There is a cycle in average global temperature along with a long term trend that goes back the 160 years for which there is data, as is shown below for the data from the National Oceanic and Atmospheric Administration (NOAA) which goes back to 1880.

The data from the Goddard Institute for Space Science also goes back to 1880.

The data from the Hadley Climate Research Unit (CRU) of the University of East Anglia goes back to 1850.

The cycle is an upswing of roughly 30 years and a downswing of roughly the same length. The pattern can be represented as a bent line, as in the diagram above, but a better representation would be

T(t) = A*sin(νt+φ) + Bt + C + u(t)

where T is the deviation of the average global temperature from its long term average, the temperature anomaly, and t is time (years since 1880). The parameter ν is the frequency of the cycle and φ is its phase. The variable u(t) is a random fluctuation. A, B and C are constants.

The cycle period P is such that νP=2π and thus P=2π/ν.

Using the formula for the sine of a sum, the formula for T(t) can be put into the form

T(t) = A[sin(νt)cos(φ) + cos(νt)sin(φ) + Bt + u(t)
or, equivalently
T(t) = A*cos(φ)sin(νt) + A*sin(φ)cos(νt) + Bt + u(t)

A regression of T(t) on sin(νt), cos(νt) and t would provide the information for determining A, φ and B. The only problem is that ν and P are not known. However P can be chosed to maximize the coefficient of determination (R²) for the regression equation.

Regression Results Using the NOAA Global Temperature Data

When a value of P equal to 60 years is used the value of R² is 0.853388. Values of 59 years and 61 years result in slightly lower values for R². A value of 59.6 years produces the highest value of R² which is 0.853511.

The regression equation using a period of 59.6 years is

T(t) = 0.032697891sin(νt) + 0.126883966cos(νt) + 0.00552405t −0.35641

The frequency ν is 0.10542257 radians per year or 0.016778523 cycles per year. The amplitude of the cycle, A, is found as the square root of the sum of the squares of the coefficients of sin(νt) and cos(νt). Its value is 0.13103 °C. The cosine of the phase φ is equal to the ratio of the coefficient of sin(νt) to the value of A; i.e., 0.03270/0.13103=0.24955. This means that the value of φ is 1.31858 radians. The division of φ by ν gives the phase in terms of years; i.e., 12.50761 years. Thus the regression equation for the temperature anomaly is

T(t) = 0.13103sin(ν(t+12.50761)) + 0.0055t − 0.35641

with ν=0.01678 cycles per year and t=Year−1880. As reported above the coefficient of determination is 0.853511, thus indicating that 85.35 percent of the variation in the global temperature anomaly from 1880 to 2010 is explained by a cycle and trend. The coefficient of the time variable indicates that there is a long term trend of 0.0055°C per year, or 0.55°C per century. This is quite comparable to the estimate of 0.5°C per century found elsewhere using a different method. The t-ratio for the trend coefficient is 23.2 and those of sin(νt) and cos(νt) are 2.5 and 10.3, respectively. Thus all are significantly different from zero at the 95 percent level of conficence.

Below is given the display of data along with the regression estimate.

The standard error of the estimate is 0.099°C. An interesting aspect of the display is that having a linear upward trend makes the upswings of the cycle to appear to be longer than the downswings.

An extrapolation of the regression estimate to the year 2100 is shown below.

The projected temperature for 2100 is about 1/4°C above the actual temperature for 2010.

(To be continued.)