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autocorrelation correlations between samples within a single time series

autocorrelation correlations between samples within a single time series. Neuse River Hydrograph. A) time series, d(t). d(t), cfs. time t, days.

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autocorrelation correlations between samples within a single time series

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  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

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