Effect of noise in principal component analysis with an application to ozone pollution

This thesis analyzes the effect of independent noise in principal components of k normally distributed random variables defined by a covariance matrix. We prove that the principal components as well as the canonical variate pairs determined from joint distribution of original sample affected by noise can be essentially different in comparison with those determined from the original sample. However when the differences between the eigenvalues of the original covariance matrix are sufficiently large compared to the level of the noise, the effect of noise in principal components and canonical variate pairs proved to be negligible. The theoretical results are supported by simulation study and examples. Moreover, we compare our results about the eigenvalues and eigenvectors in the two dimensional case with other models examined before. This theory can be applied in any field for the decomposition of the components in multivariate analysis.;One application is the detection and prediction of the main atmospheric factor of ozone concentrations on the example of Albany, New York. Using daily ozone, solar radiation, temperature, wind speed and precipitation data, we determine the main atmospheric factor for the explanation and prediction of ozone concentrations. A methodology is described for the decomposition of the time series of ozone and other atmospheric variables into the global term component which describes the long term trend and the seasonal variations, and the synoptic scale component which describes the short term variations. By using the Canonical Correlation Analysis, we show that solar radiation is the only main factor between the atmospheric variables considered here for the explanation and prediction of the global and synoptic scale component of ozone. The global term components are modeled by a linear regression model, while the synoptic scale components by a vector autoregressive model and the Kalman filter. The coefficient of determination, R2, for the prediction of the synoptic scale ozone component was found to be the highest when we consider the synoptic scale component of the time series for solar radiation and temperature.;KEY WORDS: multivariate analysis; principal component; canonical variate pairs; eigenvalue; eigenvector; ozone; solar radiation; spectral decomposition; Kalman filter; time series prediction...