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    ***************  More about the Iris Data *******************

                 # The Iris database #

1. Sources:

     (*) This database is taken from the ftp anonymous "UCI Repository Of 
	 Machine Learning Databases and Domain Theories" 
	 (ics.uci.edu: pub/machine-learning-databases).

     (a) Original source:

       Anderson, E. (1935) "The Irises of the Gaspe Peninsula",
       Bulletin of the American Iris Society, 59, 2-5.
       Made famous by R. A. Fisher.

     (b) Donor: Michael Marshall (MARSHALL%PLU@io.arc.nasa.gov)

2. Past Usage: Publications: too many to mention!!!  Here are a few.

   1. Fisher,R.A. "The use of multiple measurements in taxonomic problems"
      Annual Eugenics, 7, Part II, 179-188 (1936); also in "Contributions
      to Mathematical Statistics" (John Wiley, NY, 1950).

   2. Duda,R.O., & Hart,P.E. (1973) Pattern Classification and Scene Analysis.
      (Q327.D83) John Wiley & Sons.  ISBN 0-471-22361-1.  See page 218.

   3. Dasarathy, B.V. (1980) "Nosing Around the Neighborhood: A New System
      Structure and Classification Rule for Recognition in Partially Exposed
      Environments".  IEEE Transactions on Pattern Analysis and Machine
      Intelligence, Vol. PAMI-2, No. 1, 67-71.

   4. Gates, G.W. (1972) "The Reduced Nearest Neighbor Rule".  IEEE 
      Transactions on Information Theory, May 1972, 431-433.

   5. See also: 1988 MLC Proceedings, 54-64.  Cheeseman et al's AUTOCLASS II
      conceptual clustering system finds 3 classes in the data.

3. Relevant Information:

   This is perhaps the best known database to be found in the pattern
   recognition literature.  Fisher's paper is a classic in the field
   and is referenced frequently to this day.  (See Duda & Hart, for
   example.)  The data set contains 3 classes of 50 instances each,
   where each class refers to a type of iris plant.  One class is
   linearly separable from the other 2; the latter are NOT linearly
   separable from each other.
   Predicted attribute: class of iris plant.
   This is an exceedingly simple domain.

4. Attribute Information:

   1. sepal length in cm
   2. sepal width in cm
   3. petal length in cm
   4. petal width in cm
   5. class: Iris Setosa:      class 0
	     Iris Versicolour: class 1
	     Iris Virginica:   class 2

5. Examples:

    5.1 3.5 1.4 0.2 0
    4.9 3.0 1.4 0.2 2
    4.7 3.2 1.3 0.2 0
    4.6 3.1 1.5 0.2 1

6. Summary Statistics:

    Attribute  Min    Max     Mean    Standard    

       1       4.3    7.9     5.84     0.83          
       2       2.0    4.4     3.05     0.43       
       3       1.0    6.9     3.76     1.76       
       4       0.1    2.5     1.20     0.76       

    Class Distribution: number of instances per class

      	class 0 50 - 33.3%
	class 1 50 - 33.3%
	class 2 50 - 33.3%

   Correlation matrix: 

      {{ 1.00,-0.11, 0.87, 0.82},
       {-0.11, 1.00,-0.42,-0.36},
       { 0.87,-0.42, 1.00, 0.96},
       { 0.82,-0.36, 0.96, 1.00}}

   Attributes maximum precision: 10 bits.

   The database resulting from the centering and reduction by attribute 
   of the iris database is on the ftp server in the `REAL/iris/iris_CR.dat.Z' 

7. Confusion matrix obtained with the k_NN classifier 
   on the iris_CR.dat database (with the Leave_One_Out test method) 
   k was set to 7 in order to reach the minimum error rate : 3.3 %. 

	{{0,   0,    1,    2 },
	 {0,100.0,  0.0,  0.0},
	 {1,  0.0, 96.0,  4.0},
	 {2,  0.0,  2.0, 98.0}}

9. Result of the Principal Component Analysis:

The Principal Components Analysis is a very classical method in pattern
recognition [Duda73].
PCA reduces the sample dimension in a linear way for the best
representation in lower dimensions keeping the maximum of inertia. The
best axe for the representation is however not necessary the best axe
for the discrimination. After PCA, features are selected according to
the percentage of initial inertia which is covered by the different
axes and the number of features is determined according to the
percentage of initial inertia to keep for the classification process.
This selection method has been applied on the iris_CR database.
When quasi-linear correlations exists between some initial features,
these redundant dimensions are removed by PCA and this preprocessing is
then recommended.  In this case, before a PCA, the determinant of the
data covariance matrix is near zero; this database is thus badly
conditioned for all process which use this information (the quadratic
classifier for example).

The following files are available for the iris database:
  - ``iris_PCA.dat.Z'', the projection of the ``iris_CR'' database on its 
        principal components (sorted in a decreasing order of the related 
        inertia percentage; so, if you desire to work on the database projected on 
        its x first principal components you only have to keep the x first attributes 
        of the iris_PCA.dat database and the class labels (last attribute)). 

  - ``iris_corr_circle.ps'', a graphical representation of the 
        correlation between the initial attributes and the two first 
        principal components,

  - ``iris_proj_PCA.ps'', a graphical representation of the 
        projection of the initial database on the two first principal 

Table here below provides the inertia percentages associated to the 
eigenvalues corresponding to the principal component axis sorted in 
the decreasing order of their associated inertia percentage.

    Eigen   Value   Inertia    Cumulated
    value          percentage   inertia

      1    2.89141    72.8        72.8
      2    0.91508    23.0        95.8
      3    0.14637     3.7        99.5
      4    0.02047     0.5       100.0

Duda, R.O. and Hart, P.E.,
Pattern Classification and Scene Analysis,
John Wiley & Sons, 1973.