Example 1. Classification by means of Total Fuzzy Similarities

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Lecture IV
Example 1. Classification by means of Total Fuzzy
Similarities
Example 2. Highway driving fluency
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Example 3. Determining Athletes Aerobic and
Anaerobic Thresholds
100 metres sprinter has to run a short distance
very fact, therefore, he has to have
much training in the anaerobic zone (where his
pulse close to maximal value), while a long
distance runner needs endurance, thus, he needs
training in the aerobic zone.
It is important for an athlete to test his
aerobic and anaerobic thresholds regularly, these
tests can be done e.g. on a running track
ergometer see Picture 1.
Aerobic and anaerobic thresholds are functions of
blood lactate mmol/l, ventilation CO2 l/min
and O2 uptake . They, in turn, are functions
of heartbeat b/min. A typical test starts with
a 3 minutes worm up pulse around 100 beat/min,
then the load is increased every 2 - 3 minutes
and blood lactate, ventilation, CO2, O2
uptake and heartbeat are measured. A test lasts
until volitional exhaustion pulse near 200
b/min, this takes usually 20-30 minutes A
typical test protocol, see Picture 2.
Lactate Ventilation Pulse Order
Aerobic lowest value ? VE increasing max - 40 /- lactate, VE
threshold ?0.2 mmol/l O2 highest 10 /- 5 b/min pulse
Anaerobic rapid increasing VE clearly ? max - 20 /- lactate, VE
threshold 3 mmol/l VCO2 ? 5 /- 3 b/min VCO2,
    O2 decreasing O2 decreasing O2
    VE/VO2 ?   VE/VO2
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Clearly, all these consepts are fuzzy and can be
expressed by fuzzy intervals. They are context
dependent, too. For example Maximal pulse 40
depends on a respective measurement. We used the
following membership function
To mimic a skilled sport medicine spacialists
action when she/he is determing an aerobic
threshold, we need only one rule (namely that one
given above!) Indeed, we count the degree of
total fuzzy similarity between each measured
value and that given be the rule.
Examples open Matlab7 gtgt cd C\Matlab7\Mika\hemi
gtgt aek Samples 33, 78
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Example 4. Signalized isolated pedestrian
crossing (See Picture 3.)
As long as there are no pedestrians, vehicles
have green signal. If a pedestrian pushes a
button and no cars are approaching the
pedestrian will have immediately green signal.
In case there are vehicles approaching and
pedestrians waiting, then vehicless
green depends on the following factors how
long time have pedestrians been waiting for a
short/long/too long time how many vehicles are
approaching few/some/many what is the shortes
cap between approaching vehicles
short/large The situation is updated after
every half second.
Experienced traffic engineers described the above
fuzzy set as follows (See Picture 2).
There are 18 rules in our fuzzy IF-THEN rule base
(green extension is prefered) This corresponds to
all possible rules (3x3x2 18), therefore, the
rule base is complete.
An example of a rule (given by traffic
engineers) IF (pedestrians waiting time
short) weight 1 AND (approaching
vehicles few) weight 2 AND
(gab between approaching vehicles short)
weight 1
THEN (extend vehicles
green signal)
The output is always a crisp action (red or
green). In a 50-50 situation it is green.
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An example (assuming we would have only three
rules)
1
1
1
3/8
short
some
short
½
½
3
12
16
2
3
4
1
5
1
2
3
5
24
28
sec

sec
0
1
1
1
3/4
long
large
9/16
½
½
some
3
12
16
2
3
4
1
5
1
2
3
5
24
28
sec

sec
1
1
1
many
too long
short
16/40
½
½
3
12
16
2
3
4
1
5
1
2
3
5
24
28
sec

sec
0.1
Pedestrians waiting time 14 sec
Vehicles 4
Gab between 2 sec
The actual traffic situation is most similar to
the 2. rule, thus...
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Example 5. More complicated traffic signal
control (See....)
Traffic flow on the main street (phase A) is from
two to ten times more intensive than traffic
flow from the other direction. Normally the phase
order is A-B-C-A, however, if there are very
few or no vehicles in the next phase B or C, then
these phases can be skipped. Thus, the order
can be e.g. A-C-A-B-C or A-B-A-B-C. Thus, the
first task is to determine the right phase order.
Here the IF-THEN rules have a form IF V(B) is
medium AND V(C) is over saturated THEN phase
is C Corresponding to Step 3 of the
Algorithm, if the maximal total similarity is
not unique, the phase with the longest waiting
time will be fired. The second task is to
control As green ending, that is, the decision
of the moment when the green of the first signal
group of phase A can be terminated, so that the
first signal group of phase B or C can be
started. These rules are of the following
form IF A is a Few AND Queue is Medium THEN
Extension is Short A Similarity Model and a
Mamdani style fuzzy model was constructed at
H.U.T, Lab of Traffic and Transportation. The
results were compared
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These fuzzy models ware tested by a traffic
simulator HUTSIM and the results were as
follows
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Example 6. Real-Time Reservoir Operation
Lake Päijänne is located in the Southern
part of Finland. Its water runs to the Golf
of Finland via River Kymijoki. Each year
Päijänne is frozen at least 5 months and
lots of snow is accumulated. In spring
floods caused by melting snow would be
typical if Päijänne was not regulated. The
water reference level is a function of date
given by law. Based e.g. on snow water
equivalent, human experts are able to
regulate several dams such that water level.
can be kept close to the reference level
The task was to create a formal control system
to do the same job.
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A control system was created at Helsinki
University of Technology, Laboratory of Water
Resource Management by Tanja Dudrovin. The model
consists of two real-time submodels the first
submodel sets up a reference water level (WREF)
for each time step. Given this reference level,
the observed water level (W), and the observed
inflow (I), the second submodel makes the
decision on how much water should be released
from the reservoir during the next time
step. For the snowmelt season, WREF value is
dependent on the snow water equivalent (SWE) and
can be inferred for each time step with the
rules IF SWE is smaller than average/average/lar
ger than average/much larger than average THEN
WREF is high/middle/low/very low. In the second
sub model, the rules have a form IF W is very
low/low/objective/high/very high AND I is very
small/small/large/very large THEN release is
exceptionally small/very small/small/quite
small/quite large/large/very large/exceptionally
large. Membership functions see Picture 4.
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To calibrate the corresponding fuzzy set, a data
of real control actions collected during
1975-1985 was used. RESULTS The model was
tested using data from the years 1985-1996. The
Sugeno method was chosen for comparison against
the Total Fuzzy Similarity. With both methods the
system was kept the same as much as
possible. To apply the Sugeno method the,
defuzzification was performed using a weighted
average. The performances of the two methods
were almost indistinguishable. With the Total
Fuzzy Similarity the water level targets during
the summer were sometimes better fulfilled, but
the release tended to fluctuate more, and the
limitation on change in release was more
relevant. The model performance was generally
good, but the model did not capture expert
thinking in the most exceptional circumstances.
(- later the model was completed by an extra
subsystem to do the job.)