I would like to forecast the non-stationary time series, involving several crucial a-priori assumptions following from studying of instances of such series.
-
I've constructed time-averaged one-point probability distribution function approximated by normal distribution. $$\hat p(x) = \frac{1}{\sqrt{2\pi \sigma^2_{\infty}}} \exp\left(-\frac{x^2}{2\sigma^2_{\infty}}\right)$$
From this point of view, I want the forecast $z_t(l)$ not to exceed this when $l \to \infty$. To put it in other words, variance of $z_t(l)$ must be bounded. -
The average two-point probability distribution function $\hat p(x_i,i;x_j,j)$ also has been constructed, which led to identification of autocorrelation function. $\rho(j) \approx A j^{-\alpha} $ provided $0<\alpha<0.5$.
At first, Box-Jenkins identification process led me to $ARIMA(0,1,3)$ model, however
-
I can't have bounded variance until $d \ne 0$ (which follows from equations for BJ weights $\psi_j$). At the same time, I can't use $d=0$ since initial autocorrelation decreases slowly (which is probably evidence of non-stationarity according BJ). This is the main obstacle to me.
-
Visually, simulation of $ARIMA(0,1,3)$ does not coincide with behaviour of my samples. And correlations of first difference of the series are in the bad agreement with correlations following from the model.
-
The analysis of residuals shows significant correlations starting lag 3. This is why my initial statement about $ARIMA(0,1,3)$ is incorrect.
Trying to fit different $ARIMA(p,0,0)$ models, I see that there is significant residual correlations close to the lag $p$ for every $p$. It may assume that I need $ARIMA(\infty,0,q)$ model (as limiting choice), for instance fractional ARIMA.
From [1] I've learned about Fractional $ARIMA(p,d,q)$ models which are $ARIMA(\infty,0,q)$ in effect.
-
I've not found any GNU R packages with support of missing values for this. Missing values seems to be a kind of challenge.
-
The publications on fractional ARIMA are quite rare. Are such fractional models really used? Maybe there is a good replacement of ARIMA models for my needs? The forecasting is not my major, I have only pragmatic interest.
-
From different literature (for instance [2]), I learned that it is practically impossible to decide between fractional ARIMA and models with "level shift". However, I have not found the package for GNU R to fit 'level shift' models.
[1]: Granger, Joyeux.: J. of time series anal. vol. 1 no. 1 1980, p.15
[2]: Grassi, de Magistris.: "When long memory meets the Kalman filter: A comparative study", Computational Statistics and Data Analysis, 2012, in press.
Update: to render my own progress and to answer @IrishStat
My statement about two-point probability distribution is incorrect in general. Constructed in this way function will depend on full series length. So, there is a little to extract from this. At least, parameter named $\alpha$ will depend on full series length.
Lists 2 and 3 also have been updated.
My data is available as dat file here.
At the current moment, I doubt between FARIMA and level shifts, and I still can't find appropriate software to check this options. This is also my first experience with model identification, so any help will be appreciated.
Best Answer
I have never seen a model like Box-Jenkins identification process led me to ARIMA(0,1,3) model BUT i had never seen a black swan until I went to Australia. Please post your data as it may suggest the need for
If your data is confidential, simply scale it.
OK having received your data (some 80000 readings), I selected 805 observations starting at point 6287 and obtained.