The original data can be seen in Figures 2 onwards. The following figure shows the wavelet transform.
To show the procedure followed, consider the result at scale 10. We will look at the significant upward or downward "signals".
Scale 10 is found to contain six significant upward or downward transitions. The figure above shows their locations. From left to right, they are
There is a relatively good relationship between these indicators of subsequent performance.
Let us have a look at the cumululative effect of these up and down signals. We will code these as +1 and -1 (unweighted). Furthermore, we will simplify the analysis by taking all scales into account without any weighting for the different scales. For scales 1 to 11, we find the numbers of significant up and down "signals" to be (respectively): 206, 1000, 632, 347, 162, 73, 46, 25, 12, 6, 1. The following figure shows a plot of the input data, and a plot of the cumulative up/down nowcasts. The number of up/down signals considered is 206 + 1000 + ... + 1 = 2510. We recall that the number of values in the input data set is 6160, and that the nowcasts were carried out from time point 1000 onwards implying that the 2510 significant up/down signals were extracted from a total of 5161 nowcast values.
We recall also that this is simplified version of our results in that we are taking all up and down signals as equally important (+1 and -1), and furthermore all scales as equally important.
To test the tracking, and therefore genuine forecast, ability of the signficant nowcasts, with reference to the previous figure, we will look at how the significant up/down signals are related to the one-step ahead upward and downward movements in the original time series. The unweighted movement up or down will be used (coded +1 or -1).
Significant nowcast up, 1-step ahead original data up:
Significant nowcast down, 1-step ahead original data down: 716
Significant nowcast up, 1-step ahead original data down: 269
Significant nowcast down, 1-step ahead original data up: 240
Cases 1 and 2 here lead to a correct tracking result of 73.5%. We had significant nowcasts in 48.6% of all possible time-points.
If we consider cases 1, 2 and 4 above as being desirable, then the correct tracking result is 86.0%.
Data from Mark Savage.