Lecture 12

1. Machine Learning

  • Definition
  • Basic Paradigm
    • Observe set of examples: training data
    • Infer something about process that generated that data
    • Use inference to make predictions about previously unseen data: test data
  • Procedures
    • Representation of the features
      • separate people with features(man/woman, educated/not, etc.)
    • Distance metric for feature vectors
      • make feature vectors can be calculated in a same range.
    • Objective function and constraints
    • Optimization method for learning the model
    • Evaluation method

1.1. Supervised Learning

  • Start with set of feature vector/value pairs
  • Goal: find a model that predicts a value for a previously unseen feature vector
  • Regression models predict a real
    • As with linear regression
  • Classification models predict a label (chosen from a finite set of labels)

1.2. Unsupervised Learning

  • Start with a set of feature vectors
  • Goal: uncover some latent structure in the set of feature vectors
  • Clustering the most common technique
    • Define some metric that captures how similar one feature vector is to another
    • Group examples based on this metric

1.3. Difference between Supervised and Unsupervised

  • with label, we can classify the data to two clusters by wight or height, or four clusters by wight and height, which is Supervised Learning
  • without label, to figure out how to clustering the data, is Unsupervised Learning.

1.4. Choose Feature Vectors

  • Why should careful?
    • Irrelevant features can lead to a bad model.
    • Irrelevant features can greatly slow the learning process.
  • How?
    • signal-to-noise ratio (SNR)
      • Think of it as the ratio of useful input to irrelevant input.
    • The purpose of feature extraction is to separate those features in the available data that contribute to the signal from those that are merely noise.

1.5. Distance Between Vectors

Minkowski Metric

  • dist(X1,X2,p)=(k1lenabs(X1kX2k)p)1/pdist(X1, X2, p)=(\displaystyle\sum_{k-1}^{len}abs(X1_{k}-X2_{k})^p)^{1/p}
  • p = 1: Manhattan Distance
  • P = 2: Euclidean Distance

     def minkowskiDist(v1, v2, p):
         """Assumes v1 and v2 are equal-length arrays of numbers 
            Returns Minkowski distance of order p between v1 and v2""" 
         dist = 0.0 
         for i in range(len(v1)):
             dist += abs(v1[i] - v2[i])**p 
         return dist**(1.0/p)
    
  • For example:
    • To compare the distance between star and circle and the distance between cross and circle
    • Use Manhattan Distance, they should be 3 and 4
    • Use Euclidean Distance, they should be 3 and 2.8 = 22+22\sqrt{2^2+2^2}
Using Distance Matrix for Classification
  • Procedures
    • Simplest approach is probably nearest neighbor
    • Remember training data
    • When predicting the label of a new example
      • Find the nearest example in the training data
      • Predict the label associated with that example
  • To predict the color of X

    • The closest one is pink, so X should be pink
  • K-nearest Neighbors

    • Find K nearest neighbors, and choose the label associated with the majority of those neighbors.
    • Usually, we use odd number. This sample, we use k = 3
  • Advantages and Disadvantages of KNN

    • Advantages
      • Learning fast, no explicit training
      • No theory required
      • Easy to explain method and results
    • Disadvantages
      • Memory intensive and predictions can take a long time
      • Are better algorithms than brute force
      • No model to shed light on process that generated data
  • For Example

    • To predict whether zebra, python and alligator are reptile or not.
    • Calculate the distances, we got:
      • The closest three animals to alligator are boa constrictor, chicken and dark frog, and two of them are not reptile, so alligator is not reptile.
      • But we know alligator is reptile. So what's wrong?
      • We notice, all of the features are 0 or 1, except number of legs, which gets disproportionate weight.
        • So, Instead of number of legs, we say "has legs." And then this becomes a one.
      • The closest three animals to alligator are boa constrictor, chicken and cobra, and two of them are reptile, so alligator is reptile.
  • A More General Approach: Scaling

    • Z-scaling
    • Interpolation

      • Map minimum value to 0, maximum value to 1, and linearly interpolate
      def zScaleFeatures(vals):
        """Assumes vals is a sequence of floats"""
        result = pylab.array(vals)
        mean = float(sum(result))/len(result)
        result = result - mean
        return result/stdDev(result)
      
      def iScaleFeatures(vals):
        """Assumes vals is a sequence of floats"""
        minVal, maxVal = min(vals), max(vals)
        fit = pylab.polyfit([minVal, maxVal], [0, 1], 1)
        return pylab.polyval(fit, vals)
      

1.6. Clustering

  • Partition examples into groups (clusters) such that examples in a group are more similar to each other than to examples in other groups
  • Unlike classification, there is not typically a “right answer”
    • Answer dictated by feature vector and distance metric, not by a ground truth label

Optimization Problem

  • Clustering is an optimization problem. The goal is to find a set of clusters that optimizes an objective function, subject to some set of constraints.
  • Given a distance metric that can be used to decide how close two examples are to each other, we need to define an objective function that
    • Minimizes the distance between examples in the same clusters, i.e., minimizes the dissimilarity of the examples within a cluster.
  • To compute the variability of the examples within a cluster
    • First compute the mean(sum(V)/float(len(V)), more precisely the Euclidean mean) of the feature vectors of all the examples in the cluster. , V is a list of feature vectors.
    • Compute the distance between feature vectors
      • variability(c)=ecdistance(mean(c),e)2\text{variability}(c)=\displaystyle\sum_{e \in c}\text{distance}(\text{mean}(c), e)^2
  • The definition of variability within a single cluster, c, can be extended to define a dissimilarity metric for a set of clusters, C:
    • dissimilarity(C)=ecvariability(c)\text{dissimilarity}(C)=\displaystyle\sum_{e \in c}\text{variability(c)}
  • It's NOT the optimization problem to find a set of clusters, C, such that dissimilarity(C) is minimized. Because it can easily be minimized by putting each example in its own cluster.
  • We could put a constraint on the distance between clusters or require that the maximum number of clusters is k. Then to find the minimum between clusters.
K-means Clustering
  • Constraint: exactly k non-empty clusters
  • Use a greedy algorithm to find an approximation to minimizing objective function
  • Algorithm

      randomly chose k examples as initial centroids
      while true:
          create k clusters by assigning each
              example to closest centroid
          compute k new centroids by averaging
              examples in each cluster
          if centroids don’t change:
              break
    
  • Unlucky Initial Centroids

    • k=4, Initial Centroids:
    • Result:
    • Mitigating Dependence on Initial Centroids

        best = kMeans(points)
        for t in range(numTrials):
            C = kMeans(points)
            if dissimilarity(C) < dissimilarity(best):
            best = C
        return best
      

1.7. Wrapping Up Machine Learning

  • Use data to build statistical models that can be used to
    • Shed light on system that produced data
    • Make predictions about unseen data
  • Supervised learning
  • Unsupervised learning
  • Feature engineering
  • Goal was to expose you to some important ideas
    • Not to get you to the point where you could apply them
    • Much more detail, including implementations, in text

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