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Linear least squares regression is a method of fitting a linear model to a set of data by minimizing the sum of squared errors between the observed and predicted values. The solution of the linear least squares problem can be obtained by solving the normal equations, which are given by

ATAx=ATb,

where A is the matrix of explanatory variables, b is the vector of response variables, and x is the vector of unknown coefficients.

However, if the matrix A has two features that are perfectly linearly dependent, then the matrix ATA will be singular, meaning that it does not have a unique inverse. This implies that the normal equations do not have a unique solution, and the linear least squares problem is ill-posed. In other words, there are infinitely many values of x that can satisfy the normal equations, and the linear model is not identifiable.

This can be an issue for the linear least squares regression model, as it can lead to instability, inconsistency, and poor generalization of the model. It can also cause numerical difficulties when trying to solve the normal equations using computational methods, such as matrix inversion or decomposition. Therefore, it is advisable to avoid or remove the linearly dependent features from the matrix A before applying the linear least squares regression model.

Linear least squares (mathematics)

Linear Regression in Matrix Form

Singular Matrix Problem

 XGBoost is a built-in Amazon SageMaker machine learning algorithm that should be used for modeling the credit card fraud detection problem. XGBoost is an algorithm that implements a scalable and distributed gradient boosting framework, which is a popular and effective technique for supervised learning problems. Gradient boosting is a method of combining multiple weak learners, such as decision trees, into a strong learner, by iteratively fitting new models to the residual errors of the previous models and adding them to the ensemble. XGBoost can handle various types of data, such as numerical, categorical, or text, and can perform both regression and classification tasks. XGBoost also supports various features and optimizations, such as regularization, missing value handling, parallelization, and cross-validation, that can improve the performance and efficiency of the algorithm.

XGBoost is suitable for the credit card fraud detection problem for the following reasons:

The problem is a binary classification problem, where the goal is to predict whether a transaction is fraudulent or not, based on the information from new transactions. XGBoost can perform binary classification by using a logistic regression objective function and outputting the probability of the positive class (fraudulent) for each transaction.

The problem involves a large and imbalanced dataset of historical data labeled as fraudulent. XGBoost can handle large-scale and imbalanced data by using distributed and parallel computing, as well as techniques such as weighted sampling, class weighting, or stratified sampling, to balance the classes and reduce the bias towards the majority class (non-fraudulent).

The problem requires a high accuracy and precision for detecting fraudulent transactions, as well as a low false positive rate for avoiding false alarms. XGBoost can achieve high accuracy and precision by using gradient boosting, which can learn complex and non-linear patterns from the data and reduce the variance and overfitting of the model. XGBoost can also achieve a low false positive rate by using regularization, which can reduce the complexity and noise of the model and prevent it from fitting spurious signals in the data.

The other options are not as suitable as XGBoost for the credit card fraud detection problem for the following reasons:

Seq2seq: Seq2seq is an algorithm that implements a sequence-to-sequence model, which is a type of neural network model that can map an input sequence to an output sequence. Seq2seq is mainly used for natural language processing tasks, such as machine translation, text summarization, or dialogue generation. Seq2seq is not suitable for the credit card fraud detection problem, because the problem is not a sequence-to-sequence task, but a binary classification task. The input and output of the problem are not sequences of words or tokens, but vectors of features and labels.

K-means: K-means is an algorithm that implements a clustering technique, which is a type of unsupervised learning method that can group similar data points into clusters. K-means is mainly used for exploratory data analysis, dimensionality reduction, or anomaly detection. K-means is not suitable for the credit card fraud detection problem, because the problem is not a clustering task, but a classification task. The problem requires using the labeled data to train a model that can predict the labels of new data, not finding the optimal number of clusters or the cluster memberships of the data.

Random Cut Forest (RCF): RCF is an algorithm that implements an anomaly detection technique, which is a type of unsupervised learning method that can identify data points that deviate from the normal behavior or distribution of the data. RCF is mainly used for detecting outliers, frauds, or faults in the data. RCF is not suitable for the credit card fraud detection problem, because the problem is not an anomaly detection task, but a classification task. The problem requires using the labeled data to train a model that can predict the labels of new data, not finding the anomaly scores or the anomalous data points in the data.

XGBoost Algorithm

Use XGBoost for Binary Classification with Amazon SageMaker

Seq2seq Algorithm

K-means Algorithm

[Random Cut Forest Algorithm]

Area Under the ROC Curve (AUC) is a metric that is best suited to score the model for the given scenario. AUC is a measure of the performance of a binary classifier, such as a model that predicts whether a credit card transaction is valid or fraudulent. AUC is calculated based on the Receiver Operating Characteristic (ROC) curve, which is a plot that shows the trade-off between the true positive rate (TPR) and the false positive rate (FPR) of the classifier as the decision threshold is varied. The TPR, also known as recall or sensitivity, is the proportion of actual positive cases (fraudulent transactions) that are correctly predicted as positive by the classifier. The FPR, also known as the fall-out, is the proportion of actual negative cases (valid transactions) that are incorrectly predicted as positive by the classifier. The ROC curve illustrates how well the classifier can distinguish between the two classes, regardless of the class distribution or the error costs. A perfect classifier would have a TPR of 1 and an FPR of 0 for all thresholds, resulting in a ROC curve that goes from the bottom left to the top left and then to the top right of the plot. A random classifier would have a TPR and an FPR that are equal for all thresholds, resulting in a ROC curve that goes from the bottom left to the top right of the plot along the diagonal line. AUC is the area under the ROC curve, and it ranges from 0 to 1. A higher AUC indicates a better classifier, as it means that the classifier has a higher TPR and a lower FPR for all thresholds. AUC is a useful metric for imbalanced classification problems, such as the credit card transaction dataset, because it is insensitive to the class imbalance and the error costs. AUC can capture the overall performance of the classifier across all possible scenarios, and it can be used to compare different classifiers based on their ROC curves.

The other options are not as suitable as AUC for the given scenario for the following reasons:

Precision: Precision is the proportion of predicted positive cases (fraudulent transactions) that are actually positive. Precision is a useful metric when the cost of a false positive is high, such as in spam detection or medical diagnosis. However, precision is not a good metric for imbalanced classification problems, because it can be misleadingly high when the positive class is rare. For example, a classifier that predicts all transactions as valid would have a precision of 0, but a very high accuracy of 99%. Precision is also dependent on the decision threshold and the error costs, which may vary for different scenarios.

Recall: Recall is the same as the TPR, and it is the proportion of actual positive cases (fraudulent transactions) that are correctly predicted as positive by the classifier. Recall is a useful metric when the cost of a false negative is high, such as in fraud detection or cancer diagnosis. However, recall is not a good metric for imbalanced classification problems, because it can be misleadingly low when the positive class is rare. For example, a classifier that predicts all transactions as fraudulent would have a recall of 1, but a very low accuracy of 1%. Recall is also dependent on the decision threshold and the error costs, which may vary for different scenarios.

Root Mean Square Error (RMSE): RMSE is a metric that measures the average difference between the predicted and the actual values. RMSE is a useful metric for regression problems, where the goal is to predict a continuous value, such as the price of a house or the temperature of a city. However, RMSE is not a good metric for classification problems, where the goal is to predict a discrete value, such as the class label of a transaction. RMSE is not meaningful for classification problems, because it does not capture the accuracy or the error costs of the predictions.

ROC Curve and AUC

How and When to Use ROC Curves and Precision-Recall Curves for Classification in Python

Precision-Recall

Root Mean Squared Error

 The F1 score is a measure of the harmonic mean of precision and recall, which are both important for fraud detection. Precision is the ratio of true positives to all predicted positives, and recall is the ratio of true positives to all actual positives. A high F1 score indicates that the classifier can correctly identify fraudulent transactions and avoid false negatives. The true positive rate is another name for recall, and it measures the proportion of fraudulent transactions that are correctly detected by the classifier. A high true positive rate means that the classifier can capture as many fraudulent transactions as possible.

Fraud Detection Using Machine Learning | Implementations | AWS Solutions

Detect fraudulent transactions using machine learning with Amazon SageMaker | AWS Machine Learning Blog

1. Introduction — Reproducible Machine Learning for Credit Card Fraud Detection