Descriptive and Exploratory Methods L eon Bottou largely copied - PowerPoint PPT Presentation

Descriptive and Exploratory Methods L´ eon Bottou largely copied from Mireille Summa-Gettler lectures (in french) COS 424 – 3/23/2010

Agenda Classification, clustering, regression, other. Goals Parametric vs. kernels vs. nonparametric Probabilistic vs. nonprobabilistic Representation Linear vs. nonlinear Deep vs. shallow Explicit: architecture, feature selection Explicit: regularization, priors Capacity Control Implicit: approximate optimization Implicit: bayesian averaging, ensembles Loss functions Operational Budget constraints Considerations Online vs. offline Exact algorithms for small datasets. Computational Stochastic algorithms for big datasets. Considerations Parallel algorithms. Today’s topic fits poorly in this picture. L´ eon Bottou 2/86 COS 424 – 3/23/2010

Introduction Predictive methods – Construct models using examples (the training set). – Hope that it works well for future situations (e.g. on a testing set.) Descriptive methods – Describe the distribution of examples. – Investigate the geometry of the data. – Hope to acquire insights about the underlying phenomenon. L´ eon Bottou 3/86 COS 424 – 3/23/2010

A catalog of descriptive methods Clustering methods – K-means, K-medioids, Gaussian mixtures. . . – Hierarchical clustering. . . Projection methods �� – Principal component analysis (PCA) [Hotelling, 30s] �� – Correspondence analysis (CA) [Benzecri, 60s] �� – Multiple correspondence analysis (MCA) �� – Canonical correlation analysis (CCA), . . . Embedding methods – Kernel PCA – Locally linear embbedding (LLE) – ISOMAP L´ eon Bottou 4/86 COS 424 – 3/23/2010

I. Principal Component Analysis L´ eon Bottou 5/86 COS 424 – 3/23/2010

Sparkling water springs Observations – 21 sparkling water springs in France. Continuous variables – 8 ion concentrations (calcium, magnesium, . . . ) – price per liter. Categorical variables – Total minerality (low, medium, high) – Compliance with regulations (yes, no) – Region (Alps, Auvergne, Languedoc, . . . ) L´ eon Bottou 6/86 COS 424 – 3/23/2010

Sparkling water springs �� ,��-�� )�� ,��' �� !�� !�"�� #�� $��$ �� %� �� % ��#� �� .��$ %�� # �� # �� & '�� (�$�� $/�� # �� #�� % �� 0��1��1�� & '�� )�� # �� % � �� 2�1��"�� %�� % �� '�� (�$�� # �� & '�� 1��*3�� # �� %�� % �� '�-�-�� % �� %� �% �� & '�� (�$�� 4�-5�� #%� �� $��$ �� 1�� %� �� # �� #� �� (�$�� *6�-�' %� %� #� �� %� �� $��$ �� *+�� # �� # �% �� % �� *�� # ��%� �� %� �� *7�� #% �� % �� $��$ �� 8�� # ��& �� (�$�� 8��9�� % �� 8��$'*�-�� ## � % ��% � �� $��$ �� :��&�� # � � �� (�$�� Active Supplementary variables variables L´ eon Bottou 7/86 COS 424 – 3/23/2010

Elementary planes Pairwise graphs are not informative L´ eon Bottou 8/86 COS 424 – 3/23/2010

Approximate a data cloud by its projection High dimensional cloud. Low dimensional projection. �� L´ eon Bottou 9/86 COS 424 – 3/23/2010

Some projections are more informative � �� The main idea of PCA is the determination of a good projection. L´ eon Bottou 10/86 COS 424 – 3/23/2010

One data table, two data clouds �� L´ eon Bottou 11/86 COS 424 – 3/23/2010

PCA projection of the 21 rows L´ eon Bottou 12/86 COS 424 – 3/23/2010

PCA projection of the 8 columns L´ eon Bottou 13/86 COS 424 – 3/23/2010

Summary Principal component analysis – Table of n observations represented by p continuous variables. – Cloud of n row-points (observations) in dimension p . – Cloud of p column-points (variables) in dimension n . – Search the “best” projection for each cloud. Interpretation – Identify similar observations. – Identify similar variables. Best projection ? L´ eon Bottou 14/86 COS 424 – 3/23/2010

Distance Distances – A good projection reveals whether two points were close or distant. – We would like to use the convenient Euclidian distance. – Variables often have very different numerical ranges. �� Correlation PCA Covariance PCA – Normalize the mean and – Normalize the mean of each variable, x ij = ( z ij − ¯ z j ) , but standard deviation of each variable, x ij = ( z ij − ¯ z j ) /σ j . not the standard deviation. – This is the default and this is – This is sometimes useful. what we discuss today. L´ eon Bottou 15/86 COS 424 – 3/23/2010

Descriptive and Exploratory Methods L eon Bottou largely copied - PowerPoint PPT Presentation

Descriptive and Exploratory Methods L eon Bottou largely copied from Mireille Summa-Gettler lectures (in french) COS 424 3/23/2010 Agenda Classification, clustering, regression, other. Goals Parametric vs. kernels vs. nonparametric

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