San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
USA

Trends and the Stochastic Structure
of Temperature and Other Variables
Which are the Cumulative Sum
of Random Disturbances Which
are Serially Correlated

This material is concerned with the objective determination of trends for data which is generated by a process given by


T[t] = T[t-1] + U[t]
where
U[t] = λU[t-1] + V[t]
 

where V[t] is a random variable uncorrelated with V[s] for s≠t. Such data will appear to have trends even if the expected value of U[t] is zero for all t. This is illustrated below: (Click on REFRESH to get a new sample and new time series.)

Temperature statistics have such a stochastic structure because the rate of change of temperature T for some region is given by


C(dT/dt) = W(t)
 

where T and t are temperature and time, respectively, W(t) is the net energy inflow and C is the heat capacity coefficient of the region. Thus


T(t+h) = T(t) + (1/C)∫tt+hW(s)ds
or, equivalently
T(t+h) = T(t) + Uh(t)
 

This stochastic structure also applies to many climate variables such as humidity and soil moisture.

When there is no variation in the parameters of the probability distribution of Uh(t) the temperature reaches a level so that the average value of the temperature changes is zero.

Below is shown the data for the average global temperature from 1855 to 2003.

An examination of the changes in global temperature reveals that the changes in temperature are negatively correlated.

Consider the general formula for U(t) in terms of V(s) for s≤t. The first few terms reveals the pattern:

U(t) = λU(t-1) + V(t)
but
U(t) = λU(t-1) + V(t)
so
U(t) = λ²U(t-2) + λV(t-1) + V(t)

Similarly

U(t) = λ³U(t-3) + λ²V(t-2) + λV(t-1) + V(t)
or, equivalently
U(t) = λ³U(t-3) + V(t) + λV(t-1) + λ²V(t-2)
and the full pattern is
U(t) = λtU(0) + V(t) + λV(t-1) + λ2V(t-2) + λ2V(t-2) + λ3V(t-3) + … + λt-1V(1)

Writing out all of the expressions for U(s) for s≤t

U(t) = λtU(0) + V(t) + λV(t-1) + λ2V(t-2) + λ3V(t-3) + … + λt-1V(1)
U(t-1) = λt-1U(0)             + V(t-1) + λV(t-2) + λ2V(t-3) + λ3V(t-4) + … + λt-2V(1)

(To be continued.)


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