Case Study: Iris Data

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    #unsup.
     #better
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Unsupervised classification

Unsupervised classification (clustering, mixture modelling) of Fisher's (Anderson's) Iris data.

The data type Flower literally defines the type of the data-set, e.g. for I/O. The species (String) is dropped (unzip5 ... zip4) before unsupervised classification. Recall that unsupervised classification requires a weighted estimator, here estimator4, if it is to be unbiased.


module Main where
import StatisticalModels

type SepalLength = Double                        -- @0, defining the attributes
type SepalWidth  = Double                        -- @1
type PetalLength = Double                        -- @2
type PetalWidth  = Double                        -- @3

                                       -- a datum is a 4-tuple (+ species name)
type Flower = (SepalLength, SepalWidth, PetalLength, PetalWidth, String)

dataAcc = 0.1                                      -- data measurement accuracy

variate4 m0 m1 m2 m3 =        -- 4 x model -> 4-variate model, (m0, m1, m2, m3)
  let nlP (d0,d1,d2,d3) =
        nlPr m0 d0 + nlPr m1 d1 + nlPr m2 d2 + nlPr m3 d3
  in MnlPr (msg1 m0 + msg1 m1 + msg1 m2 + msg1 m3)
           nlP
           (\() -> "[v4 " ++ show [m0,m1,m2,m3] ++ "]")

estVariate4Weighted e0 e1 e2 e3 ds ws =          -- 4 x estimator -> estimator4
  let (d0s, d1s, d2s, d3s) = unzip4 ds
  in variate4 (e0 d0s ws) (e1 d1s ws) (e2 d2s ws) (e3 d3s ws)
                                           -- NB weighted for mixture modelling

estimator4 = estVariate4Weighted                               -- our estimator
  (estNormalWeighted 0 10 0.1 5 dataAcc)
  (estNormalWeighted 0 10 0.1 5 dataAcc)
  (estNormalWeighted 0 10 0.1 5 dataAcc)
  (estNormalWeighted 0 10 0.1 5 dataAcc)            -- i.e. 4 indep. attributes

-- ------------------------------------------LA--24--Nov--2004--CS--Uni--York--

analyse est dataSet n =                                         -- 1..n classes
 let
  nlys m =
   if m > n then putStrLn( "--")
   else
   let mx = estMixture (take m (repeat est)) dataSet
   in putStrLn( show m ++ ": " ++ show mx )
   >> putStrLn( "message = "
      ++ show(msgBase 2 mx dataSet ) ++ " = " ++ show(msg1Base 2 mx)
      ++ " + " ++ show(msg2Base 2 mx dataSet) ++ " bits" )
   >> nlys (m+1)
 in nlys 1

main = print topline
 >> putStrLn "-- Fisher's / Anderson's Iris data --"
 >> readFile "data/iris.data"                            -- connect data file
 >>= \str ->
 let flowers = read str :: [Flower]                      -- read input
     (d0s, d1s, d2s, d3s, d4s) = unzip5 flowers
     dataSet  = zip4 d0s d1s d2s d3s                     -- NB. discard species
 in
 >> analyse estimator4  dataSet  4                       -- model @0...@3
 >> putStrLn "-- finish --"
-- ----------------------------------------------------------------------------

24 Nov 2004, C.Sci., U.York
[data/iris.data]
#classes message length (bits)
1 3113.8
2 2648.8
3 2573.6
4 2551.4
C.Sci., U.York, 24/11/2004
 
There are actually 3 species (the unsupervised classification algorithm is not told that), Iris-setosa (IS), Iris-versicolor (IVe), and Iris-virginica (IVi):
- @0 @1 @2 @3
Iris setosa N(5.0,0.35) N(3.42,0.38) N(1.46,0.17) N(0.24,0.11)
Iris versicolor N(5.94,0.52) N(2.77,0.31) N(4.26,0.47) N(1.33,0.20)
Iris virginica N(6.59,0.64) N(2.97,0.32) N(5.55,0.55) N(2.03,0.27)
IVe and IVi obviously overlap a great deal on @0, @1 and @2, and IS has notably smaller values than IVe & IVi for @2 and @3.
 
The slightly lower message length for 4 classes may be due to correlation between the 4 attributes (which could perhaps be "accounted for" by a suitable model).

A Better Model

A custom model and estimator were created to address correlations between "size" attributes.
Bayesian linear model
that is <@0, @1|@0, @2|@0, @3|@2> (also see [Bayesian nets])
custom4Model m0 fm1 fm2 fm3 =                        --    @0
 let nlP (d0,d1,d2,d3) =                             --   /  \
       nlPr m0 d0 + condNlPr fm1 d0 d1               -- @1    @2
       + condNlPr fm2 d0 d2 + condNlPr fm3 d2 d3     --        \
 in MnlPr (msg1 m0 + msg1 fm1 + msg1 fm2 + msg1 fm3) --         @3
    nlP
    ...and a show method...

estCustom4Weighted e0 e1 e2 e3 ds ws = ...

custom4 = estCustom4Weighted
 (estNormalWeighted 0 10 0.1 5 dataAcc)     -- @0
 (estLinear1Weighted (-5) 5 0.1 5 dataAcc)  -- @1|@0
 (estLinear1Weighted (-5) 5 0.1 5 dataAcc)  -- @2|@0
 (estLinear1Weighted (-5) 5 0.1 5 dataAcc)  -- @3|@2

main = ...
 >> analyse custom4 dataSet 4            -- cluster with the new model
Note the use of the normal distribution for @0 and of the linear regression for @1|@0, @2|@0 and @3|@2 above.
This model gave lower message lengths all round:
#classes message length (bits)
1 2898.8
2 2447.7 = 123.2+2324.5
3 2450.0
4 2450.2
11/11/2005
A two-class mixture is best; the three- and four-class mixtures were degenerate with the extra classes evaporating. When the "secret knowledge" of the three species and their memberships was used to create a "true" three-class mixture with this estimator, it gave a two-part message length of 2452.0 bits (= 173.3 + 2278.6), which is 4.3 bits worse than the 2-class mixture found without this knowledge. The 2nd-part, data|model, is better for three species v. two classes but the worse two-part total suggests that the the more complex "true" model would be a case of overfitting for the limited information given to the mixture modeller. Could a botanist confidently infer the existence of three species given only the four measurements from just 150 specimens?
Fisher 1936: "... I.setosa is a "diploid" species with 38 chromosomes, I.virginica is a "tetraploid" with 70, and I.versicolor ... is hexaploid. [Randolph suggested that IVe] is a polyploid hybrid of the two other species. ..." Note that the present data does not include this information.

Supervised Classification

Fisher used the Iris data for a case study in supervised classification; this is completely different from unsupervised classification. A classification-tree is one tool that can be used for supervised classification: From training-data, learn a function-model so that when given measurements from a new flower you can predict which species it most likely belongs to.

data Species = IS | IVe | IVi deriving (Bounded, Enum, Eq, Show)

string2Species "Iris-setosa"     = IS
string2Species "Iris-versicolor" = IVe
string2Species "Iris-virginica"  = IVi

op = map string2Species d4s           -- i.e. strings -> Species
ct = estCTree estMultiState splits dataSet op
The program yields this tree:
{CTfork @3<|>=1.0[
   {CTleaf mState[0.98, 0.01, 0.01]},
   {CTfork @3<|>=1.8[
      {CTleaf mState[0.01, 0.89, 0.10]},
      {CTleaf mState[0.01, 0.03, 0.96]}
]} ]}
tree :  62.7 = 28.7 + 34.0 bits
null : 237.7 = 150 * log2(3) bits
Recall that only the predicted variable, here Species, is compressed in supervised classification.

Notes

As given, the data happen to be grouped -- first IS then IVe then IVi. This is how Fisher presented it. Unsupervised classification makes no assumption of similarity between data that are near each other in the data-set; it should give the same answer if the data-set is randomly permuted. But if it is appropriate to make such an assumption, that proximity implies similarity, then you have a [segmentation problem].
 
Search the [bib'] for [Fisher Iris c193x].
The data set is available at the UCI machine-learning repository (2004).
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© L. Allison   http://www.allisons.org/ll/   (or as otherwise indicated),
Faculty of Information Technology (Clayton), Monash University, Australia 3800 (6/'05 was School of Computer Science and Software Engineering, Fac. Info. Tech., Monash University,
was Department of Computer Science, Fac. Comp. & Info. Tech., '89 was Department of Computer Science, Fac. Sci., '68-'71 was Department of Information Science, Fac. Sci.)
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