## Stepwise Forward Selection Algorithm From Scratch

In this post, I will walk you through the Stepwise Forward Selection algorithm, step-by-step. We will develop the code for the algorithm from scratch using Python and use it for feature selection for the Naive Bayes algorithm we previously developed. We will then run the algorithm on a real-world data set, the iris data set (flower classification) from the UCI Machine Learning Repository. Without further ado, let’s get started!

# What is the Stepwise Forward Selection Algorithm?

One of the fundamental ideas in machine learning is the Curse of Dimensionality – as the number of attributes in the training set grows, you need exponentially more instances to learn the underlying concept. And the more instances you have, the more computational time is required to process the data.

In other words, adding attributes improves performance, to a point. Beyond that point, you need to have exponentially more instances in order for performance to improve. This phenomenon is commonly known as the Hughes Effect (or Hughes Phenomenon).

In the above image, n = number of attributes and m = number of instances.

One technique for combatting the Curse of Dimensionality is known as Stepwise Forward Selection (SFS). SFS involves selecting only the most relevant attributes for learning and discarding the rest. The metric that determines what attribute is “most relevant” is determined by the programmer. In this project, I will use the classification accuracy of Naïve Bayes.

Because SFS is just a method to create an optimal attribute subset, it needs to be wrapped with another algorithm (in this case Naïve Bayes) that does the actual learning. For this reason, SFS is known as a wrapper method.

The goal of SFS is to reduce the number of features that a machine learning algorithm needs to examine by eliminating unnecessary features and finding an optimal attribute subset.

Suppose we have attributes A1 through Ak which are used to describe a data set D.

`A = <A1,…Ak>`

SFS begins with an empty attribute set:

`A0 = <>`

It then selects an attribute from A, one at a time, until the performance (e.g. as measured by classification accuracy) of the target learning algorithm (e.g. Naïve Bayes) on a subset of the training data fails to improve performance.

# Stepwise Forward Selection Implementation

The first thing we need to do is preprocess the iris data set so that it meets the input requirements of both algorithms. Below is the required data format. I downloaded the data into Microsoft Excel and then made sure I had the following columns:

Columns (0 through N)

• 0: Instance ID
• 1: Attribute 1
• 2: Attribute 2
• 3: Attribute 3
•  …
• N: Actual Class

The program I developed (code is below) then adds 2 additional columns for the testing set.

• N + 1: Predicted Class
• N + 2: Prediction Correct? (1 if yes, 0 if no)

Here is what your data set should look like:

The iris data set contains 3 classes of 50 instances each (150 instances in total), where each class refers to a different type of iris flower. There are 4 attributes in the data set. Here is what an iris flower looks like in case you are interested:

Binarization is optional, but I prefer to do it since it speeds up the attribute selection process. You need to go through each attribute, one by one. Attribute values greater than the mean of the attribute need to be changed to 1, otherwise set it to 0. Here is the general Excel formula:

`=IF(B2>AVERAGE(B\$2:B\$151),1,0)`

Here is how the data looks after the binarization:

Save the file as a csv file (comma-delimited), and load it into the program below (Python).

Here is a link to the final data set I used.

## Stepwise Forward Selection Algorithm in Python, Coded From Scratch

Naïve Bayes was wrapped with SFS for feature selection. Classification accuracy was used to select the optimal attribute at each step. Once the optimal attribute subset was obtained, Naïve Bayes was run on the complete training set (67% of instances) and testing set (33% of instances), and classification accuracy was calculated.

Here is the code. Just copy and paste it into your favorite IDE. Don’t be scared at how long the code is. I use a lot of comments so that you know what is going on at each step:

```import pandas as pd # Import Pandas library
import numpy as np # Import Numpy library

# File name: sfs_naive_bayes.py
# Date created: 6/10/2019
# Python version: 3.7
# Description: Implementation of Naive Bayes which uses Stepwise Forward
# Selection (SFS) for feature selection. This code works for multi-class
# classification problems (e.g. democrat/republican/independent)
# Calculate P(E1|CL0)P(E2|CL0)P(E3|CL0)...P(E#|CL0) * P(CL0) and
# P(E1|CL1)P(E2|CL1)P(E3|CL1)...P(E#|CL1) * P(CL1) and
# P(E1|CL2)P(E2|CL2)P(E3|CL2)...P(E#|CL2) * P(CL2), etc. and predict the
# class with the maximum result.
# E is an attribute, and CL means class.
# Only need class prior probability and likelihoods to make a prediction
# (i.e. the numerator of Bayes formula) since denominators are same for both
# the P(CL0|E1,E2,E3...)*P(CL0) and P(CL1|E1,E2,E3...)*P(CL1), etc. cases
# where P means "probability of" and | means "given".

# Required Data Set Format:
# Columns (0 through N)
# 0: Instance ID
# 1: Attribute 1
# 2: Attribute 2
# 3: Attribute 3
# ...
# N: Actual Class

# This program then adds 2 additional columns for the testing set.
# N + 1: Predicted Class
# N + 2: Prediction Correct? (1 if yes, 0 if no)

################ INPUT YOUR OWN VALUES IN THIS SECTION ######################
ALGORITHM_NAME = "SFS-Wrapped Naive Bayes"
DATA_PATH = "iris_dataset.txt"  # Directory where data set is located
TEST_STATS_FILE = "iris_dataset_naive_bayes_test_stats.txt"#Testing statistics
TEST_OUT_FILE = "iris_dataset_naive_bayes_test_out.txt" # Testing output
# Show functioning of the program
TRACE_RUNS_FILE  = "iris_dataset_naive_bayes_trace_runs.txt"
SEPARATOR = ","  # Separator for the data set (e.g. "\t" for tab data)
TRAINING_DATA_PRCT = 0.67 # % of data set used for training. Default 0.67
testing_data_prct = 1 - TRAINING_DATA_PRCT # % of data set used for testing
SEED = 99  # SEED for the random number generator. Default: 99
#############################################################################

# Open a new file to save trace runs
outfile3 = open(TRACE_RUNS_FILE,"w")

# Naive Bayes algorithm. Accepts a Pandas dataframeas its parameter
def naive_bayes(pd_data):
# Create a training dataframe by sampling random instances from original data.
# random_state guarantees that the pseudo-random number generator generates
# the same sequence of random numbers each time.
pd_training_data = pd_data.sample(frac=TRAINING_DATA_PRCT, random_state=SEED)

# Create a testing dataframe. Dropping the training data from the original
# dataframe ensures training and testing dataframes have different instances
pd_testing_data = pd_data.drop(pd_training_data.index)

######COMMENT OUT THESE TWO LINES BEFORE RUNNING ################
#pd_training_data = pd_data.iloc[:20] # Used for testing only, gets 1st 20 rows
#pd_testing_data = pd_data.iloc[20:22,:] # Used for testing only, rows 20 & 21

# Calculate the number of instances, columns, and attributes in the
# training data set. Assumes 1 column for the instance ID and 1 column
# for the class. Record the index of the column that contains
# the actual class
no_of_instances_train = len(pd_training_data.index) # number of rows
no_of_columns_train = len(pd_training_data.columns) # number of columns
no_of_attributes = no_of_columns_train - 2
actual_class_column = no_of_columns_train - 1

# Store class values in a column, sort them, then create a list of unique
# classes and store in a dataframe and a Numpy array
unique_class_list_df = pd_training_data.iloc[:,actual_class_column]
unique_class_list_df = unique_class_list_df.sort_values()
unique_class_list_np = unique_class_list_df.unique() #Numpy array
unique_class_list_df = unique_class_list_df.drop_duplicates()#Pandas df

# Record the number of unique classes in the data set
num_unique_classes = len(unique_class_list_df)

# Record the frequency counts of each class in a Numpy array
freq_cnt_class = pd_training_data.iloc[:,actual_class_column].value_counts(
sort=True)

# Record the frequency percentages of each class in a Numpy array
# This is a list of the class prior probabilities
class_prior_probs = pd_training_data.iloc[:,actual_class_column].value_counts(
normalize=True, sort=True)

pd_testing_data = pd_testing_data.reindex(
columns=[*pd_testing_data.columns.tolist(
), 'Predicted Class', 'Prediction Correct?'])

# Calculate the number of instances and columns in the
# testing data set. Record the index of the column that contains the
# predicted class and prediction correctness (1 if yes; 0 if no)
no_of_instances_test = len(pd_testing_data.index) # number of rows
no_of_columns_test = len(pd_testing_data.columns) # number of columns
predicted_class_column = no_of_columns_test - 2
prediction_correct_column = no_of_columns_test - 1

######################### Training Phase of the Model ########################
# Create a an empty dictionary
my_dict = {}

# Calculate the likelihood tables for each attribute. If an attribute has
# four levels, there are (# of unique classes x 4) different probabilities
# that need to be calculated for that attribute.
# Start on the first attribute and make your way through all the attributes
for col in range(1, no_of_attributes + 1):

# Record the name of this column
colname = pd_training_data.columns[col]

# Create a dataframe containing the unique values in the column
unique_attribute_values_df = pd_training_data[colname].drop_duplicates()
# Create a Numpy array containing the unique values in the column
unique_attribute_values_np = pd_training_data[colname].unique()

# Calculate likelihood of the attribute given each unique class value
for class_index in range (0, num_unique_classes):

# For each unique attribute value, calculate the likelihoods
# for each class
for attr_val in range (0, unique_attribute_values_np.size) :
running_sum = 0

# Calculate N(unique attribute value && class value)
# Where N means "number of" and && means "and"
# Go through each row of the training set
for row in range(0, no_of_instances_train):
if (pd_training_data.iloc[row,col] == (
unique_attribute_values_df.iloc[attr_val])) and (
pd_training_data.iloc[row, actual_class_column] == (
unique_class_list_df.iloc[class_index])):
running_sum += 1

# With N(unique attribute value && class value) as the numerator
# we now need to divide by the total number of times the class
# appeared in the data set
try:
denominator = freq_cnt_class[class_index]
except:
denominator = 1.0

likelihood = running_sum / denominator

# Add a new likelihood to the dictionary
# Format of search key is
# <attribute_name><attribute_value><class_value>
search_key = str(colname) + str(unique_attribute_values_df.iloc[
attr_val]) + str(unique_class_list_df.iloc[
class_index])
my_dict[search_key] = likelihood

# Print the likelihood table to the console
# print(pd.DataFrame.from_dict(my_dict, orient='index'))

################# End of Training Phase of the Naive Bayes Model ########

################# Testing Phase of the Naive Bayes Model ################

# Proceed one instance at a time and calculate the prediction
for row in range(0, no_of_instances_test):

# Initialize the prediction outcome
predicted_class = unique_class_list_df.iloc[0]
max_numerator_of_bayes = 0.0

# Calculate the Bayes equation numerator for each test instance
# That is: P(E1|CL0)P(E2|CL0)P(E3|CL0)...P(E#|CL0) * P(CL0),
# P(E1|CL1)P(E2|CL1)P(E3|CL1)...P(E#|CL1) * P(CL1)...
for class_index in range (0, num_unique_classes):

# Reset the running product with the class
# prior probability, P(CL)
try:
running_product = class_prior_probs[class_index]
except:

# Calculation of P(CL) * P(E1|CL) * P(E2|CL) * P(E3|CL)...
# Format of search key is
# <attribute_name><attribute_value><class_value>
# Record each search key value
for col in range(1, no_of_attributes + 1):
attribute_name = pd_testing_data.columns[col]
attribute_value = pd_testing_data.iloc[row,col]
class_value = unique_class_list_df.iloc[class_index]

# Set the search key
key = str(attribute_name) + str(
attribute_value) + str(class_value)

# Update the running product
try:
running_product *= my_dict[key]
except:
running_product *= 0

# Record the prediction if we have a new max
# Bayes numerator
if running_product > max_numerator_of_bayes:
max_numerator_of_bayes = running_product
predicted_class = unique_class_list_df.iloc[
class_index] # New predicted class

# Store the prediction in the dataframe
pd_testing_data.iloc[row,predicted_class_column] = predicted_class

# Store if the prediction was correct
if predicted_class == pd_testing_data.iloc[row,actual_class_column]:
pd_testing_data.iloc[row,prediction_correct_column] = 1
else:
pd_testing_data.iloc[row,prediction_correct_column] = 0

print("-------------------------------------------------------")
print("Learned Model Predictions on Testing Data Set")
print("-------------------------------------------------------")

# Print the revamped dataframe
print(pd_testing_data)

# accuracy = (total correct predictions)/(total number of predictions)
accuracy = (pd_testing_data.iloc[
:,prediction_correct_column].sum())/no_of_instances_test
# Return classification accuracy
return accuracy
####################### End Testing Phase ######################################

# Stepwise forward selection method accepts a Pandas dataframe as a parameter
# and then returns the optimal dataframe from that dataframe.
def sfs(pd_data):

# Record the name of the column that contains the instance IDs and the class
instance_id_colname = pd_data.columns[0]
no_of_columns = len(pd_data.columns) # number of columns
class_column_index = no_of_columns - 1
class_column_colname = pd_data.columns[class_column_index]

# Record the number of available attributes
no_of_available_attributes = no_of_columns - 2

# Create a dataframe containing the available attributes by removing
# the Instance and the Class Column
available_attributes_df = pd_data.drop(columns = [
instance_id_colname, class_column_colname])

# Create an empty optimal attribute dataframe containing only the
# Instance and the Class Columns
optimal_attributes_df = pd_data[[instance_id_colname,class_column_colname]]

# Set the base performance to a really low number
base_performance = -9999.0

# Check whether adding a new attribute to the optimal attributes dataframe
# improves performance
# While there are still available attributes left
while no_of_available_attributes > 0:
# Set the best performance to a low number
best_performance = -9999.0

# Initialize the best attribute variable to a placeholder
best_attribute = "Placeholder"

# For all attributes in the available attribute data frame
for col in range(0, len(available_attributes_df.columns)):

# Record the name of this attribute
this_attr = available_attributes_df.columns[col]

# Create a new dataframe with this attribute inserted
temp_opt_attr_df = optimal_attributes_df.copy()
temp_opt_attr_df.insert(
loc=1,column=this_attr,value=(available_attributes_df[this_attr]))

# Run Naive Bayes on this new dataframe and return the
# classification accuracy
current_performance = naive_bayes(temp_opt_attr_df)

# Find the new attribute that yielded the greatest
# classification accuracy
if current_performance > best_performance:
best_performance = current_performance
best_attribute = this_attr

if best_performance > base_performance:
base_performance = best_performance

# Add the best attribute to the optimal attribute data frame
optimal_attributes_df.insert(
loc=1,column=best_attribute,value=(
available_attributes_df[best_attribute]))

# Remove the best attribute from the available attribute data frame
available_attributes_df = available_attributes_df.drop(
columns = [best_attribute])

# Print the best attribute to the console
print()
print(str(best_attribute) + " added to the optimal attribute subset")
print()

# Write to file
outfile3.write(str(
best_attribute) + (
" added to the optimal attribute subset") + "\n")

# Decrement the number of available attributes by 1
no_of_available_attributes -= 1

# Print number of attributes remaining to the console
print()
print(str(no_of_available_attributes) + " attributes remaining")
print()
print()

# Write to file
outfile3.write(str(no_of_available_attributes) + (
" attributes remaining") + "\n" + "\n")

else:
print()
print("Performance did not improve this round.")
print("End of Stepwise Forward Selection.")
print()
outfile3.write("Performance did not improve this round." + "\n")
outfile3.write("End of Stepwise Forward Selection." + "\n")
break

# Return the optimal attribute set
return optimal_attributes_df

def main():
# Read the full text file and store records in a Pandas dataframe

# Used for feature selection of large data sets only
#pd_full_data_set = pd_full_data_set.sample(frac=0.10, random_state=SEED)

# Runs SFS on the full data set to find the optimal attribute data frame
optimal_attribute_set = sfs(pd_full_data_set)

# Write dataframe to a file

# Open a new file to save the summary statistics
outfile2 = open(TEST_STATS_FILE,"w")

# Write to a file
outfile2.write("----------------------------------------------------------\n")
outfile2.write(ALGORITHM_NAME + " Summary Statistics (Testing)\n")
outfile2.write("----------------------------------------------------------\n")
outfile2.write("Data Set : " + DATA_PATH + "\n")

# Write the relevant stats to a file
outfile2.write("\n")
outfile2.write("Number of Test Instances : " +
str(round(len(optimal_attribute_set.index) * testing_data_prct)) + "\n")

# Run Naive Bayes on the optimal attribute set
accuracy = naive_bayes(optimal_attribute_set)
accuracy *= 100

# Write accuracy to the file
outfile2.write("Accuracy : " + str(accuracy) + "%\n")

# Print statistics to console
print()
print()
print("-------------------------------------------------------")
print(ALGORITHM_NAME + " Summary Statistics (Testing)")
print("-------------------------------------------------------")
print("Data Set : " + DATA_PATH)

# Print the relevant stats to the console
print()
print("Number of Test Instances : " +
str(round(len(optimal_attribute_set.index) * testing_data_prct)))

# Print accuracy to the console
print()
print("Accuracy : " + str(accuracy) + "%")
print()

# Create a dataframe containing the optimal attributes by removing
# the Instance and the Class Column
instance_id_colname = optimal_attribute_set.columns[0]
no_of_columns = len(optimal_attribute_set.columns) # number of columns
class_column_index = no_of_columns - 1
class_column_colname = optimal_attribute_set.columns[class_column_index]
optimal_attribute_set = optimal_attribute_set.drop(columns = [
instance_id_colname, class_column_colname])

# Write a list of the optimal attribute set to a file
outfile2.write("Optimal Attribute Subset : " + str(list(
optimal_attribute_set)))

# Print a list of the optimal attribute set
print("Optimal Attribute Subset : " + str(list(optimal_attribute_set)))
print()
print()

# Close the files
outfile2.close()
outfile3.close()

main() # Call the main function
```

# Output Statistics of Stepwise Forward Selection on the Iris Data Set

This section shows the results for the runs of Stepwise Forward Selection wrapped with Naïve Bayes on the iris data set.

## Winnow2 Algorithm From Scratch | Machine Learning

In this post, I will walk you through the Winnow2 machine learning algorithm, step-by-step. We will develop the code for the algorithm from scratch using Python. We will then run the algorithm on a real-world data set, the breast cancer data set from the UCI Machine Learning Repository. Our goal is to be able to predict if a patient has breast cancer or not based on ten different attributes. Without further ado, let’s get started!

# What is the Winnow2 Algorithm?

The Winnow2 algorithm was invented by Nick Littlestone in the late 1980s. Winnow2 is an example of supervised learning. Supervised learning is the most common type of machine learning.

In supervised learning, you feed the machine learning algorithm data that contains a set of attributes (also known as x, predictors, inputs, features, etc.) and the corresponding classification (also known as label, output, target variable, correct answer, etc.). Each instance (also known as data point, example, etc.) helps the Learner (the prediction mathematical model we are trying to build) learn the association between the attributes and the class. The job of the Learner is to create a model that enables it to use just the values of the attributes to predict the class.

For example, consider how a supervised learning algorithm would approach the breast cancer data set that has 699 instances. Each instance is a different medical patient case seen by a doctor.

For each instance, we have:

• 9 attributes: clump thickness, uniformity of cell size, uniformity of cell shape, marginal adhesion, single epithelial cell size, bare nuclei, bland chromatin, normal nucleoli, and mitoses.
• 2 classes: malignant (breast cancer detected) or benign (breast cancer not detected)

The end goal of a classification supervised learning algorithm like Winnow2 is to develop a model that can accurately predict if a new patient has breast cancer or not based on his or her attribute values. And in order to do that, the Learner must be trained on the existing data set in order to learn the association between the 9 attributes and the class.

In Winnow2, we need to preprocess a data set so that both the attributes and the class are binary values, 0 (zero) or 1 (one). For example, in the example I presented above, the class value for each instance is either 0 (benign – breast cancer not detected) or 1 (malignant – breast cancer detected).

# Algorithm Steps

The Winnow2 algorithm continuously updates as each new instance arrives (i.e. an online learning algorithm). Here is how Winnow2 works on a high level:

Step 1: Learner (prediction model) receives an instance (data point):

Since Winnow2 can only work with 0s and 1s, the attribute values need to have already been preprocessed to be either 0 or 1. For example, using the breast cancer data example, attributes like clump thickness need to first be binarized so that they are either 0 or 1. One-hot encoding is the method used in this project in order to perform this binarization.

Step 2: Learner makes a prediction of the class the instance belongs to based on the values of the attributes in the instance

For example, does the Learner believe the patient belongs to the breast cancer class based on the values of the attributes? The Learner predicts 0 if no (benign) or 1 if yes (malignant).

Step 3: Is the Prediction Correct?

The Learner is told the correct class after it makes a prediction. If the learner is correct, nothing happens. Learning (i.e. amending the mathematical model) only occurs when the Learner gets an incorrect answer.

There are two ways for the prediction to be incorrect:

In order to “learn”, a process called “Promotion” (to be explained in the next section) takes place when the Learner incorrectly predicts 0. This situation is a “false negative” result. A false negative occurs when the Learner predicts a person does not have a disease or condition when the person actually does have it.

“Demotion” takes place when the Learner incorrectly predicts 1. This situation is a “false positive” result. A false positive (also known as “false alarm”) occurs when the Learner predicts a person has a specific disease or condition when the person actually does not have it.

## Promotion and Demotion: Building the Prediction Model One Instance at a Time

To understand the promotion and demotion activities of Winnow2, we need to first examine the Learner’s mathematical model, the tool being used to predict whether an instance belongs to the benign (0) class or malignant (1) class.

When the Learner receives an instance (“Step 1” from the previous section), it runs the attributes through the following weighted sum:

where:

• d is the total number of attributes
• wi is the weighting of the ith attribute
• xi is the value of the ith attribute in binary format
• f(x) is the weighted sum (e.g. w1x1 + w2x2 + w3x3 + …. + wdxd)

The Learner then predicts an instance’s class (e.g. 1 for malignant and 0 for benign in our breast cancer example) as follows where:

• h(x) is the predicted class
• Ɵ is a constant threshold (commonly set to 0.5)

As mentioned before, if the learner makes an incorrect prediction, either promotion or demotion occurs. In both promotion and demotion, the weights wi are modified by using a constant parameter α which is any value greater than 1 (commonly set to 2).

Initially all the weights wi for each attribute are set to 1. They are then adjusted as follows:

## Winnow2 Example Using Real Data

Let’s take a look at the Winnow2 algorithm using an example with real data. We have three different attributes (which have already been converted from real numbers into binary form) and a class variable (which has also been converted into binary form). Our goal is to develop a model that can use just those three attributes to predict if a medical patient has breast cancer or not.

• 3 attributes: clump thickness (x1), uniformity of cell size(x2), uniformity of cell shape (x3),
• 2 classes (labels): malignant (breast cancer detected) or benign (breast cancer not detected)

Remember:

• All weights w for each attribute are initially set to 1. This is known as the “weight vector.”
• Ɵ = 0.5. This is our threshold.
• α = 2
• d = 3 because there are 3 attributes.

Here is the original data set with the attributes (inputs) and the class (output):

Here is what we have after we run Winnow2. We proceeded row-by-row in the original data set (one instance at a time), which generated this table:

# Winnow2 Implementation

I implemented Winnow2 from scratch in Python and then ran the algorithm on a real-world data set, the breast cancer data set from the UCI Machine Learning Repository.

## Preprocessing the Example Data Set – Breast Cancer

The breast cancer data set contains 699 instances, 10 attributes, and a class – malignant or benign. I transformed the attributes and classes into binary numbers (making them discrete is fine too, but they cannot be continuous) so that the algorithms could process the data properly and efficiently. If attribute value was greater than 5, the value was changed to 1, otherwise it was 0. I also changed Benign from 2 to 0 and Malignant from 4 to 1.

There were 16 missing attribute values in the data set, each denoted with a “?”. I chose a random number between 1 and 10 (inclusive) to fill in the data.

Finally, for each run, data sets were split into 67% for training and 33% for testing. Summary statistics were then calculated.

Here is the format the input data needs to take in the raw text/csv file. Attributes can be numerical or text, but in this example the were all numerical:

Columns (0 through N)

• 0: Instance ID
• 1: Attribute 1 (in binary)
• 2: Attribute 2 (in binary)
• 3: Attribute 3 (in binary)
• N: Actual Class (in binary)

• N + 1: Weighted Sum (of the attributes)
• N + 2: Predicted Class (in binary)…Weighted Sum > 0? (1 if yes; 0 if no)
• N + 3: True Positive (1 if yes; O if no)
• N + 4: False Positive (1 if yes; 0 if no)
• N + 5: False Negative (1 if yes; 0 if no)
• N + 6: True Negative (1 if yes; 0 if no)
• N + 7: Promote (1 if yes; 0 if no)  [for training set only]
• N + 8: Demote (1 if yes; 0 if no) [for training set only]

Here is a link to the input file after all that preprocessing was performed.

Here is a link to the output file after the algorithm below was run on the input file.

Here is a link to the final weights after the algorithm below was run on the input file.

## Winnow2 Algorithm in Python, Coded From Scratch

Here is the code. Don’t be scared at how long the code is. I included a bunch of comments so that you know what is going on at each step. I recommend you copy and paste all this code into your favorite IDE. If there is a term or a piece of code that you have never seen before, look it up on Google (even the pros who have been at this a long time have to look things up on Google all the time! This is how you learn).

```import pandas as pd # Import Pandas library
import numpy as np # Import Numpy library

# File name: winnow2.py
# Date created: 5/31/2019
# Python version: 3.7
# Description: Implementation of the Winnow2 machine learning
# algorithm invented by Nick Littlestone. Used for 2-class classificaiton
# problems (e.g. cancer/no cancer....spam/not spam, etc...)
# Nick Littlestone (1988). "Learning Quickly When Irrelevant Attributes
# Abound: A New Linear-threshold Algorithm", Machine Learning 285–318(2)

# Required Data Set Format:
# Columns (0 through N)
# 0: Instance ID
# 1: Attribute 1 (in binary)
# 2: Attribute 2 (in binary)
# 3: Attribute 3 (in binary)
# ...
# N: Actual Class (in binary)

# N + 1: Weighted Sum (of the attributes)
# N + 2: Predicted Class (in binary)...Weighted Sum > 0? (1 if yes; 0 if no)
# N + 3: True Positive (1 if yes; O if no)
# N + 4: False Positive (1 if yes; 0 if no)
# N + 5: False Negative (1 if yes; 0 if no)
# N + 6: True Negative (1 if yes; 0 if no)
# N + 7: Promote (1 if yes; 0 if no) [for training set only]
# N + 8: Demote (1 if yes; 0 if no)  [for training set only]

################ INPUT YOUR OWN VALUES IN THIS SECTION ######################
ALGORITHM_NAME = "Winnow2"
THETA = 0.5   # This is the threshold constant for the Winnow2 algorithm
ALPHA = 2.0    # This is the adjustment constant for promotion & demotion
DATA_PATH = "breast_cancer_dataset.txt"  # Directory where data set is located
TRAIN_WEIGHTS_FILE = "breast_cancer_winnow2_train_weights.txt" # Weights of learned model
TRAIN_OUT_FILE = "breast_cancer_winnow2_train_out.txt" # Training phase of the model
TEST_STATS_FILE = "breast_cancer_winnow2_test_stats.txt" # Testing statistics
TEST_OUT_FILE = "breast_cancer_winnow2_test_out.txt" # Testing phase of the model
SEPARATOR = ","  # Separator for the data set (e.g. "\t" for tab data)
CLASS_IF_ONE = "Malignant" # If Class value is 1 (e.g. Malignant, Spam, etc.)
CLASS_IF_ZERO = "Benign"  # If Class value is 0 (e.g. Benign, Not Spam, etc.)
TRAINING_DATA_PRCT = 0.67 # % of data set used for training
testing_data_prct = 1 - TRAINING_DATA_PRCT # % of data set used for testing
SEED = 99  # SEED for the random number generator. Default: 99
#############################################################################

# Read a text file and store records in a Pandas dataframe

# Create a training dataframe by sampling random instances from original data.
# random_state guarantees that the pseudo-random number generator generates
# the same sequence of random numbers each time.
pd_training_data = pd_data.sample(frac=TRAINING_DATA_PRCT, random_state=SEED)

# Create a testing dataframe. Dropping the training data from the original
# dataframe ensures training and testing dataframes have different instances
pd_testing_data = pd_data.drop(pd_training_data.index)

# Convert training dataframes to Numpy arrays
np_training_data = pd_training_data.values
np_testing_data = pd_testing_data.values

#np_training_data = pd_data.values # Used for testing only
#np_testing_data = pd_data.values  # Used for testing only

################ Begin Training Phase #####################################

# Calculate the number of instances, columns, and attributes in the data set
# Assumes 1 column for the instance ID and 1 column for the class
# Record the index of the column that contains the actual class
no_of_instances = np_training_data.shape[0]
no_of_columns = np_training_data.shape[1]
no_of_attributes = no_of_columns - 2
actual_class_column = no_of_columns - 1

# Initialize the weight vector. Initialize all weights to 1.
# First column of weight vector is not used (i.e. Instance ID)
weights = np.ones(no_of_attributes + 1)

# Create a new array that has 8 columns, initialized to 99 for each value
extra_columns_train = np.full((no_of_instances, 8),99)

# Add extra columns to the training data set
np_training_data = np.append(np_training_data, extra_columns_train, axis=1)

# Make sure it is an array of floats
np_training_data = np_training_data.astype(float)

# Build the learning model one instance at a time
for row in range(0, no_of_instances):

# Set the weighted sum to 0
weighted_sum = 0

# Calculate the weighted sum of the attributes
for col in range(1, no_of_attributes + 1):
weighted_sum += (weights[col] * np_training_data[row,col])

# Record the weighted sum into column N + 1, the column just to the right
# of the actual class column
np_training_data[row, actual_class_column + 1] = weighted_sum

# Set the predicted class to 99
predicted_class = 99

# Learner's prediction: Is the weighted sum > THETA?
if weighted_sum > THETA:
predicted_class = 1
else:
predicted_class = 0

# Record the predicted class into column N + 2
np_training_data[row, actual_class_column + 2] = predicted_class

# Record the actual class into a variable
actual_class = np_training_data[row, actual_class_column]

# Initialize the prediction outcomes
# These variables are standard inputs into a "Confusion Matrix"
true_positive = 0   # Predicted class = 1; Actual class = 1 (hit)
false_positive = 0  # Predicted class = 1; Actual class = 0 (false alarm)
false_negative = 0  # Predicted class = 0; Actual class = 1 (miss)
true_negative = 0   # Predicted class = 0; Actual class = 0

# Determine the outcome of the Learner's prediction
if predicted_class == 1 and actual_class == 1:
true_positive = 1
elif predicted_class == 1 and actual_class == 0:
false_positive = 1
elif predicted_class == 0 and actual_class == 1:
false_negative = 1
else:
true_negative = 1

# Record the outcome of the Learner's prediction
np_training_data[row, actual_class_column + 3] = true_positive
np_training_data[row, actual_class_column + 4] = false_positive
np_training_data[row, actual_class_column + 5] = false_negative
np_training_data[row, actual_class_column + 6] = true_negative

# Set the promote and demote variables to 0
promote = 0
demote = 0

# Promote if false negative
if false_negative == 1:
promote = 1

# Demote if false positive
if false_positive == 1:
demote = 1

# Record if either a promotion or demotion is needed
np_training_data[row, actual_class_column + 7] = promote
np_training_data[row, actual_class_column + 8] = demote

# Run through each attribute and see if it is equal to 1
# If attribute is 1, we need to either demote or promote (adjust the
# corresponding weight by ALPHA).
if demote == 1:
for col in range(1, no_of_attributes + 1):
if(np_training_data[row,col] == 1):
weights[col] /= ALPHA
if promote == 1:
for col in range(1, no_of_attributes + 1):
if(np_training_data[row,col] == 1):
weights[col] *= ALPHA

# Open a new file to save the weights
outfile1 = open(TRAIN_WEIGHTS_FILE,"w")

# Write the weights of the Learned model to a file
outfile1.write("----------------------------------------------------------\n")
outfile1.write(" " + ALGORITHM_NAME + " Training Weights\n")
outfile1.write("----------------------------------------------------------\n")
outfile1.write("Data Set : " + DATA_PATH + "\n")
outfile1.write("\n----------------------------\n")
outfile1.write("Weights of the Learned Model\n")
outfile1.write("----------------------------\n")
for col in range(1, no_of_attributes + 1):
colname = pd_training_data.columns[col]
s = str(weights[col])
outfile1.write(colname + " : " + s + "\n")

# Write the relevant constants used in the model to a file
outfile1.write("\n")
outfile1.write("\n")
outfile1.write("-----------\n")
outfile1.write("Constants\n")
outfile1.write("-----------\n")
s = str(THETA)
outfile1.write("THETA = " + s + "\n")
s = str(ALPHA)
outfile1.write("ALPHA = " + s + "\n")

# Close the weights file
outfile1.close()

# Print the weights of the Learned model
print("----------------------------------------------------------")
print(" " + ALGORITHM_NAME + " Results")
print("----------------------------------------------------------")
print("Data Set : " + DATA_PATH)
print()
print()
print("----------------------------")
print("Weights of the Learned Model")
print("----------------------------")
for col in range(1, no_of_attributes + 1):
colname = pd_training_data.columns[col]
s = str(weights[col])
print(colname + " : " + s)

# Print the relevant constants used in the model
print()
print()
print("-----------")
print("Constants")
print("-----------")
s = str(THETA)
print("THETA = " + s)
s = str(ALPHA)
print("ALPHA = " + s)
print()

# Print the learned model runs in binary form
print("-------------------------------------------------------")
print("Learned Model Runs of the Training Data Set (in binary)")
print("-------------------------------------------------------")
print(np_training_data)
print()
print()

# Convert Numpy array to a dataframe
df = pd.DataFrame(data=np_training_data)

# Replace 0s and 1s in the attribute columns with False and True
for col in range(1, no_of_attributes + 1):
df[[col]] = df[[col]].replace([0,1],["False","True"])

# Replace values in Actual Class column with more descriptive values
df[[actual_class_column]] = df[[actual_class_column]].replace([0,1],[CLASS_IF_ZERO,CLASS_IF_ONE])

# Replace values in Predicted Class column with more descriptive values
df[[actual_class_column + 2]] = df[[actual_class_column + 2]].replace([0,1],[CLASS_IF_ZERO,CLASS_IF_ONE])

# Change prediction outcomes to more descriptive values
for col in range(actual_class_column + 3,actual_class_column + 9):
df[[col]] = df[[col]].replace([0,1],["No","Yes"])

# Rename the columns
df.rename(columns={actual_class_column + 1 : "Weighted Sum" }, inplace = True)
df.rename(columns={actual_class_column + 2 : "Predicted Class" }, inplace = True)
df.rename(columns={actual_class_column + 3 : "True Positive" }, inplace = True)
df.rename(columns={actual_class_column + 4 : "False Positive" }, inplace = True)
df.rename(columns={actual_class_column + 5 : "False Negative" }, inplace = True)
df.rename(columns={actual_class_column + 6 : "True Negative" }, inplace = True)
df.rename(columns={actual_class_column + 7 : "Promote" }, inplace = True)
df.rename(columns={actual_class_column + 8 : "Demote" }, inplace = True)

# Change remaining columns names from position numbers to descriptive names
for pos in range(0,actual_class_column + 1):
df.rename(columns={pos : pd_data.columns[pos] }, inplace = True)

print("-------------------------------------------------------")
print("Learned Model Runs of the Training Data Set (readable) ")
print("-------------------------------------------------------")
# Print the revamped dataframe
print(df)

# Write revamped dataframe to a file
################ End Training Phase #####################################

################ Begin Testing Phase ######################################

# Calculate the number of instances, columns, and attributes in the data set
# Assumes 1 column for the instance ID and 1 column for the class
# Record the index of the column that contains the actual class
no_of_instances = np_testing_data.shape[0]
no_of_columns = np_testing_data.shape[1]
no_of_attributes = no_of_columns - 2
actual_class_column = no_of_columns - 1

# Create a new array that has 6 columns, initialized to 99 for each value
extra_columns_test = np.full((no_of_instances, 6),99)

# Add extra columns to the testing data set
np_testing_data = np.append(np_testing_data, extra_columns_test, axis=1)

# Make sure it is an array of floats
np_testing_data = np_testing_data.astype(float)

# Test the learning model one instance at a time
for row in range(0, no_of_instances):

# Set the weighted sum to 0
weighted_sum = 0

# Calculate the weighted sum of the attributes
for col in range(1, no_of_attributes + 1):
weighted_sum += (weights[col] * np_testing_data[row,col])

# Record the weighted sum into column N + 1, the column just to the right
# of the actual class column
np_testing_data[row, actual_class_column + 1] = weighted_sum

# Set the predicted class to 99
predicted_class = 99

# Learner's prediction: Is the weighted sum > THETA?
if weighted_sum > THETA:
predicted_class = 1
else:
predicted_class = 0

# Record the predicted class into column N + 2
np_testing_data[row, actual_class_column + 2] = predicted_class

# Record the actual class into a variable
actual_class = np_testing_data[row, actual_class_column]

# Initialize the prediction outcomes
# These variables are standard inputs into a "Confusion Matrix"
true_positive = 0   # Predicted class = 1; Actual class = 1 (hit)
false_positive = 0  # Predicted class = 1; Actual class = 0 (false alarm)
false_negative = 0  # Predicted class = 0; Actual class = 1 (miss)
true_negative = 0   # Predicted class = 0; Actual class = 0

# Determine the outcome of the Learner's prediction
if predicted_class == 1 and actual_class == 1:
true_positive = 1
elif predicted_class == 1 and actual_class == 0:
false_positive = 1
elif predicted_class == 0 and actual_class == 1:
false_negative = 1
else:
true_negative = 1

# Record the outcome of the Learner's prediction
np_testing_data[row, actual_class_column + 3] = true_positive
np_testing_data[row, actual_class_column + 4] = false_positive
np_testing_data[row, actual_class_column + 5] = false_negative
np_testing_data[row, actual_class_column + 6] = true_negative

# Convert Numpy array to a dataframe
df = pd.DataFrame(data=np_testing_data)

# Replace 0s and 1s in the attribute columns with False and True
for col in range(1, no_of_attributes + 1):
df[[col]] = df[[col]].replace([0,1],["False","True"])

# Replace values in Actual Class column with more descriptive values
df[[actual_class_column]] = df[[actual_class_column]].replace([0,1],[CLASS_IF_ZERO,CLASS_IF_ONE])

# Replace values in Predicted Class column with more descriptive values
df[[actual_class_column + 2]] = df[[actual_class_column + 2]].replace([0,1],[CLASS_IF_ZERO,CLASS_IF_ONE])

# Change prediction outcomes to more descriptive values
for col in range(actual_class_column + 3,actual_class_column + 7):
df[[col]] = df[[col]].replace([0,1],["No","Yes"])

# Rename the columns
df.rename(columns={actual_class_column + 1 : "Weighted Sum" }, inplace = True)
df.rename(columns={actual_class_column + 2 : "Predicted Class" }, inplace = True)
df.rename(columns={actual_class_column + 3 : "True Positive" }, inplace = True)
df.rename(columns={actual_class_column + 4 : "False Positive" }, inplace = True)
df.rename(columns={actual_class_column + 5 : "False Negative" }, inplace = True)
df.rename(columns={actual_class_column + 6 : "True Negative" }, inplace = True)

df_numerical = pd.DataFrame(data=np_testing_data) # Keep the values in this dataframe numerical
df_numerical.rename(columns={actual_class_column + 3 : "True Positive" }, inplace = True)
df_numerical.rename(columns={actual_class_column + 4 : "False Positive" }, inplace = True)
df_numerical.rename(columns={actual_class_column + 5 : "False Negative" }, inplace = True)
df_numerical.rename(columns={actual_class_column + 6 : "True Negative" }, inplace = True)

# Change remaining columns names from position numbers to descriptive names
for pos in range(0,actual_class_column + 1):
df.rename(columns={pos : pd_data.columns[pos] }, inplace = True)

print("-------------------------------------------------------")
print("Learned Model Predictions on Testing Data Set")
print("-------------------------------------------------------")
# Print the revamped dataframe
print(df)

# Write revamped dataframe to a file

# Open a new file to save the summary statistics
outfile2 = open(TEST_STATS_FILE,"w")

# Write to a file
outfile2.write("----------------------------------------------------------\n")
outfile2.write(ALGORITHM_NAME + " Summary Statistics (Testing)\n")
outfile2.write("----------------------------------------------------------\n")
outfile2.write("Data Set : " + DATA_PATH + "\n")

# Write the relevant stats to a file
outfile2.write("\n")
outfile2.write("Number of Test Instances : " +
str(np_testing_data.shape[0])+ "\n")

tp = df_numerical["True Positive"].sum()
s = str(int(tp))
outfile2.write("True Positives : " + s + "\n")

fp = df_numerical["False Positive"].sum()
s = str(int(fp))
outfile2.write("False Positives : " + s + "\n")

fn = df_numerical["False Negative"].sum()
s = str(int(fn))
outfile2.write("False Negatives : " + s + "\n")

tn = df_numerical["True Negative"].sum()
s = str(int(tn))
outfile2.write("True Negatives : " + s + "\n")

accuracy = (tp + tn)/(tp + tn + fp + fn)
accuracy *= 100
s = str(accuracy)
outfile2.write("Accuracy : " + s + "%\n")

specificity = (tn)/(tn + fp)
specificity *= 100
s = str(specificity)
outfile2.write("Specificity : " + s + "%\n")

precision = (tp)/(tp + fp)
precision *= 100
s = str(precision)
outfile2.write("Precision : " + s + "%\n")

recall = (tp)/(tp + fn)
recall *= 100
s = str(recall)
outfile2.write("Recall : " + s + "%\n")

neg_pred_value = (tn)/(tn + fn)
neg_pred_value *= 100
s = str(neg_pred_value)
outfile2.write("Negative Predictive Value : " + s + "%\n")

miss_rate = (fn)/(fn + tp)
miss_rate *= 100
s = str(miss_rate)
outfile2.write("Miss Rate  : " + s + "%\n")

fall_out = (fp)/(fp + tn)
fall_out *= 100
s = str(fall_out)
outfile2.write("Fall-Out : " + s + "%\n")

false_discovery_rate = (fp)/(fp + tp)
false_discovery_rate *= 100
s = str(false_discovery_rate)
outfile2.write("False Discovery Rate : " + s + "%\n")

false_omission_rate = (fn)/(fn + tn)
false_omission_rate *= 100
s = str(false_omission_rate)
outfile2.write("False Omission Rate  : " + s + "%\n")

f1_score = (2 * tp)/((2 * tp) + fp + fn)
s = str(f1_score)
outfile2.write("F1 Score: " + s)

# Close the weights file
outfile2.close()

# Print statistics to console
print()
print()
print("-------------------------------------------------------")
print(ALGORITHM_NAME + " Summary Statistics (Testing)")
print("-------------------------------------------------------")
print("Data Set : " + DATA_PATH)

# Print the relevant stats to the console
print()
print("Number of Test Instances : " +
str(np_testing_data.shape[0]))

s = str(int(tp))
print("True Positives : " + s)

s = str(int(fp))
print("False Positives : " + s)

s = str(int(fn))
print("False Negatives : " + s)

s = str(int(tn))
print("True Negatives : " + s)

s = str(accuracy)
print("Accuracy : " + s + "%")

s = str(specificity)
print("Specificity : " + s + "%")

s = str(precision)
print("Precision : " + s + "%")

s = str(recall)
print("Recall : " + s + "%")

s = str(neg_pred_value)
print("Negative Predictive Value : " + s + "%")

s = str(miss_rate)
print("Miss Rate  : " + s + "%")

s = str(fall_out)
print("Fall-Out : " + s + "%")

s = str(false_discovery_rate)
print("False Discovery Rate : " + s + "%")

s = str(false_omission_rate)
print("False Omission Rate  : " + s + "%")

s = str(f1_score)
print("F1 Score: " + s)

###################### End Testing Phase ######################################
```

## Output Statistics of Winnow2 on the Breast Cancer Data Set

Here is a link to a screenshot of the summary statistics:

## What Is a Maximum Spanning Tree?

In this post, I will explain the concept of a maximum spanning tree.

# What is a Spanning Tree?

Let’s say we have a graph G with three nodes, A, B, and C. Each node represents an attribute. For example, for a classification problem for breast cancer, A = clump size, B = blood pressure, C = body weight.

Graph G:

A spanning tree is a subset of the graph G that includes all of the attributes with the minimum number of edges (that would have to be 2 because a tree with just one edge would only connect at most 2 attributes). In the graph above, there are three spanning trees. All spanning trees in this graph G must have the same number of attributes (3 in total) and edges (2 in total).

Spanning Tree 1:

Spanning Tree 2:

Spanning Tree 3:

# What is a Maximum Spanning Graph?

OK, so we have our spanning trees. Now, imagine that each edge has a weight. This weight would be some number. Weighted graphs look like this:

The graph above could has three spanning trees, subsets of the graph G that include all of the attributes with the minimum number of edges.

Which one of those spanning graphs is the “maximum spanning graph?”…the one that, when you add up the weights of each edge of the spanning graph, delivers the greatest result. The answer to that is our maximum spanning tree.

Here is the maximum spanning tree:

Since the Attribute Designated as the Root Is Arbitrary, Is It Safe to Assume That This Choice Does Not Affect the Model Effectiveness?

Yes, it is safe to assume that. The graph doesn’t change, and Kruskal’s algorithm, the algorithm for finding the maximum spanning tree in a graph doesn’t care what the root is…it just wants to find the largest edge at each step that doesn’t produce a cycle.

The number of maximum spanning trees in a graph G remains constant. Whether you start at C, B, and E, doesn’t matter. The graph is what it is…unless of course you decide to add a new attribute…but then it would be a different graph with a whole other set of spanning trees.