Derivation of Some Least Squares Formulas

  One can use the method of least squares to derive vector formulas for best fits. For a linear fit one can define the deviation, δ_k, as the vertical distance of data point k from a line and the variance, V, as the sum of the squares of the deviations. The formula for estimating the best slope through a set of data going through a particular point is derived as as shown below. The best slope is assumed to be that for which the sum of the squares of the deviations or the variance is a minimum. From the theory of maxima and minima in calculus we know that for the minimum the derivative of the variance is zero.

In the above the chosen point that the line goes through is (t_p,ΔT_p). The equation for the line is  ΔT(Δt) =  ΔT_p + s Δt where Δt = t - t_p. A vector Σ whose components are all 1 is needed to include the scalars t_p and ΔT_p in the vectors defining δ and Δt.

  One can do something similar to find vector formulas for the best straight line through a set of data.

Merry Christmas.

Polar Warming

  NOAA Climate has recently pointed out that the arctic is warming at a faster rate than the global average. Temperature anomaly data is available from the NOAA ftp site for 30° intervals of latitude and can be used to compute global and arctic tracks for comparison.

 The anomaly baseline for the ftp data is 1971-2000 so the vertical scale is different from NOAA's plot. The arctic track appears to have started increasing about 1970, a little earlier that the global track of the anomaly mean. The arctic anomaly data also shows much more variation in its values than the antarctic data.

This may be due to the fact that the arctic has an ocean ice pack and the antarctic a continental ice sheet. The antarctic currently appears to be cooling.

Tracking Global Land-Ocean Warming

  I tweaked the procedure a little for the global land-ocean anomaly by using a dotted line for the linear projection at the end of the plot.

The computed slopes seem to be a little more sensitive to changes in the step size, q, than the anomaly track.

Tracking Global Land Warming

  I did the same tracking calculation for the global land anomaly with the last few values obtained using the last estimate of the slope. Again something seems to have happened in the late 70's.

The tracking method seems to work better than the successive residual polynomial fit method with better behavior at the beginning and end of the plot.

Tracking Global Ocean Warming

  One can use the formula for estimating slopes to track changes in the global ocean anomaly. I fit a straight line to the first five years of data to get a starting anomaly and slope. Then I stepped forward one year along that track to get a new value for the anomaly, estimated the slope for the next 5 year interval (Δp) and repeated the process. The fit for the last five years was a linear projection based on the last slope found. Here is the calculation for the repeated steps.

The warming indicates the rate is not steady as the track anomaly and slope show.

There seems to have been a slight jump in the rate of ocean warming after the mid 70s but it's difficult to characterize the fluctuations.

Modified Formula for the Slope

  The formula for the slope in the last blog assumed that Δt_k = t_k - t₈₀₄ so that Δt₀ = 0. The formula can be modified to give the best slope through an arbitrary point (t_p,ΔT_p) is as follows.

This vector formula is similar to method used to estimate the derivative of f(t) using Δf/Δt. The dot products are needed since division is not defined for vectors.

A Check on the Same Rate Calculation Using a Different Method

  Instead of doing a search to find the slope which gives the best fit for same starting point to the global warming record one can use a formula to estimate the best slope. The formula for s_min below uses vector notation and the products are dot products. A function is needed to compute s_min since one has to use vectors whose lengths are determined by the length of the period specified by m _j.

The formula gives values which are nearly identical to those found by the search as the plot shows. The calculation using the formula takes less time.