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Contents 1 April 16th, 10 1.1 Fitting algorithm 1.2 Some Comparisons 1.3 t0 1.4 A different fit

April 16th, 10

I have implemented a 3-parameter fit into the VDC Cluster tracking code in the Analyzer software. First, I will address how the fit works:

Fitting algorithm

I have implemented a Gauss - Newton fitting algorithm that does a least squares regression. A functional form is provided with unknown parameters which require an initial "guess".

The basic function I am using is:

y = m*(x + x0) + b

Where y is the wire position, x is the drift distance, x0 = t0*DriftVelocity, and b is the intercept.

My initial guesses are

m_0 = (y[last] - y[first])/(x[last] - x[first])
b_0 = -1.0
t0_0 = -1.0E-09

The first / last notation above corresponds to the two most positionally distant wires in a cluster with the two greatest drift times. I choose the b_0 was chosen arbitrarily, but the t0_0 was chosen as negative because my test data was mostly positive t0... In other words, I wanted to make sure my fit was giving reasonable results, and iterating from negative to positive correctly.

The iterative process involves solving for b in

J^T J b = J^T r

Where J is the Jacobian matrix with dimensions of (n of cluster hits)X(n of parameters in fit). J^T is the Jacobian's transpose, r is the vector of the sum of squares, and b is a vector containing the iterative adjustment to be made to the fit parameters.

Some Comparisons

I have processed 10000 events for one ep elastic scattering event from my thesis (Experiment e03104, run 1047). The electron and proton are detected in coincidence, and I have plotted the calculated missing Energy from both the simple 2-parameter calculation and the 3-parameter fit below:

And now with the 3-parameter fit:

I also have the missing transverse momentum for the 2-parameter calculation [1], and the 3-parameter fit [2].

The 3-parameter fit usually converges in ~4 iterations.

The time to analyze these same 10000 events are:

2-parameter calc:   Tracking  : Real Time =   0.34 seconds Cpu Time =   0.34 seconds
3-parameter fit:    Tracking  : Real Time =  15.63 seconds Cpu Time =  15.82 seconds

Here's a few output lines to see how the 2-parameter calc and the 3-parameter fit compare:

2-parm calc-  m: 0.59994   b: -0.391813
2-parm-calc-  m: 0.595642   b: -0.391208
3-parm fit-   r: 2.38445e-05   m: 0.679953  m_init: 0.728562   b: -0.390504   t0: -5.1947e-08  it_conv: 4
3-parm fit-   r: 4.43636e-11   m: 0.588742  m_init: 0.728562   b: -0.391197   t0: 2.12367e-09  it_conv: 4
3-parm fit-   r: 1.91814e-06   m: 0.787639  m_init: 0.728562   b: -0.391328   t0: -4.9091e-08  it_conv: 4
3-parm fit-   r: 7.9447e-05   m: 0.416289  m_init: 0.728562   b: -1   t0: 2.95753e-05  it_conv: 4

2-parm calc-  m: 0.799215   b: 0.0421269
2-parm calc-  m: 0.805859   b: 0.0420217
3-parm fit-   r: 8.56512e-05   m: 0.287817  m_init: 0.933519   b: 0.0455402   t0: 1.98799e-07  it_conv: 4
3-parm fit-   r: 4.32118e-06   m: 0.801151  m_init: 0.933519   b: 0.0420393   t0: 1.01897e-09  it_conv: 4
3-parm fit-   r: 5.01798e-08   m: 0.628181  m_init: 0.933519   b: 0.0421143   t0: 5.43537e-08  it_conv: 4
3-parm fit-   r: 1.74321e-05   m: 0.819345  m_init: 0.933519   b: 0.0418019   t0: -8.27862e-09  it_conv: 4

In the first case the 2-parameter and the 3-parameter fit would choose a very similar slope. In the second case, it is very different.

t0

Ok, lets look at what the fit t0 looks like:

Now we can look at the comparison per arm in the u and v planes:

A different fit

One possibility is to fit the distance calculation (a 3rd order slope parametrization) at the same time as fitting the 3-parameter linear fit. This allows distance correction to be a function of t0 as well, and the slope to be represented more accurately in the distance correction.

Currently the fitting algorithm takes ~100 iterations to converge. This could be improved with a faster converging algorithm, like the Levenberg–Marquardt algorithm.

I have fitted this function and recieved the following results:

Here is the t0 for the right arm: