# Statistical Analysis Techniques

## Principal Component Analysis

Principal Component Analysis (PCA) is a statistical technique that is used to reduce the dimensions of a data set while retaining most of the variability that the original data conveyed. Some good examples for high dimensionality analytical data would be projects involving polycyclic aromatic hydrocarbons (PAHs) or polychlorinated biphenyls (PCBs).  In data sets with many dimensions or variables, it can be difficult to clearly identify trends and relationships. PCA establishes independent dimensions, or vectors, in the directions that display the greatest variance to allow the most important discriminating factors to be determined. The data is thus represented in fewer variables, but the variance is retained, allowing for the most effective visualization of the data and discrimination between samples. Chemistry Matters uses PCA as one of its statistical models for sample analytics.

Principal Component Analysis works by projecting plotted data points onto a vector—a Principal Component—that is made to best fit the original data points. This vector is therefore the most precise representation of the highest number of data points possible on a single line. A second principal component (PC2) is then fit, orthogonal to the first principal component (PC1), that represents the next highest amount of variance among the data. PC2 must be orthogonal so that PC1 and PC2 are uncorrelated and, when used together to display the data, can express the maximum amount of variance in the data possible in only two dimensions.

The original data points are plotted from observations of variables used to describe various samples. The collected environmental data that is used in PCA is often transformed, or normalized, to proportions or percentages so that the data is properly scaled for analysis. Though the PCA graph of the samples no longer expresses specific measurements of variables—for example, concentrations—the graph shows the relationship between samples and uses the most representative variables for this display to optimize the data trends reflected.

The data on the PCA graph can be further analyzed through clustering, a process that groups samples that are similar to each other together and can help to explain the environmental occurrences being investigated. The PCA graph is then read as simply as samples found in the same area of the graph being more similar to each other compared to samples found in a different area.

Left graphic shows PCA graph accounting for 52.6% of data variance. Right graphic shows groups of chemicals identified through Hierarchical Cluster Analysis highlighted on PCA graph, reflecting the variance in the data.

## Hierarchical Cluster Analysis

Hierarchical Cluster Analysis is a means of establishing groups in a set of data. HCA allows the samples analyzed by Chemistry Matters to be sorted using an algorithm that identifies similarities between samples, then between subsequent groups of samples.  This allows for the clustering or grouping of like samples – samples that belong to the same family.

Hierarchical Cluster Analysis takes samples and arranges them in a dendrogram to display how closely the observations are related. Dendrograms are composed of various branches, dividing the samples into small groups that form larger classifications until all data is contained within one branch.

Before the data is clustered, it must be normalized. The original data that this method is applied to takes the form of a variety of measured observations for different samples. These various observations may be scaled differently and must be adjusted to allow for proper comparison.

Clustering is executed using an algorithm that analyzes the distance between clusters to determine which are most similar. Each data point or sample is placed in an individual cluster to start, then grouped with the next closest cluster, forming a branch. The clusters on each branch are compared and new branches are formed between the closest related clusters until all data is under the same branch. From this point, the appropriate number of clusters to represent the specific data set can be chosen by cutting the dendrogram at any point. Longer lines, or branches, in a dendrogram indicate more variance between samples connected by the branch, while shorter branches indicate that the contained samples are more similar.

Left graphic shows dendrogram produced with HCA, with samples listed on the left and branches relating samples on the right. Right graphic shows groups of chemicals identified based on the similarities communicated through HCA.

## Overview of Statistical Analysis

PCA used in tandem with HCA allows the variables that are responsible for the clusters to be visualized. Clusters are defined using an algorithm but meaning can be found in the data by having PCA reveal the strongest relationships that exist among the variables and determining what variables are behind the trends observed in the field.

PCA and HCA are two statistical analysis techniques within the realm of unsupervised machine learning. Machine learning encompasses algorithms and statistical models that allow computer systems to find structure in data that can be used to predict data based on a cycle of input, system processing, output, and user feedback. Unsupervised machine learning is applied to unlabeled data, allowing the user to develop their own conclusion based on the analysis of data by the machine. To contrast, in supervised machine learning, data is labelled, and the machine is able to come to the same type of conclusion that the user would develop in unsupervised learning.

PCA and HCA are forms of unsupervised machine learning, as they are used to enhance insight into the data so the user can more effectively make sense of the data. At Chemistry Matters, our team of expert chemists and data analysts uses these methods to be able to see data and relationships more clearly and apply their comprehensive knowledge of chemistry to answer the questions brought to them by clients. This data analysis is conducted in cases involving geoforensics, environmental forensics, arson investigations, and biomonitoring. Data sets typically consist of measured concentrations of samples from rivers, soil, and other matrices. Statistical analysis can then aid in identifying the composition of samples in different locations, changes in concentration and sources of chemicals detected, etc. The visualization of data that PCA and HCA provide is essential for identifying and comprehending relationships in collected data and clearly communicating what these relationships reveal about the environmental concerns brought to CMI.  At CMI, we commonly use PCA and HCA in tandem with source apportionment models (receptor modelling) to allocate contaminants to responsible parties.