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.

The Global Ocean Warming Rate Appears To Be 1°C per Century

  I recomputed the warming rates with a starting point of Jan 1947 and the values stabilize at 1°C per century. This appears to be the current rate of global ocean warming as determined by least squares slope method.

As the length of the period for the least squares estimate increases the value becomes less sensitive to the fluctuations in the anomaly.

A Steady Increase in the Mean Global Ocean Warming

  In a blog two weeks ago I used a global ocean warming rate based on a least squares fit of the slope a line through the fixed starting point of the record. There is an advantage for using the fixed starting point since it allows the comparison of periods of different lengths. This allows us to avoid the comparison of apples and oranges. As the length of the period over which the rate is determined increases one might expect it to approach some fixed value if there was some long term average for the global warming rate and its fluctuations were bounded. This definition however shows a nearly steady rate of increase since about 1945.

The changes in CO₂ emissions do not seem to have significantly affected this change in the rate of global ocean warming.

Why Does Successive Residual Polynomial Fits Work?

  To see why the successive residual polynomial fit works so well we can compute the polynomials for the sequence of fits and measure the distances between the fits and the original curve represented by the data points. A simple definition of the distance between the curves represented by two sets of points, f and g, is the square root of the sum of the squares of the distances for each set of points.

The calculation of the successive fits proceeds as follows.

A plot of the distances shows that the successive fits steadily approach the original data points.

The root mean square error is another measure of the distance between two sets of points. One needs to be careful about the density of points and that could be taken into account by including a weight function in the sum of the squares. There may also be some error correction of the coefficients for the fit polynomial taking place as one gets closer to the desired function. Blunders may be eliminated by removing unusually large deviations from the fit.

NIST Saturation Vapor Pressure Curve Fit

  I tested the successive residual polynomial fit method against NIST saturation vapor pressure data to see how it works with more accurate data. It's easiest to fit the log of the pressure, logp, as a function of the inverse of the temperature, U, with a polynomial fit. See Karapetoff, Engineering Applications of Higher Mathematics (1916) for more details on fit procedures. The best fit that I was able to get was for a 35 degree polynomial which added roughly 3 digits of precision to the values of the pressure in the NIST tables.

δ is the error in log(p) for the fit and appears to be just a round off error distorted by the log function. The temperature, T, ranges from the triple point of water to its critical temperature.