter of our instance space X. In our case, n is the number of system functions in our system. Note that n affects the number of possible qualitative states and thus the size of the instance space X. The difficulty of learning a concept that has been selected from Cn depends on the size ICn I of Cn. Let 1
An= lg Cl An may be viewed as the minimum number of bits needed to specify an arbitrary element of Cn. If An is polynomial in n, Cn is said to be polynomialsized. We are interested in characterizing when a class C of concepts is easily learnable from examples.
3.1.2
PAC Learnability
Stated informally, PAC learnability is the notion that the concept acquired by the learner should closely approximate the concept being taught, in the sense that the acquired concept should perform well on new data drawn according to the same probability distribution as that according to which the examples used for learning are drawn. We define Pn as the probability distribution defined on the instance space Xn according to which the examples for learning are drawn. The performance of the acquired concept c' is measured by the probability that a new example drawn according to Pn will be incorrectly classified by c'. For any hypothesized concept c' and a target concept c, the error rate of c' is defined as:
error(c')= Prepn[c'(x) $ c(x)] llg x = log2 x
29
where the subscript x E P, indicates that x is drawn randomly according to P,. We would like our learning algorithm A to be able to produce a concept whose error rate is less than a given accuracy parameter
with 0 < e < 1. However, we cannot expect this
to happen always, since the algorithm may suffer from "bad luck" in drawing a reasonably representative set of examples from Pn for learning.
Thus we also include a confidence
parameter 6 with 0 < 6 < 1 and require that the probability that A produces an accurate answer be at least 1  6. As
and
approach zero, we expect A to require more examples
and computation time. We would like the learning algorithm to satisfy the following properties: * The number of examples needed is small. * The amount of computation performed is small. * The concept produced is accurate, i.e. the error rate
is arbitrarily small.
* The algorithm produces such an accurate concept most of the time, i.e. with a probability of 1  6 that is arbitrarily close to 1. The concept class C is said to be efficiently PAC learnable if there exists an algorithm A for learning any concept c E C that satisfies the above criteria. To define PAC learnability formally, we say that a concept class C is efficiently PAC learnable if there exists an algorithm A and a polynomial s(.,., 6, all probability distributions P, on X,
) such that for all n, , and
and all concepts c E Cn, A will with probability
at least 1  6, when given a set of examples of size m = s(n, l, ~) drawn according to P, output a c' E Cn such that error(c') < e. Furthermore, A's running time is polynomially
bounded in n and m. The hypothesis c' E C of the learning algorithm is thus approximately correct with high probability, hence the acronym PAC for Probably Approximately Correct.
3.1.3
Proving PAC Learnability
One approach of PAC learning due to Blumer et al. [3] is as follows: Draw a large enough set of examples according to P, and find an algorithm which, given the set of examples, outputs any concept c E Cn consistent with all the examples in polynomial time. If there
exists such an algorithm for the concept class C, C is said to be polynomialtime identifiable. 30
Formally, we say that C is polynomialtime identifiable if there exists an algorithm A and a polynomial p(A, m) which, when given the integer n and a sample S of size m, produces in time p(A,, m) a concept c E C consistent with S, if such a concept exists. This leads to an interesting interpretation of learning. Learning can be seen as the act of finding a pattern in the given examples that allows compression of the examples. We can measure simplicity by the size of the concept space of the algorithm, or equivalently by An, the minimum number of bits needed to specify a concept c among all the concepts in C,. An algorithm that finds a succinct hypothesis consistent with the given examples is called an Occam algorithm. This formalizes a principle known as Occam's Razor which equates learning with the discovery of a "simple" explanation for the observed examples. A concept that is considerably shorter than the examples is likely to be a good predictor for future data [29]. Next we look at how large a set of examples is "large enough" for PAC learning.
[3]
includes a key theorem on this issue. Theorem
Given a concept c E Cn and a set of examples S of c of size m drawn according
to P, the probability is at most ICnl (1 e)m that there exists a hypothesis c' E C, such that:
* the error of c' is greater than , and · c' is consistent with S. Proof
If c' is a single hypothesis in C, of error greater than
, the chance that c' is
consistent with a randomly drawn set of examples S of size m is less than (1 
)m .
Since C, has ICnI members, the chance that there exists any member of C, satisfying both conditions above is at most

ICE(1 E)m . ·
Using the inequality x
< e x
we can prove that 1
m > (ln Cnl+In E
31
1
implies that
ICnl(l E)m < Thus if m satisfies the above lower bound, the probability is less than 6 that a hypothesis
in Cn which is consistent with S will turn out to have error greater than e. Therefore to PAC learn a concept, the learning algorithm need only examine m examples where m has a lowerbound as follows: 2
m=
Q((In lCn + n))
Recall if Cn is polynomialsized, then An and therefore In ICn is polynomial in n. Therefore
m is polynomial in n,
and
To conclude, an algorithm that draws m examples according to Pn and outputs any concept consistent with all the examples in polynomial time is a PAC learning algorithm. Thus if Cn is polynomialsized and polynomialtime identifiable, then it is efficiently PAC learnable.
3.1.4
An Occam Algorithm for Learning Conjunctions
In [37] Valiant provides an algorithm for PAC learning the set of boolean formulae in conjunctive normal form (CNF) where each clause contains at most k literals. This set of boolean formulae is known as k CNF. The algorithm is capable of PAC learning from positive examples only. In Section 3.3.2 we will map the problem of identifying a QSIM model consistent with a given set of examples to the problem of identifying a kCNF consistent with a given set of examples. First we calculate the number of examples needed. The number of conjunctions over the boolean variables x1,..., Xn is 3n since each variable either appears as a positive or negative literal, or is absent entirely. Applying the formula for the lower bound in the previous section, we see that a sample of size O(n + in
is sufficient to guarantee that
the hypothesis output by our learning algorithm is e accurate with confidence of at least 1 J.
The algorithm starts with the hypothesis conjunction which contains all the literals: c' = X1A
1
A..
2A = Q(B) denotes B is a lower bound of A.
32
A
An
Upon each positive example x, the algorithm updates c' by deleting the literal xi if xi = 0 in the example, and deleting the literal xi if xi = 1 in the example. Thus the algorithm
deletes any literal that contradicts the data. This can be seen as starting with the most specific concept and successively generalizing the concept upon each positive example given. Since the algorithm takes linear time to process each example, given m examples with
m as calculated above, the running time is bounded by mn and hence is bounded by a polynomial in n,
1
and . Therefore this is an efficient PAC learning algorithm for the class
of kCNF.
3.2
The GENMODEL Algorithm
GENMODEL is a program for generating qualitative models from example behaviors [5]. Given a set of qualitative states describing a system behavior, GENMODEL outputs all
QSIM constraints consistent with the state history. Together, these constraints form the most specific QSIM model given the example behavior.
The Algorithm Input: * A set of system functions, Functions.
* A set of units for the system functions, Units * A set of landmark lists for the system functions, Landmarks.
* A set of qualitative states, States. Output: A qualitative model which consists of all constraints that are consistent with the state
history and dimensionally correct, Model. Functions used: search() A function for searching corresponding values from a set of qualitative states. dimcheck()
A function for checking dimensional compatibility of functions within a
constraint.
33
begin
Constraints+0; Correspondings  search(States); for each f, f2 in Functions such that fl
# f2 do
for each predicate2 in {inv, deriv, invderiv, M +, M} do
fl, f2, Units) then if dimcheck(predicate2, add predicate2(fl,f 2) to Constraints;
for each f,
f2,
f3 in Functions such that f # f2#f3 do
for each predicate3 in {add, mult} do
fl, f2, f3, Units) then if dimcheck(predicate3, add predicate3(fl,f2,f3) to Constraints; for each s in States do for each c in Constraints do if not check(c, s, Landmarks, Correspondings) then delete c from Constraints;
reduce(Constraints);
Model+ Constraints; end
Figure 31: GENMODEL algorithm. check() A function for checking validity of a constraint given a qualitative state and
sets of corresponding values. reduce() A function for removing redundancy from constraints.
For example, since
M+(A, B) and M+(B, A) specify the same relationship, one of them can be removed. Method:
* Search the entire state history States for sets of corresponding values. * Generate the initial search space by constructing all dimensionally correct constraints with different permutations of system functions in Functions.
* Successivelyprune inconsistent constraints upon each qualitative state in States. * Remove redundancy from the remaining constraints * Output the result as a qualitative model. The entire algorithm is shown in Figure 31.
34
3.3
QSIM Models are PAC Learnable
From Section 3.1.3 we conclude that to prove that the concept class of QSIM models is PAC learnable, it suffices to prove that the class is polynomialsized and that it is polynomialtime identifiable. The following two sections provide these proofs.
3.3.1
The Class of QSIM Models is PolynomialSized
To show that the concept class of QSIM models is polynomialsized, we begin by noting that in the QSIM formalism there are five kinds of twofunction constraints (inv, deriv,
invderiv,
M+ and M), and two kinds of threefunction constraints (add and mult).
Therefore with n system functions, the number of possible QSIM constraints N is as follows:
N = 5n(n  1) + 2n(n  1)(n 2) Therefore, the number of possible QSIM models is:
IQSIMModels(n)l= 2N = 25n(n1 )+2n(n1)(n2) since each QSIM constraint can either be present or absent in the model. This implies that:
lg (QSIMModels(n)I) = N = O(n3 ) Therefore the concept class of QSIM models is polynomialsized. To PAC learn a QSIM model, we need m examples where m is calculated as follows:
m=Q 3.3.2
((5n(n  1) + 2n(n  1)(n  2))ln2 + In ))
QSIM Models are PolynomialTime Identifiable
In this section we show that GENMODEL is an algorithm for efficiently constructing a QSIM model consistent with a given set of examples.
We prove this by mapping the
problem of identifying a QSIM model consistent with a given set of examples to the problem of identifying a kCNF consistent with a given set of examples. The algorithm presented in Section 3.1.4 then yields an algorithm for identifying a QSIM model consistent with a
35
given set of examples in polynomial time. We show that this algorithm is in fact identical to GENMODEL. We view each QSIM model as a conjunction of QSIM constraints, and each QSIM constraint as a boolean variable.
Then learning QSIM models is equivalent to learning
monotone conjunctions 3 with N boolean variables, where N is the number of possible QSIM constraints as calculated in the previous section. Now the algorithm starts with the hypothesis of a monotone conjunction which contains all N of the boolean variables, i.e. all possible QSIM constraints:
C'=xlA
Ax,n
The qualitative states provided for learning constitute the positive examples.
Upon
each positive example x, the algorithm updates c' by deleting the boolean variable xi if the corresponding QSIM constraint is inconsistent with the example. Since each boolean variable xi corresponds to a QSIM constraint, the algorithm prunes any constraint that is
inconsistent with each qualitative state. This can be seen as starting with the most specific model and successively generalizing the model upon each qualitative state given. This is
identical to the approach taken by GENMODEL. Now it remains to show that GENMODEL takes polynomial time to process each qualitative state. We review each step taken by GENMODEL in learning a QSIM model: * Search the entire state history for sets of corresponding values. For m qualitative
states, there are at most m sets of corresponding values, and the search takes O(m) time. * Generate the initial search space by constructing all constraints with different permutations of system functions. For n system functions, this takes O(n3 ) time as shown in the previous section.
* Successivelyprune inconsistent constraints upon each qualitative state. Checking for consistency of a constraint with a qualitative state involves:  Checking landmark values and directions of change. This takes linear time. 3
Monotone conjunctions are ones with positive literals only.
36
 Checking corresponding values. Since there are at most m sets of corresponding values, this takes O(m) time.
* Remove redundancy from the remaining constraints. Since we started off with O(n3 ) constraints, there are at most the same number of constraints remaining in the final model. Therefore, removing redundancy takes O(n3 ) time.
* Output the result as a qualitative model. Therefore for GENMODEL the processing time of each qualitative state is polynomial in
m and n. Since the algorithm takes polynomial time to process each qualitative state, given m states with m as calculated above, the running time is bounded by p(m, n) where p(, ) is a polynomial in the two arguments. Therefore the running time is bounded by a polynomial in n,
and L 6
60
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Figure 63: Patient 1: Filtered Signals (L = 121)
L=121 mult(HR, ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct)
75
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Figure 64: Patient 1: Filtered Signals (L = 241)
L=241 M+(ABPM,
RPP)
Correct given that HR was constant due to betablockers.
increased due to increased Enflurane dosage.
mult(HR, ABPM, RPP) (Correct) mnult(HR, SV, CO) (Correct)
76
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Figure 65: Patient 1: Filtered Signals (L = 361)
L=361 M (ABPM, VC) VC increased to compensate for decreasing ABPM due to increased Enflurane dosage.
mult(HR, ABPM, RPP) (Correct)
77
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·
1000
65 0
tsec
1500
2000
1000 Vsec
4
y 1,v
4.2 1 '
0o
z
4
1
&R~0
A
1000
1500
.
2000
1000
t/sec
t/sec
In_ IllllI Ill
I 1.6
I UUVVVV
a
a. Cc 8000
5 1.4
w
Ernnn .1 II BI
500
1000
1500
.
4 ni 
I
2000
t/sec
_
500
1000
t/sec
Figure 66: Patient 1: Filtered Signals (L = 481)
L=481 M+(ABPM,
RPP)
Correct given that HR was constant due to betablockers.
increased due to increased Enflurane dosage.
M+(CVPM, VC) Both increased  FrankStarling Law of the Heart.
M(ABPM, SV) (Spurious) M (SV, RPP) (Spurious) mult(HR, SV, CO) (Correct)
78
ABPM
70
7_ f

n 60 a 6 500
1000
1500
"8
2000
1000
o00
t/sec z
70 I ""'
:1!'1
1000
1500
L ,'
0o
1000
x1
I
nI
oo
2000
1500
2000
i~~~~~~
II
1000
1500
.
!00
2000
1000
t/sec
t/sec
xrnnrn_ I BB BI BB]
1 1
IVVI
t
13 8000
n l1a Ern v.vgoo
.B .
1500
Vsec
4 ."
a
2000
I'
2000
t/sec "
1500
> 400 
___
o.o
2000
.
ApJ' .LL
z
.
1500
t/sec
..
W
.. .I
1000
1500
I .u
1.4~ I
2000
t/sec
.
'
500
1000
t/sec
Figure 67: Patient 1: Filtered Signals (L = 601)
L=601 M+(ABPM,RPP)
Correct given that HR was constant due to betablockers.
increased due to increased Enflurane dosage.
M+(CVPM, VC) Both increased  FrankStarling Law of the Heart. M+ (HR, CO) Both decreased because of increased Enflurane dosage.
invderiv(ABPM, RPP) (Spurious) inv_deriv(ABPM, VC) (Spurious)
mult(HR,ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct)
79
ABPM
6.1.2
Patient 2
The patient was a 65yearoldblind and partially deaf lady who was admitted in acute left ventricular failure and found to have significant aortic valve regurgitation. The operation performed was to replace the patient's aortic valve and to check a mitral valve installed approximately 27 years ago. The data segment was taken when the patient was just coming off bypass and was artificially paced. 4 nn I I V IE
2
50
CD
0
.I
I
9000
9500
10000
10500
9o600
_
k_
9500
i/sec Jl
M 100
Iz zi r
I
LII TrrrL
n ·
I
9000
9500
10000
9000
10500
9500
i/sec x 106
10000 ·
10
IL JL
0
E
n'Y·1
· YU~y.
i
=h
9500
·
~·YI·
i
10000
900
10500
9500
t/sec
10000
10500
i/sec
x 104
'V
4I
a
JI
Er
J w
0 9500
10500
t/sec ·
1
9000
10500
x 104
n
Cv
10000 i/sec
10000
In
"6oo
10500
9500
10000
10500
t/sec
i/sec
Figure 68: Patient 2: Original Signals. Note the relatively constant heart rate due to artificial pacing.
80
^A "All Jl
c.
_
> 10t
m
0
r 9ooo00
500
·
9500
n, Y..
1000_ 7$y
,..
9500
10000
... 9'0o 900
10500
t/sec
00
X 104 i
rv _
9500
10500
10000
10500
10000
10500
_
10000
10500
9ooo00
9500
t/sec
t/sec
"IB "
n
I " .
i
L n~
J.u
'
m
I; 0.6
nE 8000
J
0 nA [
I
6000
10000 t/sec
IB
9500
9ooo00
nn I II
·
0 0.5
5 I
I
U, 500
i75
10
10500
t/sec
Vsec
'9o00
10000
9500
10000
10500
9600
inn 9500
t/sec
t/sec
Figure 69: Patient 2: Filtered Signals (L = 61)
L=61 M+(SV, CO) Correct given that HR was constant due to artificial pacing.
mult(HR, SV, CO) (Correct)
81
/~s Ar, 35
50
i
m3
000oo
9500
10000
n_
0
5j9000
10500
9500
10000
10500
10000
10500
10000
10500
10000
10500
t/sec
t/sec
800
Ir 78
> 600 (0 o
JII·II BI
900o
9500
10000
'100oo
10500
t/sec
9500
t/sec
4~~o°
x 104 r
0o 6
U> 0.5
9000
vv'
i iq
9500
10000
10500
9'000
9500
t/sec 9000
a. n n

t/sec ·
0.8 0.6
8000
0
7000 9000
9500
10000
10500
0.4
9000
t/sec
9500
t/sec
Figure 610: Patient 2: Filtered Signals (L = 121)
L=121 M+(SV, CO) Correct given that HR was constant due to artificial pacing.
82
55
·
·
Jr., _
m rn 50
> 4_
0
,I
4! 93000 ^
/_//
·
·
9500
10000
105500
9500
t/sec
10000
10500
t/sec
0OU
800_
78
60 0
II
.
.

OO_
·· _ I7'9 00
Jl III
9500
10000
I I
9000
10500
9500
Vsec
10000
10500
10000
10500
Vsec
4
,X 10
4i .
II
0 0.5
06
AII _
!
1 9000
9500
10000
10500
90ooo
9500
t/sec 9000

t/sec 0.8
.
.
a. a 8000
0.6
_VO i I VVV I.......
.
.

9000
w III ~rll
9500
10000
_
9U i00
10500
t/sec
9500
10000
t/sec
Figure 611: Patient 2: Filtered Signals (L = 241)
L=241 M+(SV, CO) Correct given that HR was constant due to artificial pacing.
83
10500
_'
i

50
aC2
t
4 1j.
§oo
9500
10000
_
9000
10!500
9500
t/sec 80
700
.
.
78 '9'600
9500
10000
1.
.
.
9500
10000
10500
10000
10500
10000
10500
t/sec
I
_
9%00
0 0.5 .l
9500
10000
9'00
10500
9500
t/sec
t/sec ..  .no V...
8000 AL
a
7nA
.
.
900
,x 104
o6

500
10500
t/sec I
10500
i 600
lm
J
10000
t/sec
0J
" 'A
_
, vv 9000
0.6
9500
10000
10500
IL
''6oo
t/sec
9500
t/sec
Figure 612: Patient 2: Filtered Signals (L = 361)
L=361 No constraints were obtained.
84
A_ U
2 50
4 , j,
900
9500
10000
10500
900oo
t/sec
m
9500
9EvD fUU
I 78
10000
10500
/sec .
.
j 650
9000
9500
10000
l I I t w9ooO
10500
.
9500
t/sec
2
.
.
10500
t/sec
104
7x
.
10000
.
.
C I6
,1 Il
9o00
9500
10000
10500
9o00
9500
t/sec nOUUU
10500
10000
10500
0.8
0
a. Ir
10000
t/sec
0.6
nn J ~
IIB
Wa 0C.n
__
9000
9500
10000
10500
i JA
_ _
Vo00
t/sec
9500
t/sec
Figure 613: Patient 2: Filtered Signals (L = 481)
L=481 No constraints were obtained.
85
·
0 
2
2
50 6o
o000
9500
10000
9'00
10500
9500
t/sec
10000
10500
10000
10500
t/sec 7n.. III
80
78
Cn650.
'9000
9500
10000
,n.^ 9VooO
10500
9500
t/sec
t/sec
14
I
·
.
.
o6 I
9000
9500
10000
I
10500
I
9000
9500
t/sec
10500
10000
10500
t/sec
nnn II
·i · I
...
aa
10000
OvVV
i
w 0.6
a: Inn
a
I
9000
9500
10000
..n .A
_
9000
10500
t/sec
9500
t/sec
Figure 614: Patient 2: Filtered Signals (L = 601)
L=601 M (ABPM, SV) (Spurious)
86
6.1.3
Patient 3
The patient was a 47yearold gentleman who had a recent large myocardial infarction and recurrent episodes of left ventricular failure since then.
His exercise tolerance was very
poor and he developed a left ventricular aneurism following the myocardial infarction. The
operation performed was insertion of coronary artery bypass grafts and excision of the ventricular aneurism. The data segment was taken when the surgery was just starting. 4n_ lBtI.
n
80
0
0
o 1000
1500
LLSU·Y" ·
1000
2000
i/sec
I
WU 
aI
0 1 1Eli _UU9660
rY 1000
1500
2000
1500
2000
i/sec 2000
·

0
,
W.'pi IL.

'.......
1500
'WVVKo 0
2000
1000
t/sec
/sec
x 105
d~ I E B
IV
ILI.
,..L
0500
1000
1500
0
0
O.1
2000


I
'oo00
1000
t/sec
1500
2000
1500
2000
MJ,
0
11.5i
04
!
1500
II
t/sec
x 10 4
1000
I W l1
I

/ L J
i
500
2000
1000
t/sec
t/sec
Figure 615: Patient 3: Original Signals
87
7r I 
to,
m 60 500 50;
1000
1500
2000
t/sec
4 2 500
1000
1
.
> 300
'1
200
500
t/sec L
x 134 ._
113~~~~~~
sX
1500
2000
1500
2000
1
° 0.5
n v
__ I
500
·
1000
1500
'iSy
I~~~~~~
2000
500
t/sec H 2
5000 5000
1 .
1000
1000
t/sec
10000
II lJ
1000
t/sec
02 o
0
2000
400
I
a
1500
t/sec
1500
2000
t/sec
O_ 500
.
.
1000
1500
t/sec
Figure 616: Patient 3: Filtered Signals (L = 61)
L=61 No constraints were obtained.
88
2000
70II I co 60
·
2a. 0
_
4 i/
n 9
Ern 500oo
1000
1500
2000
500
1000
t/sec 400 
100
IX
2000
.
.
> 300
0JII
1000
1500
BI
'"oo
2000
1000
t/sec 4
x 104
1500 t/sec
1000
t/sec 11
"
aaa: 5000
1 uJ
!
1000
1500
2000
I
!
00oo
I
I
1000
1500
t/sec
t/sec
Figure 617: Patient 3: Filtered Signals (L = 121)
L=121 M + (HR, RPP)
2000
Bi
oo
2000
nnnr
0
1500
0 0.5 i
500
2000
_
I
1000
BlBB "I
1500
t/sec
32 (D
B
1500
t/sec
Correct due to slowly rising ABPM.
mult(HR,ABPM,RPP) (Correct) mult(HR, SV, CO) (Correct)
89
2000
70 
>4
a 60
EL
co
I
O rn.
1000
oo00
1500
2000
500
1000
t/sec
1500
2000
1500
2000
1500
2000
t/sec
70
barn
6JDU
I 60
> 300
T
r n 
500
1500
1000
2000
1000
t/sec I
I
1
t/sec
X 1 ()4
4_ ]r
(>)0
00 23
yo1

12
1000
5oo
1500
2000
o00
1000
t/sec Onnf J uv ]]
~l
t/sec
_
.¢'
a_ a 6000 n'
4000 
500
5i
·
·
1000
1500
0 ·
2000
500
t/sec
1000
1500
t/sec
Figure 618: Patient 3: Filtered Signals (L = 241)
L=241 AM+(HR, RPP)
Correct due to slowly rising ABPM.
mult(HR, ABPM, RPP) (Correct) nmult(HR,CVPM, RPP) (Spurious) ,mult(HR, SV, CO) (Correct)
90
2000
70
a. 4
60
L
rEn
oo
0 .
.
1000
1500
0
1000
0oo
2000
,,
/U
400
3 60
> 300
[
500oo
1000
1500
2000
1500
2000
1500
2000
t/sec
t/sec
1500
ok4 )O sV0
2000
/
1000
t/sec
t/sec
x 104 1
0
>0.5
O 2
t
_
II
oo
1000
1500
goo
2000
1000
t/sec annnII
I
t/sec eII
I
a
L 6000 4000 500
 1
0
I
1000
1500
!
500
2000
1000
1500
t/sec
t/sec
Figure 619: Patient 3: Filtered Signals (L = 361)
L=361 mult(HR,ABPM,RPP) (Correct) mult(HR, SV, CO) (Correct)
91
2000
'=
·
/U
6
2
·
a4
60
O i
... ~l!

ffi
1000
1500
2000
1000
t/sec 7^
2000
1500
2000
400 
/U
C> 300
60 cn
Inn 'oo
_
III
5'oo
1000
1500
2000
I~
I
1000
/sec ex
t/sec
104
1
0 0.5
02 i
1500
t/sec
I
i
500
1000
1500
OL 0
2000
1000
t/sec
I
1500
2000
1500
2000
i/sec
8000
H 1.5
f
6000A1~1 ^Ill
LU a
1
500V 500
1000
1500
2000
iJ i
r,
_
500
t/sec
1000
t/sec
Figure 620: Patient 3: Filtered Signals (L = 481)
L=481 M+(HR, RPP)
Correct due to slowly rising ABPM.
92
9 6
aa. 60
a
0 o00
1000
1500
4

(
1000
2000
t/sec 70
400 
1500
2000
t/sec
.
.
>c 300 I
JII
1000
1500
2000
'
1000
oo
t/sec
10)4
,x Li
0 0.5
02 I
1000
1500
2000
1000
t/sec 1.EIr
·
0 6000
1 C \r I V.0
5
AAlto
II IB
1000
1500
2000
2000
_ 1000
t/sec
i/sec
Figure 621: Patient 3: Filtered Signals (L = 601)
L=601 M + (HR, RPP)
1500
f
I
0
500
2000
t/sec c
8000 
III
1500
t
I
I
go 0
2000
1
.
.
1500 /sec
Correct due to slowly rising ABPM.
mult(HR,ABPM, RPP) (Correct)
93
6.1.4
Patient 4
The patient was a 60yearold gentleman with a history of angina and breathlessness on exercise.
He had been a noninsulin dependent diabetic for many years.
His coronary
angiography showed severe triple vessel disease with an occluded right coronary artery and severely impaired left ventricular function. He was on betablockers (Atenolol) and calcium channel blockers (Nicardipine).
The data segment was taken when the operation had just started, and the period was relatively uneventful.
1 \~m~ · aco 70
8
0
IwFn#ljvI
cJ
1500
1000
2000
2500
1500
1'600
t/sec 1·
rn
2500
nrnr
'JII
 100
L,,,,1, ,,,, ,
n 100oo I
2000
t/sec I·
Al
lf00
, A
111TAP"
BI
1500
tsec
2000
1000
2500
1500
2000
2500
t/sec
,,x 105
o_
.I kill.
Ii
IAMI
ULWL
'I'T'F7TrIq7rrI1TWi"7 1500
1000
J
L&dIdlILL..
2000
·
_c.
2500
1000
1500
t/sec Ax
z
tsec
2000
2500
104
a 1
0
II.
1.L I
1000
nFTL .
4~u1LLJX 1500
2000
a 1.5 10002500
1000
t/sec
1500
2000
2500
t/sec
Figure 622: Patient 4: Original Signals. Note the relatively constant heart rate due to the effect of betablockers.
94
11
a
eo
100
1500
2000
0 6_
2500
o000
1500
t/sec 4hi Or
44
*
,
.
800
.
1500
2000
8000
2500
1500
t/sec x 10
0 0
2000
2500
900
c)
A
2500
I UUU
:
I
1000oo
2000 t/sec
t/sec
4 1
.
.
J.i
0>0.5 AI
fl
1000
1500
2000
2500
1ib00
1500
t/sec
2000
2500
2000
2500
t/sec
""' rcnn
_
I
4·
a.
a 5000n
4500 1000
w
0
1500
2000
2500
J q
B
_
1 boo
t/sec
1500
t/sec
Figure 623: Patient 4: Filtered Signals (L = 61). Note that the trends of the relatively constant heart rate are amplified.
L=61 No constraints were obtained.
95
i/\i
n. M
·
6CI.I
a: "`
'i000
1500
2000
44 l
G"
A,1
ZL.4
__
1o00
1500
2000
~~~~ ~ ~
6 C
A
2500
L
10 0
t/sec
AC ··
!
a
n'n iii
900 M
2500
I...
2000
2500
2000
2500
t/sec
vJ
VVI
.B.
(n
1500
1500
100
t/sec
t/sec
x 104
0 0.5
00 5
·
Z
1000
1500
2000
25 00
1000
1500
t/sec PA I II
1 .,II
11
2500
So
w
0
...
1000
2000
I
I
5000 I~.
2500
t/sec
rRt5 OIL
2000
1500
2000
2500
4 I
I
b 00 110
t/sec
1500 t/sec
Figure 624: Patient 4: Filtered Signals (L = 121)
L=121 M+(SV, CO) Correct given that HR was constant due to betablockers.
M+(CVPM, /T) (Spurious) mult(HR, SV, CO) (Correct)
96
6.
oh> 900
r44.5
no 
AA
·
0)~r!I
I
1000
1500
2000
1000
2500
1500
t/sec
2000
2500
2000
2500
2000
2500
t/sec
1
x 104 O 0.5
0) 5 Jl
2000
t/sec
AI
1000
1500
2000
1000
2500
1500
t/sec
t/sec
r Ann 0Q 0 5000 aI
A
Jl
"1
UJ
0
An
1000
1500
2000
2500
4
i
t/sec
B
boo
1500 t/sec
Figure 628: Patient 4: Filtered Signals (L = 601)
L=601 mult(HR, ABPIM, RPP) (Correct) mnult(HR, SV, CO) (Correct)
100
6.1.5
Patient 5
The patient was a 66yearoldgentleman with a fairly long history of angina and a proven inferior infarct 3 months before the operation. He had very poor exercise tolerance, developing severe ischemia after very moderate exercise. His catheterization showed severe triple vessel disease with reasonably good left ventricular function. He was hypertensive and was treated with betablockers (Atenolol). He was also a noninsulin dependent diabetic.
The data segment was taken quite some time after the surgery had started. Before the period, the initial lightness of anesthesia caused a sharp rise in ABP from 90 mmHg systolic up to 160 mmHg systolic and that was sustained for several minutes. During the period, the depth of anesthesia (Enflurane) and analgesia (Alfentanil) and the dosage of GTN were
increased to bring the ABP back down. 80
·

1
2a.
m 60
C)
":
An
gl
Ill
_
'o0(
3500
0
4000
4500
30
t/sec I'll
Jtc
r
I.
I III
I
i/sec
_
et3 :s0oo0
I
4000
!
!

;·C
I
3500
X 104
3'000
4500
·
3500
t/sec 106
x
I
·
·
I
3000
3500
B
4000
4500
4000
4500
t/sec ·
5
>r, S
4000
4500
3000
t/sec
3500 t/sec
x 104
a Ec
0
 r
J _,^
C3
!
3000
3500
4000
4500
30
t/sec
t/sec
Figure 629: Patient 5: Original Signals. Note the relatively constant heart rate due to the effect of betablockers.
101
2a
A
60
50
3
0
40 300
3500
4000
4500
3'boo
t/sec
zir
1
3500
'1I IIIBI nn
7: .hh1
300
__
I
·
3500
4000
·
II
4500
3ooo00
3500
t/sec x 104
1
o0
4500
.
.
> 0.5 ll
I
30ooo
3500
4000
_
3600
4500
3500
t/sec
I
I VVV
.
a 6000
rnn _
...
4000
t/sec
1
cc
4500
> 500 iiri?
Jl
4000
t/sec
30
II
is\
300( 0
3500
4000
4500
4000
4500
t/sec
4000
2.2 4500
3000
t/sec
3500
t/sec
Figure 630: Patient 5: Filtered Signals (L = 61). Note that the trends of the relatively
constant heart rate are amplified.
L=61 M(CVPM,
AT) (Spurious)
M+(SV, CO) Correct given that HR was constant due to betablockers.
M+(CVPM, AT) (Spurious)
102
"I n.l
50
a0
n
A Jlm Fr
:
a;3 C 

3500
3o00
I
I
3500
4000
4500
4000
4500
4000
4500
4000
4500
^~~~~~
4000
3000
4500
t/sec ._
__
bV
I
t/sec 600
·
I 300(0
\
> 400
ii7
""
l
nnhl_
3500
4000
Jl I 1 LUUv
4500
3000
3500
t/sec
t/sec
x10 4
1
t.
oO 4
> 0.5
,
A
J
3000
3500
4000
4500
3v00
3500
t/sec 7000 0
 6000
mnn ..


t/sec w
\N \
L 2.2
UJ
JJ..J
300JU_
300( 0
3500
4000
4500
3000
t/sec
3500 t/sec
Figure 631: Patient 5: Filtered Signals (L = 121)
:L=121 .No constraints were obtained.
103
0u

·
·
·
·
I
Im
an 50 co
0
3000
3500
0
4000
4500


3500
3'b00
I.3000
> 400 4000
3500
V
onn I
4500
3oo000
·
·
3500
4000
tsec x104
1
·
0 0.5
0
I
3000
3500
4000
3'o00
4500
!
3500
4000
4500
4000
4500
Vsec
t/sec ·

 6000
w
(r I
4500
t/sec
04
7000
4500
600
60
a5 I 58
6
4000 t/sec
t/sec
2.2
0
II
VVUV
3000
3500
4000
4500
3000
3500
t/sec
t/sec
Figure 632: Patient 5: Filtered Signals (L = 241)
L=241 M+(SV, CO) Correct given that HR was constant due to betablockers.
mult(HR, SV, CO) (Correct)
104
I
f!
r
0
0 I_ 50s0
1000
1500
2000
o00
Iv
1500
2000
t/sec
n
·· ·
1000
t/sec 1000
> 500
50 A 00oo
n 5o 0
·
1000
1500
2000
·
1000
2000
1500
2000
1500
2000
tsec
Vsec
4
1500
104
1
0.5
2
I 0oo !
0
1500
1000
2000
1000
t/sec
t/sec _ n'' 1·04
a a13 1r
2
0. LU
.
I
vU 500
1000
1500
2000
I
_
goo
1000
t/sec
/sec
Figure 639: Patient 6, Segment 1: Filtered Signals (L = 241)
L=241 M(HR, SV) (Spurious)
invderiv(ABPM, RPP) (Spurious) invderiv(ABPM,
VC) (Spurious)
111
_nr
,....
Ivv
0
a. U
a
m
_
c,\ 'goo ..
1000
1500
500
2000
1000
1500
2000
1500
2000
1500
2000
1500
2000
t/sec
t/sec 100
4 ffnnn I .... Ivvv
50
u) 500 l
I
1000
goo
1500
500
2000
1000
t/sec
t/sec
X 104
4
1
0(D
o 0.5 n 0oo
1000
1500
2000
00oo
1000
t/sec fx
t/sec
10 4
e·
a a 1 n
2
A1L
500
q
1000
1500
2000
I
_
goo
t/sec
1000
t/sec
Figure 640: Patient 6, Segment 1: Filtered Signals (L = 361)
L=361 M + (HR, RPP) Both dropped because of increased depth of anesthesia.
mult(HR, ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct)
112
A
·4El nnlI
I UV
a.
>2
M
n,
...
0
__

oo
1000
1500
2000
0
1000
t/sec
1500
2000
t/sec
100 
1000
I 50
> 500 I
1000
o60
1500
1000
2000
Vsec
1500
2000
Vsec
4
10
I!
0
82
>0 0.5
III
II
oo00
1000
1500
2000
500
5
I
I
1000
1500
20 00
1500
2000
t/sec
t/sec
4 x o10
I
J
a.
1 0
I. C.,4·
.
500
I~~~~~~~~~~~~~~~~
1000
1500
oo
2000
t/sec
1000 i/sec
Figure 641: Patient 6, Segment 1: Filtered Signals (L = 481)
L=481 M + (HR, RPP) Both dropped because of increased depth of anesthesia. M+(ABPM,
AT) Both dropped because of vasodilating effect of GTN.
invderiv(ABPM, RPP) (Spurious) inv deriv(ABPM, VC) (Spurious) mult(HR, ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious)
mult(HR,CVPM,RPP) (Spurious) mult(HR, SV, CO) (Correct)
113
11Il1
4
_rr
I UU
L a
2
m
1
n
I, IJ90)
I
1000
1500
2000
o00
1000
100
2000
1500
2000
1500
2000
1500
2000
UU 
a:
I 50 I!
1500 t/sec
t/sec
> 400
206
I'
1500
1000
2000
1000
t/sec
t/sec 4 n x 10
4_ I
o 0.5t II I I
III
g00
1000
1500
00
2000
1000 t/sec
t/sec ,x 104 m
_
!a1
1
'
, I
500
1000
1500
2000
500oo
t/sec
1000
t/sec
Figure 642: Patient 6, Segment 1: Filtered Signals (L = 601)
L=601 t+ (ABPM, CO) Both dropped because of vasodilation and increased venous tone caused by increased GTN dosage.
M + (HR, RPP) Both dropped because of increased depth of anesthesia. M + (ABPM, AT) Both dropped because of vasodilating effect of GTN. IM+(CO,AT) Both dropped because of vasodilation and increased venous tone caused by increased GTN dosage.
invderiv(ABPM, RPP) (Spurious) invderiv(ABPM, VC) (Spurious)
mult(HR,ABPM, RPP) (Correct) mult(HR,CVPM,RPP) (Spurious) mult(HR, SV, CO) (Correct)
114
6.2.2
Segment 2
The increased GTN dosage successfully lowered ABP and improved ischemia. In this segment, GTN dosage was reduced. 80

r
> a211*uun~~unrIII ~~~~~~
m 60 rn
IL 4 L
I
2000oo
2500
"'fij
0D 3000
V~
t/sec
An
ii I i 500 An JfiJ
ZUL
I
10( I
Fiji
.1 1,..
)
APW
U
.. .. .
2500
I,.,~..
2000
3000
10
.1
.
3
0
>C 
2oo00
2500

ri
r
3000
'ooo
t/sec x 10 nI
C 2(500
  

,7 Onr11
2500 t/sec
30100
C 1. ,4TJ
II
3000
2500 t/sec
t/sec x
,L111~1
_
200UU
..
2( 000
00
3000
2500
b0oo
t/sec
LL. 2500 t/sec
1.
Ff
1
3000
4;nl
200
i 2500 t/sec
Figure 643: Patient 6, Segment 2: Original Signals
115
30(30
60
c
50 > AN
0 C
illI
,l'l I, ....
2500 t/sec
3000
o00
2500
3000
t/sec 1000
I
,,vv
szj7C
I 50
,I '" 2v00 2000 Il
2 15
_
2000
2.5
> 500 r A'00
%
2500 t/sec
!
3000
x 1(4
2500 t/sec
3000
2500 t/sec
3000
2500
3000
2
0U I
1 2000
I
"v
A
2500 t/sec
A
,I~1111E II II III
Q. 5000 2000
20C00
I 1.2
r
iii
[
2v
3000
L 1.1
w 
0
·
2500 t/sec
3000
4i
_
2b00
t/sec
Figure 644: Patient 6, Segment 2: Filtered Signals (L = 61)
L=61 mult(HR, SV, CO) (Correct)
116
vuv ...
__
·
~~~~~~~~~~~~~~~~~2.5
0EL
 50
2
O An

2000
.~~~~~~~~~~~
2500 t/sec
15
3000
ZO00
n,
I
I~
IUU
1000
50
> 500
'1
2500 t/sec
3000
2500 t/sec
3000
2500
3000
A
00
2500 t/sec
2O 90 20(
3000
Ax 104
2
4 r
01 I
o O22 i i !
II
2d00
2500
2000
30i 00
t/sec
t/sec
100( U cc

5 00 
,
1 .0
 1.1
\
2000
2500
30100
12_ 20( 0O
t/sec
2500
t/sec
Figure 645: Patient 6, Segment 2: Filtered Signals (L = 121)
L=121 M+ (HR, RPP) Correct given relatively constant ABPM.
mult(HR, ABPM, RPP) (Correct)
117
3000
hi
2.5
f
a
50 500 nIl _
'200oo
2500
3000
2600
/sec A
/sec
x 10 4
_.__ i
1.5
02
0
c
1 n l
I!
2500 t/sec
2000
I_
3UUU
3000
t/sec
I
.
L 0r
w
2000
2500 t/sec
~~~~~~~~~~~~~~~~~~~~~ 2500 30 00
ooo
3000
\1
1.1 1!

2b00
. 2500 t/sec
Figure 646: Patient 6, Segment 2: Filtered Signals (L = 241)
L=241 deriv(SV, CO) (Spurious)
118
3000
> 'L Cl
2.5
2
50 O
2500 t/sec
2000
3000
I 5, '00oo
3000
t/sec hl\h
60
I UUU
I 40\
> 500 r%
~Ct
2500 t/sec
3000
20(30
00 2
0 
2i 00
600( II=
1
n 1 ,
.
2500 t/sec
:oo
3000
2500
 1.2
0 400(
r
* 1.1



w
0 2500
3000
t/sec 
200C I 2( 00
3000
m · .1
1
_
_
2500 t/sec
X 10 4 ,__ _
.,
2500
3000
1 !
2b00
t/sec

~
2500 t/sec
Figure 647: Patient 6, Segment 2: Filtered Signals (L = 361)
L=361 'mult(HR, SV, CO) (Correct)
119

3000
I
J
> a
co 50 It
2
0o
cl
_,
B
0oo
2500 t/sec
'oo
3000
.4I nfrf .....
n 40 T'
c 500
_
2500 t/sec
n 2v0(00
3000
21
1 .,
2500 t/sec
2C)00
I
n0
4000
n
l~~~~~~~~~~~~~~~~~~~
2500 t/sec
3000
I"
oo u2U000
3000
l~~~~~~~~~~~~~~~~
2000
3000
1.5
0
o 2
2000 
2500
t/sec
x 104
6000 
3000
l B
B
2VOI 00
2500
t/sec
60
on
t !
4
i
2500 t/sec
3000
2500 t/sec
3000
I
1.1 .4 B
2b00
Figure 648: Patient 6, Segment 2: Filtered Signals (L = 481)
L=481 nM+ (HR, VC) Both were relatively steady. deriv(SV, CO) (Spurious)
120
rr
55
~~ ~
~
~
~
~
~
~
~
~
2.
am 50
2
E >
_
AC
2000
_ _
_
_
_ _
_
0 4
_
2500 t/sec
I__
3000
'200
2500 t/sec
3000
2500 t/sec
3000
2500 t/sec
3000
2500 t/sec
3000
f'n

Du
I UW0
> 500
:40 on 00
2500 t/sec
n 2'000
3000
1
x 104 j2
O
_.
co2
A>r
2000
2500 t/sec
u60o
3000
 nx
a.
a 4000

1= n
1.1
LU 2500 t/sec
2000
3000
2b00
Figure 649: Patient 6, Segment 2: Filtered Signals (L = 601)
L=601 M+ (HR, RPP) Both continued to drop because of increased depth of anesthesia. 1M(ABPM, HR) ABPM started to rise moderately following decreased GTN dosage. M (ABPM, RPP)
ABPM started to rise moderately following decreased GTN dosage.
deriv(SV, CO) (Spurious)
mult(HR, ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious)
mult(HR,CVPM,RPP) (Spurious) mult(HR, CVPM, VC) (Spurious)
121
6.2.3
Segment 3
The patient condition was stabilized. Depth of anesthesia was decreased. This period was uneventful. QU
a.
0
4
1 .
a.
70
0 ,imnflM]bhII
iJlnlyrJlj\/i*N*J/JC/C 3500 i/sec
0ooo
4000
3500
3bl00
4000
t/sec Annr.
20
UUUVV
100
> 2000 ~.L. _..L...,. =.. _ n II,'
n1111 3500 i/sec
3V00
4000
3'600
.
AL. J AM
. ~,~. ~.A.
3500 t/sec
40100
x 105
4I
0
02 A,
tr~~~~ulr~~A11 0
b
3500
4000
3'000
t/sec 4
I'
4
4
I.

a.
rI 3ooo00
I.
I
I
6.1
w0
rd 3500
3500
40100
t/sec
X 104 n
Ir
.I..

3600
a 1I
I
4000
I
'3600
t/sec
3500
t/sec
Figure 650: Patient 6, Segment 3: Original Signals
122
40100
r
ml I li
2.
r
Z.b
2
>
2
_
4
3500 t/sec
'300
4000
1I nn
3500 t/sec
4000
3500 t/sec
4000
3500
4000
nnnrr
I UU
Iv I 50T"',
> 500 n
I m _
I! I'
3500 t/sec
3v000
4000
a300
104
1U I
0o
1
_
5
o 0.5 
Il I
30(00
II ll l VVVV
c 5000 CC Ai
S~~~~~~~~~~
3500 t/sec
II
4000
t/sec
r
3r00
3v000
1. I r H
I
0 3500 t/sec
4000
1 1 =
3000
3500
t/sec
Figure 651: Patient 6, Segment 3: Filtered Signals (L = 61)
L=61 mult(HR, ABPjM, RPP) (Correct) rmult(HR, SV, CO) (Correct)
123
4000
Fij
20 I
a 54 &J in,
2
01I
!
3500
0oo
4000
.
'3000
4
100
V
> 500
n1
I! 3ooo00
*.l lI
3500
3oo00
4000
tsec
4000
1
o 0.5
5
n
I
remN I
I!
3Vooo
3500
4000
3oo00
t/sec
3500
4000
t/sec
Iv 100CEn
a. a
3500
t/sec
x 104 1u I
o
4000
nn
1II I
50
0
3500
t/sec
t/sec
4,..
ro
I
500)0
1
L
3'6000
3500
4000
t/sec
1.1 =
3000
3500
t/sec
Figure 652: Patient 6, Segment 3: Filtered Signals (L = 121)
L=121 M (CVPM, AT) (Spurious) M + (HR, RPP)
Both rose because of decreased depth of anesthesia.
mult(HR,ABPM, RPP) (Correct) mult(HR, CVPM, RPP) (Spurious) mult(HR, SV, CO) (Correct)
124
4000
11~~~~~~~~nL
"I
00
 p
t.0
54 A6
'300
4000
'3000
i/sec
3500
4000
i/sec
~r
I1uuu
u
50

I
1 ri
3500
> 500 A
n
3'000
3500
4000
300
t/sec
3500
4000
t/sec
x 104
4
IV
o 0.5
05 'U
n
n
3000
3500
4000
3V000
t/sec
3500
I
4000
t/sec
4I AAAA

0V
I
f 5000 A
A
2 0
3000
3500
4000
3000
t/sec
3500
t/sec
Figure 653: Patient 6, Segment 3: Filtered Signals (L = 241)
L=241 M + (HR, CO) Both rose because of decreased depth of anesthesia. M+(RPP, VC) Both rose because of decreased depth of anesthesia.
mult(HR, ABPM, RPP) (Correct) mult(HR, CVPM, RPP) (Spurious) mnult(HR, CVPM, VC) (Spurious) mult(HR, SV, CO) (Correct)
125
4000
_IIA
r.I.
53 ~
>.
'
rI
'3000
1 no u
C,
Ui
n X 10
4000
_
*Il
3500 t/sec
3500 t/sec
50,
I
·
4000
o
n rooo

2
0 3500 t/sec
3000

I
I
r I3
I
50 D
40100
3 30oo
3500 /sec
4000
4
1
0
>0.5
(D 5
I!I
300oo
3500 t/sec
40100
i
3'000
3500 t/sec
4000
3500
4000
II nnn on 1I Ilil uuuq
a. 0
)
I2r 0/l
500( D
III
i
I
ri
i1
33000
3500
40 DO
3000
t/sec
t/sec
Figure 654: Patient 6, Segment 3: Filtered Signals (L = 361)
L=361 M+(CO, RPP) Both rose because of decreased depth of anesthesia. AM+(HR, CO) Both rose because of decreased depth of anesthesia. M + (HR, RPP)
Both rose because of decreased depth of anesthesia.
mult(HR, ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious) mult(HR, CVPM, RPP) (Spurious) mult(HR, CVPM, VC) (Spurious) mult(HR, SV, CO) (Correct)
126
r
2.EIh
f a ll.
r m
53
I
_
ob00 rrrI I
2
i
3500 t/sec
4000
U '3000
> 500
50 A\
_
300
3500
4000
03_
'00
tsec
I
3500
40 00
t/sec
.n_ 0(D
4000
1000
IrV
··· .11
i
3500
t/sec
I JUU
I
02a. 1 r, _
X 104
1
0 0.5
0
n
i I
300oo
3500
40100
3ooo00
3500
4000
t/sec
t/sec 10000
a
5000
0. 5000 I= nIl
_
3000
w2 3500
] 4C100
o
3000
t/sec
3500
t/sec
Figure 655: Patient 6, Segment 3: Filtered Signals (L = 481)
L=481 M+(HR, CO) Both rose because of decreased depth of anesthesia.
AlM+ (HR, RPP) M+(HR, SV) M+(HR, VC)
M+(CO,RPP) M+(CO,VC) M+(RPP, VC) M+(SV, CO)
M+(SV,RPP) M+(SV, VC)
mult(HR,ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious)
mult(HR,CVPM,RPP) (Spurious) mult(HR, CVPM, VC) (Spurious) 127
4000
rnult(HR, SV, CO) (Correct)
128
,Fat
I
W
11

a.
53
0 ;R
4
L
%3oo
l
3500
f
4000
3500
t/sec 4 IIIIAnn
100
Ir
50
_
C,,
0o
30 00
3500
500 I
n 30oo
4000
t/sec
3500
4000
/sec
0I X 104
1

JrI
o
4000
t/sec
0. 5 O
3000
A
3500
3'00
40(20
t/sec
3500
4000
t/sec
4I nnnn .. .. .I.
Jl
a aC 5000
2I D n' ..v I
I I=
3o00
3500
3000
40( 20
t/sec Figure 656: Patient
6, Segment
3:
3500 t/sec
Filtered Signals (L = 601)
L=601 A+ (CO, RPP)
Both rose because of decreased depth of anesthesia.
Ml+(CO, VC) ATM+(HR,CO)
M+(HR,RPP) MI+(HR, SV)
M+(HR, VC) M+(RPP, VC) M+(SV, CO)
M+(SV, RPP) M+(SV, VC)
mult(HR, ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious) mult(HR, CVPM, RPP) (Spurious) mult(HR, CVPM, VC) (Spurious) 129
4000
rnult(HR, SV, C'O) (Correct)
130
6.2.4
Segment 4
The patient condition was steady. This period was uneventful. 200
10C I I
am
l
> 100
50
0,
I
J
4000
4500
5000
4o00
,n
J
,
t/sec
A I
4500
. .
5000
rnn ·. '
I
4V00
Ix
5000
4500
/sec
/sec
105
J
0C. 0
.4
_c, 40oo
0 o or
 .r 
rlr
. w .n
.
I
A I
5000
Ir
I500
rr
t.~'~
· · ·· vUUU
I 10o
nL a 1
,,,
t/sec
20CII
Z
A R
4500
4500 t/sec
5000
4oo
4500
5000
i/sec
x 1o04 I
W1~tZ
4000
4500
I
5 1
w
0
A
VUoo
5000
4500
i/sec
i/sec
Figure 657: Patient 6, Segment 4: Original Signals
131
5000
60I nI IF

E
5
2C 4( I ~0boo
10TV · [
·
Jt
I
l

4500 t/sec
5000
4000

1UUU

500 A
5000
4u00v 4000
I
4500
5000
t/sec
I
Iv
o
4500 t/sec
X 104
I
0
5000
t/sec
50 >1' 4000
4500
1 r

5 I'O. n
u _ 4000
x
rr
0
mr
4500 t/sec
4000
5000
nn
III II
5000
4500 t/sec
5000
4500 t/sec
5000
J
V
i1 w
a
I I.
40000
4500 t/sec
5000
4000
Figure 658: Patient 6, Segment 4: Filtered Signals (L = 61)
L=61 nmult(HR, ABP,
RPP) (Correct)
mult(HR, SV, CO) (Correct)
132
In nJ
10
.1 X 0
Q 40 m
L
I'l '4000 ,..l
5
o 4500
5000
4'000
t/sec
5000
t/sec 1000
10 I I JV
a
Lo 500
VIj
,I
·I
4500
'4.000
5000
4 00 4000
t/sec x10
0o
1
0 0.5
5
nJJ J
4000
4500
5000
W l_

4Vo00
t/sec nnn JJ JJ
5000
4
f
.J
4500
t/sec
lU
Er a:
4500
4500
5000
t/sec
_

2
6000 A
~J
nn JJ JJ
0 I.
4000
4500 t/sec
n
4000
5000
4500
t/sec
Figure 659: Patient 6, Segment 4: Filtered Signals (L = 121)
L=121 mult(HR, ABP.M, RPP) (Correct) mult(HR, SV, CO) (Correct)
133
5000
nf, ~E
1
a
03 50
0
Ar! oo00
4500
5000
5v
5 n 4v000
t/sec
j
OU
r70 70
4000
4500
5000
A,f~.
0
50 0 Ai1
4500
4000
5000
t/sec
t/sec
4 x 1C0
I
5000
100
lI
c,\
I,, UV
4500 t/sec
h'1
1
I I
0 !
Q
A
'L L_~
·
v 4000
4500
·  i4 _
5000
4000 boo
t/sec anrar _ III II
AAn 4000
5000
4500
5000
2
~I
a 6000
4500 t/sec
U
4500
4000
5000
4000
t/sec
t/sec
Figure 660: Patient 6, Segment 4: Filtered Signals (L = 241)
L=241 invderiv(ABPM, RPP) (Spurious) mnult(HR, SV, CO) (Correct)
134
55

2a
rn50
4
0
Ar
DO 401 Gu
5000
4500 t/sec
9
4000
5000
t/sec 1000

500
>
70
ni
40400
4500
5000
I
4600
t/sec b
4500
4500
5000
/sec
x 104 m
1
_
0 Or n_
· 4
i
4vo00
4500
5000
_
4000
t/sec 
cc
1.'rI
I
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Figure 661: Patient 6, Segment 4: Filtered Signals (L = 361)
L=361 No constraints were obtained.
135
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Figure 662: Patient 6, Segment 4: Filtered Signals (L = 481)
L=481 M+(ABPM, SV) Both decreased overall.
M+(RPP, VC) (Spurious)
136
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Figure 663: Patient 6, Segment 4: Filtered Signals (L = 601)
L=601 M (ABPM, CO) Both decreased overall. AI (ABPM, SV) Both decreased overall. M+ (SV, CO) Both decreased overall.
mult(HR, ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct)
137
5000
6.2.5
Segment 5
The patient experienced low ABP post bypass due to poor cardiac performance secondary to a technically poor graft and possibly hypovolemia. Inotropic therapy (Dobutamine) was given which potentially caused the patient to develop myocardial ischemia. Blood infusion was performed to bring ABP back up. 100
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138
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Figure 665: Patient 6, Segment 5: Filtered Signals (L = 61)
L=61 mnult(HR, ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct)
139
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Figure 666: Patient 6, Segment 5: Filtered Signals (L = 121)
LJ=121 mult(HR, SV, CO) (Correct)
140
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Figure 667: Patient 6, Segment 5: Filtered Signals (L = 241)
L=241 M + (ABPM, CO) Both dropped initially because of poor cardiac performance and hypovolemia, and started to rise following blood infusion.
M+(ABPM, SV) Both dropped initially because of poor cardiac performance and hypovolemia, and started to rise following blood infusion.
M+(SV, CO) mult(HR, ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct)
141
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L=361 M+(ABPM,
CO() Both dropped initially because of poor cardiac performance and hypov
olemia, and started to rise following blood infusion.
142
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L=481 No constraints were obtained.
143
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Figure 670: Patient 6, Segment 5: Filtered Signals (L = 601)
L=601 A/+ (SV, CO) Both dropped because of hypovolemia. MA(HR, CO) HR increased both as a compensatory response to decreasing CO due to hypovolemia, and as a response to inotropic therapy.
A1(HR, SV) invderiv(SV,
CO) (Spurious)
mult(HR, ABPiM, RPP) (Correct) mult(HR, SV, CO) (Correct)
144
6.2.6
Segment 6
ABP gradually rose as a result of switching to another inotrope (Adrenaline) and blood infusion. 80
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Figure 672: Patient 6, Segment 6: Filtered Signals (L = 61)
L=61 mult(HR, SV, CO) (Correct)
146
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L=121 mult(HR,ABPM, RPP) (Correct) mult(HR, SV, CO) (Correct)
147
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Figure 674: Patient 6, Segment 6: Filtered Signals (L = 241)
L=241 mult(HR,ABPM, RPP) (Correct)
148
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Figure 675: Patient 6, Segment 6: Filtered Signals (L = 361)
L=361 M+(SV, CO) Both rose after blood infusion.
deriv(SV, CO) (Spurious) rmult(HR, SV, CO) (Correct)
149
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Figure 676: Patient 6, Segment 6: Filtered Signals (L = 481)
L=481 M+(SV, CO) Both rose following blood infusion.
deriv(SV, CO) (Spurious)
mult(HR,ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious)
150
3000
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t/sec
Figure 677: Patient 6, Segment 6: Filtered Signals (L = 601)
L=601 M(HR,
AT) HR decreased following blood infusion. Lagging effect of vasoconstriction
caused AT to increase. M (HR, SV) HR decreased and SV increased following blood infusion.
M+(SV, AT) deriv(ABPM, RPP) (Spurious) deriv(ABPM, VC) (Spurious) deriv(SV, CO) (Spurious)
mult(HR, ABPM, RPP) (Correct) mult(HR, ABPM, VC) (Spurious) mult(HR, CVPM, VC) (Spurious) mult(HR, SV, CO) (Correct)
151
Chapter 7
Discussion The goal of this thesis is to develop a system for learning qualitative models from physiological signals. In Chapter 1 we mentioned two potential applications for such a system. First, the system could be a useful tool for the knowledge acquisition task of inducing useful models from vast amounts of data. Second, the system could be incorporated into an intelligent patient monitoring system to perform adaptive model construction for diagnosis in a dynamic environment. In this chapter, we evaluate the performance of the system in knowledge acquisition and identify sources of error. We then examine its applicability in diagnostic patient monitoring.
7.1
Validity of Models Learned
The results in Chapter 6 show that reasonable qualitative models can be learned from raw clinical data. Model constraints in our "gold standard" model constructed in Section 4.2 and other useful constraints showed up repeatedly in the models learned from our clinical data. These include:
* constraints valid in general such as mult(HR, SV, CO) and mult(HR, ABPM, RPP). * constraints valid in specific patient conditions, possibly representing compensatory
mechanisms, such as M (HR, CO) and M (CO, AT) in hypovolemia. * constraints valid under the effect of certain drugs. For example, M+ (SV, CO) showed
up in patients on betablockers because of their steady heart rate, and M+ (ABPM, AT) showed up in patients with an increased dosage of GTN causing vasodilation. 152
7.1.1
Model Variation Across Time
As discussed in Chapter 3, GENMODEL learns a qualitative model by creating an initial search space of all possible QSIM constraints, and successively pruning inconsistent con
straints upon each given system state. Therefore if the system changes within the modeling period (in our case 16.7 minutes) resulting in a different underlying model, neither the old model nor the new model may be obtained. Constraints in the old model are pruned be
cause they are inconsistent with the states after the system change. Constraints in the new model are pruned before the system change because they are inconsistent with the previous system. This may explain cases when we obtain very few or no model constraints.
For
example, if a patient is previously stable with an increasing relationship between the heart
rate (HR) and the cardiac output (CO):
M+(HR, CO) but develops hypovolemia in the middle of a modeling process, resulting in a decreasing
cardiac output and a compensatory mechanism involvingan increasing heart rate, the new valid constraint is:
M(HR, CO) But this will not appear in the final model because at the onset of hypovolemia, this constraint has already been pruned by GENMODEL according to states corresponding to the
previously stable condition. Furthermore, the previously valid constraint M+(HR, CO) will be pruned because it is now inconsistent with the system states corresponding to hypovolemia.
7.1.2
Model Variation Across Different Levels of Temporal Abstraction
The models learned in Chapter 6 varied across different levels of temporal abstraction, represented by the filter length L. For example, constraints which involve the skintocore temperature gradient AT representing the level of vasoconstriction in the body, generally appeared only under large values of L, i.e. in coarser time scales. This means the response of AT generally lags behind the responses of other parameters.
This may be due
to the considerable heat capacity of the body causing a delay in the measurable effects of
153
vasoconstriction. In general, we observe that fewer model constraints were learned with decreasing L or finer time scales. This can be due to the following reasons: * As discussed in Section 5.2.4, smaller values of L and therefore smaller values of a correspond to larger cutoff frequencies in the lowpass Gaussian filters, and larger band
widths in the bandpass filtering operation equivalent to the cascade of the Gaussian filter with the differentiator. This reduces the amount of noise rejection achieved, and results in noise sensitivity problems in detecting zero crossing points and therefore less accurate segmentation. This additional amount of noise may have caused correct constraints to be pruned, resulting in fewer or even no constraints left in the final model. * Smaller values of L correspond to faster processes which may have more dynamic models. As discussed in Section 7.1.1, a system change within a modeling period can
cause constraints belonging to both the previous and the current model to be pruned, resulting in a smaller model or even one with no constraints.
7.1.3
Model Variation Across Different Levels of Fault Tolerance
We observe that in general the size of the model learned increases with increasing levels of fault tolerance. A fault tolerance level of q7means that GENMODEL allows for inconsistent states up to a fraction a constraint.
of the total number of states in the system behavior before pruning
Therefore with larger 7, fewer constraints will be pruned and the resulting
model will contain more constraints. An indication of r7being set too high is that conflicting constraints start to appear. For example, in Patient 5 with L = 61 (Figure 630), both
M+(CVPM, AT) and
M(CVPM, AT) appear in the model learned. This is because both CVPM and AT are relatively steady and contain only few inc and dec segments which distinguish between the M+ and M154
constraints. Within a high level of tolerance, the distinction is obscured.
7.1.4
Sources of Error
False Positives In the models learned, we observe that spurious constraints sometimes appeared in the resulting model. For example, in Patient 1 (L = 601), we obtained the spurious constraint:
inv.deriv(ABPM,RPP) This can be due to several possible reasons: * The waveforms are relatively smooth with few critical points. This results in a system behavior with few states, corresponding to few examples for learning.
Now it is
relatively probable that these examples are consistent with the incorrect constraint. For instance, if whenever ABPM decreases, RPP is positive, then the above incorrect invderiv
constraint will be learned.
* The level of fault tolerance is set too high resulting in incorrect constraints not being
pruned. False Negatives We observe that even constraints that are generally valid in all conditions, such as
mult(HR, SV, CO) did not appear in every model learned. There are several possible reasons for this: * Since few states are available in the data segment resulting in few examples for learning, if these examples are corrupted by noise, the correct constraint will be pruned. * Values corrupted by noise are recorded as corresponding values by the system. This problem is explained further in Section 8.1.1. * The level of fault tolerance is set too low resulting in correct constraints being pruned.
155
Landmark Values Temporal abstraction refers to how close two times have to be before we label them as the same distinguished time point. Similarly we have to decide how close two function values have to be before we label them as the same landmark value. If the tolerance is set too low, we may amplify trends of relatively steady signals. This is the case in the heart rate signals of Patients 4 and 5 (Figures 623 and 630) which are relatively steady due to the effect of betablockers. The fluctuations within 23 beats per minute are amplified into a series of inc (increasing) and dec (decreasing) segments. The whole segment might well have been labeled as std (steady) if we had set the tolerance appropriately.
7.2
Applicability in Diagnostic Patient Monitoring
From the models shown in Chapter 6, we see that constraints of models learned do track changes in patient condition over time. For example, the following changes were tracked:
Compensatory mechanisms during shock e.g. the constraints M(HR, CO) and M (CO, AT) learned when the patient experienced hypovolemia. Effects of drugs e.g. the constraint M+ (SV, CO) tracked the effect of betablockers because of the patient's relatively constant heart rate, and the constraint M+ (ABPM, AT) tracked the effect of an increased dosage of GTN causing vasodilation.
7.2.1
A LearningBased Approach to Diagnostic Patient Monitoring
Since model constraints learned track patient condition over time, we might be able to build a diagnostic patient monitoring system based on our learning system. The patient
monitoring system continually learns models from patient data and detects changes in the models learned. Diagnoses are made based on these changes. This learningbased approach to diagnostic patient monitoring is summarized in Figure 71. The traditional historybased approach to diagnostic patient monitoring goes in the opposite direction.
It generates histories based on different models. These histories are
matched with the patient data. Diagnoses are based on models corresponding to histories
that best match the patient data. This approach can be achieved by a hypothesizetestrefine cycle as shown in Figure 71 [6].
156
Learningbased
Historybased Approach:
Approach: Diagnosis
Refine
Figure 71: Two approaches to diagnostic patient monitoring. In the learningbased approach, models are continually learned from the patient data. In the historybased approach,
a hypothesizetestrefine cycle is used to generate models that best match the patient data. In each approach, diagnoses are made based on the current model. The learningbased approach may be more efficient since the hypothesis model is generated directly from the patient data and there is no need for a hypothesizetestrefine cycle.
7.2.2
Generating Diagnoses Based on Models Learned
We need to be careful in making diagnoses based on models learned. The same model constraint can correspond to different diagnoses in different contexts. For example,
* The constraint M + (HR, CO) can correspond to a healthy person performing physical exercise, when the rise in heart rate increases the cardiac output. It can also correspond to a hypovolemic condition when the compensatory effects of increased heart rate, increased myocardial contractility and vasoconstriction successfully increase the cardiac output, but the patient is still in hypovolemia. * The constraint M (HR, CO) can correspond to a person in hypovolemic shock when the increasing heart rate is incapable of compensating for the decreasing cardiac output. The same constraint can also be obtained when blood infusion is performed in response to the hypovolemia. In this case, the cardiac output will increase because of
157
the blood infusion, and the heart rate will decrease to normal. Therefore different patient conditions may result in the same model constraint. However, this may not be a problem for applying the learning approach to diagnostic patient monitoring for the following reasons: * Most changes in patient condition will involve changes in many parameters and in
multiple constraints. * If only few parameters are monitored, a semiquantitative representation may be used. This may allow more fine grained distinctions between patient conditions.
158
Chapter 8
Conclusion and Future Work This thesis describes work in learning qualitative models from clinical data corrupted with
noise and artifacts. The learning system consists of three parts: Frontend Processing removes artifacts and smooths the signals according to the level of temporal abstraction desired. Segmenter segments the signals at critical points producing a qualitative behavior of the system. GENMODEL
learns a qualitative model from the qualitative behavior.
We tested the system on clinical data obtained from six patients during cardiac bypass surgery. Useful model constraints were obtained, representing the following categories of information:
General Physiological Knowledge e.g. mult(HR, SV, CO), mult(HR, ABPM, RPP). Compensatory Mechanisms During Shock e.g. M(HR, CO) and M(CO, AT) representing the mechanisms of tachycardia and vasoconstriction to compensate for a hypovolemic shock.
Effects of Drugs e.g. M+(SV, CO) representing the effect of betablockers producing a steady heart rate, and M+(ABPM, AT) representing the effect of vasodilators such as GTN causing vasodilation which decreases both the arterial blood pressure and the skintocore temperature gradient.
159
The learning system might be used in a diagnostic patient monitoring system to learn models from patient data. Based on these models, diagnoses of the patient condition can be made. This provides a learningbased approach to diagnostic patient monitoring, which is different from the traditional historybased approach. The learningbased approach may be more efficient because it avoids the hypothesizetestrefine cycle.
8.1
Future Work
Promising directions for future work include:
Frontend Processing * Investigate wavelet transforms for multiresolution signal analysis in achieving different levels of temporal abstraction in the resulting qualitative behavior [35].
GENMODEL * Investigate more robust ways of dealing with data corrupted by artifacts, e.g. examine the applicability of results in PAC learning from data corrupted with
attribute noise. * Investigate methods of using corresponding value information in noisy data. (See Section 8.1.1.)
Patient Monitoring * Investigate more precisely the relationship between levels of temporal abstraction and delays in physiological responses. For example, the system can be tested in the domain of pharmacokinetics to relate the level of temporal abstraction at which appropriate constraints are learned to the onset of action of different drugs. * Investigate the context information necessary for accurate diagnoses based on learned models.
8.1.1
Using Corresponding Values in Noisy Learning Data
In Section 3.4.4 we mentioned that since we are learning from noisy signals, we have to allow for errors in the qualitative state history produced from the signals by the frontend 160
processor and the segmenter.
We accommodated this noise in the qualitative behavior
by introducing fault tolerance into GENMODEL. Specifically, we set a noise level r7 to a
fraction of the total number of qualitative states, such that a constraint is not pruned until it has failed for this many states. The noisy learning data introduce another difficulty in GENMODEL  that of locating corresponding values. When two or more functions reach landmark values in the same qualitative state, these landmark values are noted to be corresponding values. However, if the learning data is noisy, we do not know which states are valid and which are not. If we continue to label all landmark values which coexist in the same state as corresponding values, we will come up with a number of incorrect corresponding value sets. These incorrect sets will consistently cause constraints that check for corresponding values (i.e. M + , M, add, mult) to fail and ultimately be pruned, even if they are valid constraints. This is in fact the approach currently taken in our learning system, and can be regarded as a source of error. One simple solution to this problem is to give up corresponding value checking and simply rely on directionofchange consistency for checking monotonic constraints.
Without
the magnitude information provided by the corresponding values, a certain level of performance can still be achieved, as in attempts of Bratko et al [4] and Varsek [38] in learning the Utube model. However, omitting corresponding value checking results in much weaker forms of qualitative constraints, which no longer correspond to Kuipers' original definitions in [20]. For example, without corresponding values, the monotonic constraints M + and .A1 no longer require a onetoone mapping between values of their function arguments, as they do before (described in Section 2.1). Now if we have M + (a, b), if a increases steeply to the same value a(1l) in two different states, b can increase steeply to a large value b(l) in one state, and increase only slightly to a small value b(2) in the other state, as long as it increases at all. Magnitudes no longer need to correspond. (Figure 81)
Using landmark intervals to represent corresponding value information in noisy
learning data In the previous section we mentioned that exact corresponding values may be difficult to obtain from noisy learning data because we do not know which states are valid and which are not. Here we propose using landmark value intervals together with landmark values to allow 161
a(1
ne
I
b(1
b(2 t(O)
t(1l)
t(2)
t(3)
t(4)
Time
Figure 81: The two functions a and b shown obey directionofchange consistency for the
qualitative constraint M+(a, b), but not magnitude consistency. for an error tolerance in the corresponding values obtained. We show that this is analogous
to using a boundingenvelopeto represent a monotonic function in semiquantitative methods [22, 16].
Error tolerance in corresponding values can be achieved by allowing a point to cor
respond to a region which represents an error band within which the true corresponding landmark value may exist. A landmark value a(ti) corresponds to a closed landmark value interval [b(t2 ), b(t3 )] if and only if there exists a corresponding pair between the landmark value a(ti) and a value within the closed interval [b(t2 ), b(t 3)]. Furthermore, we assume that the noise in the learning data is of a reasonable magnitude, such that a value can be mislabelled by a range of at most one landmark value. Therefore, a landmark value interval can extend by at most two landmark values (one landmark value
greater than and one less than the true value). Searching for corresponding sets is similar to before. We proceed down the qualitative state history, looking for landmark values reached in the same state.
But now, instead
of simply recording each set of values, we check first to see if one of the corresponding values already exists within a corresponding set. If this is the case, and if the other new corresponding value is adjacent to the existing corresponding value in the landmark list, then we extend the existing value into an interval to include the new value. However, if the
162
new corresponding value differs from the existing corresponding value by more than one landmark value, we can conclude that a onetoone mapping does not exist between the two functions. Therefore, we can prune all monotonic constraints (M+ and M) between the two functions, and form a new set of corresponding values for constraint checking with add and mult, since add and mult do not require a onetoone mapping between values of its argument functions. For example, if we have corresponding values (pl, qi), and we come across the state
QS(p,t3) = QS(q,t3) = < q2, std > Then if q and q2 are adjacent to each other in the landmark list, then we have the new
corresponding pair (P1, [ql, q2]) assuming q < q2.
However, if ql and q2 are not adjacent to each other in the landmark list, then we can
prune the followingconstraints M+ (p, q) M (p, q) and form the new corresponding value set (pl, q2, .. )
for constraint checking with add and mult. New constraint rules can be added to accommodate intervals in corresponding pairs. For example, a rule similar to Kuipers' Proposition B.1 in [20] can be added as follows: Suppose M + (f, g), with corresponding values (p, [ql, q2]), and
QS(f, t1 , t2 ) = < (p,p'), dec >, QS(g, t1 , t 2 ) = < (q, q'), dec >, where q E [ql, q2] Then one of the following two possibilities must be true at t 2: (1)
f(t 2 ) =p, g(t2) E [ql, q2];
(2) f(t2) > p, g(t2) > q1. 163
b(t)
estimated function upper envelope

boundary
/
lower envelope
,
boundary
b(l). b(O)
a(O)
a(t)
Corresponding value pair: (a(O),[b(O),b(1)])
Figure 82: Incorporating intervals into corresponding value sets is analogous to adding a bounding envelope to monotonic functions. Incorporating intervals into corresponding values is analogous to bounding monotonic functions in semiquantitative methods [22, 16]. The bounding envelope represents a region which bounds all real behaviors of the given system with a certain probability p. Therefore if we have M+(a, b), i.e. a = f(b) where f(x) is a monotonically increasing function, then if we know at t(O) a is at a(O), we can be reasonably sure (with confidence p) that b lies within the envelope of f(a(O)). This is analogous to the corresponding value pair (a(O), [b(O),b(l)]) where [b(O),b(l)] represents the envelope of f(a(O)) (Figure 82).
164
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