1. Def.: Time Series Time series is a collection or sequence of numbers that represent the state of any system as a function of time or space or any other “ ordering ” independent variable.

2. Scalar Time Series Data e.g., Sealevel Pressure e.g., Bottom Temperature (almost periodic) (transient)

3. Vector Time Series Data Speed (length of “ stick ” ) and direction (orientation of “ stick ” ) as a function of time Ocean Currents

4. Matrix Time Series Data Spatial pattern of “ some scalar ” Reflectance at 645 nm (red part of visible spectrum) of ice, water, and land from MODIS July-26, 2009 11:40 UTC

5. Matrix Time Series Data Spatial patterns that vary with time

6. MEASUREMENT = SIGNAL + NOISE How to separate?

7. Ensembles are a set of realizations x i (t) of a random process {x(t)} Experiment-4 Experiment-3 Experiment-2 Experiment-1 tt+τTime t and lag τ tree-1, drifter-1 tree-2, drifter-2 tree-3, drifter-3 tree-4, drifter-4 x 1 (t) x 2 (t) x 4 (t) x 3 (t)

8. Qualities of time series: Sine wave Sine wave plus random noise Narrow-band random noise Wide-band random noise

9. Probability Density Functions: Sine wave Sine wave plus random noise Narrow-band random noise Wide-band random noise (histograms)

10. Auto-correlation functions Sine wave Sine wave plus random noise Narrow-band random noise Wide-band random noise 0 Lag Time  Lagged Auto-Correlation R xx (  )

11. Auto-spectral Density Sine wave Sine wave plus random noise Narrow-band random noise Wide-band random noise