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Package smile.timeseries

Time series analysis.

See: Description

Package smile.timeseries Description

Time series analysis. A time series is a series of data points indexed in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data.

Methods for time series analysis may be divided into two classes: frequency-domain methods and time-domain methods. The former include spectral analysis and wavelet analysis; the latter include auto-correlation and cross-correlation analysis. In the time domain, correlation and analysis can be made in a filter-like manner using scaled correlation, thereby mitigating the need to operate in the frequency domain.

The foundation of time series analysis is stationarity. A time series {r_t} is said to be strictly stationary if the joint distribution of (r_t1,...,r_tk) is identical to that of (r_t1+t,...,r_tk+t) for all t, where k is an arbitrary positive integer and (t1,...,tk) is a collection of k positive integers. In other word, strict stationarity requires that the joint distribution of (r_t1,...,r_tk) is invariant under time shift. This is a very strong condition that is hard to verify empirically. A time series {r_t} is weakly stationary if both the mean of r_t and the covariance between r_t and r_t-l are tim invariant, where l is an arbitrary integer.

Intuitively, a stationary time series is one whose properties do not depend on the time at which the series is observed. Thus, time series with trends, or with seasonality, are not stationary — the trend and seasonality will affect the value of the time series at different times. On the other hand, a white noise series is stationary — it does not matter when you observe it, it should look much the same at any point in time. Note that a time series with cyclic behaviour (but with no trend or seasonality) is stationary.

Differencing is a widely used data transform for making time series stationary. Differencing can help stabilize the mean of the time series by removing changes in the level of a time series, and so eliminating (or reducing) trend and seasonality. In addition, transformations such as logarithms can help to stabilise the variance of a time series.

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