1. The Problem. To recapitulate the argument, suppose we want totasetofnmeasurements y to a set

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1 Paul Aver CBX 9, 74 October 8, 99 Applied Fitting Theor III Non-Optimal Least Squares Fitting and Multiple Scattering Section : The Problem Section 2: Non-Optimal Least Squares Fitting Section 3: Non-Optimal Fitting with Constraints Section 4: Simple Application to Multiple Scattering Section 5: Summar of Multiple Scattering Formulas. The Problem In the rst paper in this series, I gave a general overview of least squares tting theor and showed that if the dependence of the measurements on the parameters can be linearized, the parameters which minimize the 2 function can be obtained rather easil, even in the presence of constraints. To recapitulate the argument, suppose we want totasetofnmeasurements to a set of m parameters through the relation l = f l () for l n. If the f l () are nonlinear we can expand them about an approximate solution = A : l = f l ( A )+(@f l =@ i )( i, Ai ). This linearization permits us to dene the 2 statistic as 2 =(,f A ( A ),A(, A )) t V, (, A) t V, (, A) (, f A ( A ), A( A, )) where =, f A ( A ), A li l ()=@ ia is a constant matrix, =, A is the new vector of unknowns, and V, is the inverse of the covariance matrix of the measurements. Since the 2 measures how much the measurements \miss" the function, the solution we want is that which minimizes 2, 2 =@ i = 0. The solution was found to be = V A A t V, with covariance matrix V, where V = V A =(A t V, A),. The Gauss- Markov theorem states that the parameters obtained b this procedure are both unbiased and have minimum uncertainties, i.e., the are the best parameters that can be determined b an method. So far, so good.

2 Under normal circumstances the measurment errors l are independent of each other, i.e., the weight matrix V, can be written in diagonal form: V, = 0 = = = 2 n C A ; which is obtained b inversion of the diagonal covariance matrix V ij = h i j i = 2 i ij. Thus it is a trivial matter to t an arbitrar number of measurements. There are situations, however, where the data measurements l are not independent of each other, so that movement of a particular measurement will cause the simultaneous adjustment of other measurements. An example of this is multiple scattering, in which a single scattering event at a particular point aects the drift distance measurements at all subsequent points. In this case the deviation of the measurements from the original track will have an independent component due to random measurement errors and a correlated component due to the presence of the multiple scattering event. The correlations among the data points can be computed and put into the measurement covariance matrix V. The problem now is obtaining V, : the inversion of the covariance matrix can be prohibitivel time consuming when the number of measurements n is large. In the CLEO central detector, for instance, a track can have as man as 7 measurements. Inverting a 7 7 matrix at least once per track to account for multiple scattering is just not ver practical, unless one has a lot of CPU time to waste. 2. Non-Optimal Least Squares Fitting Several attempts have been made in the literature to account for multiple scattering eects in track tting. The methods emploed range from the mundane to the elegant, but a common goal links all of them: nding a set of track parameters whose errors are as close as possible to the optimal values calculated in Section, but without the high cost in CPU time which that technique entails. I call this the \optimal" approach. 2

3 For most tracks inclusion of multiple scattering eects in the tting algorithm does not improve the parameters of the t signicantl, except for ver slow tracks. A large amount of programming and CPU time is invested in making marginal improvements in the track parameters. In fact, what is man times desired is not the best possible parameters, but parameters whose errors are well understood. Knowledge of these errors is important when the are used as input to kinematic tting or when a sensitive lifetime measurement depends criticall on uncertainties in the vertex resolution. In both cases, lack of understanding of the parameter errors and their correlations can give erroneous phsics results, especiall when those results are based on a quoted number of standard deviations. I start b noting that the procedure whereb one looks for parameters which minimize the 2 function is merel one possible estimation scheme although it is the optimal one and that others are possible. Assume, for instance, that an approximation for V, exists which we call V ~,. V ~, will almost certainl be diagonal, but this assumption is not necessar in what is to follow. We dene a modied form of the 2 function ~ 2 =(,f A,A) t V ~, (,f A,A) (, A) t V ~, (, A) The parameters obtained b minimizing ~ 2 are given b = V ~ A A t V ~,, where V ~ A = (A t V ~, A),. The covariance matrix must be calculated from the denition V h t i = ~ V A A t ~ V, ht i ~ V, A ~ V A = ~ V A A t ~ V, V ~ V, A ~ V A : This equation collapses to V = V A when ~ V, = V,. Notice what has been accomplished here: I have produced an estimate of the tting parameters and their associated covariance matrix without inverting the measurement covariance matrix V. Furthermore, the estimate is unbiased (the parameters converge to the true values if the experiment is repeated indenitel) and the covariance matrix and hence the errors are known exactl. The price paid for this estimation scheme is that the parameter errors are no longer the best possible; that would require the minimization of the original 2 function. However, if V ~, is a reasonable representation of V, then the \non-optimal" estimate ma in fact be close to optimal. 3

4 3. Non-Optimal Fitting with Constraints The eect of constraints can easil be taken into account b the standard Lagrange Multiplier method where the \initial" unconstrained values and their covariance matrix are obtained b the non-optimal tting procedure. This procedure works exactl the same as if the parameters were obtained b the normal optimal least squares technique. The reason is because constraints act onl to move the tted parameters awa from their unconstrained values and the relative motion of each of the parameters is governed b the covariance matrix of the t, which has absorbed all the information regarding details of the measurement errors and correlations. This point was discussed in Section 5 of the rst paper in this series, CBX 9{ Simple Application to Multiple Scattering Particles moving through a detector suer innumerable collisions which alter the trajector of the particle b a stochastic process. In general, a particle traversing material of length x and a radiation length L R will be deected in an particular plane b an random angle whose r.m.s. value can be calculated from h 2 0:04 2 x i = = Hx; p L R where p is the momentum in GeV/c and is the velocit in units of c. Note that this is the width in each of two planes perpendicular to each other and ling along the ight path. I am onl accounting for standard multiple scattering eects that can be described b Gaussian statistics; single scattering theor is too complicated to incorporate in least squares tting, at least b me. Now consider a particle moving in a solenoidal detector (having a magnetic eld in the z direction) from a point near the center and assume that the drift distance to a wire is measured in ever laer. If the particle has sucient momentum so that there is not much bending, we can equate this situation to a simple geometr in which n planar drift chamber are arraed along the x axis which is dened to be the direction of the particle in the r, plane (i.e., x is reall a measurement of the radius r and measures the 4

5 drift distance). The covariance matrix of the measurements can be represented b the sum V = V 0 + V D + V C, where V 0 = diag( 2 i ) is the component due to independent measurement errors, V D is due to multiple scattering at discrete surfaces and V C is a result of continuous multiple scattering in gas. Let's consider discrete scatterers rst. If material of thickness t is present atx=x S, the track will be deected stochasticall from its nominal ight path b an angle s, causing the drift distance measurements at plane i to be changed b an amount i = s (x i, x S ). The covariance matrix element (V D ) ij can then be calculated from its denition, viz. (V D ) ij = h i j i = h 2 s i(x i, x S )(x j, x S ) Ht(x i, x S )(x j, x S ); and is 0 if x i x S or x j x S. Note that t should reect the total path length of the track through the scatterer; this is expeciall important for tracks having signicant components in the z direction. Continuous scattering is slightl more complicated. Assume the presence of gas in the region x G x x G2, where L = x G2, x G. If we divide up the path into small regions x the total scattering angle and deviation up to a point x in the gas volume is = = NX l NX l x = NX (N, l) l x; where x =(x,x G)=N.Thus the covariance matrix element for points x i and x j (where x j x i ) in the gas volume is (V G ) ij = XN i N j X (N i, l)(n j, l 0 )h l l 0ix 2 = l = 0 Z x i x G = H 2 (x i, x G) 2 (x j, x G, 3 (x i, x G)); H(x i, x)(x j, x)dx where I have used h l l 0i = Hx ll 0. 5

6 If x i and x j both are beond the gas region the above equations have to be modied to XN G = l XN G XN G = l x + (x, x G2) = (N G, l) l x + (x, x G2); where N G = L=x. In this case the covariance matrix is (V D ) ij = HL 3 L2 + 2 L[(x i, x G2)+(x j,x G2)]+(x j,x G2)(x i, x G2) : If x j is beond the gas region and x i is inside it the covariance matrix becomes (V D ) ij = H 2 (x i, x G) 2 (L, 3 (x i, x G)+(x j,x G2)): x i and x j are interchanged in the above equation if x i is beond the gas region and x j is inside of it. 5. Summar of Multiple Scattering Formulas A summar of the contributions to the multiple scattering matrix V = V 0+V D +V C is shown below (remember that x j and x i are dened such that x j x i ).. Measurement error: (V 0) ij = 2 i ij where i is the measurement error of laer i. 2. Scattering from discrete material (thin) at x = x S and thickness t (remember that H =(0:04=p) 2 =L R is the square of the r.m.s. scattering angle per unit length in a plane 6

7 and L R is the radiation length): (V D ) ij = Ht(x i, x S )(x j, x S ) for x i ;x j x S =0 otherwise 3. Scattering in a gas region x G x x G2; L= x G, x G2: (V G ) ij = H 2 (x i, x G) 2 (x j, x G, 3 (x i, x G)) for x G x i ;x j x G2 (V D ) ij = H 2 (x i, x G) 2 (L, 3 (x i, x G)+(x j,x G2)) for x G x i x G2; x j x G2 (V D ) ij = H 2 (x j, x G) 2 (L, 3 (x j, x G)+(x i,x G2)) for x G x j x G2; x i x G2 (V D ) ij = HL 3 L2+ 2 L[(x i, x G2)+(x j,x G2)] + (x j, x G2)(x i, x G2) for x i ;x j x G2 =0 otherwise 7