PIECEWISE LINEAR SERIES

At the end, the solution that minimizes the residual sum of squares is chosen. For each of these cases one gets the best fit, that satisfies the desired conditions between two consecutive linear trends. One starts with the case of one breakpoint, up to a maximum of twelve breakpoints. With the creation of the A matrix completely automated, the system of equation (1) can be consecutively solved for every combination of possible solutions, that obeys the minimum imposed distance between two consecutive breakpoints. The results are still the same if instead of a minimum 15 year period ones uses 10, 20 or even 30 years. , representing the mean world temperature, with the conditions of a minimum 15 year interval between breakpoints and of changing sign between two consecutive trends, leads to the results they have obtained.

PIECEWISE LINEAR SERIES

Applying this methodology to the time series used by Karl et al. The method computes the best combination of continuous line segments that minimize the residual sum of squares, subjected to a pair of conditions: (a) the interval between breakpoints must equal or exceed a given value, (b) two consecutive trends must obey one or more imposed conditions. resulted of an eye inspection of the time series, the number and location of the breakpoints are simultaneously optimized. The methodology proposed here is a development of that second approach, where instead of arbitrarily fixing the number of line segments, which in Karl et al. Using this approach they were able to evaluate the partial trends, a better overall trend, and most of all they identified three breakpoint years: 1910, 19. In order to clearly separate the two periods of warming, they devised two approaches: one based on Haar Wavelets, which was able to identify three discontinuities in the time series, and the second (the preferred one) consisting of the minimization of the residual sum of squares of all possible combinations of four line segments representing time intervals of 15 years or more, and constrained to have their end points intersecting at the year of change point.

Even an eye inspection revealed that the mean warming obtained by fitting a straight line did not occur in a persistent way, but in two sustained periods, one beginning around 1910 and the other starting in the mid 1970s. pointed out that a linear trend is not adequate to describe its low frequency behavior. Analyzing the global temperature time series for the period 1880 till 1997, Karl et al. Many of these studies were motivated by the quest for anthropogenic climate changes in the last century. During the last decade there have been a large number of papers discussing long term linear tendencies of climate parameters, such as precipitation, temperature and the NAO index, to mention just a few.

Piecewise linear