Direction MET Significance

We are trying to capture the likelihood that the MET is due mis-measured jets (as opposed to a LSP). To a very good approximation, the RMS of the jet resolution is

sigma = A*sqrt (ET)

where A is a constant equal to about 1 sqrt(GeV) and we will take to be unity. So,

sigma^2 = ET

For several jets aligned along the same direction, the variance of the distribution of total energy is just the sum of variances:

sigma_total^2 = sum (sigma_i ^ 2) = sum (ET_i)

Now, for an event with multiple balanced jets and no true MET, the MET distribution is centered around zero and the variance is equal to the variance of the distribution of energy. From this distribution, we would like to calculate a statistical significance for the MET of a single event, i.e. a measure of how unlikely it is for jet mis-measurement to be responsible for the MET. One good measure of the statistical significance is the number of standard deviations it is from the center

MET / sigma_total = MET / sqrt (sum (ET_i) ) = METsignificance

This is how they define the normal MET significance variable. An alternate measure is called the p-value:

pval = N * \int_(-x)^(x){ exp( -x^2/(2*sigma^2) )}

where N is the normalization

N = sigma^-2 * (2*pi)^(-1/2)

The p-value of an event is the probability of getting a result at least as extreme as that event (given a distribution). The p-value and the significance are equivalent variables for measuring the likelihood that the MET is due to jet mis-measurement, in the sense that they are related by a monotomic function.

All of the above applied to a 1-D distribution, where all the jets are aligned. Now lets consider when the jets are not aligned. In this case, the total energy (and hence the MET) is distributed in the (2-D) transverse plane. The distribution is vectoral. It turns out that when a 2-D distribution is constructed by adding together a bunch of vectors whose lengths are normally distributed, the total distribution is completely specified by two eigen-directions, each with an associated variance. In the simple case of two perpendicular jets, the directions and variances of the vector distribution are just the corresponding directions and variance of the jets. For the general case of many jets in random directions, it turns out that the distribution is completely characterized by two perpendicular directions and two associated variances. (I'm not totally confident that the two direction are neccessarily perpendicular. I'm going to assume that they are for now). Basically, no matter how many jets (of whatever phi's and true-pt's) are added together, the resulting distribution looks like an elliptical 2D gaussian.

Many details are available on the wikipedia page on Multivariate normal distribution.

Given a distribution, what we need to do come up with a notion of significance or p-value for the MET (both norm and direction) in the event. From what I can tell, the significance is easy to calculate, but the p-value is not, for a 1-D distribution. On the other hand, the p-value is relatively easy to calculate for a 2-D distribution, but the significance (when defined as distance/sigma) is not directly meaningful. So let's calculate the p-value of an event for our 2-d normal distribution. Assume that the distribution has variances sigma_x and sigma_y in the x and y directions, respectively, and that our event has coordinate (a,b). Then the p-value is the integral of the area (actually, volume) under the distribution and within the ellipse which contains the point (a,b). This is

pval = N * \int_(elipse){ exp( -r^2/2 ) )}

= 1-exp(-R^2/2)

where

r^2 = x^2/sigma_x^2 + y^2/sigma_y^2

R^2 = a^2/sigma_x^2 + b^2/sigma_y^2

Clearly, the p-value, exp(R^2), R^2, and R are all related by monotomic function. If we momentarily simplify back to the 1-D case (equivalent to b=0, I think), then we see that R is just the normal significance variable. So, for the full 2-D MET distribution, I propose we define a new variable I'll call the directional MET significance (DMS) which is equal to R:

DMS^2 = MET_x^2/sigma_x^2 + MET_y^2/sigma_y^2

Here, the x and y directions are the eigen direction of the distribution determined by the jet pt's and phi's in the event.

How are the direction and variances calculated? I'm not totally sure yet, but it shouldn't be too hard. I think you basically just need to invert a matrix made by summing up the jets.

-- CharlesRiedel - 16 Sep 2008