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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

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 U_{h}(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)