Predictions for logL and chi2 Fits

Last Update: 09-Sep-05

Index

Types of Fits Performed
Form of Basic PDF
Normalization of Multivariate PDFs
Chi2 Definitions's
Variables Used for Each Fit Type
Source-by-Source PDF definitions

Types of Fits Performed

logL(Lxy): Event-by-Event logL Fits using Lxy, Pt(meas) & σ(Lxy)
logL(VPDL): Event-by-Event logL Fits using VPDL
(added in v6.3)
chi2(VPDL): χ² fits to binned asymmetry using VPDL

Form of the Basic PDF used in the Fit

The base PDF for event i (logL fits) or bin-i (chi2 fits) is a sum over sources (a).

PDF_i

x_i

H_a(z_1,i),H_a(z_2,i)...

where:

f_a = the fraction of source a
F_a = the PDF for source a
x = the (single) variable used for plotting the prediction
z = the vector of other event/bin variables that contribute to the PDF
p = the vector of parameters that depend on the source
H(z_j) = PDF of single variable - used for proper normalization of PDF

Sources fall into three main categories

Combinatoric: sources (like c-cbar) that have no lifetime or mixing
Long-Lived, No Mixing: e.g. B^±
Mixing: e.g. B_s, B_d

Normalization of Multivariate PDFs

See the PDG probability writeup.

Two Variables

Joint pdf, F(x,y) is given in terms of the conditional pdf, F(x|y) (pdf of x given a fixed y).

F(x,y) = F(x|y) F(y)

Assumption for Three Variables

Considering y,z to be independent variables.

F(x,y,z) = F(x|y,z) F(y|z) F(z) = F(x|y,z) F(y) F(z)

log(Lxy) Fits

For the logL(Lxy) fits, the PDFs for the independent variables are:

H(P) = histogram of P_t(meas) for given source
H(σ) = histogram of σ(L) - same for all sources
- Note: this factorizes out of total PDF = sum of PDFs for all sources
H(T) = 1

log(VPDL) Fits

For the logL(Lxy) fits, the PDFs for the independent variables are:

H(σ) = histogram of σ(VPDL) for a given source
- Note: only really need H(σ) for those sources with resolution convolution or for the case of combinatoric bgrc (cc), which uses σ_x explicitly
- However: the code is a lot more robust if H(σ) is used for all sources any time smeared VPDL is requested
- If there is no resolution convolution for a source, then the PDF does not depend on σ_x and the effect of ∫ dσ_x H(σ) is just to multiply the PDF for that source by 1.
H(T) = 1

Asymmetry Chi2 Definition

The predicted asymmetry for bin-i is constructed using the source-by-source predictions for the number of events in that bin, constructed from the PDFs.

N_unm(i)

N_mix(i)

The Predicted Asymmetry is then:

N_unm(i) - N_mix(i)

N_unm(i) + N_mix(i)

Variables Used in the Fits

See also the mapping of these variables onto arrays in the code for different fit inputs.

L_xy(meas) = Lx_i,a =
M_a Lxy_i / P_i VPDL of bin i [1]

Fit x - plot var z - other var's p - as required by src

logL(Lxy)

P_t(meas) = P

σ(Lxy) = σ

T = Tag: +1=unmixed, -1=mixed

P(mis-tag) = probability of incorrectly tagging event (mixing sources)

P(mix) = probability of source to appear in mixed sample (non-mixing sources)

M = mass of this source

K = K-factor (usually a convolution variable)

τ = lifetime of this source

Δm = oscillation frequence for this source

logL(VPDL)

σ_x;i,a =
M_a σ_i(Lxy) / P_i

P(mis-tag) = probability of incorrectly tagging event (mixing sources)

P(mix) = probability of source to appear in mixed sample (non-mixing sources)

K = K-factor (usually a convolution variable)

τ = lifetime of this source

Δm = oscillation frequence for this source

chi2(VPDL)

<σ_x>

P(mis-tag) = probability of incorrectly tagging event (mixing sources)

P(mix) = probability of source to appear in mixed sample (non-mixing sources)

K = K-factor (usually a convolution variable)

τ = lifetime of this source

Δm = oscillation frequence for this source

Other Variables of Interest

K-Factor: K = P_t(meas) / P_t(B) (or 1 if no B present)
This has PDF H(K)
Dilution: D = 1 - 2w (w = mis-tag prob)
Prob for this source to appear in Mixed Sample:
p(T) = 1 - P_mix - T = +1 (unmixed tag)
p(T) = P_mix - T = -1 (mixed tag)
Fration: f_a = fraction of source a in sample

Notes

VPDL of bin-i is numerically the same for each source. This is fine except for the fact that the data to which the prediction is fit uses a fixed M(def) in calculating its VPDL. Thus, if the data was composed only of B⁺, but the data VPDL was made using M(B_s) the lifetime returned by the fit would be wrong.

Source-by-Source PDF Definitions

In the code - all PDF are derived from the logL(Lxy) PDF by redefining the array of input variables and parameters (see variables in the code).

Fit PDF: Combinatoric (cc) Bgrd

logL(Lxy) F_cc(L|P,T,σ,T ; P(mix))
= p(T) H(P) Gauss(L;σ) = (p(T)H(P)/sqrt(2π)σ) exp(-L²/2σ²)

logL(VPDL) F_cc(x|σ_x,T ; P(mix))
= p(T) H(σ_x) Gauss(x;σ_x) = (p(T)H(σ_x)/sqrt(2π)σ_x) exp(-x²/2σ_x²)

chi2(VPDL) Binned predictions for T = +1/-1:
n_cc(bin-i|<σ_x>,T ; P(mix))
= ∫_bin dx F_cc(x|<σ_x>,T ; P(mix))

Fit PDF: Long Lifetime - No Mixing

logL(Lxy) F_life(L|P,σ,T ; M,K,τ,P(mix))
= p(T) H(P) ∫ dK H(K) ∫ dL^t R(L^t-L;σ) F_exp(L^t|P ; M,K,τ)

F_exp = (M K / cτ P) exp(-(M K / cτ P) L^t)

logL(VPDL) F_life(x|σ_x,T ; K,τ,P(mix))
= p(T) H(σ_x) ∫ dK H(K) ∫ dx^t R(x^t-x;σ_x) F_exp(x^t ; K,τ)

F_exp = (K / cτ) exp(-(K / cτ) x^t)

chi2(VPDL) Binned predictions for T = +1/-1:
n_life(bin-i|<σ_x>,T ; K,τ,P(mix))
= ∫_bin dx F_life(x|<σ_x>,T ; K,τ,P(mix))

Fit PDF: Mixing

logL(Lxy) F_mix(L|P,σ,T ; M,K,P(mis-tag),τ,Δm)
= H(P) ∫ dK H(K) ∫ dL^t R(L^t-L;σ) F_osc(L^t|P,T ; M,K,P(mis-tag),τ,Δm)

F_osc = (1/2) (M K / cτ P) exp(-(M K / cτ P) L^t) (1 + T D cos(Δm t) )
where t = K M L^t / P

logL(VPDL) F_mix(x|σ_x,T ; K,P(mis-tag),τ,Δm)
= H(σ_x) ∫ dK H(K) ∫ dx^t R(x^t-x;σ_x) F_osc(x^t|T ; K,P(mis-tag),τ,Δm)

F_osc = (1/2) (K / cτ) exp(-(K / cτ) x^t) (1 + T D cos(Δm t) )
where t = K x^t

chi2(VPDL) Binned predictions for T = +1/-1:
n_mix(bin-i|<σ_x>,T ; K,P(mis-tag),τ,Δm)
= ∫_bin dx F_mix(x|<σ_x>,T ; K,P(mis-tag),τ,Δm)

Fit	x - plot var	z - other var's	p - as required by src
logL(Lxy)	P_t(meas) = P σ(Lxy) = σ T = Tag: +1=unmixed, -1=mixed	P(mis-tag) = probability of incorrectly tagging event (mixing sources) P(mix) = probability of source to appear in mixed sample (non-mixing sources) M = mass of this source K = K-factor (usually a convolution variable) τ = lifetime of this source Δm = oscillation frequence for this source
logL(VPDL)	σ_x;i,a = M_a σ_i(Lxy) / P_i	P(mis-tag) = probability of incorrectly tagging event (mixing sources) P(mix) = probability of source to appear in mixed sample (non-mixing sources) K = K-factor (usually a convolution variable) τ = lifetime of this source Δm = oscillation frequence for this source
chi2(VPDL)	<σ_x>	P(mis-tag) = probability of incorrectly tagging event (mixing sources) P(mix) = probability of source to appear in mixed sample (non-mixing sources) K = K-factor (usually a convolution variable) τ = lifetime of this source Δm = oscillation frequence for this source

Fit	PDF: Combinatoric (cc) Bgrd
logL(Lxy)	F_cc(L\|P,T,σ,T ; P(mix)) = p(T) H(P) Gauss(L;σ) = (p(T)H(P)/sqrt(2π)σ) exp(-L²/2σ²)
logL(VPDL)	F_cc(x\|σ_x,T ; P(mix)) = p(T) H(σ_x) Gauss(x;σ_x) = (p(T)H(σ_x)/sqrt(2π)σ_x) exp(-x²/2σ_x²)
chi2(VPDL)	Binned predictions for T = +1/-1: n_cc(bin-i\|<σ_x>,T ; P(mix)) = ∫_bin dx F_cc(x\|<σ_x>,T ; P(mix))
Fit	PDF: Long Lifetime - No Mixing
logL(Lxy)	F_life(L\|P,σ,T ; M,K,τ,P(mix)) = p(T) H(P) ∫ dK H(K) ∫ dL^t R(L^t-L;σ) F_exp(L^t\|P ; M,K,τ) F_exp = (M K / cτ P) exp(-(M K / cτ P) L^t)
logL(VPDL)	F_life(x\|σ_x,T ; K,τ,P(mix)) = p(T) H(σ_x) ∫ dK H(K) ∫ dx^t R(x^t-x;σ_x) F_exp(x^t ; K,τ) F_exp = (K / cτ) exp(-(K / cτ) x^t)
chi2(VPDL)	Binned predictions for T = +1/-1: n_life(bin-i\|<σ_x>,T ; K,τ,P(mix)) = ∫_bin dx F_life(x\|<σ_x>,T ; K,τ,P(mix))
Fit	PDF: Mixing
logL(Lxy)	F_mix(L\|P,σ,T ; M,K,P(mis-tag),τ,Δm) = H(P) ∫ dK H(K) ∫ dL^t R(L^t-L;σ) F_osc(L^t\|P,T ; M,K,P(mis-tag),τ,Δm) F_osc = (1/2) (M K / cτ P) exp(-(M K / cτ P) L^t) (1 + T D cos(Δm t) ) where t = K M L^t / P
logL(VPDL)	F_mix(x\|σ_x,T ; K,P(mis-tag),τ,Δm) = H(σ_x) ∫ dK H(K) ∫ dx^t R(x^t-x;σ_x) F_osc(x^t\|T ; K,P(mis-tag),τ,Δm) F_osc = (1/2) (K / cτ) exp(-(K / cτ) x^t) (1 + T D cos(Δm t) ) where t = K x^t
chi2(VPDL)	Binned predictions for T = +1/-1: n_mix(bin-i\|<σ_x>,T ; K,P(mis-tag),τ,Δm) = ∫_bin dx F_mix(x\|<σ_x>,T ; K,P(mis-tag),τ,Δm)