1. (a) Measuring the memory in a time series with auto-correlation function (ACF)

2. (a) Shifting the time series by one time step gives pairs of observations We calculate the (auto-)correlation at lag 1 r = 0.45

3. If we have sufficient data we can shift the time series also by two or m time steps. The shifting is also called lag. r = 0.13

4. (a) Using an number of different shifts (‘lags’) we obtain a Auto-correlation function (ACF)

5.  acf(x) Null hypothesis r-values that mark the significance level (at 5%) Example with a random sample from White Noise

6. . “White noise” “Red noise” Autocorrelation Function The mean, variance autocovariance (and thus the ACF) are not changing over time.

7. “Red noise” “White noise” The mean, variance autocovariance (and thus the ACF) are not changing over time.

8.  We have studied already many times correlations between two time series (e.g. temperature records from Albany and New York Central Park)  This was done without a time lag.  But we can shift one time series by one time step, 2 or m time steps and then calculate the correlation  => Cross Correlation Function (ccf)

9. Article in Nature 2012.  Paleoclimate temperature reconstruction from temperature sensitve ‘proxies’

10.  CO2 curve  ftp://aftp.cmdl.noaa.gov/products/trends/co2 /co2_mm_mlo.txt ftp://aftp.cmdl.noaa.gov/products/trends/co2 /co2_mm_mlo.txt