Need To Use Weighted Fits For The Drift Function

  The frequency table indicates that not all the data points for the expected next value of the anomaly have the same number of points contributing to the estimate. The points near the center of the plot are more accurate than those at the ends. Weighted least squares polynomial fits give a better fit near for the central portion of the graph. A derivation of the procedure that I use for function fits is shown below. For polynomial fits the functions are just the powers of x. The data points are (x_k,y_k) which have weights w_k associated with them.

The function φ2a uses this method to find the coefficients of the polynomial which gives the fit with the least variance. The weights used for the x values are the sum of the frequencies in its column of the frequency table. For ordinary least squares polynomial fits the weights used are w_k = 1 for each data point and so W is just the identity matrix and consequently it is not needed to find the coefficients.

In the plot above most of the data to the left of x = 0 is prior to the beginning of 1980 while that to the right is after that. The data points between x = -1 and x = 1 have greater weights and less uncertainty in the estimated expected values. The estimates are better to the left since they are based on about 100 years of data while those to the right are based on about 33 years of data. It appears to be the same curve on both sides of the center and more data would help confirm this. The fit does not appear to be symmetrical about the stable point but there is a relatively flat section just to the right of it. The curvature is present in the most accurately known portion of the drift function just to the left of center.