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Induction of Qualitative Trees

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### Induction of Qualitative Trees

Dorian Šuc and Ivan Bratko

AI Lab

Faculty of Computer and Information Sc.

University of Ljubljana, Slovenia

Overview

- Discovering qualitative relations in numerical data
- Qualitative trees
- The QUIN algorithm
- Experiments with QUIN
- Application of QUIN to skill reconstruction in systems control

Qualitative vs. quantitative models

- Less detailed than quantitative modes
- Often easier to understand

- Abstractions of numerical models:
- numerical values qualitative values
- real functions qualitative constraints

- Our motivation: application in reconstruction
- of control skill (behavioral cloning)

- Applied to control of crane, acrobot, bike, ...

Example: behaviour of gas

Quantitative law:

Pressure * Volume / Temperature = const.

Qualitative law expressed by QCF:

Pressure = M+,-(Temperature, Volume)

Program QUIN QUalitative INduction

Numerical examples

QUIN

Qualitative tree

Qualitative tree: similar to decision tree,

qualitative constraints in leaves

Induced qualitative tree for functionz=x2-y2

x

> 0

£

0

y

y

> 0

> 0

£

£

0

0

-,+

-,-

+,+

+,-

z=

(

x,y)

z=

(

x,y)

z=

(

x

y)

z=

(

x,y)

M

M

M

,

M

Z monotonically increasing with X and

monotonically decreasing with Y

Qualitatively Constrained Functions, QCF

Ms1, ..., sm: R m --> R, si= + or -

Signs si indicate directions of change:

If si = + then:

function monotonically increases in i-th attribute

Function “positively related” to i-th attr.

si = -: function “negatively related” to i-th att.

QCF consistency with examples

- Each pair of examples (e,f) defines a qualitative change vector q with respect to no-change threshold
- A QCF is consistent with (e,f) if QCF permits q

QCF ambiguity

- A QCF may be consistent with qualitative change vector q and ambiguous w.r.t. q
- QCF is ambiguous w.r.t. q if QCF also permits other qualitative changes in class then those in q

Error-cost of QCF

Weighted by proximity of concerned examples

- Error-cost of a QCF w.r.t. an example set defined as weighted encoding length
- Error-cost of a QCF considers:
encoding of QCF

+ encoding of inconsistent predictions by QCF

+ encoding of ambiguous predictions by QCF

Outline of QUIN algorithm

- Top-down greedy algorithm to induce qualitative tree
- For every possible split, find the “most consistent” QCF (min. error-cost) for each subset of examples
- Select the best split according to MDL

Heuristics in QUIN

- Finding best QCF for a set of examples exponential in # attributes
- Greedy heuristic to find a “good” QCF; complexity quadratic in # attributes
- In error-cost computation: sum over k nearest neighbours only

Learning QCFs

Pres = 2 Temp / Vol

Temp Vol Pres

315.00 56.00 11.25

315.00 62.00 10.16

330.00 50.00 13.20

300.00 50.00 12.00

300.00 55.00 10.90

- For each pair of examples form a qualitative change vector

Learning QCFs

QCF Incons. Amb.

M+(Temp)

M-(Temp)

M+(Vol)

M-(Vol)

M+,+(Temp,Vol)

M+,-(Temp,Vol)

M-,+(Temp,Vol)

M-,-(Temp,Vol)

QCF Incons. Amb.

M+(Temp) 3 1

M-(Temp)

M+(Vol)

M-(Vol)

M+,+(Temp,Vol)

M+,-(Temp,Vol)

M-,+(Temp,Vol)

M-,-(Temp,Vol)

QCF Incons. Amb.

M+(Temp) 3 1

M-(Temp) 2,4 1

M+(Vol) 1,2,3 /

M-(Vol) 4 /

M+,+(Temp,Vol) 1,3 2

M+,-(Temp,Vol) / 3,4

M-,+(Temp,Vol) 1,2 3,4

M-,-(Temp,Vol) 4 2

qTemp=neg

qVol=neg

qPres=pos

Select QCF with minimal

QCF error-cost

ep-QUIN, simplified QUIN

ep-QUIN

- uses every pair of examples to evaluate a QCF, not just near neighbours
- does not weigh examples by proximity
- does not search for best QCF heuristically
- performance inferior to QUIN

Problem with ep-QUIN, example

- 12 learning examples that correspond to 3 linear functions

ep-QUIN does not consider the locality of qual. changes

Induced qual. tree does not correspond to the intuition

QUIN does it better

- Heuristic QUINalgorithm considers the locality and consistency of qualitative change vectors

QUIN considers the proximity of examples

Qualitative change vectors of near-by points weigh more

QUIN finds 3 groups of examples

Experimental evaluation

- On a set of artificial domains:
- Results by QUIN better than ep-QUIN
- QUIN can handle noisy data well
- QUIN finds qualitative relations corresponding to our intuition

- QUIN in skill reconstruction:
- QUIN used to induce qual. control strategies from examples of the human control performance
- Experiments in the crane domain

Experimental evaluation in artificial domains

- A set of artificial domains: real functions with up to 4 arguments
- Examples uniformly distributed over the attributes
- 2 irrelevant attributes added
- Noise added, various levels of noise

Artificial test domains

- Sin c = sin( x/ 10)
- SinLn c = x/10 + sign(x) sin( x/ 10)
- Poli c = ln( 104 + |(x+16) (x+5) (x-5) (x-16)| )
- Signs c =
- sign(u+0.5) (x-10)2, if v 0
- sign(u-0.5) (y+10)2, otherwise
- QuadA c = x2 – y2
- QuadB c = (x-5)2 - (y-10)2
- SQuadB c = sign(u) ((x-5)2 - (y-10)2)
- YSinX c = y sin( x/ 10)

Noise in the class variable (Sin)

Normally distributed noise with std.dev. 0.5 is added to y=sin(x). Qualitative trees induced by QUIN have qualitative consistency over 90%

Experimental evaluation in artificial domains

y=sin( x/10), x [-20, 20]

Minima of QUIN’s cost-error (at x=-15, -5, 5, 15) divide space into intervals with monotonic target function

X=5

Noise curves (Sin):consistency and ambiguity

Application in behavioral cloning

- Domain: crane control
- Goal: effective and comprehensible clones

Container crane

Control forces: Fx, FLState: X, dX, , d, L, dL

Based on previous work of Urbancic(94)

Control task: transport the load from the start to the goal position

QUIN in skill modeling, crane domain

- Qualitative trees induced from execution traces for rope and trolley control
- Traces of 2 operators with different control styles

desired_velocity = f(X, ,d)

First the trolley velocity is increasing

X < 20.7

yes

no

From about middle distance from the goal the trolley velocity is decreasing

M+(X)

X < 60.1

yes

no

At the goal reduce the swing of the rope (by acceleration of the trolley when the rope angle increases)

M-(X)

M+()

Crane control: comparing operators

Enables comparison of differences in control styles

Operator S

Operator L

X < 20.7

X < 29.3

yes

yes

no

no

M+(X)

M+,+,-(X, , d)

X < 60.1

d < -0.02

yes

yes

no

no

M-(X)

M+()

M-(X)

M-,+(X,)

- Induced control strategies:
- Comprehensible and very successful
- Enable insight into individual differences
- in control styles
- QUIN able to detect very subtle aspects of human tacit skill (aspects earlier believed absent)

Related work in qualitative reasoning

- In qualitative reasoning: Our QFC’s inspired by qualitative proportionalities (Q+) in QPT (Forbus) and monotonicity relations (M+) in QSIM (Kuipers)
- In learning qualitative models of dynamic systems: Mozetic; Coiera; Bratko et al.; Varsek; Richards et al.; Dzeroski, Todorovski;
- Distinguishing features of QUIN: models of static systems, qualitative trees, takes numerical examples directly

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