The classical Fourier analysis of time series data can be used to detect periodic trends
that are of sinusoidal shape. However, this analysis can be misleading when time series
trends are not sinusoidal. When the time series process of interest is binary or
categorical-valued data, it might be more reasonable that the time process be represented
by a square or rectangular form of functions instead of sinusoidal functions. The WalshFourier
analysis takes this approach using a square form of functions.
The Walsh-Fourier analysis is based on the Walsh functions. The Walsh functions are a
square form of functions that take on only two values + 1 and -1. But, unlike sinusoidals,
the Walsh functions are not periodic. Harmuth (1969) introduced the term sequency to
describe generalized frequency to identify functions that are not periodic, such as Walsh
functions. The term sequency is interpreted as the nun1ber of zero crossings or sign
changes per unit time. While the Walsh-Fourier analysis is reasonable in theory for
binary or categorical-valued time series data, the interpretation of sequency is often
difficult.
In this dissertation, using a sequence of periodic functions, we develop the theory and
method that can be applied to binary or categorical-valued data where patterns more
naturally follow a rectangular shape. The theory parallels the Fourier theory and leads to
a "Fourier-like" data transform that is specifically suited to the identification of
rectangular trends.
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