Testing the Delta Phenomenon, Part 6

I have now finished the testing of the Intermediate Term solution for the S&P 500 and summary results are shown below, giving the test period for each separate test conducted (15 in total) along with the test statistic of the actual data and the p-value percentage, which should be read as the percentage of random permutations (500,000 in total for each test) that were as good as or better than the actual test statistic in question for that period. The period for each test covers the time span for one complete cycle of the solution, i.e. from point 1 to point 12 of the solution. The entire test period more or less corresponds with the period for the earlier Medium Term solution tests, and as in those previous tests, the data for the period is out of sample data, i.e. not "seen" by the solution prior to the tests being conducted.

5th March 2007 (4445) to 25th June 2007 (4557)
[1] 3.5
[1] 0.3668 %

3rd July 2007 (4565) to 22nd Oct 2007 (4676)
[1] 4.75
[1] 3.3826 %

30th Oct 2007 (4684) to 17th Feb 2008 (4794)
[1] 4.583333
[1] 2.8366 %

25th Feb 2008 (4802) to 14th June 2008 (4912)
[1] 3.916667
[1] 0.9496 %

17th June 2008 (4915) to 10th Oct 2008 (5030)
[1] 3.333333
[1] 0.2472 %

18th Oct 2008 (5038) to 5th Feb 2008 (5148)
[1] 3.333333
[1] 0.2524 %

9th Feb 2009 (5152) to 4th June 2009 (5267)
[1] 5.083333
[1] 4.7866 %

5th June 2009 (5268) to 2nd Oct 2009 (5387)
[1] 5.333333
[1] 5.477 %

9th Oct 2009 (5394) to 29th Jan 2010 (5506)
[1] 4.25
[1] 1.9232 %

2nd Feb 2010 (5510) to 4th June 2010 (5632)
[1] 4.230769
[1] 1.306 %

31st May 2010 (5628) to 21st Sept 2010 (5741)
[1] 3.583333
[1] 0.4506 %

23rd Sept 2010 (5743) to 28th Jan 2011 (5870)
[1] 3.538462
[1] 0.2902 %

31st Jan 2011 (5873) to 21st May 2011 (5983)
[1] 4.083333
[1] 1.5542 %

31st May 2011 (5993) to 8th Sept 2011 (6093)
[1] 4.272727
[1] 3.2184 %

16th Sept 2011 (6101) to 2nd Feb 2012 (6240)
[1] 3.928571
[1] 0.2982 %

As can be seen, if 5 % is taken as the level of statistical significance only one test fails to reject the null hypothesis, and 7 out of the remaining 14 are statistically significant at the 1 % level. I also repeated the test several times over the entire data period, encompassing a total of 182 separate turning points with a test statistic of 4.10989 for these 182 points. In these repeated tests (for a total of a few million permutations) not a single permutation was as good as or better than the given test statistic! These results are much better than I had anticipated and I therefore consider Delta to have passed these tests as well. For readers' interest a plot of the period in which the null hypothesis is not rejected is shown below,

where the yellow lines are my identification of the actual turning points and the white, red and green lines are the turning points projected by the solution.

Now that this series of tests is complete, and Delta has passed them, what does it all mean? This will be the subject of my next post.