Unobserved Component Models

Abstract

For many years, the problem of filtering and the extraction of component signals from noisy time series data has occupied the minds of mathematicians and statisticians from a variety of disciplines. Indeed, three of the most notablemathematicians of the Twentieth Century - Wiener (1941), Kolmogorov (1941), and Kalman (1960) - have all made central contributions to the development of filter and signal extraction theory. A useful concept in filtering, smoothing and signal extraction is the Unobserved Component (UC) model, where the observed variable y(k) is related to a number of components which represent different perceived features of the data, usually differentiated by their characteristic spectral properties. For example, consider the two well known time series in Figure 5.1: the monthly atmospheric carbon dioxide (CO2) measurements in Mauna Loa in Hawaii, over the period 1958-2000; and the monthly airline passenger data over the period 1949-1960. Both of these time series have two rather obvious characteristics: both are ‘nonstationary’ time series, exhibiting a pronounced upward ‘trend’ and an ‘annual cycle’; and, in the airline passenger case, this annual cycle is growing in amplitude over the observed time period. Note also that atmosphericCO2 series has some missing monthly measurements which, as we shall see, present no real difficulty when these data are analysed using an appropriate UC model.