Statistical-Learning-Toolkit

A comprehensive Statistical Learning Toolkit featuring a diverse set of classifiers based on Bayesian decision theory, parameter and density estimation techniques, and dimensionality reduction procedures.

1. Installation

$ git clone https://github.com/Jishnu8/Statistical-Learning-Toolkit $ pip install -r requirements.txt

2. Supported Features

This statistical learning toolkit has a wide range of functionalities. Following is a brief description on the methods that are currently supported:

2.1. Dimensionality Reduction Techniques

• Multiple Discriminant Analysis (MDA): A dimensionality reduction procedure which aims to minimize the loss of "classifiability" by keeping the lower dimensional data best seperated in the least squares sense. This method poses an additional requirement, necessitating the availability of class labels for the provided data.

• Principal Component Analysis (PCA): A dimensionality reduction procedure which aims to minimize the loss of "data fidelity", measured in the least square sense.

2.2. Parameter Estimation Procedures

• Maximum Likelihood Estimation (MLE): A method of estimating the parameters of an assumed probability distribution by maximizing the likelihood of obtaining the observed data as a function of the unknown parameters. Here, the unknown parameters are considered to be fixed, deterministic parameters. Currently, this technique is only supported for Gaussian distributions.

• Bayesian Parameter Estimation: Here, the unknown parameters are considered to be random variables with known/assumed prior pdfs. This approach estimates the parameter values by updating a prior belief about model parameters (i.e., prior distribution) with new evidence (i.e., observed data) via a likelihood function, resulting in a posterior distribution. An estimate of the unknown parameter can be made in a number of ways from the posterior distribution (e.g. mean, mode, median). Currently, this technique is only supported for Gaussian distributions.

2.3. Classifiers

• Bayes Classifier: When the prior and class conditional density functions are known, Bayes decision rule (which predicts the posterior probabilities) leads to minimum error-rate classification. Bayes classifier can also be easily represented as a discriminant function given by $gi(x) = ln(P(x/ωi) + ln(P(w_i))$. Evidently, in practical scenarios, priors and class conditional densities are rarely known. The most common technique is to assume the class conditional density as a normal distribution with unknown parameters (which are estimated using a parameter estimation technique), leading to the following 3 discriminant functions:

  • Normal Discriminant 1 ($Σ_i = σ^2I. $): In this case, only the mean of the normal density unknown. The assumption is that the features are statistically independent and each feature has the same variance regardless of class.
  • Normal Discriminant 2 (Σi = Σ): In this case, it is assumed that all classes have the same, but arbitrary covariance matrix. Hence, only class specific means and a common covariance matrix are parameters to be estimated.
  • Normal Discriminant 2 (Σi = Arbitrary): In this case, no restriction are made. Class specific means and covariance matrices are parameters to be estimated.

• K-Nearest Neighbors (KNN): This non-parametric, supervised learning classifier assigns a class label to a data point based on the majority class among its k nearest neighbors in the feature space. Proximity is measured by some relevant distance metric in the feature space, and is most commonly the Minkowski metric.

• PCA Approximation Method: This is a custom method which is both extremly efficient and accurate, created and supported by this toolkit. Given the training set, it involves finding the principal components (using PCA) for each class. Given the eigenspaces associated with each class, a test sample is first projected onto the eigenspace of each class and then reconstructed back to its original dimension. The prediction is made based on the class for which this reconstruction closely aligns with the original test sample, measured by the mean square error.

2.4. Pipelines

The general paradigm for most classification tasks involves dimensionality reduction, parameter estimation (if Bayes classifiers are used) and the usage of the classifier itself. For all classifiers mentioned above (except for the PCA Approximation Method which does not follow this general paradigm), this toolkit supports a single method to perform classification with the desired procedures of each part of the pipeline.

3. Usage

The usage of this toolkit is fairly straightforward as it has been well documented. Below are sample usages of how to set up the pipelines for classification using the normal discriminant, KNN and PCA Approximation method. Details on how to use a classifier, a parameter estimation method, or a dimentionality reduction technique independently can be found in the documentation provided for each method.

We first provide the code for importing the MNIST datasets. The code for the subsquent pipelines are also given below.

```python import tensorflow.keras as tk

(xtrain, ytrain), (xtest, ytest) = tk.datasets.mnist.loaddata() xtrain = xtrain.reshape((xtrain.shape[0], xtrain.shape[1] * xtrain.shape[2])) xtest = xtest.reshape((xtest.shape[0], xtest.shape[1] * x_test.shape[2])) ```

3.1. Normal Discriminants

python dim_reduction = "MDA" projected_dim = 2 param_estimation = "MLE" discriminant_type = "1" display_conf = True normal_discriminant_classify(x_train, y_train, x_test, y_test, dim_reduction=dim_reduction, projected_dim=projected_dim, discriminant_type=discriminant_type, param_estimation=param_estimation, display_conf=display_conf)

3.2. K-Nearest Neighbors (KNN)

python dim_reduction = "MDA" projected_dim = 2 neighbours=3 metric_ord = 2 display_conf = True knn_classify(x_train, y_train, x_test, y_test, dim_reduction=dim_reduction, projected_dim=projected_dim, neighbours=neighbours, metric_ord=metric_ord, display_conf=display_conf)

3.3. PCA Approximation Method

python display_conf = True projected_dim = 2 pca_approximation_method_classify(x_train, y_train, x_test, y_test, dim=projected_dim, display_conf=display_conf)

4. Call for Contributions

This statistical learning toolkit is in active development. Despite featuring a wide range of functionalities, it is apparent that it currently represents only a tiny fraction of the extensive spectrum of available statistical learning methods. Hence, any helpful comments and improvements are highly encouraged. To do so, please open an issue in this Github page.

5. Citation

If you use or extend this work, please consider citing it as below:

@software{Parayil_Shibu_Statistical_Learning_Toolkit_2024, author = {Parayil Shibu, Jishnu}, month = jan, title = {{Statistical Learning Toolkit}}, url = {https://github.com/Jishnu8/Statistical-Learning-Toolkit}, version = {1.0.0}, year = {2024} }