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CY3G2 Modern Heuristics
Lecture 9

• Dr. Gillian Walker
• Room 182
• g.c.walker_at_reading.ac.uk

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Problem of last Week

• There are five houses, each of a different colour
  and inhabited by women of different
  nationalities, with different pets, favourite
  drinks and cars. Moreover
• The Englishwoman lives in the red house.
• The Spaniard owns the dog.
• The woman in the green house drinks cocoa.
• The Ukrainian drinks eggnog.
• The green house is immediately to the right of
  the ivory house.
• The owner of the Toyota car also owns snails.
• The owner of the Ford lives in the yellow house.

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Problem of last week

• The man in the middle house drinks milk.
• The Norwegian lives in the first house on the
  left.
• The woman who owns the Chevrolet lives in the
  house next to the house where the woman owns a
  fox.
• The Ford owners house is next to the house where
  the horse is kept.
• The Mercedes-Benz owner drinks orange juice.
• The Japanese drives a Volkswagen.
• The Norwegian lives next to the blue house.
• The question is .... Who owns the zebra?
  Furthermore, who drinks water?

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Solution of last week.

• Establish a table representing all the
  information, or implied information we were
  given.
• Fill in the direct information first.
• Sentence 1 states the englishwoman lives in the
  red house
• Sentence 5 states that the green house is
  immediately to the right of the red house
• So.... House 1 isnt red, house 3 4 or 4 5
  are ivory and green.

House 1 2 3 4 5
Colour Blue
Drink Milk
Country Norwegian
Car
Pet
5
Solution of last week.

• So House 1 must be yellow. Therefore the
  Norwegian owns the ford and the horse is kept in
  house two.
• From here there are two possibilities. The only
  possible sequences for the colours of houses 3, 4
  and 5 are
• ivory, green, red or red ivory green.

House 1 2 3 4 5
Colour Yellow Blue
Drink Milk
Country Norwegian
Car Ford
Pet Horse
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Solution of last week.

• Consider ivory, green red.
• We can now infer more information The
  englishwoman lives in house 5, as it is red.
  Cocoa is drunk in house 4 because it is green.
  The Ukrainian must live in house 2 because he
  drinks eggnog. The Englishwoman owns a Mercedes
  and drinks orange juice. Our information is

House 1 2 3 4 5
Colour Yellow Blue Ivory Green Red
Drink Eggnog Milk Cocoa Orange
Country Norwegian Ukrainian English
Car Ford Mercedes
Pet Horse
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Solution of last week.

• Consider ivory, green red.
• This wont lead to a solution!
• Who owns the Toyota? The Japanese owns the
  Volkswagen, it isnt the Ukrainian because the he
  owns a horse not snails, and it cant be the
  Spaniard because she owns a dog.
• We have reached a contradiction. So we will try
  the other colour combination.

House 1 2 3 4 5
Colour Yellow Blue Ivory Green Red
Drink Eggnog Milk Cocoa Orange
Country Norwegian Ukrainian English
Car Ford Mercedes
Pet Horse
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Solution of last week.

• Consider red, ivory, green
• The Ukrainian drinks eggnog and so must live in
  house 2 and 4.
• If the Ukrainian lives in house 4 then the
  Spaniard (who owns a dog) must live in house 5
  and the Japanese must live in house 2.
• Also Orange juice must be drunk in house 2 whose
  inhabitant drives a Mercedes. This is a
  contradiction because the Japanese owns a
  Volkswagon SO the Ukrainian must live in house 2.

House 1 2 3 4 5
Colour Yellow Blue Red Ivory Green
Drink Milk Cocoa
Country Norwegian English
Car Ford
Pet Horse
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Solution of last week.

• Consider red, ivory, green
• The owner of the Mercedes drinks orange juice and
  must live in house 4.
• The Japanese owns a Volkswagen must live in house
  5.
• The Spaniard (who owns a dog) lives in house 4

House 1 2 3 4 5
Colour Yellow Blue Red Ivory Green
Drink Eggnog Milk Orange Cocoa
Country Norwegian Ukrainian English Spaniard Japanese
Car Ford Mercedes Volkswagen
Pet Horse Dog
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Solution of last week.

• The Toyota owner also owns snails. So he must
  live in house 3.
• The Chevrolet owner is in house 2 and the fox
  house 1.
• So the Japanese owns the zebra and
• The Norwegian drinks the water.

House 1 2 3 4 5
Colour Yellow Blue Red Ivory Green
Drink Eggnog Milk Orange Cocoa
Country Norwegian Ukrainian English Spaniard Japanese
Car Ford Chevrolet Toyota Mercedes Volkswagen
Pet Fox Horse Snails Dog
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Summary

• In the last lecture we looked at how we could
  further develop evolutionary algorithms to deal
  with problem constraints.
• This lecture we are going to discuss how to tune
  our algorithms to our problems.
• In addition we are going to look at
  hybridisation.
• Finally we are going to test to make sure we have
  the best solution.

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Tuning

• Each problem solving technique we have looked at
  has had parameters, in general the more
  complicated the technique the more parameters
  were involved
• Hill climbing (size of local search space)
• Tabu (how to impliment memory structure)
• Simulated annealing (Temperature parameter and
  how to cool)
• Evolutionary algorithms have even more
  parameters Population size, who breeds, how they
  breed, who passes on to the next generation,
  mutation probabilities, termination conditions
  etc.

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Tuning

• How do we tune these parameters to
• Solve a given problem in the shortest possible
  time?
• Solve as many problems with the one algorithm as
  possible?
• You have experience of this from looking at the
  Sudoku code.
• Trial and Error is tedious and time consuming.
• There are loads of parameters to do.
• Once you have optimised one parameter this may
  affect the optimum value of another parameter and
  so on.
• Trying all possible combinations is practically
  impossible and it is difficult to know when you
  have the optimum parameter combination.

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Tuning

• There are two approaches to parameter tuning
• Tuning parameters before you run the algorithm.
• Tune parameters during the running of the
  algorithm.
• The latter option inevitably leads you to the
  concept of control.
• Parameter control can take the form of
• Deterministic parameter control (search over
  time)
• Adaptive parameter control (use feedback from
  the environment to determine optimum parameters
  (credit based))
• Self-Adaptive run an evolutionary algorithm
  inside the evolutionary algorithm for each of the
  parameters evolve the parameters.

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No Free Lunch

• The No Free Lunch theorem states that
• All algorithms that search for an extreme of a
  cost function perform exactly the same, according
  to any performance measures, when averaged over
  all possible cost functions. Wolpert
  and Macready 96
• No single optimum search algorithm exists for
  blind search.
• Optimisation of individual algorithms has to be
  related to the problem.

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Hybridisation

• No single algorithm can be the best approach to
  solve every problem.
• We need to incorporate knowledge from our problem
  to help us solve it, otherwise all we have is a
  blind search.
• One way to try to achieve a more automated
  algorithm is to hybridise evolutionary algorithms
  with other more standard approaches (hill
  climbing or greedy methods) to problem solving.
• Improve individual solutions with a local search
  and replace them in the solution population to
  compete with the rest.
• Seed the initial population with solutions which
  are found with standard techniques.
• The only restriction to hybridisation is your
  imagination.

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Hybrid Systems

• Evolutionary algorithms are flexible and can be
  extended by including diverse concepts and
  alternative approaches
• Incorporate local searches to improve solutions
  in a population.
• Use Lamarckian and Baldwin ideas of evolution to
  handle constraints.
• Incorporate control parameters so that the
  evolutionary algorithm can tune itself.
• Introduce memory into a proportion, or the whole
  population so as to better deal with time-varying
  environments.
• This is just the beginning.
• The most common problem with hybrid systems is
  that the designer gets carried away and
  incorporates too many concepts.

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Hybrid Enhancements

• Some hybrid enhancements are
• Incorporate memory
• Temperature
• Mating
• Attractiveness
• Subpopulations
• Gender
• These lead to the possibility of other hybrid
  enhancements If we have a subpopulation
• We could run different evolutionary algorithms on
  each sub population (island)
• From time to time we could move solutions from
  one sub population to the next (migration)

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Hybrid enhancements

• These enhancements in turn lead to new problem
  parameters
• How many subpopulations should there be?
• Should all the subpopulations be the same size?
• Should the same or different evolutionary
  algorithms be used on each subpopulation? (you
  could tune each algorithms parameters
  specifically to the population in question)
• What topology should you use to connect the
  subpopulations?
• What are the mechanisms for migration?
• How often should migration occur?
• Who should migrate? (Best, worse or random
  solutions?)
• What are the rules for immigration. (do we
  replace solutions or increase the population?)

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How do we know if we have a good solution?

• If we blindly accept the inputs to our algorithm
  often it will blindly generate the outputs.
• Garbage In Garbage Out
• Evolutionary based algorithms are designed to
  produce optimal or near optimal solutions.
• The final population (whatever your termination
  criteria) will likely be a population of good
  solutions.
• As you have seen each run of an evolutionary
  algorithm took a different number of iterations
  to generate the solution, and the final
  population in each case will be different too.
• There are a number of statistical methods to
  determine the deviation of the final population,
  or a subset of the final population to give us a
  measure of the quality of the solution.

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Standard Deviation

• Standard deviation is the most common measure of
  statistical dispersion.
• The standard deviation is the square root of the
  variance. This means it is the root mean square
  (RMS) deviation from the average.
• This gives a measure of dispersion that is
• A non-negative number
• Has the same units as the data.
• A distinction is made between the standard
  deviation of a whole population or of a random
  sample of the population, or a subpopulation.

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Students t-test

• "Student" was the pen name of William Sealy
  Gosset, a statistician for Guinness brewery in
  Dublin, Ireland.
• Gosset was hired as a result of an innovative
  policy of Claude Guinness to recruit the best
  graduates from Oxford and Cambridge for the
  application of biochemistry and statistics to
  Guinness's industrial processes. Gosset published
  the t-test in Biometrika in 1908, but was forced
  to use a pen name by his employer who regarded
  the fact that they were using statistics as a
  trade secret. In fact, Gosset's identity was
  unknown not only to fellow statisticians but to
  his employer - the company insisted on the
  pseudonym so that it could turn a blind eye to
  the breach of its rules.
• Gosset invented the t-statistic to enable the
  quality of beer brews to be monitored in a
  cost-effective manner.
• Today, it is more generally applied to the
  confidence that can be placed in judgements made
  from small samples.

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Students t-test

• The test compares the means of two treatments (or
  final populations of two evolutionary algorithms)
  even if they have different numbers of
  replicates.
• The t-test compares the actual difference between
  two means in relation to the variation in the
  data (expressed as the standard deviation of the
  difference between the means.)

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Students t-test

• SE standard error of the difference.
• Take the variance for each group and divide it
  by the number of people in that group. Add these
  two values and then take their square root.

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The Analysis of Variance (ANOVA)

• ANOVA is a family of general techniques used to
  test the hypothesis that the means among two or
  more groups are equal, R. Fisher 1920s
• - under the assumption that the sampled
  populations are normally distributed.
• Multiple t-tests are not useful as the number of
  groups for comparison grows. As the number of
  comparison pairs grows, the more likely we are to
  observe things that happen only 5 of the time
• Thus P0.05 for one pair cannot be considered
  significant.
• ANOVA puts all the data into one number (F) and
  gives us one P for the null hypothesis.

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Significance

• In statistical hypothesis testing, two hypotheses
  are stated, only one of which can be true.
• The null hypothesis, is what is presumed to be
  true.
• The alternative hypothesis, is will be
  considered true only if the facts are strong
  enough.
• The statistical hypothesis testing procedure
  (e.g. t-test) produces a value,
• If the t value that is calculated is greater than
  the threshold chosen for statistical significance
  (usually the 0.05 level), then the null
  hypothesis that the two groups do not differ is
  rejected in favour of the alternative hypothesis,
  which typically states that the groups do differ.
  

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Confusion Matrix

• The confusion matrix is a visualization tool used
  in supervised learning.
• Each column of the matrix represents the
  instances in a predicted class, while each row
  represents the instances in an actual class.
• One benefit of a confusion matrix is that it is
  easy to see if the system is confusing two
  classes (i.e. commonly mislabelling one as an
  other).
• Often used in the diagnosis of diseases.

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Confusion Matrix

• True Positive an individual classified as
  positive by the test and verified by the gold
  standard
• True Negative an individual classified as
  negative by the test and verified by the gold
  standard
• False-Positive and False-Negative also used
• Sensitivity True Positive Decisions
• All Gold Standard Positives
• Specificity True Negative Decisions
• All Gold Standard Negative

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Problem of the Week

• Day of the week of January the 1st.
• Which day of the week appears more often as the
  first day of a year - Saturday or Sunday?