A Journey Towards Mont Blanc

SVD unfolding June 22, 2011

Currently I am working on unfolding on ttbar system mass distribution. I decided to use Singular Value Decomposition (SVD) method for unfolding. Unfolding is basically to go back to truth level distribution taking into account the detector effects which distorts the distribution. Simple inverting matrix method has been shown not to give reasonable result due to statistical fluctuation for ttbar system mass. This fluctuation comes from the small coefficient of decomposed singular values. In order to suppress the fluctuating components, we regularize the solution assuming that truth distribution must be smooth. For this reason,  the regularization can also be called smoothing procedure. The small singular value components are suppressed by multiplying  Si²/(Si²+Sk²)  to 1/Si. Then if Si is small and Sk is high, the component becomes suppressed. That’s the basic idea.

Recently what I found is that with k=1, the unfolded distribution is supposed to be same as the training truth distribution. I tested. I confirmed that it is true. But I failed to have right normalization. The unfolded distribution is half smaller than what it is supposed to be.

I studied more. I realized that the scale factor, Si²/(Si²+Sk²) becomes 0.5 if you set k to be 1. So this results in the unfolded distribution is half smaller than it should be. However, we are not using k=1. Setting k=1 will give useless unfolded result which is exactly the same as the training truth distribution. In order to have reasonable result, we should set k to be larger than 2 at least.

Determining k term is not easy. From original paper, the plot of log di as a function of i seems to give some idea where we can set k to be. Another way to determine the k term is to compare unfolded result with truth distribution which is not used for producing response matrix and get chi2. I was able to obtain the plot of log di as a function of i. I will have to test with another generator as well to follow the second way.