Transformation of Input Space using Statistical Moments: EA-Based Approach Ahmed Kattan: Um Al Qura University, Saudi Arabia Michael Kampouridis: University of Kent, UK Yew-Soon Ong: Nanyang Technological University, Singapore
Transcript
Slide 1
Transformation of Input Space using Statistical Moments:
EA-Based Approach Ahmed Kattan: Um Al Qura University, Saudi Arabia
Michael Kampouridis: University of Kent, UK Yew-Soon Ong: Nanyang
Technological University, Singapore Khalid Mehamdi: Um Al Qura
University, Saudi Arabia
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The problem Standard Regression models are presented with
Observational data of the form (x i, y i ) i=1n Each x i denotes a
k-dimensional input vector of design variables and y is the
response. When k n, high variance and over-fitting become a major
concern.
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The problem High dimensional regression problem Regression
Model Poor approximation
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Solutions Curse of dimensionality is solved by: RReduce number
of dimensions by selecting important features (e.g., PCA,
FDA,..etc.) TTransformation of input space (e.g., GP, FFX,..etc.)
Majority of work in this topic has been done for classification
problems. The idea of transforming input space to reduce the number
of design variables in the regression problems to improve
generalisation is relatively little explored thus far.
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Contributions of this work Contributions A novel evolutionary
approach to transform the high-dimensional input space of
regression models using only statistical moments. analysis to
understand the impact of different statistical moments on the
evolved transformation procedure dramatically improve LRs
generalisation and make it competitive to other state-of-the-art
regression models.
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The proposed transformation (x i, y i ) (z i, y i )
Transformation x1x1,,, xkxk x0x0 z1z1 znzn z0z0 We transform the
input vector x into and vector called z. The z is smaller than x
and easier to be approximated by standard regression models.
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The proposed transformation We used standard Genetic
Algorithm
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Genetic Algorithm Population representation
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Genetic Algorithm Search operators Crossover in which two
individuals exchange statistical moments and their parameters,
randomly. op 0 op 1 op 2 op g a0a2a3a7a5a8a0a2a3a7a5a8 a 2 a 3 a 4
a 2 a 7... a0a2a7a0a2a7 a0a5a6a7a9a0a5a6a7a9 . op 0 op 1 op 2 op g
a0a2a3a7a5a8a0a2a3a7a5a8 a 2 a 3 a 4 a 2 a 7... a0a2a7a0a2a7
a0a5a6a7a9a0a5a6a7a9 .
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Genetic Algorithm Search operators Aggressive mutation operator
that replaces a statistical moment and its parameters, randomly
selected, with another randomly selected moments from the pool of
statistical moments. op 1 op 2 op g a0a2a3a7a5a8a0a2a3a7a5a8 a 2 a
3 a 4 a 2 a 7... a0a2a7a0a2a7 a0a5a6a7a9a0a5a6a7a9 . a4a3a9a4a3a9
op 0 New op 0
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Genetic Algorithm Search operators Smooth mutation operator
where a parameter of a randomly selected statistical moment is
mutated into a new parameter. op 0 op 1 op 2 op g
a0a2a3a7a5a8a0a2a3a7a5a8 a 2 a 3 a 4 a 2 a 7... a0a2a7a0a2a7
a0a5a6a7a9a0a5a6a7a9 . a4a4
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Genetic Algorithm Fitness measure We used average prediction
errors of Linear Regression (LR) as a fitness measure for GA. LR is
a very simple algorithm where it considers the family of linear
hypotheses:
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Genetic Algorithm Fitness measure Why LR ? Hence, given these
features LR can push the GAs evolutionary process to linearly align
the transformed inputs with their outputs and minimise the
dimensionality of the new space.
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Genetic Algorithm Fitness measure The GA aims to minimise the
following fitness function:
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Genetic Algorithm Training Two disjoint sets: training and
validation. LR: two-folds cross-validation approach. The best
individual in each generation is further tested with the validation
set. We select the individual that yields the best performance on
the validation set across the run.
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Empirical tests We tested the effects of the transformation
procedure on LR and compared the results against five regression
models, namely: 1.RBFN 2.RBFN + PCA 3.Kriging 4.Kriging + PCA 5.LR
6.LR + PCA 7. piecewise LR 8.Genetic Programming 9.Genetic
Programming + PCA
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Empirical tests F1 = Rastrigin functionF2 = Schwefel function
We tested 5 benchmark functions
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Empirical tests F5 = Dixon & Price function F3 =
Michalewicz function F4 = Sphere function
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Empirical tests For each test function, we trained all
regression models to approximate the given function when the number
of variables is 100 variables. 500 variables. 1000 variables.
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Empirical tests
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Approximation Quality Sphere function for 2 variables
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Empirical tests LR approximate the Sphere function after input
transformation
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Learn from evolution
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It is clear from the heat maps that each problem has its unique
characteristics. Interestingly, there is a consensus among all maps
that the following operators do not contribute to the construction
of good transformation procedures. copy copy intercept.
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Learn from evolution Also, all maps agree that the following
are important across all problems. Average Deviation Geometric Mean
Min Max We still do not have a full understanding of the effect of
these moments on the transformed space. In future research we will
focus on this aspect.
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Conclusions In this work we presented: A novel evolutionary
approach to transform the high-dimensional input space of
regression models using only statistical moments. analysis to
understand the impact of different statistical moments on the
evolved transformation procedure. dramatically improve LRs
generalisation and make it competitive to other state-of-the-art
regression models. We hope our results will inspire other
researchers to build a deeper understanding to discover relations
between straight statistical momnets on making good
transformation