Astronomy · Observatories, surveys and archives

Catalogues, detection limits and selection

Treat catalogue entries as documented measurements, then distinguish selection probability, completeness and purity with original examples whose denominators remain explicit.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

Back to the chapter

After reading, you can

Read a catalogue row together with its quantity definitions, observing context and quality information.
Explain how selection changes the distribution of an observed sample.
Calculate completeness and purity for a declared known-truth example and state what the calculation cannot establish.

A catalogue row needs a data model

An astronomical catalogue is a documented collection of entries and measured or derived quantities. A row might describe a detection, a combined source estimate or a candidate selected for a particular purpose. Its relation to a physical object is part of the catalogue’s data model.

Read the release, quantity definitions, units, band, observing epoch, coordinate conventions and uncertainty information. A derived temperature or distance is not the same kind of quantity as a detector count. Its meaning depends on the inputs and the method that produced it.

One row need not represent one isolated star. Crowded sources can blend; one physical object can be split into detections; measurements in different bands can be matched incorrectly. The 2MASS explanatory notes provide concrete examples of crowding, artefacts, merging and deblending limitations.

Quality flags identify specified concerns under the release’s definitions. A flag is a reason to investigate its meaning, not a universal verdict that a row is unusable. An unflagged value is likewise not a guarantee of physical truth or perfect measurement.

Inclusion probability can change the apparent class mixture

A selection function describes the probability of inclusion under declared observations and processing. It can depend on position, brightness, colour, angular extent, crowding and other relevant properties. It belongs to a specified sample and measurement design, not to an object in all possible observing circumstances.

Consider an original synthetic parent population containing exactly 100 objects of class A and 100 of class B. Stipulate inclusion probabilities 0.8 for A and 0.2 for B. The expected selected counts are 100 × 0.8 = 80 and 100 × 0.2 = 20.

Class A is half the parent population, but its ratio of expected selected counts is 80 / (80 + 20) = 0.8. Equal parent counts can therefore produce an unequal selected mixture. The plotted values are expectations, not promised observed counts; random realised samples can differ.

The ratio of expected counts is not in general the expectation of a random realised sample fraction. Here it supplies a transparent teaching comparison. Applying such a model to data would require evidence for the selection probabilities and the eligible population, with uncertainty retained.

Original stipulated parent counts and inclusion probabilities. Selected counts are expectations, not actual survey observations.
Class	Parent objects	Inclusion probability	Expected selected objects
A	100	0.8	80
B	100	0.2	20

Equal parents, unequal expected selection

Scroll sideways to see the full diagram. Its explanation is below.

Original synthetic count model, not an observed catalogue. Group A and group B each contain a stipulated parent count of 100 objects, giving 200 parent objects in total. Inclusion probabilities are dimensionless: 0.8 for A and 0.2 for B. Labelled arrows point from the parent counts through those probabilities to the expected selected counts. Expected selected A is 100 × 0.8 = 80 objects; expected selected B is 100 × 0.2 = 20 objects, giving an expected selected total of 100. The two stacked count panels share exactly the same zero-based horizontal scale, 0 to 100 objects, with 5 drawing pixels per object. Bars begin at x = 160 and have widths 500 and 500 pixels for the two stipulated parent counts, then 400 and 100 pixels for the expected selected counts. A has a solid outline and B a dashed outline as well as different colours. No uncertainty interval or measured count is plotted. Actual selected counts fluctuate. The ratio of expected selected counts, 80/(80+20) = 0.8, differs from the parent fraction 100/(100+100) = 0.5; it is not generally the expectation of a random sample fraction. A correct nonzero selection model is required for inverse-probability reasoning; zero inclusion probability, model error, correlations and measurement error can prevent simple recovery. No real survey, observed abundance or universal correction is claimed.

A denominator determines what a quality fraction means

Completeness asks what fraction of eligible true objects were included. Purity, also called reliability under some stated conventions, asks what fraction of selected candidates are genuine members of the desired class. The denominators differ.

In a known-truth synthetic test, stipulate 100 eligible true objects. A loose cut includes 80 of them and 40 false candidates. Its completeness is 80/100 = 0.8, while its purity is 80/(80 + 40) = 2/3, about 66.7 percent.

A stricter cut, chosen as a subset of that loose selection in this example, includes 60 true objects and 10 false candidates. Completeness is 60/100 = 0.6; purity is 60/70 = 6/7, about 85.7 percent. Here the stricter choice improves purity while losing true objects. That tradeoff is a result of this stipulated test, not a theorem about every cut.

Real estimates need a validation strategy, such as suitable comparison data or injection-and-recovery tests, together with matching rules and uncertainty. The truth labels stipulated here are not obtained merely by reading the candidate list. Completeness and contamination can vary across brightness, position, morphology and observing conditions.

Known-truth synthetic test with 100 eligible true objects. Purity counts genuine selected objects among all candidates.
Selection	True included	False included	Completeness	Purity
Loose	80	40	80/100 = 80%	80/120 ≈ 66.7%
Strict subset	60	10	60/100 = 60%	60/70 ≈ 85.7%

A larger list does not automatically repair missing information

A catalogue’s footprint is one part of selection. Overlapping fields must be counted as a union, and repeated entries must be related to their source identities. Adding counts without that accounting can count the same opportunity or object more than once.

For a probability example, stipulate two independent inclusion opportunities with probabilities 0.5 and 0.4. Inclusion at least once has probability 0.5 + 0.4 − 0.5 × 0.4 = 0.7, not 0.9. Multiplication for the overlap is justified only by the declared independence. Dependent selections require their actual joint model.

With known nonzero selection probabilities, inverse-probability methods can help estimate population quantities under appropriate assumptions. The expectation example gives 80/0.8 + 20/0.2 = 200 when applied to its expected counts. This illustrates the model; it does not validate a correction for an actual catalogue.

If a region or class has zero inclusion probability, the catalogue contains no direct sample of that missing component to recover by division. Uncertain selection, measurement error and dependence also need modelling. The retained survey-selection Foundation develops a deeper version of this relationship rather than promising an automatic cure.

Use catalogue documentation as part of the scientific argument

The Gaia Data Release 3 collaboration paper illustrates that related catalogue products can use different input choices and selection thresholds, with distinct completeness and contamination limitations. Its historical examples support the method of reading a release; they do not establish the latest release or current mission status.

The Subaru survey design and 2MASS cautionary notes similarly show that depth, observing conditions and processing affect what enters a list. Their documentation is useful precisely because it describes a particular measurement and release, rather than an ideal census of everything that exists.

A missing entry therefore does not by itself establish that a physical source is absent. An entry does not by itself establish every proposed classification or derived property. A population claim must explain how observations, selection, quantity definitions and inference connect.

The next lesson follows that connection into archives. Reusing a catalogue or image requires enough metadata and provenance to identify its inputs, processing and limits, including relationships between products that share observations.

Worked examples

Separate the parent mixture from expected selection

In the stipulated 100 A and 100 B population, what follows from inclusion probabilities 0.8 and 0.2?

Expected selected A = 100 × 0.8 = 80; expected selected B = 100 × 0.2 = 20.
The parent A fraction is 100/200 = 0.5.
The ratio of expected selected A counts to expected total is 80/100 = 0.8.

Different inclusion probabilities change the selected mixture. These are expectations, and they do not establish a universal class abundance.

Check which question a percentage answers

Why does the loose known-truth test have 80 percent completeness but only about 66.7 percent purity?

Completeness divides the 80 included true objects by all 100 eligible true objects.
Purity divides those same 80 genuine candidates by all 120 selected candidates.
The 40 false candidates enter the purity denominator, not the eligible-truth denominator.

The two fractions answer different questions about the same selection. Neither can replace the other.

Try the reasoning

Does absence from a catalogue prove that a source does not exist?
Show the answer and reasoning
No.
Coverage, sensitivity, crowding, masking and processing or selection can prevent inclusion.
What is the strict synthetic cut’s completeness if it includes 60 of 100 eligible true objects?
Show the answer and reasoning
60 percent.
The denominator is the 100 eligible true objects, not its 70 selected candidates.
May two inclusion probabilities always be multiplied to calculate their overlap?
Show the answer and reasoning
No.
That product requires independence; a general union uses the actual joint inclusion probability.
Can inverse weighting of selected entries alone recover a class with zero inclusion probability?
Show the answer and reasoning
No.
There are no sampled entries from that component, and division by zero does not supply missing evidence.

What to carry forward

Catalogue entries obtain their meaning from quantity definitions, context and release documentation.
Selection can reshape an observed sample before any population inference is made.
Completeness and purity need distinct denominators and an appropriate validation model.

References

Teaching resources and research supporting this lesson’s explanations and examples.