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Published byCornelius Ross Modified over 9 years ago
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autocorrelation correlations between samples within a single time series
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A) time series, d(t) time t, days d(t), cfs Neuse River Hydrograph
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high degree of short-term correlation whatever the river was doing yesterday, its probably doing today, too because water takes time to drain away
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A) time series, d(t) time t, days d(t), cfs Neuse River Hydrograph
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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
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A) time series, d(t) time t, days d(t), cfs Neuse River Hydrograph
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moderate degree of year-lagged correlation what ever the river was doing this time last year, its probably doing today, too because seasons repeat
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A) time series, d(t) time t, days d(t), cfs Neuse River Hydrograph
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1 day3 days30 days
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autocorrelation in MatLab
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Autocovariance = Autocorrelation x sdev^2 3130 CFS 2
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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.
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symmetric about zero corr(x,y) = corr(y,x) Autocovariance of Neuse River Hydrograph
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peak at zero lag a point in time series is perfectly correlated with itself Autocovariance of Neuse River Hydrograph
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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
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negative correlation at lag of 182 days points separated by a half year are negatively correlated Autocovariance of Neuse River Hydrograph
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positive correlation at lag of 360 days points separated by a year are positively correlated Autocovariance of Neuse River Hydrograph
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A) B) repeating pattern the pattern of rainfall approximately repeats annually Autocovariance of Neuse River Hydrograph
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autocorrelation in MatLab
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autocovariance related to convolution
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Important Relation #1 autocorrelation is the convolution of a time series with its time-reversed self. This is symmetric of course.
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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)
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Summary time lag 0 frequency0 rapidly fluctuating time series narrow autocorrelation function wide spectrum
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Summary time lag 0 frequency0 slowly fluctuating time series wide autocorrelation function narrow spectrum
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End of Review
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Part 1 correlations between time-series
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scenario discharge correlated with rain but discharge is delayed behind rain because rain takes time to drain from the land
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time, days rain, mm/day dischagre, m 3 /s
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time, days rain, mm/day dischagre, m 3 /s rain ahead of discharge
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time, days rain, mm/day dischagre, m 3 /s shape not exactly the same, either
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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
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this defines the cross-covariance
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cross-correlation in MatLab
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just a generalization of the auto-covariance different times in the same time series different times in different time series
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like autocorrelation, it is similar to a convolution
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As with auto-correlation, two important properties #1: relationship to convolution #2: relationship to Fourier Transform
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As with auto-correlation two important properties #1: relationship to convolution #2: relationship to Fourier Transform cross-spectral density
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Example aligning time-series a simple application of cross-correlation
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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
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u(t) v(t) two similar time-series, with a time shift (this is simple “test” or “synthetic” dataset)
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cross-correlation
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maximum time lag find maximum
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In MatLab
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compute cross- correlation
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In MatLab compute cross- correlation find maximum
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In MatLab compute cross- correlation find maximum compute time lag
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u(t) v(t+t lag ) align time series with measured lag
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A) B) solar insolation and ground level ozone (this is a real dataset from West Point NY)
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B) solar insolation and ground level ozone note time lag
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C) maximum time lag 3 hours
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Coherence a way to quantify frequency-dependent correlation
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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
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time, years 123 123 wind speed temperature
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time, years 123 123 wind speed temperature summer hot and windy winters cool and calm
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time, years 123 123 wind speed temperature heat wave not especially windy cold snap not especially calm
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in this case times series correlated at long periods but not at short periods
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Scenario B in a hypothetical region plankton growth rate and precipitation correlate at periods of a few weeks but they do not correlate seasonally
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time, years 123 123 growth rate precipitation
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time, years 123 123 plant growth rate precipitation summer drier than winter growth rate has no seasonal signal
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time, years 123 123 plant growth rate precipitation growth rate high at times of peak precipitation
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in this case times series correlated at short periods but not at long periods
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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 )
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Fourier transform route to Coherence
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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
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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.
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