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Unformatted text preview: ion of autocorrelation is by plotting the auto‐
correlation as a function of lag time, that is autocorrelation function (acf).
U(t), U(t+1); U(t), U (t+2); U(t), U(T+3);….. U(t), U(t+6);……
U(t), U(t+24);……… From statistics, autocorrelation can be established as
Tauto (t)=f1* Tauto (t1) + f2* Tauto (t2) + … (2). Stationarity
Stationary: time series have no statistical properties that change through time (no mean and variance change, no trend). Otherwise, nonstationarity. Annual Cycle Non‐stationarity in the mean can be removed by subtracting the average value for each hour, day, or by fitting and removing as analytical function, such as sinusoid, polynomials. (2). Stationarity
Stationary: time series have no statistical properties that change through time (no mean and variance change, no trend). Otherwise, nonstationarity. Annual Cycle (3). Time series model
It is important to identify variability of a climatological time series, and then to develop a model for the time series.
One method: component decomposition.
Most climatological time series can be decomposed into an additive time series that may contain:
(1) Cycles: an annual cycle, a diurnal cycle, etc.
(2) auto‐correlation,
(3) a linear trend,
(4) a random component.
Time series model:
T(t) = T cycle (t) + T auto (t) + T random (t) + α × t Time series model:
T(t) = T cycle (t) + T auto (t) + T random (t) + α × t
From statistics, autocorrelation can be established as:
Tauto (t)=f1* Tauto (t1) + f2* Tauto (t2) + f3* Tauto (t3) … The cycle is a form of non‐stationary and can be
modeled as a series of sine and cosine functions
(harmonics in a Fourier series). Based on Fourier
analysis in calculus, any well‐behaved continuous
function can be described by an infinite Fourier series.
Not Required:
T=a0+∑n[ancos(2πnt/N)+bnsin(2πnt/N)] Time series model:
T(t) = T cycle (t) + T auto (t) + T random (t) + α × t
The cycle can be modeled as a series of sine and cosine functions (non‐stationarity part)
Tcycle =a0+∑k[akcos(2πkt/n)+bkSin(2πkt/n)]
From statisti...
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This document was uploaded on 01/22/2014.
 Winter '14

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