autocorrelation correlations between samples within a single time series

1. autocorrelationcorrelations between samples within a single time series

2. Neuse River Hydrograph A) time series, d(t) d(t), cfs time t, days

3. high degree of short-term correlationwhatever the river was doing yesterday, its probably doing today, toobecause water takes time to drain away

4. Neuse River Hydrograph A) time series, d(t) d(t), cfs time t, days

5. low degree of intermediate-term correlationwhatever the river was doing last month, today it could be doing something completely differentbecause storms are so unpredictable

6. Neuse River Hydrograph A) time series, d(t) d(t), cfs time t, days

7. moderate degree of year-lagged correlationwhat ever the river was doing this time last year, its probably doing today, toobecause seasons repeat

8. Neuse River Hydrograph A) time series, d(t) d(t), cfs time t, days

9. 1 day 3 days 30 days

10. autocorrelation in MatLab

11. Autocovariance = Autocorrelation x sdev^2 CFS2 1 3 30

12. Autocovariance of Neuse River Hydrograph The decay around 0 lag is like a composite or typical feature of the time series (a blend of the positive and negative excursions). Periodicities show up as repeating long-range autocorrelations.

13. Autocovariance of Neuse River Hydrograph symmetric about zero corr(x,y) = corr(y,x)

14. Autocovariance of Neuse River Hydrograph peak at zero lag a point in time series is perfectly correlated with itself

15. Autocovariance of Neuse River Hydrograph falls off rapidly in the first few days lags of a few days are highly correlated because the river drains the land over the course of a few days

16. Autocovariance of Neuse River Hydrograph negative correlation at lag of 182 days points separated by a half year are negatively correlated

17. Autocovariance of Neuse River Hydrograph positive correlation at lag of 360 days points separated by a year are positively correlated

18. Autocovariance of Neuse River Hydrograph repeating pattern A) B) the pattern of rainfall approximately repeats annually

19. autocorrelation in MatLab

20. autocovariance related to convolution

21. Important Relation #1autocorrelation is the convolution of a time series with its time-reversed self. This is symmetric of course.

22. Important Relation #2Fourier Transform of an autocorrelationis proportional to thePower Spectral Density of time seriesRecall FT(a*b) = FT(a) x FT(b)

23. Summary rapidly fluctuating time series time narrow autocorrelation function lag 0 wide spectrum 0 frequency

24. Summary slowly fluctuating time series time wide autocorrelation function lag 0 narrow spectrum 0 frequency

25. End of Review

26. Part 1correlations between time-series

27. scenariodischarge correlated with rainbut discharge is delayed behind rainbecause rain takes time to drain from the land

28. rain, mm/day time, days dischagre, m3/s time, days

29. rain, mm/day time, days rain ahead of discharge dischagre, m3/s time, days

30. rain, mm/day time, days shape not exactly the same, either dischagre, m3/s time, days

31. treat two time series u and v probabilistically p.d.f. p(ui, vi+k-1) with elements lagged by time (k-1)Δt and compute its covariance

32. this defines the cross-covariance

33. cross-correlation in MatLab

34. just a generalization of the auto-covariance different times in different time series different times in the same time series

35. like autocorrelation, it is similar to a convolution

36. As with auto-correlation,two important properties #1: relationship to convolution #2: relationship to Fourier Transform

37. As with auto-correlationtwo important properties #1: relationship to convolution #2: relationship to Fourier Transform cross-spectral density

38. Examplealigning time-seriesa simple application of cross-correlation

39. central idea two time series are best alignedat the lag at which they are most correlated, which is the lag at which their cross-correlation is maximum

40. two similar time-series, with a time shift (this is simple “test” or “synthetic” dataset) u(t) v(t)

41. cross-correlation

42. find maximum maximum time lag

43. In MatLab

44. In MatLab compute cross-correlation

45. In MatLab compute cross-correlation find maximum

46. In MatLab compute cross-correlation find maximum compute time lag

47. align time series with measured lag u(t) v(t+tlag)

48. solar insolation and ground level ozone (this is a real dataset from West Point NY) A) B)

49. solar insolation and ground level ozone B) note time lag

50. maximum C) time lag 3 hours