Linear Analysis for Dimensionality Reduction in Python

Reducing the number of input variables for a predictive model is referred to as dimensionality reduction.

Fewer input variables can result in a simpler predictive model that may have better performance when making predictions on new data.

Linear Discriminant Analysis, or LDA for short, is a predictive modeling algorithm for multi-class classification. It can also be used as a dimensionality reduction technique, providing a projection of a training dataset that best separates the examples by their assigned class.

The ability to use Linear Discriminant Analysis for dimensionality reduction often surprises most practitioners.

In this tutorial, you will discover how to use LDA for dimensionality reduction when developing predictive models.

After completing this tutorial, you will know:

• Dimensionality reduction involves reducing the number of input variables or columns in modeling data.
• LDA is a technique for multi-class classification that can be used to automatically perform dimensionality reduction.
• How to evaluate predictive models that use an LDA projection as input and make predictions with new raw data.

Let’s get started.

Tutorial Overview

This tutorial is divided into four parts; they are:

1. Dimensionality Reduction
2. Linear Discriminant Analysis
3. LDA Scikit-Learn API
4. Worked Example of LDA for Dimensionality

Dimensionality Reduction

Dimensionality reduction refers to reducing the number of input variables for a dataset.

If your data is represented using rows and columns, such as in a spreadsheet, then the input variables are the columns that are fed as input to a model to predict the target variable. Input variables are also called features.

We can consider the columns of data representing dimensions on an n-dimensional feature space and the rows of data as points in that space. This is a useful geometric interpretation of a dataset.

In a dataset with k numeric attributes, you can visualize the data as a cloud of points in k-dimensional space …

— Page 305, Data Mining: Practical Machine Learning Tools and Techniques, 4th edition, 2016.

Having a large number of dimensions in the feature space can mean that the volume of that space is very large, and in turn, the points that we have in that space (rows of data) often represent a small and non-representative sample.

This can dramatically impact the performance of machine learning algorithms fit on data with many input features, generally referred to as the “curse of dimensionality.”

Therefore, it is often desirable to reduce the number of input features. This reduces the number of dimensions of the feature space, hence the name “dimensionality reduction.”

A popular approach to dimensionality reduction is to use techniques from the field of linear algebra. This is often called “feature projection” and the algorithms used are referred to as “projection methods.”

Projection methods seek to reduce the number of dimensions in the feature space whilst also preserving the most important structure or relationships between the variables observed in the data.

When dealing with high dimensional data, it is often useful to reduce the dimensionality by projecting the data to a lower dimensional subspace which captures the “essence” of the data. This is called dimensionality reduction.

— Page 11, Machine Learning: A Probabilistic Perspective, 2012.

The resulting dataset, the projection, can then be used as input to train a machine learning model.

In essence, the original features no longer exist and new features are constructed from the available data that are not directly comparable to the original data, e.g. don’t have column names.

Any new data that is fed to the model in the future when making predictions, such as test dataset and new datasets, must also be projected using the same technique.

Linear Discriminant Analysis

Linear Discriminant Analysis, or LDA, is a linear machine learning algorithm used for multi-class classification.

It should not be confused with “Latent Dirichlet Allocation” (LDA), which is also a dimensionality reduction technique for text documents.

Linear Discriminant Analysis seeks to best separate (or discriminate) the samples in the training dataset by their class value. Specifically, the model seeks to find a linear combination of input variables that achieves the maximum separation for samples between classes (class centroids or means) and the minimum separation of samples within each class.

… find the linear combination of the predictors such that the between-group variance was maximized relative to the within-group variance. […] find the combination of the predictors that gave maximum separation between the centers of the data while at the same time minimizing the variation within each group of data.

— Page 289, Applied Predictive Modeling, 2013.

There are many ways to frame and solve LDA; for example, it is common to describe the LDA algorithm in terms of Bayes Theorem and conditional probabilities.

In practice, LDA for multi-class classification is typically implemented using the tools from linear algebra, and like PCA, uses matrix factorization at the core of the technique. As such, it is good practice to perhaps standardize the data prior to fitting an LDA model.

For more information on how LDA is calculated in detail, see the tutorial:

• Linear Discriminant Analysis for Machine Learning

Now that we are familiar with dimensionality reduction and LDA, let’s look at how we can use this approach with the scikit-learn library.

LDA Scikit-Learn API

We can use LDA to calculate a projection of a dataset and select a number of dimensions or components of the projection to use as input to a model.

The scikit-learn library provides the LinearDiscriminantAnalysis class that can be fit on a dataset and used to transform a training dataset and any additional dataset in the future.

For example:

...
# prepare dataset
data = ...
# define transform
lda = LinearDiscriminantAnalysis()
# prepare transform on dataset
lda.fit(data)
# apply transform to dataset
transformed = lda.transform(data)

The outputs of the LDA can be used as input to train a model.

Perhaps the best approach is to use a Pipeline where the first step is the LDA transform and the next step is the learning algorithm that takes the transformed data as input.

...
# define the pipeline
steps = [('lda', LinearDiscriminantAnalysis()), ('m', GaussianNB())]
model = Pipeline(steps=steps)

It can also be a good idea to standardize data prior to performing the LDA transform if the input variables have differing units or scales; for example:

...
# define the pipeline
steps = [('s', StandardScaler()), ('lda', LinearDiscriminantAnalysis()), ('m', GaussianNB())]
model = Pipeline(steps=steps)

Now that we are familiar with the LDA API, let’s look at a worked example.

Worked Example of LDA for Dimensionality

First, we can use the make_classification() function to create a synthetic 10-class classification problem with 1,000 examples and 20 input features, 15 inputs of which are meaningful.

The complete example is listed below.

# test classification dataset
from sklearn.datasets import make_classification
# define dataset
X, y = make_classification(n_samples=1000, n_features=20, n_informative=15, n_redundant=5, random_state=7, n_classes=10)
# summarize the dataset
print(X.shape, y.shape)

Running the example creates the dataset and summarizes the shape of the input and output components.

(1000, 20) (1000,)

Next, we can use dimensionality reduction on this dataset while fitting a naive Bayes model.

We will use a Pipeline where the first step performs the LDA transform and selects the five most important dimensions or components, then fits a Naive Bayes model on these features. We don’t need to standardize the variables on this dataset, as all variables have the same scale by design.

The pipeline will be evaluated using repeated stratified cross-validation with three repeats and 10 folds per repeat. Performance is presented as the mean classification accuracy.

The complete example is listed below.

# evaluate lda with naive bayes algorithm for classification
from numpy import mean
from numpy import std
from sklearn.datasets import make_classification
from sklearn.model_selection import cross_val_score
from sklearn.model_selection import RepeatedStratifiedKFold
from sklearn.pipeline import Pipeline
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
from sklearn.naive_bayes import GaussianNB
# define dataset
X, y = make_classification(n_samples=1000, n_features=20, n_informative=15, n_redundant=5, random_state=7, n_classes=10)
# define the pipeline
steps = [('lda', LinearDiscriminantAnalysis(n_components=5)), ('m', GaussianNB())]
model = Pipeline(steps=steps)
# evaluate model
cv = RepeatedStratifiedKFold(n_splits=10, n_repeats=3, random_state=1)
n_scores = cross_val_score(model, X, y, scoring='accuracy', cv=cv, n_jobs=-1, error_score='raise')
# report performance
print('Accuracy: %.3f (%.3f)' % (mean(n_scores), std(n_scores)))

Running the example evaluates the model and reports the classification accuracy.

Note: Your results may vary given the stochastic nature of the algorithm or evaluation procedure, or differences in numerical precision. Consider running the example a few times and compare the average outcome.

In this case, we can see that the LDA transform with naive bayes achieved a performance of about 31.4 percent.

Accuracy: 0.314 (0.049)

How do we know that reducing 20 dimensions of input down to five is good or the best we can do?

We don’t; five was an arbitrary choice.

A better approach is to evaluate the same transform and model with different numbers of input features and choose the number of features (amount of dimensionality reduction) that results in the best average performance.

LDA is limited in the number of components used in the dimensionality reduction to between the number of classes minus one, in this case, (10 – 1) or 9

The example below performs this experiment and summarizes the mean classification accuracy for each configuration.

# compare lda number of components with naive bayes algorithm for classification
from numpy import mean
from numpy import std
from sklearn.datasets import make_classification
from sklearn.model_selection import cross_val_score
from sklearn.model_selection import RepeatedStratifiedKFold
from sklearn.pipeline import Pipeline
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
from sklearn.naive_bayes import GaussianNB
from matplotlib import pyplot

# get the dataset
def get_dataset():
X, y = make_classification(n_samples=1000, n_features=20, n_informative=15, n_redundant=5, random_state=7, n_classes=10)
return X, y

# get a list of models to evaluate
def get_models():
models = dict()
for i in range(1,10):
steps = [('lda', LinearDiscriminantAnalysis(n_components=i)), ('m', GaussianNB())]
models[str(i)] = Pipeline(steps=steps)
return models

# evaluate a give model using cross-validation
def evaluate_model(model, X, y):
cv = RepeatedStratifiedKFold(n_splits=10, n_repeats=3, random_state=1)
scores = cross_val_score(model, X, y, scoring='accuracy', cv=cv, n_jobs=-1, error_score='raise')
return scores

# define dataset
X, y = get_dataset()
# get the models to evaluate
models = get_models()
# evaluate the models and store results
results, names = list(), list()
for name, model in models.items():
scores = evaluate_model(model, X, y)
results.append(scores)
names.append(name)
print('>%s %.3f (%.3f)' % (name, mean(scores), std(scores)))
# plot model performance for comparison
pyplot.boxplot(results, labels=names, showmeans=True)
pyplot.show()

Running the example first reports the classification accuracy for each number of components or features selected.

Note: Your results may vary given the stochastic nature of the algorithm or evaluation procedure, or differences in numerical precision. Consider running the example a few times and compare the average outcome.

We can see a general trend of increased performance as the number of dimensions is increased. On this dataset, the results suggest a trade-off in the number of dimensions vs. the classification accuracy of the model.

The results suggest using the default of nine components achieves the best performance on this dataset, although with a gentle trade-off as fewer dimensions are used.

>1 0.182 (0.032)
>2 0.235 (0.036)
>3 0.267 (0.038)
>4 0.303 (0.037)
>5 0.314 (0.049)
>6 0.314 (0.040)
>7 0.329 (0.042)
>8 0.343 (0.045)
>9 0.358 (0.056)

A box and whisker plot is created for the distribution of accuracy scores for each configured number of dimensions.

We can see the trend of increasing classification accuracy with the number of components, with a limit at nine.

We may choose to use an LDA transform and Naive Bayes model combination as our final model.

This involves fitting the Pipeline on all available data and using the pipeline to make predictions on new data. Importantly, the same transform must be performed on this new data, which is handled automatically via the Pipeline.

The code below provides an example of fitting and using a final model with LDA transforms on new data.

# make predictions using lda with naive bayes
from sklearn.datasets import make_classification
from sklearn.pipeline import Pipeline
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
from sklearn.naive_bayes import GaussianNB
# define dataset
X, y = make_classification(n_samples=1000, n_features=20, n_informative=15, n_redundant=5, random_state=7, n_classes=10)
# define the model
steps = [('lda', LinearDiscriminantAnalysis(n_components=9)), ('m', GaussianNB())]
model = Pipeline(steps=steps)
# fit the model on the whole dataset
model.fit(X, y)
# make a single prediction
row = [[2.3548775,-1.69674567,1.6193882,-1.19668862,-2.85422348,-2.00998376,16.56128782,2.57257575,9.93779782,0.43415008,6.08274911,2.12689336,1.70100279,3.32160983,13.02048541,-3.05034488,2.06346747,-3.33390362,2.45147541,-1.23455205]]
yhat = model.predict(row)
print('Predicted Class: %d' % yhat[0])

Running the example fits the Pipeline on all available data and makes a prediction on new data.

Here, the transform uses the nine most important components from the LDA transform as we found from testing above.

A new row of data with 20 columns is provided and is automatically transformed to 15 components and fed to the naive bayes model in order to predict the class label.

Predicted Class: 6

Summary

In this tutorial, you discovered how to use LDA for dimensionality reduction when developing predictive models.

Specifically, you learned:

• Dimensionality reduction involves reducing the number of input variables or columns in modeling data.
• LDA is a technique for multi-class classification that can be used to automatically perform dimensionality reduction.
• How to evaluate predictive models that use an LDA projection as input and make predictions with new raw data.

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