# Discriminant Analysis (2)

```Discriminant Analysis: A
Powerful Tool for
Classification
Discriminant Analysis (DA) is a statistical technique that enables the
effective classification of observations or categories based on a specific
set of predictor variables. By finding linear combinations of variables that
best differentiate between different groups, DA empowers businesses,
researchers, and organizations to make data-driven decisions and
predictions through accurate classification.
Ea
by Ebru Bekar
The Fundamentals of
Discriminant Analysis
Discriminant Functions
Discriminant functions are
Maximizing Between-Group
Differences
mathematical equations used to
The primary objective of the
classify observations into different
discriminant function is to
groups based on predictor
maximize the differences between
variables. These functions are
the means of predictor variables
derived from the characteristics of
among different groups. This
the data and aim to maximize the
ensures a good separation in the
difference between groups while
feature space of groups and
minimizing differences within
facilitates accurate classification of
groups.
observations.
Minimizing Within-Group Variability Establishing Decision Boundaries
Another aim of the discriminant
The discriminant function balances
function is to minimize the
between maximizing the difference
variability of observations within
between different classes and
each group. By reducing within-
minimizing within-group variability.
group variability, the discriminant
As a result, it establishes decision
function can better distinguish
boundaries that effectively classify
between groups and improve the
observations into their respective
accuracy of classification.
groups.
Real-World Applications of
Discriminant Analysis
Market Segmentation
Medical Diagnosis
Credit Risk Assessment
Determining consumer
Classifying patients
Predicting whether a
segments based on
into different disease
credit applicant will
demographic
categories based on
default or not based
characteristics,
symptoms, test
on financial indicators.
behaviors, or
results, or biomarkers.
preferences.
Linear Discriminant Analysis
Discriminant Analysis
(QDA)
Choosing Between LDA and QDA
Linear Discriminant Analysis (LDA)
The choice between LDA and
LDA is a classification technique
QDA depends on the specific
that assumes multivariate normal
characteristics of the dataset and
distribution within each group of
the underlying distribution of the
predictor variables. LDA aims to
data. LDA is preferred when the
find a linear combination of
assumption of equal covariance
features that best separates the
matrices holds true, while QDA
groups while maximizing the
may be more suitable when this
variance between groups and
assumption is violated or when
minimizing within-group variance.
there is non-linearity in the data.
1
2
Analysis (QDA)
QDA is a similar classification
method to LDA but relaxes the
assumption of homogeneous
covariance matrices between
groups. Unlike LDA, QDA allows
for different covariance matrices
for each group, allowing it to
capture complex relationships
3
Key Assumptions and
Considerations in Discriminant
Analysis
1
Multivariate
Normal
Distribution
2
Homogeneity
of Covariance
Matrices
3
Independenc
e of
Observations
This assumption
This assumption
This assumption
states that
states that
assumes that
observations are
predictor
covariance
independent
matrices of
within and
multivariate
predictor
between groups.
normal
variables are
Violations of this
distribution within
equal across all
assumption, such
each group.
groups.
as serial
Violations of this
Violations of this
correlation or
assumption can
assumption can
clustering of
observations, can
parameter
discriminant
estimates and
functions and
parameter
The Discriminant Analysis Workflow
Data Preparation
Collect and clean the data, dealing with missing values, outliers, and errors.
Feature Selection
Select relevant predictor variables for analysis, considering their
importance and contributions.
Data Transformation
Standardize or normalize variables to ensure equal contributions
Model Training
Develop discriminant functions using appropriate techniques (e.g.,
LDA or QDA) based on the features of the data.
Model Evaluation
Assess the performance of the discriminant model using crossvalidation or other validation techniques.
Interpretation
Interpret the results, including the significance of predictor variables
Validation Techniques for
Discriminant Analysis
Leave-One-Out
Cross-Validation
(LOOCV)
K-Fold Cross-Validation Bootstrap Validation
Divide the dataset into k
Create multiple
Train the model on all
subsets, train the model
bootstrap samples
data except one
on k-1 subsets, and test
using repeated
observation and test it
it on the remaining
sampling with
on the remaining
subset. Repeat this
replacement from the
observation, repeating
process k times, using a
dataset. Train the model
this process for each
different subset as the
on each bootstrap
observation.
test set each time.
sample and evaluate its
performance on the
original dataset.
Challenges and Limitations in
Discriminant Analysis
1
Assumption
Violations
2
HighDimensional
Problems
3
Overfitting
When the
When there are
Overfitting occurs
assumptions of
more variables
when a discriminant
multivariate normality,
than
model captures noise
homogeneity of
observations, the
or random fluctuations
covariance matrices,
analysis
in the training data,
and independence of
becomes more
resulting in poor
observations are
challenging, and
generalization
violated, the results of
techniques like
performance on
discriminant analysis
dimensionality
unseen data.
reduction may be
necessary.
The Future of Discriminant Analysis
As machine learning techniques
continue to evolve, the integration
of DA with these methods can lead
to enhanced classification
performance and the ability to
Handling High-Dimensional Data
handle more complex, non-linear
Researchers
relationships. are exploring ways to
adapt DA to effectively handle highdimensional datasets, where the
number of features far exceeds the
number of observations, expanding
Hybrid Approaches
its applicability in the era of big
Combining
DA with other statistical
data.
and machine learning techniques,
such as regularization methods
and ensemble learning, can lead to
more robust and accurate