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Computational Intelligence:
Methods and Applications
Lecture 16
Model evaluation and ROC
Source: Włodzisław Duch; Dept. of Informatics, UMK;
Google: W Duch
Confusion matrices
For n observations, including n+ form class w+ and n- from other
classes labeled w- prediction of a model M are counted and put in
a confusion matrix (rows=true, columns=predicted by model M),
 T  w
1
P w | wM    w n
n
 w- n-
w


  w P
w P
 - -
wPP--
wr
w-
wr  M 
nn--
n r
n- r
w w 
 
P r P    w TP FN
P- r P-   w- FP TN
n
n-




wr


R P 
R- P- 
P (P-) is the priori probability of class w (w-) estimated from data.
FP and FN
Confusion matrix, including possibility of rejection (don’t know answer):
TP  P
P  true | predicted   
 FP  P-
FN  P- 
TN  P-- 
This notation is used especially in medical applications:
P is a hit or true positive (TP); P/P is a true positive rate;
P-- is a hit or true negative (TN); P--/P- is a true negative rate
P- false alarm, or false positive (FP); ex. healthy predicted as sick.
P- is a miss, or false negative (FN); ex. sick predicted as healthy.
Accuracy and errors
Models are frequently evaluated on the basis of their accuracy.
Elements of Pij depend on the evaluated model, Pij(M)
Accuracy A( M )  P ( M )  P-- ( M )
Error rate L( M )  P- ( M )  P- ( M )
Rejection rate R( M )  P r ( M )  P- r ( M ) 
 1 - L( M ) - A( M )
Accuracy for class k and balanced accuracy used for unbalanced data
(i.e. data with small
Ak M  Pkk M Pk
number of samples in
some classes).
1
 
BM  
 
A  M   A  M 

2

-
What is better?
Rank 4 classifiers producing the following confusion matrices:
0.25 0.25
0.4
P w | wM 1   
P  w | wM 2   

0.25 0.25
 0.1
0.25 0.5 
0.4
P w | wM 3   
P  w | wM 4   

 0.0 0.25
0.2
Accuracy 0.5
0.1
0.4
0.0
0.4
Accuracy 0.8
But which one is better for given accuracy?
M1 or M3? M2 or M4?
Ordering is meaningless because there are two parameters, not
one! For example, accuracy for class 1 and class 2.
Other evaluation measures
In medical applications: accuracy for class + is called sensitivity:
what percentage of really sick people does this test recognize?
Accuracy for class - is called specificity:
is this test specific to class + or does it always says + ?
In information retrieval sensitivity is called recall: what % of truly relevant
information has been recalled?
Precision measures % of really relevant info among all recalled info.
F-measure is the harmonic mean of recall and precision.
S  M   P|  M   P  M  / P
sensitivity, recall
S-  M   P-|-  M   P--  M  / P-
specificity
Prec  M   P  M  / P  M   P  M  /  P  M   P-  M  
F  M   2Recall  Prec  Recall  Prec  F - measure
Example
N=100 observations, for x=0 let 10 be from w, and 30 from w- classes,
and for x=1 there are 40 and 20 cases from w, and w- classes:
10 40
 0.1 0.4 
N w , x   
; P w , x   


30
20
0.3
0.2




P  x  0   0.4; P  x  1  0.6; P w   P w-   0.5;
1/ 4 2 / 3
P w | x   P w , x  / P  x   

3/
4
1/
3


MAP rule for this data is:
if x=0 then select w- because P(w-|x=0) >P(w|x=0)
if x=1 then select w because P(w|x=1) >P(w-|x=1)
If x=0 then 30 w- vectors are assigned to w-, or P(w-,w-)=0.3 and 10 w
vectors assigned to w- class, so P(w,w-)=0.1; P(w,w)=0.2; P(w-,w)=0.4
Example
If x=1 then 30 w vectors are assigned to w, or P(w,w)=0.4
and 20 w- vectors assigned to w class, so P(w-,w)=0.2
Therefore MAP decisions lead then to the confusion matrix:
with P=0.5, P- =0.5 and
Accuracy = 0.4+0.3=0.7,
Error = 0.1+0.2=0.3
0.4 0.1
P wi | w j  M    

0.2
0.3


Sensitivity =0.4/0.5=0.8 (recall)
Specificity =0.3/0.5=0.6
Balanced accuracy = (0.8+0.6)/2 = 0.7
Precision = 0.4/0.6 = 2/3 = 0.67
F-measure = 2*0.8*2/3 *1/(0.8+2/3) = 16/22 = 8/11=0.73
Error functions
Accuracy of the model M is usually maximized, it is a sum of TP+TN, or a
combination of sensitivity and specificity with the class priors as weights:
A M   P  M   P--  M   P S  M   P- S-  M 
This is obvious: class + has P+ fraction of all cases and sensitivity S+ is
the accuracy for this class, same for class -, so the sum gives accuracy.
Equivalently the error ratio may be minimized:
L  M   1 - A M   P-  M   P-  M 
Sometimes we would like to make decisions if we are quite sure that they
are correct. Confidence in model M may be increased by rejecting some
cases. Error=sum of the off-diagonal confusion matrix P(wi,wMj) elements
(true vs. predicted), accuracy=sum (trace) of diagonal elements, so
E ( M ; )   L  M  - A  M     P(wi , wMj ) - Tr P(wi , wMj )  -1
i j
More error functions
This combination for 2 classes (ex. one class against all the rest) is:
E( M ; )    P-  P-  -  P  P--   -1
Rejection rate, or the fraction of the samples that will not be classified is:
R( M )  1 - L( M ) - A( M )
Minimization of the error-accuracy is thus equivalent to error + rejection:
min E ( M ;  )  min (1   ) L( M )  R( M )
M
M
For   0 this is equal to error + rejection; for large  minimization of this
error function over parameters of the model M reduces the sum of FP and
FN errors, but at a cost of growing rejection rate; for example if the model
M {x >5 then w1 else w2 } makes 10 errors, all for x[5,7] then leaving
samples in this range unclassified gives M’ {x >7 then w1 or x 5 then w2 },
no errors, but lower accuracy, higher rejection rate and high confidence.
Errors and costs
Optimization with explicit costs:
assume that FP (false alarms) cost a times more than FN (misses).
min E ( M ;a )  min P- ( M )  a P- ( M )
M
M
 min P (1 - S  ( M )) - P r ( M ) 
M
a  P- (1 - S - ( M )) - P- r ( M )
For a = 0 this is equivalent to maximization of
min E ( M ;a  0)  min P ( M )  P r ( M )
M
and for large a to the maximization of
min E ( M ;a
M
1)  min P-- ( M )  P- r ( M )
Lifts and cumulative gains
Technique popular in marketing, where cumulative gains and “lifts” are
graphically displayed: lift is a measure of how effective model predictions
are = (results obtained with)/(without the predictive model).
Ex: is Xi likely to respond? Should I send him an offer?
Suppose that 20% of people respond to your offer.
Sending this offer randomly to N people gives Y0=0.2*N replies.
A predictive model (called “response model” in marketing),
P(w|X;M) uses information X to predicts who will respond.
Order predictions from the most likely to the least likely:
P(w|X1;M) > P(w|X2;M) ... > P(w|Xk;M)
The ideal model should put those 20% that will reply in front, so that the
number of replies Y(Xj) grows to Y0=0.2*N for j=1 .. Y0.
In the ideal case cumulative gain will be then a linear curve reaching Y0
and then remaining constant; lift will then be the ratio Y(Xj)/0.2*j.
Cumulative gains chart
There is no ideal model, so check your predictions P(w|X;M) against
reality. If the prediction was true plot next point one unit to the right, one
unit up; if prediction was false plot it just one unit to the right. The vertical
axis contains P+ portion of all data samples Y0 = the number of all that
responded (in this case Y0=1000) out of all N=5000 people contacted.
Rank P(w|X) True
1
0.98
+
2
0.96
+
3
0.96
-
4
0.94
+
5
0.92
-
........................
See more here
1000
0.08
-
ROC intro
Receiver Operator Characteristic evaluate TP (or P++ rate) as a function
of some threshold; for example using the likelihood ratio:
P  X | w1  P w1   P  X | w 2  P w 2 
P  X | w1  P w2 
 X 


P  X | w2  P w1 
the threshold may be treated as a
variable measuring confidence;
for 1D distributions it is clear that for
a large threshold only positive cases
will be left, but their proportion S+
(recall, sensitivity) decreases to zero.
What is the optimal choice? Depends on the ratio of false alarms (FP,
false positives, P-) we are willing to accept, or proportion of 1-S-= P/P-
ROC curves
ROC curves display (S+,1-S-), or error of class w- versus accuracy
of class w+ for different thresholds:
Ideal classifier: below some
threshold S+ = 1 (all positive
cases recognized) for 1-S-= 0
(no false alarms).
Useless classifier (blue): same
number of true positives as
false alarms for any threshold.
Reasonable classifier (red):
no errors until some threshold that allows for recognition of 0.5 positive
cases, no errors if 1-S- > 0.6; slowly rising errors in between.
Good measure of quality: high AUC, Area Under ROC Curve.
AUC = 0.5 is random guessing, AUC = 1 is perfect prediction.
Realistic ROC
ROC curve for realistic case: finite number of thresholds and data
points makes it rugged. Instead of the percentage of true positives
(recall, sensitivity) precision is sometimes used.
Each ROC point captures all information contained in confusion matrix
for some parameters (thresholds) of the model and shows for different
probabilities confidence in predictions of the classifier.
Try Yale tutorial 14 showing ROC plots.
ROC for combination of classifiers
More convex ROC curves show superiority of models for all
thresholds. Here ROC curves for two models show strong and
weak areas: combination of the results of the two models may give
a ROC curve covering the grey area.
ROC for logical rules
Rules for classifiers that give only yes/no predictions, for example
logical rules, give only one point on the ROC curves, at (S, 1-S-).
AUC = (S + S-)/2, and it is identical along S1-S-=const line.
Try the demo of AccuROC for Windows (sample output above)
http://www.accumetric.com/accurocw.htm
Computational Intelligence:
Methods and Applications
Lecture 17
WEKA/RapidMiner
Knowledge extraction
from simplest decision trees
Source: Włodzisław Duch; Dept. of Informatics, UMK;
Google: W Duch
A few data mining packages
A large number of data mining packages that include many CI models
for data analysis is available.
See long list of DM software, including large commercial packages.
GhostMiner, from Fujitsu (created by our group); please get it.
WEKA started the trend to collect many packages in one system.
RapidMiner, formerly YALE – initially a better front-end to WEKA,
includes all WEKA models, free source; please get it.
New interesting projects: see my list of software.
Orange, component-based data mining software, includes
visualizations, SOM/MDS modules.
KNIME, based on Eclipse platform, includes Weka and R-scripts,
modular data exploration platform, visual data flows.
R-project, language for statistical computing and graphics.
WEKA Project
Machine learning algorithms in Java:
I.H. Witten, E. Frank, Data Mining: Practical Machine Learning Tools
and Techniques with Java Implementations. Morgan Kaufmann 1999
Project Web page: www.cs.waikato.ac.nz/ml/weka
One of the most popular packages.
Essentially a collection of Java class libraries implementing various
computational intelligence algorithms.
ARFF data format, with data in CSV format (comma separated,
exportable from spreadsheets), and additional information about the
data, type of each feature, etc.
CLI, or command line interface (only for Unix lowers), ex:
java weka.classifiers.j48.J48 -t data/weather.arff
calls one of the methods (here J48) from the library.
WEKA Software
“Explorer GUI” for making basic calculations, recently much improved.
“Experimenter” and “Knowledge Flow” environments for performing more
complex experiments is provided, this allows for averaging over
crossvalidation results or combining different models.
On-line documentation for library classes but little description of methods.
WEKA/RM software contains:
• preprocessing filters, supervised and unsupervised
• many classification models
• rule-based models for knowledge discovery
• association rules (one method)
• regression, or numerical prediction models
• 3 clusterization (unsupervised learning) methods
• scatterogram visualization (2D)
• a collection of sample simple problems (from the UCI repository).
More WEKA/RM Software
Simple “Explorer GUI” for making basic calculations.
Rather rough “Experimenter” environment for performing more
complex experiments, such as averaging over crossvalidation results
or combining different models is provided.
On-line documentation for library classes.
WEKA/RM software contains:
• preprocessing filters, supervised and unsupervised
• classification models
• rule-based models for knowledge discovery
• association rules (one method)
• regression, or numerical prediction models
• 3 clusterization (unsupervised learning) methods
• scatterogram visualization (2D)
• a collection of sample problems (from UCI repository)
WEKA strong/weak points
Platform independent – Java!
Many projects created around it: listed here.
Free, contains large collection of filters and algorithms.
May be extended by a serious user.
But ... Java programs are not so stable as Windows programs,
there are problems with some Java versions;
rather poor visualization of data and results;
RapidMiner is a big improvement.
Simple user interface has been improved recently,
Knowledge Flow GUI changes this.
Requires tedious programming to perform experiments – RM is easier.
Algorithms are not described in details in documentation and in the book
(only in the class libraries).
RapidMiner
Like WEKA, same models + few more, easier to use.
Free, contains large collection of filters and algorithms.
May be extended by a serious user.
Download RapidMiner, start it and read the tutorial!
Includes 20 visual data exploration methods: scatter, scatter matrix,
interactive scatter 3D, parallel, 2D density, radial radviz, gradviz,
SOM (U-distance and P-density).
Unfortunately algorithms are not described in details in the
documentation and you have to study class libraries to understand what
exactly GridViz or RadViz does, or read original papers to understand
what U, U* and P matrix SOM visualization is.
Check much better descriptions of methods in Orange!
Knowledge representation
Knowledge representation is an important subject in Artificial
Intelligence, here only simple forms of knowledge are considered.
Decision rules:
prepositional rules: IF (all conditions are true) THEN facts
M-of-N rules: IF (M conditions of N are true) THEN facts
fuzzy rules: IF (conditions true to some degree) THEN
facts are true to some degree
Linguistic variables: favorite-colors, low-noise-level, young-age, etc:
• subsets of nominal or discrete values,
• intervals of numerical values, ex: teenager ={T if age<20}
• constrained subsets of numerical values.
WEKA/RM filters
Many filters that can be applied to attributes (features) or to instances
(vectors, samples), some specific to signal/time series data.
• Divided into supervised/unsupervised, attribute or instance.
• Create new attribute from existing ones using algebraic operations;
• remove instances with attribute values in some range, for example
•
•
•
•
•
•
missing values; delete attributes of specific type (ex. binary)
change nominal values to binary combinations, ex.
Xi{a,b,c,d} => (Xi1,Xi2)({0,1},{0,1})
rank the usefulness of attributes (several schemes);
evaluate usefulness of subsets of features (several schemes);
perform PCA; normalize features in many ways;
discretize attributes, define simple bins or look for more natural
discretization, for example bins created by the Minimum
Description Length (MDL) principle (called “use Kononenko”).
many others ...
Classification algorithms
Divided into:
• Bayes – versions of probabilistic Bayesian methods
• Functions – parameterized functions, linear and non-linear
• Lazy – no parameter learning, all work done when classifying
• Meta – committees, voting, boosting, stacking ... metamodels.
• Misc – untypical models, fuzzy lattice, hyperpipes, voting features
• Tree-building models, recursive partitioning
• Rule learning models
These algorithm enable:
• knowledge discovery, or data mining (trees, rules);
• predictive modeling in classification or regression tasks.
See WEKA detailed presentation:
http://prdownloads.sourceforge.net/weka/weka.ppt
Decision rules
Algorithm for knowledge discovery, or data mining, 10 rule and 10 treebased, providing knowledge in form of logical rules.
•
•
•
•
Zero-R, predicting majority class (or mean values)
One-R, simplest one-level (one attribute) decision tree.
Decision stump, one-level tree
C4.5, called here J.48, since this is Java implementation of the
version 8 of C4.5 decision tree algorithm.
• M5’ model tree learner.
• Naive Bayes tree classifier.
• PART rule learner (covering algorithm).
Prototype – based algorithms:
• Instance –based learner (IB1, IBk, ID3) nearest neighbor method
• Decision table
Regression algorithms
Regression (function) and classification algorithms include:
• Naive Bayes (2 versions)
• Linear Regression, or LDA
• Additive regression
• Logistic regression
• LWR, Locally Weighted Regression
• MLP (multi-layer perceptron) neural network,
• VPN, voted perceptron network
• SMO, or Support Vector Machine algorithm
• K*, similarity based system with algorithmic complexity
minimization.
Other algorithms
Statistical algorithms for model improvement (meta-algorithms):
• bagging,
• boosting,
• adaboost
• logit boost,
• stacking
Clusterization:
• K-means,
• Expectation Maximization,
• Cobweb
Association: find relations between attributes.
Visualization of 2D scatterograms
WEKA/RM example
Contact lenses: do I need hard, soft or none?
Very small data set, 24 instances: contact-lens.arff
What is in the database?
1. age of the patient: (1) young, (2) pre-presbyopic, (3) presbyopic
2. spectacle prescription: (1) myope, (2) hypermetrope
3. astigmatic: (1) no, (2) yes
4. tear production rate: (1) reduced, (2) normal
Class Distribution:
1. hard contact lenses: 4
2. soft contact lenses: 5
3. no contact lenses: 15
ZeroR
Zero method:
• for a small number of classes (categorical class variables) predict
the majority class;
• for numerical outputs (regression problems) predict the average.
Useful to establish the base rate, zero variance, large bias:
if any method obtains results that are worse than ZeroR serious
overfitting of data occurs.
For contact-lenses: confusion matrix
=== Confusion Matrix ===
a
0
0
0
b c <= classified as
0 5 | a = soft
0 4 | b = hard
0 15 | c = none
15 classified correctly, 62.5%
on the whole data.
What happens in 10xCV?
DT - idea
Class: {cancer,
healthy}
Features: cell body: {gray, stripes}
nuclei: {1, 2}; tails: {1, 2}
cancer
healthy
healthy
cancer
More ambitious tree
1R
1R: simplest useful tree (Holte 1993), sometimes results are good.
One level tree, nominal attributes.
1R algorithm:
for every attribute X
for every attribute value Xi:
count the class frequencies N(Xi,wj)
find the most frequent class c = arg maxj N(Ai,wj)
create a rule (majority classifier): IF Xi THEN wc
Calculate accuracy of this rule.
Select rules of highest accuracy.
Missing value ? is treated as any other nominal value.
1R example
Example taken from WEKA book: weather condition and decision to
play an in-door games (tennis); 14 examples are given
Task: find the decision rule (weather.nominal.arff).
Attribute: Outlook
Outlook = Sunny has 3
examples with No and 2
with Yes;
Outlook = Overcast has 4
examples with Yes
Rainy has 3 examples
with Yes and 2 with No;
Optimal rules: using only Outlook, or only Humidity.
Dataset is too small to evaluate accuracy, but rules are reasonable.
1R continuous
How should the continuous values be treated?
Divide the range of continuous attribute into intervals Ii(X) (discretize the
attribute);
treat intervals as nominal values, i.e. write X=Ii if XIi(X).
For each attribute X
sort all cases according to the increasing X values;
define the intervals Ii(X) where class wc dominates, maxc N(Ii(X),wc).
This should decrease the number of errors in 1R algorithm.
Problem: if the data is noisy or some examples are quite untypical, rules
should not be created!
A simpler solution: use buckets, or intervals with minimum # of elements,
admitting some “impurities”.
1R example with continuous variables
Discretization of temperature, instead of hot, mild, cool.
To avoid noise, intervals containing not less than 4 elements are used.
WEKA implementation:
Bucket size = min number
of elements in interval.
Slightly more accurate
solution is found with
B=2-4, but B=1 has only
1 error – is it good or bad?
Still it makes no sense ...
1R example with continuous variables
Discretization of humidity, instead of high/normal (small font = no):
65 70 70 70 75 80 80 85 86 90 90 91 95 96
To avoid noise, bucket containing B4 elements are used.
65 70 70 70 75 80 80 | 85 86 90 90 91 95 | 96
Rather naive, adds to bucket until non-majority class sample is found.
WEKA implementation:
Bucket size = min number
of elements in interval.
Slightly more accurate
solution is found with
B=2-4, but B=1 has only
1 error – is it good or bad?
Still it makes no sense ...
Netflix 1M$ Prize
Netflix Prize, an award of $1 million to the first person or team who
can achieve certain accuracy goals when recommending movies
based on personal preferences – announced in Oct 2006.
The company made 100 million anonymous movie ratings available to
contestants for learning.
Details for registering and competing for the Netflix Prize are at:
http://www.netflixprize.com
All members of the CI/machine learning community are invited to
participate!
10% improvement is needed, 9.65% achieved in April 2009,
RMSE <= 0.8563, achieved 0.8596.
Computational Intelligence:
Methods and Applications
Lecture 18
Decision trees
Source: Włodzisław Duch; Dept. of Informatics, UMK;
Google: W Duch
Multi-level DT
First split on the best test/attribute, then continue recursively.
Children nodes may be split using different test/attributes, sometimes
repeating those already used in the parent nodes.
General DT properties
DT: first general, later more specific decisions
node  test on attribute, selecting subsets or intervals
branching  splitting data vectors into subsets
leaves of the tree are associated with decisions (classes)
Tests: on a single attribute, or their combination
attribute = {value1,value2 ..} or attribute < value1
Criteria: maximize information gain, maximize the purity of new nodes;
maximize separability of subset vectors
Pruning: remove branches that contain only a few cases,
simple trees may generalize better (lower variance, higher bias)
evaluation optimal tree complexity on validation set.
Stop criterion: node purity, accuracy, tree complexity.
Attribute selection
Which attribute should be taken next, A1 or A2? Which test?
P+ and P- are a priori w±
class probabilities in the
node S, test divides the
S set into St and Sf.
How much information is gained by splitting? Calculate entropy!
Optimal no. of bits to code message with P+ is ~ lg2 P+, same for P-
H (ω |S)  - P lg2 P - P- lg2 PG (ω, A|S)  H (ω | S) -
St
S
H ( ω | St ) -
Sf
S
H (ω | S f )
Entropy of “pure” nodes (vectors from one class) is 0;
Max. entropy is for a node with mixed samples Pi=1/2.
Plot of H(P+|S)
for P+ =1-P-
Entropy content
Information is just negative entropy
E, thus entropy change = info gain:
H  p1 , p2 ,... pn   - pi lg2 pi
Weather example (from WEKA book)
i
Creation of a decision tree
Creation of a tree  search in the
hypothesis space for the simplest
set of hierarchical rules (most
compact tree).
ID3 tree – split criterion based on
information gain.
Bias: smaller trees are better, if equal
information gain select split with lower
number of branches.
No backtracking.
Rather robust in the presence of
noise, although local (greedy) search
does not guarantee that the final tree
will be optimal.
Decision borders
Hierarchical partitioning of feature space into hyper-rectangles.
Example: Iris flowers data, with 4 features; displayed in 2-D.
Uni and multi-variate criterions
Univariate, or
monothetic trees,
mult-variate, or
oblique trees.
Figure from
Duda, Hart & Stork,
Chap. 8
DTs are not stable
Moving just one
example slightly
may lead to quite
different trees and
space partition!
Lack of stability
against small
perturbation of data.
Figure from
Duda, Hart & Stork,
Chap. 8
Ockham’s razor
Why simple trees should be preferred?
1.
2.
3.
The number of simple hypothesis that may accidentally fit the
data is small, so chances that simple hypothesis uncover some
interesting knowledge about the data are larger.
Simpler trees have higher bias and thus lower variance, they
should not overfit the data that easily.
Simpler trees do not partition the feature space into too many
small boxes and may generalize better; complex trees may create
a separate box for each training data sample.
Still, even if the tree is small ...
sufficiently long search by pure chance may find false solution;
for small datasets with many attributes several equivalent (from
the accuracy point of view) descriptions may exist.
=> One tree is not sufficient, we need a forest of healthy trees!
Overfitting
A model H overfitts the data if:
A model H’ exists such that:
Training-error(H) < Training-error(H’)
Test-error(H) > Test-error(H’)
Model H draws conclusions (makes hypothesis) that are too
detailed for the amount of evidence available.
Accuracy as a function of
the number of tree nodes:
on the training data it may
grow up to 100%, but the
final results may be worse
than for the majority
classifier!
Random data example
Generate and label randomly data samples as class w1 or class w2,
with the a priori probability of the majority class P(w1)=p>0.5
Majority classifier makes Emaj=1-p percent of errors.
Overfitted tree that classifies correctly all training data has:
N·p nodes from w1 class
N-N·p nodes from w2 class.
A random X is assigned to w1 class with prob. p and w2 with 1-p.
Confusion matrix:
Tree error/Majority classifier error
 p2

 (1 - p) p
p(1 - p) 
2 
(1 - p) 
2 p(1 - p )
 2 p  1 for p  0.5
(1 - p )
For p=0.75 overfitted tree makes 37.5% errors,
while the majority classifier will make only 25% errors.
Some examples
Please run a few example of decision tree solutions using Rapid Miner or
WEKA on benchmark data using WEKA knowledge explorer or RM.