1. autocorrelation correlations between samples within a single time series

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

3. high degree of short-term correlation whatever the river was doing yesterday, its probably doing today, too because water takes time to drain away

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

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

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

7. moderate degree of year-lagged correlation what ever the river was doing this time last year, its probably doing today, too because seasons repeat

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

9. 1 day3 days30 days

10. autocorrelation in MatLab

11. Autocovariance = Autocorrelation x sdev^2 3130 CFS 2

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

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

15. 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 Autocovariance of Neuse River Hydrograph

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

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

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

19. autocorrelation in MatLab

20. autocovariance related to convolution

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

22. Important Relation #2 Fourier Transform of an autocorrelation is proportional to the Power Spectral Density of time series Recall FT(a*b) = FT(a) x FT(b)

23. Summary time lag 0 frequency0 rapidly fluctuating time series narrow autocorrelation function wide spectrum

24. Summary time lag 0 frequency0 slowly fluctuating time series wide autocorrelation function narrow spectrum

25. End of Review

26. Part 1 correlations between time-series

27. scenario discharge correlated with rain but discharge is delayed behind rain because rain takes time to drain from the land

28. time, days rain, mm/day dischagre, m 3 /s

29. time, days rain, mm/day dischagre, m 3 /s rain ahead of discharge

30. time, days rain, mm/day dischagre, m 3 /s shape not exactly the same, either

31. treat two time series u and v probabilistically p.d.f. p(u i, v i+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 the same time series different times in different 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-correlation two important properties #1: relationship to convolution #2: relationship to Fourier Transform cross-spectral density

38. Example aligning time-series a simple application of cross-correlation

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

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

41. cross-correlation

42. maximum time lag find maximum

43. In MatLab

44. compute cross- correlation

45. In MatLab compute cross- correlation find maximum

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

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

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

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

50. C) maximum time lag 3 hours

51. Coherence a way to quantify frequency-dependent correlation

52. Scenario A in a hypothetical region windiness and temperature correlate at periods of a year, because of large scale climate patterns but they do not correlate at periods of a few days

53. time, years 123 123 wind speed temperature

54. time, years 123 123 wind speed temperature summer hot and windy winters cool and calm

55. time, years 123 123 wind speed temperature heat wave not especially windy cold snap not especially calm

56. in this case times series correlated at long periods but not at short periods

57. Scenario B in a hypothetical region plankton growth rate and precipitation correlate at periods of a few weeks but they do not correlate seasonally

58. time, years 123 123 growth rate precipitation

59. time, years 123 123 plant growth rate precipitation summer drier than winter growth rate has no seasonal signal

60. time, years 123 123 plant growth rate precipitation growth rate high at times of peak precipitation

61. in this case times series correlated at short periods but not at long periods

62. Brute force way to get the in-phase part of coherence band pass filter the two time series, u(t) and v(t) around frequency, ω 0 compute their cross correlation (large when the time series are similar in shape) repeat for many ω 0 ’s to create a function c(ω 0 )

63. Fourier transform route to Coherence

64. The "cross-spectrum" has 2 parts “Squared Coherence” frequency-dependent power (squared covariance) between two time series (possibly in a lagged sense) "Phase difference" frequency-dependent lag between time series

65. A subtle point A single pure sinusoid, a single Fourier component of u(t), is by definition perfectly correlated (at some lag) with the same-period Fourier component of v(t). Monochromatic cross-spectral coherence is always 1! But if u and v are meaningfully, physically connected (with some lag) on some time scale, all the Fourier components with periods near that time scale will exhibit a similar lag. Alternately, if there is such a physical relationship, a single given Fourier component (frequency) will exhibit a similar lag between u and v in many realizations (such as segments of a long time series). You need to combine several degrees of freedom (Fourier components or realizations) to get meaningful cross-spectral coherence tests for a relationship.