1. Two examples

2. Canadian Lynx data 1821-1934 Annual trappings of Canadian Lynx

3. The Data (an example of a prey-predator relationship). Note sharp peaks and wide minima

4. Try Log Scale

5. Its ACF. MA models not likely. Looks like AR model with complex roots

6. PACF is confusing though. Should we try AR(2) or AR(4) or AR(7) or even AR(11)?

7. AR(2)? m = 6.686 p-value 0.0000 a 1 = 1.39 p-value 0.0000 a 2 = -0.7528 p-value 0.0000 Portmanteau test: p-value 0.0999

8. ACF for residuals in AR(2) model. Looks good? But-

9. AR(4)? m = 6.684 p-value 0.0000 a 1 = 1.272 p-value 0.0000 a 2 = -0.7005 p-value 0.0000 a 3 = 0.1413 p-value 0.3604 a 4 = -0.2061 p-value 0.0318 Portmanteau test: p-value 0.0350

10. ACF for residuals in AR(4) model, Portmanteau test is no good any longer.

11. AR(7)? m = 6.699 p-value 0.0000 a 1 = 1.269p-value 0.0000 a 2 = -0.6901p-value 0.0000 a 3 = 0.2776 p-value 0.1018 a 4 = -0.3588 p-value 0.0335 a 5 = 0.1836p-value 0.2778 a 6 = -0.213 p-value 0.1728 a 7 = 0.2316p-value 0.0177 Portmanteau test: p-value 0.0184

12. For AR(7) model, Portmanteau test is even worse: p-value 0.018

13. AR(10)-first model with good Portmanteau test m = 6.697 p-value 0.0000 a 1 = 1.243p-value 0.0000 a 2 = -0.6605p-value 0.0000 a 3 = 0.2863p-value 0.0881 a 4 = -0.3504p-value 0.0398 a 5 = 0.2112p-value 0.2192 a 6 = -0.2084 p-value 0.2269 a 7 = 0.174p-value 0.3072 a 8 = -0.1253p-value 0.4589 a 9 = 0.3683p-value 0.0194 a 10 = -0.2184p-value 0.0335 Portmanteau test: p-value 0.2812

14. ACF for residuals in AR(10) model

15. Since φ(11) looked significant, we try AR(11) as well m = 6.697 p-value 0.0000 a 1 = 1.168p-value 0.0000 a 2 = -0.5346p-value 0.0005 a 3 = 0.2515p-value 0.1121 a 4 = -0.2963p-value 0.0661 a 5 = 0.1409p-value 0.3881 a 6 = -0.1397 p-value 0.3938 a 7 = 0.05p-value 0.7618 a 8 = -0.0288p-value 0.8595 a 9 = 0.1458p-value 0.3633 a 10 = 0.2216p-value 0.1503 a 11 = -0.3758p-value 0.0003 Portmanteau test: p-value 0.8189

16. ACF for residuals in AR(11) model

17. AR(11) has the best AIC value, followed by AR(12)-AR(19) (??) Remember, the data set is 114 points long.

18. AIC values

19. Sunspots data

20. As of Yesterday …

21. We have seen this series before. We clearly see 11 years cycle. The data is too short to see longer (100+ years) cycles

22. Here is another view:

23. and another …

24. Yet another view (reversed in time)

25. Let’s consider this, once again

26. ACF. Typical behavior for AR(2) model with complex roots.

27. Its PACF. AR(2)?

28. Now, let us look at monthly data,778 points, since December 1944. Note steep accents and not as steep drops. This is a clear evidence of non-linearity in the model.

29. Its ACF (200 points, about 19 years)

30. Its PACF. First model that has reasonable Portmanteau test, is AR(13)

31. Since the data is only 60+ years long, we can see only the 11 years cycle here though longer cycles definitely exist. The model below predicts next grand minimum around 2050

32. Have a Nice Break!