Trend Estimation




Trend definition. The trend fn is the deterministic part of a non-stationary stochastic process Xn=fn+Zn with additive stationary noise Zn. Due to its stationarity, for each n the expectation value of the noise is zero E{Zn}=0. If the trend and the noise are mutually independent, then the trend represents the variation in time of the mean of the stochastic process fn=E{Xn}. If many time series {xn} obtained as realizations of {Xn} are available, then each term of the trend can be computed by a simple average.

Trend estimation. In practice more time series obtained under exactly the same conditions are rarely available. Usually we have to analyze a single time series and then the trend is estimated as the component of a time series that is “slowly changing in time”. It is implicitly assumed that the noise does not have this property, i.e., the values of the noise become uncorrelated on time intervals smaller than the time scale for which the trend values are correlated. Otherwise the slowest fluctuations of the noise are confounded with the variations due to the trend, and their separation is very difficult. In such cases the false trend is called stochastic trend.

Accuracy of trend estimation. In designing a trend estimation algorithm an essential component is the method to evaluate its accuracy for a large diversity of time series. However, the algorithms are very often tested under unrealistic conditions and on too small number of time series. Even when the Monte Carlo experiments are performed, the artificial time series are much simpler than those encountered in practice, with monotonic (linear, power-law, exponential, and logarithmic) or periodic (sinusoidal) trends. The main difficulty is to generate realistic nonmonotonic trends with a diversity of shapes comparable with that of the real time series.

Al these issues are addressed in our book

Călin Vamoş, Maria Crăciun
Automatic Trend Estimation
Springer 2012

We present original automatic algorithms for processing nonstationary time series containing a stationary noise superposed over a nonmonotonic trend which can be freely downloaded.

trendacd automatically estimates the monotonic component of a given time series.
local_extrema automatically determines the local extrema of a temporal scale from a given time series.
trendrcma automatically estimates a non-monotonic trend from a given time series.

In our book we have illustrated the functioning of the analyzed algorithms by processing time series from astrophysics, finance, biophysics, and paleoclimatology. We are interested in their abbility to estimate the trend from other time series, so those who intend to use these algorithms are invited to share with us their experiences and the possible errors that will appear.