Time series Analysis

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

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

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

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

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

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

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

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) Cross-Correlation Function

9. Paleoclimatetemperature reconstruction from temperature sensitve ‘proxies’ Cross-Correlation Function Article in Nature 2012.

10. CO2 curve • ftp://aftp.cmdl.noaa.gov/products/trends/co2/co2_mm_mlo.txt Harmonic Analysis: Estimating the annual cycle amplitude