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DISTRIBUTIONS / CONTINUOUS / ARGUS

Argus distribution

Portrait

ARGUS is a bounded continuous distribution that combines a phase-space factor with an exponential falloff at a threshold. Its typical variable is a reconstructed mass that cannot exceed a known kinematic energy or mass.

Historical trail

The name comes from the ARGUS experiment at the DORIS II collider at DESY. The function became familiar in particle physics as an empirical model for combinatorial background under a kinematic limit.

Two useful connections

  • Unlike a normal or a Crystal Ball shape, ARGUS is built around an upper endpoint. In an invariant-mass analysis it is commonly combined with a signal function, such as a Gaussian or resonance line shape, in a mixture fit.
  • The name is an acronym for the experiment, A Russian–German–United States–Swedish collaboration, and not a reference to the mythological giant Argus.

A use case

Its best-known use is modelling background events in m_ES or reconstructed invariant masses. The shape parameter changes the background curvature; the upper endpoint should come from the kinematics or analysis design.

Diagnostic advice

A fit can confuse a narrow signal with background curvature if the endpoint, detector resolution, or fitting region is not justified.

A background that vanishes at a kinematic endpoint

In a reconstructed-mass spectrum, accidental combinations may fill the lower region but cannot exceed the available energy. ARGUS was created for such background: density falls to zero at a known upper endpoint, with chi governing the curvature of that descent.

The name comes from the ARGUS particle-physics experiment and its international collaboration, not the mythological figure. In a signal fit, this component often sits beside a Gaussian or resonance line shape. The endpoint should come from kinematics or calibration; allowing the sample to move it freely can let background absorb some signal. Phitter’s version rescales support through loc and scale.

Decision guide

A good candidate when: there is a physical upper endpoint and density must fall to zero near that cutoff, as in some kinematic backgrounds.

Compare it with: Beta or a transformed extreme-value model. Inspect behaviour immediately below the endpoint: the ARGUS cutoff constrains shape, not only support.

References

Argus Distribution: equations and calculator

Distribution defintion

X∼Argus(χ,Loc,Sc)X\sim\mathrm{Argus}\left(\chi,\text{Loc},\text{Sc}\right)

Distribution domain

x∈(Loc,Loc+Sc)x\in\left(\text{Loc},\text{Loc}+\text{Sc}\right)

Parameters domain and parameters constraints

χ∈R+,Loc∈R,Sc∈R+\chi\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R},\text{Sc}\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−Ψ(χ1−z(x)2)Ψ(χ)F_{X}\left(x\right)=1-\frac{\Psi\left(\chi\sqrt{1-z(x)^2}\right)}{\Psi(\chi)}

Probability density function

fX(x)=1Sc⋅χ32π Ψ(χ)⋅z(x)1−z(x)2exp⁡(−12χ2(1−z(x)2))f_{X}\left(x\right)=\frac{1}{\text{Sc}}\cdot\frac{\chi^3}{\sqrt{2\pi}\,\Psi(\chi)}\cdot z(x)\sqrt{1-z(x)^2}\exp\left(-\frac12 \chi^2\Big(1-z(x)^2\Big)\right)

Percent point function/Sample

FX−1(u)=Loc+Sc1−2P−1(32,(1−u)P(32,χ22))χ2F^{-1}_{X}\left(u\right)=\text{Loc}+\text{Sc}\sqrt{1-\frac{2\text{P}^{-1}(\frac{3}{2},(1-u)\text{P}(\frac{3}{2},\frac{\chi^{2}}{2}))}{\chi^{2}}}

Non-central parametric moments

μk′=E[Xk]=∫LocLoc+ScxkfX(x)dx\mu'_{k}=E[X^k]=\int_{\text{Loc}}^{\text{Loc}+\text{Sc}}x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′=Loc+Scπ/8 χe−χ24I1(χ24)Ψ(χ)\mathrm{Mean}(X)=\mu'_{1}=\text{Loc}+\text{Sc}\sqrt{\pi/8}\,\frac{\chi e^{-\frac{\chi^2}{4}} I_1(\tfrac{\chi^2}{4})}{\Psi(\chi)}

Parametric variance

Variance(X)=μ2′−μ1′2=Sc2⋅(1−3χ2+χϕ(χ)Ψ(χ))−(μ−Loc)2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\text{Sc}^2\cdot\left(1-\frac{3}{\chi^2}+\frac{\chi\phi(\chi)}{\Psi(\chi)}\right)-(\mu-\text{Loc})^2

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}

Parametric median

Median(X)=Loc+Sc1−2P−1(32,12P(32,χ22))χ2\mathrm{Median}(X)=\text{Loc}+\text{Sc}\sqrt{1-\frac{2\text{P}^{-1}(\frac{3}{2},\frac{1}{2}\text{P}(\frac{3}{2},\frac{\chi^{2}}{2}))}{\chi^{2}}}

Parametric mode

Mode(X)=Loc+Sc2χ(χ2−2)+χ4+4\mathrm{Mode}(X)=\text{Loc}+\frac{\text{Sc}}{\sqrt2\chi}\sqrt{(\chi^2-2)+\sqrt{\chi^4+4}}

Additional information and definitions

Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz(x)=\left(x-\text{Loc}\right)/\text{Sc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Ψ(χ)=Φ(χ)−χϕ(χ)−12\Psi(\chi)=\Phi(\chi)- \chi \phi( \chi )-\tfrac{1}{2}
Φ(x):CDF normal standard distribution\Phi(x):\text{CDF normal standard distribution}
ϕ(x):PDF normal standard distribution\phi(x):\text{PDF normal standard distribution}
Iα(x):Modified Bessel function of the first kind of order α∈NI_{\alpha}\left(x\right):\text{Modified Bessel function of the first kind of order }\alpha\in\mathbb{N}
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma function\text{P}(a,x)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
P−1(a,y):Inverse of regularized lower incomplete gamma function\text{P}^{-1}(a,y):\text{Inverse of regularized lower incomplete gamma function}
γ(a,x):Lower incomplete gamma function\gamma(a,x):\text{Lower incomplete gamma function}
Γ(x):Gamma function\Gamma(x):\text{Gamma function}