Download CARMENES - NIR First Stage Pipeline

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7.2
NIR First Stage Pipeline (v
2.348)
Ordinary Least Squares Fit
It is easy to deduce from Monte Carlo tests that the ordinary least squares fit for data measured
with some noise in the dependent variable [18, 19, 20, 14] is estimating the slope of the line
better than estimating the slope by averaging Fowler-Pair differences. The result is summarized
in Figure 12. An ideal sequence of measurements was intialized with a constant slope. Then 2N
measurements are taken: The first N are sampled at times 1, 2, . . . N , then g ≥ 0 points are skipped
and not taken into account or simply not available, and then N final points are sampled at times
N + 1 + g, N + 2 + g, . . . 2N + g. To each data point noise with the same width was added. Then the
slope was estimated by the OLS fit on one hand and by averaging the N Fowler pairs on the other
hand. This was done 2000 times, resulting in two statistics of slope errors. The root-mean-square
of the error for the OLS fit divided by the root-mean-square of the error for the MER fit is generally
smaller than one. (The case with N = 1 pair is marginal because the fits of the slope are the same
with the two estimators, so the errors are also the same for any given noise.)
The fact that both means of deriving a slope are not equivalent is obvious: One could add actually
a constant value to both values in any measurement of a Fowler pair, which would not take any
effect on the slope estimator because that constant would cancel when the difference is calculated.
The least squares estimator would be effected because the leverage arm of the two samples relative
to the center would differ.
In conclusion there is no reason to implement a separate MER data analysis because the OLS fit
outperforms it for all combinations of sample number and intermediate gap in the sample series.
Therefore only a single fitting (merging) function pipFits ols is implemented.