The two-stage method of determining linear trend parameters for data in aggregated time sequences of periods and moments
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Regularities concerning the dynamics of variables and regularities concerning co-dependance in time can be observed providing one possesses data in the form of time sequences. The observation sequence of a studied phenomenon in consecutive time units is called a time sequence and it is a realization of a stochastic process. With regard to the topic of this article, the division of time sequences into sequences of periods and sequences of moments is significant. The time sequences of moments represent the state of a phenomenon under study in defined moments in time (eg 31 December each year), while the time sequences of periods provide information about the scale of a phenomenon in specific periods (eg years, quarters of a year, months). The most common analytical forms of a trend include a linear model (the resp. equation given). In dynamics analysis, the smaller the intervals between observations of a stochastic process, the bigger the number of observations and the more detailed information about the studied process. Regrettably, dynamic models are built on the basis of available statistical data, which not always match the intended degree of accuracy. The two-stage method of determining linear trend function parameters involves constructing a model for aggregated data (eg annual) and subsequently, using the trend parameters, determining a model for shorter, deaggregated data (of m-period trend). The paper presents formulas (taken from references and the author's own) of parameter estimates for a linear trend, which is a time sequence analysis as a realization of a continuous stochastic process. The two-stage method formulas are dependent on the kind of time sequence and the way of numbering the time variable 't'. The examples of a suggested approach to the determination of parameter estimates of a 'deaggregated' linear trend based on parameter estimates of an 'aggregated' linear trend are also given.
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