Timeseries

Question 1

Cross correlation and auto-correlation:

Answer 1

Correlation: A standardized representation of association between two variables . -1<X<1.

Cross correlation and autocorrelation are very similar, but they involve different types of correlation:
-Cross correlation:
Measure of similarity of two series as a function of the lag of one
relative to other.

R(T)= E{(X(t) - ux)(Y(t+1) -ux)} / ox*oy

Example: High cross-correlation between precipitation and discharge after four hours,
due to the process of infiltration and water flow to the discharge point.

• Autocorrelation: Correlation of a variable with itself sampled at different points in time. It shows the similarity between observations as a function of the time lag between them. Tool for finding repeating patterns such as the presence of a periodic signal obscured by noise. Periodicity can be found. Basic idea: Shift the same dataset (e.g., X) in time by one unit (time lag between subsequent observations) and calculate the correlation coefficient. Do the same after shifting by two time units and so on

R(T)= E{(X(t) - ux)(X(t+1) -ux)} / o2x

Example: Discharge has a large autocorrelation at the first hour of shift. This means that
two observations with an one-hour time lag between them will be similar. As time goes
by, autocorrelation gets smaller. Precipitation has a low auto-correlation even after only
a two-hour lag time. This means: If it rains a 2 pm it is not very likely that it will also rain
at 3 pm.

note: We assume stationarity of the time series (mean, variance, and autocorrelation structure do
not change over time). If this condition is not met in a measured time series, it must be
transformed prior to analysis.

Question 2

time series

Answer 2

• Consider one or two sequences of data points (X and Y) measured over a continuous time interval, using equal spacing between consecutive measurements, with each moment in time within the time interval having assigned one data point.