Changeset 2779 for trunk/doc/pslib/psLibADD.tex

Timestamp:

Dec 21, 2004, 11:37:08 AM (22 years ago)

Author:

Paul Price

Message:

Adding extra information on histograms, including an example of
histograms in the presence of errors.

File:

• trunk/doc/pslib/psLibADD.tex (modified) (3 diffs)

trunk/doc/pslib/psLibADD.tex

@@ -1 @@
-%%% $Id: psLibADD.tex,v 1.55 2004-12-06 21:35:43 price Exp $
+%%% $Id: psLibADD.tex,v 1.56 2004-12-21 21:37:08 price Exp $
 \documentclass[panstarrs]{panstarrs}
 
@@ -304 +304 @@
 contribute a fraction of a value, equivalent to the weight.  In the
 interests of speed, a boxcar PDF may be used to represent each input
-value (as opposed to a Gaussian), where the width is equal to 2.35
-times the error (i.e., the full width at half maximum) and each input
-value contributes constant area.  Then the mean, median, mode,
-standard deviation and quartiles are estimated in the same manner as
-above.
+value (as opposed to a Gaussian), where the boxcar width is equal to
+$2 \sqrt{2 \ln 2}$ times the error and each input value contributes
+constant area.  Then the mean, median, mode, standard deviation and
+quartiles are estimated in the same manner as above.
 
 \paragraph{Histograms}
@@ -315 +314 @@
 input values, the approach described above for the robust statistics
 is used (i.e., the histograms become probability density functions).
+
+An example may help here.  Say we have our histogram bounds being 0,
+1, 2, 3, 4, 5; and our value is $2.5 \pm 0.5$.  Then, the width of the
+contribution is $0.5 \times 2.35... \approx 1.175$.  Half the width is
+0.5875, so we will treat this value as a boxcar from $2.5 - 0.5875$ to
+$2.5 + 0.5875$.
+
+Consequently, the bins 0 to 1 and 4 to 5 get no value, because none of
+the boxcar overlaps.  The bin 1 to 2 gets 0.0875, because that's the
+fraction of the boxcar that overlaps with it; same thing with the bin
+3 to 4.  The bin 2 to 3 gets 0.825 because that's the fraction of the
+boxcar that overlaps with it.  So the single value $2.5 \pm 0.5$ makes
+the following histogram:
+
+\begin{tabular}{lr}
+Bin & Value \\ \hline
+0--1 & 0 \\
+1--2 & 0.0875 \\
+2--3 & 0.8250 \\
+3--4 & 0.0875 \\
+4--5 & 0 \\
+\end{tabular}
+
+Note that the total adds to one --- the number of values added.
 
 \subsubsection{Matrix Operations}