This is a small, independent project that uses regression modeling and principal component analysis $($PCA$).$ Your objective is to take a dataset and make it lower in dimension so that you may use Multiple Linear Regression $($MLR$)$ on the new, smaller collection of characteristics.

$($a$)$ It is necessary for you to search for a quantitative dataset that has at least five characteristics and $500$ observations. You should use PCA to find a new set of orthogonal dimensions after preprocessing the dataset:

$($i$)$ using Python from scratch; and

$($ii$)$ by putting Scikit$-$Learn to use.

$($b$)$ Second, train a regression model MLR $($you can use the sklearn libraries directly$)$ to the smaller collection of features as well as the original dataset, and evaluate the regression model's performance. You must to provide an explanation for the decision regarding the number of PCA components kept.

We need you to prepare a report $($no more than $25$ pages$)$ with special attention to the following:

$$ Abstract: A succinct synopsis of the entire work.

$$ Introduction: A thorough explanation of the significance of dimensionality reduction prior to regression model fitting.

$$ Literature Review: An overview of no more than five studies that used PCA to lessen the multicollinearity issue before MLR was put into practice.

$$ Methodology: preparation and selection of data. Application of the regression technique and PCA. You have$$to give a thorough breakdown of the methods and procedures used.

$$ Results: An explanation and assessment of the outcomes. To evaluate the accuracy of forecasts, use several performance metrics such as Mean Squared Error, R$-$Squared, modified R$-$Squared, etc.

$$ Conclusion and Limitations.