Common Motion Covariance Screen

Screen common proper motion with a supplied combined two-coordinate covariance; the relative Gaussian likelihood is not a membership probability or prior.

Description

A common-motion comparison should consider covariance as well as the difference between measured motions.

A common-motion comparison should consider covariance as well as the difference between measured motions. A Gaussian relative likelihood measures distance from zero difference, not membership or a prior probability.^[1]

Screen common proper motion with a supplied combined two-coordinate covariance; the relative Gaussian likelihood is not a membership probability or prior.

When to use Common Motion Covariance Screen

Use this tool to screen common proper motion with a supplied combined two-coordinate covariance; the relative Gaussian likelihood is not a membership probability or prior.

Prepare the source, detector or coordinate measurements. Confirm frame, epoch, angular units, axis directions and the stated RA proper-motion convention before entering values.

How to use Common Motion Covariance Screen and understand its results

Prepare the source, detector or coordinate measurements. Confirm frame, epoch, angular units, axis directions and the stated RA proper-motion convention before entering values.
Enter firstMotion, secondMotion, combinedCovariance, maximumChiSquare using the field table and calculation rules, then run the tool.
Read motionChiSquare, relativeGaussianLikelihood, passesSuppliedCriterion. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
firstMotion	First motion	list of number	Required
secondMotion	Second motion	list of number	Required
combinedCovariance	Combined covariance	list of list of number	Required
maximumChiSquare	Maximum chi square	number	Required

Enter two finite two-component east/north motion vectors, a finite symmetric positive-definite 2×2 combined covariance and positive maximum chi-square. The tool uses the explicit 2×2 inverse quadratic form and returns chi-square, relative likelihood and an inclusive threshold flag. Equal motions give zero chi-square and likelihood one.

chiSquare=Δmotionᵀ covariance⁻¹ Δmotion; relativeLikelihood=exp(−chiSquare/2)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
motionChiSquare	Motion chi square	number
relativeGaussianLikelihood	Relative gaussian likelihood	number
passesSuppliedCriterion	Passes supplied criterion	boolean

json

{
  "input": {
    "firstMotion": [
      1,
      2
    ],
    "secondMotion": [
      0,
      0
    ],
    "combinedCovariance": [
      [
        1,
        0
      ],
      [
        0,
        4
      ]
    ],
    "maximumChiSquare": 3
  },
  "output": {
    "motionChiSquare": 2,
    "relativeGaussianLikelihood": 0.36787944117144233,
    "passesSuppliedCriterion": true
  }
}

Limitations and assumptions

Covariance must already include the appropriate combined errors and correlations in squared motion units. The tool does not construct it, include prior odds or establish common physical membership.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Matrices are limited to 128 rows and 128 entries per row; other lists are limited to 100,000 entries. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors. A successfully returned false flag is an evaluated outcome; its meaning is explained above.

Alternative or Complementary methods

Use Convergent Point Motion Alignment: Compare a measured tangent-plane proper-motion direction with the great-circle direction to a supplied cluster apex; alignment does not establish membership. For a complementary task, use Astrometric Reduced Chi Square: Evaluate weighted astrometric residual chi-square and a supplied reduced-chi-square criterion; independent correctly estimated errors are assumed.

References

Common Motion Covariance Screen — tool.io

Similar or alternative tools

Convergent Point Motion Alignment
Compare a measured tangent-plane proper-motion direction with the great-circle direction to a supplied cluster apex; alignment does not establish membership.
Astrometric Reduced Chi Square
Evaluate weighted astrometric residual chi-square and a supplied reduced-chi-square criterion; independent correctly estimated errors are assumed.