12-2014

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics (PhD)

#### Department

Mathematical Sciences

Giovanni Petris

Edward Gbur

Mark Arnold

#### Third Committee Member

Avishek Chakraborty

#### Keywords

Pure sciences, Exponentially weighted moving average, Structural breaks, Time series data

#### Abstract

Outliers and structural breaks occur quite frequently in time series data. Whereas outliers often contain valuable information

about the process under study, they are known to have serious negative impact on statistical data analysis. Most obvious effect is model misspecification and biased parameter estimation which results in wrong conclusions and inaccurate predictions. Structural time series consist of underlying features such as level, slope, cycles or seasonal components. Structural breaks are permanent disruptions of one or more of these components and might be a signal of serious changes in the observed process.

Detecting outliers and estimating the location of structural breaks has progressively become monumental both as a theoretical research problem and an essential part of applied data analysis. Among numerous applications include finance, industrial manufacturing, medical informatics, severe weather prediction.

Given that these data arrive rather frequently and sequentially in time, fast reliable and accurate detection techniques are required. We propose a model from class of state-space models of the form $y_{t} = f(X_{t}, \psi, v_{t})$ and $X_{t} = g(X_{t-1}, \psi, w_{t})$ where $\big\{ X_{t} \big\}_{t\geq 0}$ is a hidden Markov state process. The inference of $\big\{ X_{t} \big\}_{t\geq 0}$ depends on the observation process $\{y_{t}\}_{t\geq 1}$ and the parameter vector $\psi$, whose elements are usually unknown.

The innovations $v_{t}$ and $w_{t}$ are conditionally \textit{Gaussian} given the precision parameter $\lambda$ and auxiliary state $\omega$.

We employ sequential Monte Carlo techniques to approximate the joint target distribution $p(X_{0:t}, \psi|y_{1:t})$. The posterior estimates for the auxiliary states $\omega$ will be used to identify outliers and structural breaks. The results prove that the algorithm is comparable to traditional and computationally expensive MCMC and superior to regular techniques such as Exponentially Weighted Moving Average (EWMA), Shewhart, and cumulative sum (CUSUM) control charts

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