Les secondes Journées sur la Correction de la Saisonnalité (Dominique Ladiray, Insee, Paris)

En guise de rappel j'adjoins un bref résumé au sujet du contenu des fichiers excel, que j'emprunte à la précédente entrée:

How to Play with the Excel-files?

Let's take the simplest exercise, Excel-file AR(1), which is an AR(1) real-time filter applied to my simulated process:

• You should see four worksheets when clicking on the above link:
  1. examples
  2. time series
  3. filter characteristics
  4. filtered time series
• The simple artificial process that I'm using -for didactical purposes only-  is generated in the second worksheet `time series'.
  • It is a cosine with white noise.
  • You could change the frequency of the cycle and the signal to noise ratio below the fields marked in yellow.
  • The periodogram is showing the simple line + noise frequency spectrum.
• The crucial filter characteristics -amplitude and time-shift- can be seen in the 3.worksheet (filter characteristics).
  • You could change the parameters of the AR(1)-filter as you like (fields in yellow)
    • pole
    • normalization
    • argument
  • Changes in amplitude and time-shift (phase divided by frequency) can be seen immediately.
  • Everything is calculated in the spread-sheet so that you may verify results straightforwardly.
  • The peak-correlation graph refers to the 4.fourth worksheet and looks for the lead/lag at which correlation of real-time estimate and signal (cycle) is maximal.
• One can straightforwardly verify two fundamental aspects
  • The time-shift function has indeed the interpretation of a delay of the real-time filter (it coincides with the peak correlation concept).
    • We talk about speed of detection.
  • The amplitude function can be related to the amount of noise that contaminates the real-time estimate. 
    • We talk about reliability of real-estimates.
• A sample example is proposed in the work-sheet called accordingly:
  • The cells correspond exactly to those in the worksheet `filter characteristics'
  • You may put the sample example into work by marking the cells in the worksheet `example'  and by copying them to the worksheet `filter characteristics'.
• The application of the one-sided filter to the data is described in the last worksheet (filtered time series).

AR(1), AR(2), ARMA(2,2)-ZPC, Model-Based Real-Time Designs

Besides AR(1) I implemented also more `practically relevant' (in the context of my artificial example) designs:

• AR(2)
• ARMA(2,2)-ZPC
• Model-based : ARMA(2,2) one-step ahead forecasting model
• I added this one on Oct 28: ARMA(4,4)_cycle and deseasonalization.
  • The time series consist of a (deterministic) cycle, noise and a seasonal component (you could change frequency, height of spectral peak and width of spectral peak of the seasonal).
  • The real-time filter is a serial linkage of an ARMA(2,2)-ZPC cycle-filter (as in the examples above) and an ARMA(2,2)-ZPC deseasonalizing filter.
  • The deseasonalizing parameters include control of: frequency, heigth of spectral peak (deepness of filter-trough) and width of peak (trough).
  • The examples in the corresponding worksheet propose pure cycle, pure deseasonalizing as well as combinations of both filters.
  • Unlike the ubiquous 1-B^s deseasonalization filter (yearly-growth for data with fundamental seasonal of length s) the proposed design is able to account for specific location(s), height(s) and width(s) of the seasonal spectral peak (see my open-source book chaps. 3 and 4).

The elaboration is similar to the AR(1)-example:

• Four spread-sheets
• Fields marked in yellow correspond to parameters
• You can put sample-examples into work by marking cells and copying them to the worksheet named `filter characteristics'
• Please try to tailor the filter to your needs by modifying parameters and by observing:
  • Filtered time series
  • Amplitude function
  • Time-shift function
  • Peak-correlation