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Feature Selection, Feature Extraction

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Need for reduction Classification of leukemia tumors from microarray gene expression data 1 – 72 patients (data points) – 7130 features (expression levels of different genes) Text mining, document classification – features are words Quantitative Structure-Activity Relationship (QSAR) – features are molecular descriptors, there exist plenty of them 1 Xing, Jordan, Karp, Feature Selection for High-Dimensional Genomic Microarray Data, 2001

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QSAR biological activity – an expression describing the beneficial or adverse effects of a drug on living matter Structure-Activity Relationship (SAR) – hypotheses that similar molecules have similar activities molecular descriptor – mathematical procedure transforms chemical information encoded within a symbolic representation of a molecule into a useful number

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Molecular descriptor adjacency (connectivity) matrix total adj. index A V – sum all a ij measure of the graph connectedness Randic connectivity indices measure of the molecular branching

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QSAR Form a mathematical/statistical relationship (model) between structural (physiochemical) properties and activity. The mathematical expression can then be used to predict the biological response of other chemical structures. descriptor biological activity

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Selection vs. Extraction In feature selection we try to find the best subset of the input feature set. In feature extraction we create new features based on transformation or combination of the original feature set. Both selection and extraction lead to the dimensionality reduction. No clear cut evidence that one of them is superior to the other on all types of task.

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Why to do it? 1.We’re interested in features – we want to know which are relevant. If we fit a model, it should be interpretable. facilitate data visualization and data understanding reduce experimental costs (measurements) 2.We’re interested in prediction – features are not interesting in themselves, we just want to build a good predictor. faster training defy the curse of dimensionality

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Feature selection (FS)

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Classification of FS methods Filter – Assess the relevance of features only by looking at the intrinsic properties of the data. – Usually, calculate the feature relevance score and remove low-scoring features. Wrapper – Bundle the search for best model with the FS. – Generate and evaluate various subsets of features. The evaluation is obtained by training and testing a specific ML model. Embedded – The search for an optimal subset is built into the classifier construction (e.g. decision trees).

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Filter methods Two steps (score-and-filter approach) 1.assess each feature individually for ist potential in discriminating among classes in the data 2.features falling beyong threshold are eliminated Advantages: – easily scale to high-dimensional data – simple and fast – independent of the classification algorithm Disadvantages: – ignore the interaction with the classifier – most techniques are univariate (each feature is considered separately)

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Scores in filter methods Distance measures – Euclidean distance Dependence measures – Pearson correlation coefficient – χ 2 -test – t-test – AUC Information measures – information gain – mutual information complexity: O(d)

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Wrappers Search for the best feature subset in combination with a fixed classification method. The goodness of a feature subset is determined using cross-validation (k-fold, LOOCV) Advantages: – interaction between feature subset and model selection – take into account feature dependencies – generally more accurate Disadvantages: – higher risk of overfitting than filter methods – very computationally intensive

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Exhaustive search Evaluate all possible subsets using exhaustive search – this leads to the optimum subset. For a total of d variables, and subset of size p, the total number of possible subsets is complexity: O(2 d ) (exponential) Various strategies how to reduce the search space. – They are still O(2 d ), but much faster (at least 1000-times) – e.g. “branch and bound” e.g. d = 100, p = 10 → ≈2×10 13

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Stochastic Genetic algorithms Simulated Annealing

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Difficulty in Searching Global Optima starting point descend direction local minima global minima barrier to local search Introduction to Simulated Annealing, Dr. Gildardo Sánchez ITESM Campus Guadalajara

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Consequences of the Occasional Ascents Help escaping the local optima. desired effect Might pass global optima after reaching it adverse effect Introduction to Simulated Annealing, Dr. Gildardo Sánchez ITESM Campus Guadalajara

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Simulated annealing Slowly cool down a heated solid, so that all particles arrange in the ground energy state. At each temperature wait until the solid reaches its thermal equilibrium. Probability of being in a state with energy E: E … energy T … temperature k B … Boltzmann constant Z(T) … normalization factor

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Cooling simulation At a fixed temperature T : – Perturb (randomly) the current state to a new state – E is the difference in energy between current and new state – If E < 0 (new state is lower), accept new state as current state – If E 0, accept new state with probability Eventually the systems evolves into thermal equilibrium at temperature T When equilibrium is reached, temperature T can be lowered and the process can be repeated Metropolis, 1953 Metropolis algorithm

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Simulated annealing Same algorithm can be used for combinatorial optimization problems: – Energy E corresponds to the objective function C – Temperature T is parameter controlled within the algorithm

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Algorithm initialize; REPEAT perturb ( config.i config.j, C ij ); IF C ij < 0 THEN accept ELSE IF exp(- C ij /T) > random[0,1) THEN accept; IF accept THEN update(config.j); UNTIL equilibrium is approached sufficient closely; T := next_lower(T); UNTIL system is frozen or stop criterion is reached

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Parameters Choose the start value of T so that in the beginning nearly all perturbations are accepted (exploration), but not too big to avoid long run times At each temperature, search is allowed to proceed for a certain number of steps, L(k). The function next_lower (T(k)) is generally a simple function to decrease T, e.g. a fixed part (80%) of current T.

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At the end T is so small that only a very small number of the perturbations is accepted (exploitation). The choice of parameters {T(k), L(k)} is called the cooling schedule. If possible, always try to remember explicitly the best solution found so far; the algorithm itself can leave its best solution and not find it again.

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Deterministic Sequential Forward Selection (SFS) Sequential Backward Selection (SBS) “ Plus q take away r ” Selection Sequential Forward Floating Search (SFFS) Sequential Backward Floating Search (SBFS)

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Sequential Forward Selection SFS At the beginning select the best feature using a scalar criterion function. Add one feature at a time which along with already selected features maximizes the criterion function. A greedy algorithm, cannot retract (also called nesting effect). Complexity is O(d)

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Sequential Backward Selection SBS At the beginning select all d features. Delete one feature at a time and select the subset which maximize the criterion function. Also a greedy algorithm, cannot retract. Complexity is O(d).

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“Plus q take away r” Selection At first add q features by forward selection, then discard r features by backward selection Need to decide optimal q and r No subset nesting problems Like SFS and SBS

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Sequential Forward Floating Search SFFS It is a generalized “plus q take away r” algorithm The value of q and r are determined automatically Close to optimal solution Affordable computational cost Also in backward disguise

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Embedded FS The feature selection process is done inside the ML algorithm. Decision trees – In final tree, only a subset of features are used Regularization – It effectively “shuts down” unnecessary features. – Pruning in NN.

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Feature extraction (FE)

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FS – indetify and select the “best” features with respect to the target task. – Selected features retain their original physical interpretation. FE – create new features as a transformation (combination) of original features. Usually followed by FS. – May provide better discriminatory ability than the best subset. – Do not retain the original physical interpretation, may not have clear meaning.

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Principal Component Analysis (PCA)

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x1x1 x2x2

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x1x1 x2x2 Make data to have zero mean (i.e. move data into [0, 0] point). centering

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x1x1 x2x2 This is a line given by equation w 0 + w 1 x 1 + w 2 x 2 This is another line w’ 0 + w’ 1 x 1 + w’ 2 x 2

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x1x1 x2x2 The variability in data is highest along this line. It is called 1 st principal component. And this is 2 nd principal component.

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x1x1 x2x2 Principal components (PC’s) are linear combinations of original coordinates. The coefficients of linear combination (w 0, w 1, …) are called loadings. In the transformed coordinate system, individual data points have different coordinates, these are called scores. w 0 + w 1 x 1 + w 2 x 2 w’ 0 + w’ 1 x 1 + w’ 2 x 2

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PCA - orthogonal linear transformation that changes the data into a new coordinate system such that the variance is put in order from the greatest to the least. Solve the problem = find new orthogonal coordinate system = find loadings PC’s (vectors) and their corresponding variances (scalars) are found by eigenvalue decompositions of the covariance matrix C = XX T of the x i variables. – Eigenvector corresponding to the largest eigenvalue is 1 st PC. – The 2 nd eigenvector (the 2 nd largest eigenvalue) is orthogonal to the 1 st one. … Eigenvalue decomposition is computed using standard algorithms: eigen decomposition of covariance matrix (e.g. QR algorithm), SVD of mean centered data matrix.

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Interpretation of PCA New variables (PCs) have a variance equal to their corresponding eigenvalue Var(Y i )= i for all I = 1…p Small i small variance data changes little in the direction of component Y i The relative variance explained by each PC is given by i / i

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How many components? Enough PCs to have a cumulative variance explained by the PCs that is >50-70% Kaiser criterion: keep PCs with eigenvalues >1 Scree plot: represents the ability of PCs to explain de variation in data

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