How can you use PCA to increase the accuracy of predictive models?

Principal Component Analysis, commonly known as PCA, is a powerful multivariate data analysis method. Its main purpose is to summarize large datasets by removing any redundancy in the data. As an additional advantage of the method, results are then presented using graphical tools.

In high energy physics this technique is not usually used, but I guess many analysis could benefit from it. In our case we just look at the importance of each variable after training a MVA tool and we eventually remove those variables. In alternative we let deep learning techniques build more complex variables starting from low level features.

We can calculate PCA from scratch using Python using following steps : (Tested code) import numpy as np from sklearn.preprocessing import StandardScaler from sklearn.decomposition import PCA # Standardize the data x= StandardScaler().fit_transform(X) # Calculate Covariance ct = np.cov(x.T) # Calculate eigen values and eigen vectors evalues,evectors = np.linalg.eig(ct) # Sort the eigen values sindex = np.argsort(evalues)[::-1] sevalues = evalues[sindex] # Sort the eigenvectors sevectors = evectors[:,sindex] # Calculate the PCA pdata = np.matmul(x,evectors)

Following are use to choose Proper number of Principal Components : 1. Kaiser Criterion. 2. Cumulative explained variance Plot 3. Scree Plot

If the % of variation captured across say the first 3-5 PCs is low, you don't need to use PCA as this suggests that there is very little redundancy in your data. When the % of variation captured by the 3-5 first PCs is relatively large (compared to the number of variables in the PCA), it is a good idea to use resampling techniques along with PCA to get a measure of uncertainty with the amount of variation captured by each PC.

Much more efficient to use methods especially designed to account for multicollinearity such as PLS regression. The response is accounted for in the extraction of the principal components and therefore PC selection coupled with cross-validation is much more straightforward and efficient.

The ?curse of dimensionality“ outlines that high-dimensional datasets need (much) more data to be modelled reliably. PCA can help us reduce the number of features while maintaining general patterns in the data, thus helping with dimensionality challenges and making models more efficient and even reliable.

It is important to notice that PCA may not be suitable in some cases, for example when dealing with non linear relationships or requiring high interpretability or in the presence of outliers in your data. In addition to that PCA is sensitive to small sample sizes, also it is unsuitable for categorical variables, and assumes normal distribution. However you could consider some alternatives to PCA based on your data and goals. 1. t-SNE for non linear relationships 2. Factor Analysis for preserving interpretability 3. Robust PCA for handling outliers 4. Multiple Correspondence Analysis for categorical data Depending on domain specifics other techniques may be more suitable.

PCA is a data reduction technique (multivariate data analysis method) used to compress several variables into a few informative dimensions. It is a technique that is used to handle correlation in a large set of variables. It is used by many non-statisticians to manage redundancy (multicollinearity) in modelling. A common mistake consists of using the say first 3-5 principal components and use them in subsequent models. However, PCs in PCA do not account for the response. It is a much better approach to use Partial Least Squares (PLS) regression to manage multicollinearity. It is also a misconception to always work on the correlation matrix, when all variables are expressed on the same scale, it may prove more informative not to scale.

PCA can be key for : 1. Dimensionality reduction: PCA reduces the number of variables, focusing on those with the most variance, improving model efficiency. 2. Overfitting Reduction: Less dimensions mean less chance of overfitting, especially in models sensitive to high-dimensional spaces. 3. Noise Reduction: By concentrating on principal components, PCA helps filter out noise, clarifying signal in data. But remember: - Reducing dimensions can sometimes lose important info. - Standardize features before applying PCA for effective results. - Some models inherently handle high dimensions well and might not benefit as much from PCA. Integrating PCA with careful consideration can significantly enhance your model's performance.