Walkthrough: Fitting a histogram (15 minutes)

I created a file with a couple of histograms in it for you to play with. Switch to your UNIX window and copy this file into your directory:1

> cp ~seligman/root-class/histogram.root $PWD

Go back to your TBrowser window. (If you’ve quit ROOT, just start it again and start a new browser.) Click on the folder in the left-hand pane with the same name as your home directory.

Double-click on histogram.root. You can see that I’ve created two histograms with the names hist1 and hist2. Double-click on hist1; you may have to move or switch windows around, or click on the Canvas 1 tab, to see the c1 canvas displayed.

Note

You can guess from the x-axis label that I created this histogram from a gaussian distribution, but what were the parameters? In physics, to answer this question we typically perform a “fit” on the histogram: you assume a functional form that depends on one or more parameters, and then try to find the value of those parameters that make the function best fit the histogram.

Right-click on the histogram and select FitPanel. Under Fit Function, make sure that Predef-1D is selected. Then make sure gaus is selected in the pop-up menu next to it, and Chi-square is selected in the Fit Settings->Method pop-up menu. Click on Fit at the bottom of the panel. You’ll see two changes: A function is drawn on top of the histogram, and the fit results are printed on the ROOT command window.2

Note

Interpreting fit results takes a bit of practice. Recall that a gaussian has 3 parameters (\(P_0\), \(P_1\), and \(P_2\)); these are labeled “Constant”, “Mean”, and “Sigma” on the fit output. ROOT determined that the best value for the “Mean” was 5.98±0.03, and the best value for the “Sigma” was 2.43±0.02. Compare this with the Mean and RMS printed in the box on the upper right-hand corner of the histogram.

Statistics questions

Why are these values almost the same as the results from the fit?

Why aren’t they identical?

On the canvas, select Fit Parameters from the Options menu; you’ll see the fit parameters displayed on the plot.

Note

As a general rule, whenever you do a fit, you want to show the fit parameters on the plot. They give you some idea if your “theory” (which is often some function) agrees with the “data” (the points on the plot).

gaussian fit

Figure 11: The resulting plot should look something like this.

xkcd mu

Figure 12: http://xkcd.com/815 by Randall Munroe. It will look nothing like this. This would be a poor fit for your function.

As a check, click on landau (which vaguely resembles the plot in Figure 12) on the FitPanel’s Fit Function pop-up menu and click on Fit again; then try expo and fit again.

Note

You may have to click on the Fit button more than once for the button to “pick up” the click.

It looks like of these three choices (gaussian, landau, exponential), the gaussian is the best functional form for this histogram. Take a look at the “Chi2 / ndf” value in the statistics box on the histogram (“Chi2 / ndf” is pronounced “kye-squared per [number of] degrees of freedom”). Do the fits again and observe how this number changes. Typically, you know you have a good fit if this ratio is about 1.3

The FitPanel is good for gaussian distributions and other simple fits. But for fitting large numbers of histograms (as you’d do in the Advanced Exercises and the Expert Exercises) or for more complex functions, you’ll want to learn the following ROOT commands.

To fit hist1 to a gaussian, type the following command:4

[] hist1->Fit("gaus")

This does the same thing as using the FitPanel. You can close the FitPanel; we won’t be using it anymore.

Go back to the browser window and double-click on hist2.

Note

You’ve probably already guessed by reading the x-axis label that I created this histogram from the sum of two gaussian distributions. We’re going to fit this histogram by defining a custom function of our own.

Define a user function with the following command:

[] TF1 func("mydoublegaus","gaus(0)+gaus(3)")

Note

Note that the internal ROOT name of the function is “mydoublegaus”, but the name of the TF1 object is func.

What does gaus(0)+gaus(3) mean? You already know that the “gaus” function uses three parameters. gaus(0) means to use the gaussian distribution starting with parameter 0; gaus(3) means to use the gaussian distribution starting with parameter 3. This means our user function has six parameters: \(P_0\), \(P_1\), and \(P_2\) are the “constant”, “mean”, and “sigma” of the first gaussian, and \(P_3\), \(P_4\), and \(P_5\) are the “constant”, “mean”, and “sigma” of the second gaussian.

Let’s set the values of \(P_0\), \(P_1\), \(P_2\), \(P_3\), \(P_4\), and \(P_5\), and fit the histogram.5

[] func.SetParameters(5.,5.,1.,1.,10.,1.)
[] hist2->Fit("mydoublegaus")

It’s not a very good fit, is it? This is because I deliberately picked a poor set of starting values. Let’s try a better set:

[] func.SetParameters(5.,2.,1.,1.,10.,1.)
[] hist2->Fit("mydoublegaus")

Note

These simple fit examples may leave you with the impression that all histograms in physics are fit with gaussian distributions. Nothing could be further from the truth. I’m using gaussians in this class because they have properties (mean and width) that you can determine by eye.

Chapter 5 of the ROOT Users Guide has a lot more information on fitting histograms, and a more realistic example.

If you want to see how I created the file histogram.root, go to the UNIX window and type:

> less ~seligman/root-class/CreateHist.C

In general, for fitting histograms in a real analysis, you’ll have to define your own functions and fit to them directly, with commands like:

[] TF1 func("myFunction","<...some parameterized TFormula...>")
[] func.SetParameters(...some values...)
[] myHistogram->Fit("myFunction")

For a simple gaussian fit to a single histogram, you can always go back to using the FitPanel.

xkcd curve fitting

Figure 13: http://xkcd.com/2048 by Randall Munroe. Here are some possibilities for fitting plots using ROOT. If you choose to read the discussion on statistics this cartoon may be funnier (or more tragic; such is the nature of physics).


1

If you’re going through this class and you can’t login to a system on the Nevis particle-physics Linux cluster, you’ll have to get the files from my web site.

If you want to get all the files from that directory at once, one way is to use this UNIX command:

wget -r -np -nH --cut-dirs=2 -R "index.html*" \
    https://www.nevis.columbia.edu/~seligman/root-class/files/

You may have to install the wget command on your system, since it’s often not installed by default.

Be aware that in that directory there are a lot of work files I created to test things. There’s more in there than just the files I reference in my tutorials.

2

What do all these options mean? The Fit Function selects which mathematical function is going to be used to fit the histogram. Predef-1D means that the function is going to come from one of ROOT’s pre-defined one-dimensional math functions; as you will learn in just a bit, you can define functions of your own. Chi-square refers a fitting method; for any fit that you’re likely to do with a FitPanel, this will be the method you’ll use.

3

If you’re not familiar with terms like “chi2” or “chi-squared” there’s a brief introduction to statistics in this tutorial.

4

What’s the deal with the arrow “->” instead of the period? It’s because when you read in a histogram from a file, you get a pointer instead of an object. This only matters in C++, not in Python. See the section on pointers for more information.

5

It may help to cut-and-paste the commands from here into your ROOT window.

Warning: For now, don’t fall into the trap of cutting-and-pasting every command from this tutorial into ROOT. Save it for the more complicated commands like SetParameters or file names like ~seligman/root-class/AnalyzeVariables.C. You want to get the “feel” for issuing commands interactively (perhaps with the tricks you’ve learned), and that won’t happen if you just type Ctrl-C/click/Ctrl-V over and over again.

When we get to The Notebook Server, you’ll start cutting-and-pasting commands into notebooks on a regular basis.