Wednesday, November 11, 2020

Solution for QSoas quiz #1: averaging spectra

This post describes the solution to the Quiz #1, based on the files found there. The point is to produce both the average and the standard deviation of a series of spectra. Below is how the final averaged spectra shoud look like:
I will present here two different solutions.

Solution 1: using the definition of standard deviation

There is a simple solution using the definition of the standard deviation: $$\sigma_y = \sqrt{<y^2> - {<y>}^2}$$ in which \(<y^2>\) is the average of \(y^2\) (and so on). So the simplest solution is to construct datasets with an additional column that would contain \(y^2\), average these columns, and replace the average with the above formula. For that, we need first a companion script that loads a single data file and adds a column with \(y^2\). Let's call this script load-one.cmds:
load ${1}
apply-formula y2=y**2 /extra-columns=1
flag /flags=processed
When this script is run with the name of a spectrum file as argument, it loads it (replaces ${1} by the first argument, the file name), adds a column y2 containing the square of the y column, and flag it with the processed flag. This is not absolutely necessary, but it makes it much easier to refer to all the spectra when they are processed. Then to process all the spectra, one just has to run the following commands:
run-for-each load-one.cmds Spectrum-1.dat Spectrum-2.dat Spectrum-3.dat
average flagged:processed
apply-formula y2=(y2-y**2)**0.5
dataset-options /yerrors=y2
The run-for-each command runs the load-one.cmds script for all the spectra (one could also have used Spectra-*.dat to not have to give all the file names). Then, the average averages the values of the columns over all the datasets. To be clear, it finds all the values that have the same X (or very close X values) and average them, column by column. The result of this command is therefore a dataset with the average of the original \(y\) data as y column and the average of the original \(y^2\) data as y2 column. So now, the only thing left to do is to use the above equation, which is done by the apply-formula code. The last command, dataset-options, is not absolutely necessary but it signals to QSoas that the standard error of the y column should be found in the y2 column. This is now available as script method-one.cmds in the git repository.

Solution 2: use QSoas's knowledge of standard deviation

The other method is a little more involved but it demonstrates a good approach to problem solving with QSoas. The starting point is that, in apply-formula, the value $stats.y_stddev corresponds to the standard deviation of the whole y column... Loading the spectra yields just a series of x,y datasets. We can contract them into a single dataset with one x column and several y columns:
load Spectrum-*.dat /flags=spectra
contract flagged:spectra
After these commands, the current dataset contains data in the form of:
lambda1	a1_1	a1_2	a1_3
lambda2	a2_1	a2_2	a2_3
in which the ai_1 come from the first file, ai_2 the second and so on. We need to use transpose to transform that dataset into:
0	a1_1	a2_1	...
1	a1_2	a2_2	...
2	a1_3	a2_3	...
In this dataset, values of the absorbance for the same wavelength for each dataset is now stored in columns. The next step is just to use expand to obtain a series of datasets with the same x column and a single y column (each corresponding to a different wavelength in the original data). The game is now to replace these datasets with something that looks like:
0	a_average
1	a_stddev
For that, one takes advantage of the $stats.y_average and $stats.y_stddev values in apply-formula, together with the i special variable that represents the index of the point:
apply-formula "if i == 0; then y=$stats.y_average; end; if i == 1; then y=$stats.y_stddev; end"
strip-if i>1
Then, all that is left is to apply this to all the datasets created by expand, which can be just made using run-for-datasets, and then, we reverse the splitting by using contract and transpose ! In summary, this looks like this. We need two files. The first, process-one.cmds contains the following code:
apply-formula "if i == 0; then y=$stats.y_average; end; if i == 1; then y=$stats.y_stddev; end"
strip-if i>1
flag /flags=processed
The main file, method-two.cmds looks like this:
load Spectrum-*.dat /flags=spectra
contract flagged:spectra
expand /flags=tmp
run-for-datasets process-one.cmds flagged:tmp
contract flagged:processed
dataset-options /yerrors=y2
Note some of the code above can be greatly simplified using new features present in the upcoming 3.0 version, but that is the topic for another post.

About QSoas

QSoas is a powerful open source data analysis program that focuses on flexibility and powerful fitting capacities. It is released under the GNU General Public License. It is described in Fourmond, Anal. Chem., 2016, 88 (10), pp 5050–5052. Current version is 2.2. You can download its source code and compile it yourself or buy precompiled versions for MacOS and Windows there.

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