1) Load the iris sample dataset into Python using a Pandas dataframe. Perform a PCA using the Scikit Decomposition component, and provide the percentage of variance explained by each of the Principal Components. Compare this to the percentage of variance explained by each of the original features. What do you observe?

2) Use Matplotlib to plot a projection of each feature onto the 1st Principal Component from the above problem against vs. the original feature itself. Which pair of features show a closer relationship to PC1 vs. the others? Why? (Hint: Think in terms of cosine distance/the angle ). Calculate the correlation coefficient between the pair of features you have selected and their projections onto PC1 - do the result agree with the visual inspection?

3) Calculate the total variance of the original features and the total variance of the four eigenvectors from the above problem. What can you say about the corresponding values? If we wished to capture > 95% of the variance of the original data, how many principal components would we be selecting? How does this number correspond to the number of dimensions we are reducing our features to?