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

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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

Example problem for QUIN

Noisy examples:

z = x2 - y2 + noise(st.dev. 50)

Qualitative patterns in data

x > 0 & y > 0 =>

z = M+,-(x,y)

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
• 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

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

Trolley control, operator S

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,)

QUIN in skill modeling

• 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