Five Parameter Design Rank

Audit rank of a position/proper-motion/parallax design from supplied epoch offsets and RA*/Dec parallax factors; conditioning and measurement weights are not assessed.

Description

A five-parameter astrometric fit requires independent information for two positions, two proper motions and parallax.

A five-parameter astrometric fit requires independent information for two positions, two proper motions and parallax. A design-rank audit checks identifiability before attempting a weighted fit.^[1]

Audit rank of a position/proper-motion/parallax design from supplied epoch offsets and RA*/Dec parallax factors; conditioning and measurement weights are not assessed.

When to use Five Parameter Design Rank

Use this tool to audit rank of a position/proper-motion/parallax design from supplied epoch offsets and RA*/Dec parallax factors; conditioning and measurement weights are not assessed.

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 Five Parameter Design Rank 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 epochOffsetsYears, raParallaxFactors, decParallaxFactors, rankTolerance using the field table and calculation rules, then run the tool.
Read designRank, designRowCount, fullFiveParameterRank. 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
epochOffsetsYears	Epoch offsets years	list of number	Required
raParallaxFactors	Ra parallax factors	list of number	Required
decParallaxFactors	Dec parallax factors	list of number	Required
rankTolerance	Rank tolerance	number	Required

Enter aligned nonempty epoch offsets in years and RA*/Dec parallax factors, plus positive rank tolerance. At most 64 epochs produce 128 design rows. The tool uses partial-pivot elimination with an absolute pivot tolerance and returns rank, row count and fullFiveParameterRank when rank is five. Epochs need not be sorted or unique.

Paired lists must be nonempty and have equal lengths.

rows per epoch=[1,0,t,0,RAfactor] and [0,1,0,t,Decfactor]

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

Output field	Meaning	Value type
designRank	Design rank	number
designRowCount	Design row count	number
fullFiveParameterRank	Full five parameter rank	boolean

json

{
  "input": {
    "epochOffsetsYears": [
      -1,
      0,
      1
    ],
    "raParallaxFactors": [
      1,
      0,
      1
    ],
    "decParallaxFactors": [
      0,
      1,
      0
    ],
    "rankTolerance": 1e-10
  },
  "output": {
    "designRank": 5,
    "designRowCount": 6,
    "fullFiveParameterRank": true
  }
}

Limitations and assumptions

It does not solve fitted parameters, assess conditioning or use measurement weights. Tolerance and column scaling influence numerical rank, so full rank alone does not guarantee a well-determined fit.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Lists are limited to 100,000 entries; the rules above may require smaller limits. 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 Astrometric Reduced Chi Square: Evaluate weighted astrometric residual chi-square and a supplied reduced-chi-square criterion; independent correctly estimated errors are assumed. For a complementary task, use Ruwe From Fit Normalization: Derive RUWE from chi-square, fit degrees of freedom, and caller-supplied magnitude/color normalization; no catalog calibration table is embedded.

References

Five Parameter Design Rank — tool.io

Similar or alternative tools

Astrometric Reduced Chi Square
Evaluate weighted astrometric residual chi-square and a supplied reduced-chi-square criterion; independent correctly estimated errors are assumed.
Ruwe From Fit Normalization
Derive RUWE from chi-square, fit degrees of freedom, and caller-supplied magnitude/color normalization; no catalog calibration table is embedded.