Diagnosing Anchor Coverage and Directionality

Why Anchor Totals Are Not Enough

An anchor set is evidence about correspondences between cells in different datasets, but its total size does not say how broadly that evidence is distributed. A set of many anchor pairs might repeatedly involve a small number of cells, leaving much of a dataset without support. Conversely, fewer pairs can cover a large fraction of cells if most pairs involve distinct endpoints. Diagnosis therefore needs at least three views: how many pairs were found, which cells participate, and where those cells occur in the local structure of each dataset.

This matters especially in reference-based mapping. The anchor relation between two datasets is a set of paired cells, but using that relation to map query cells into a reference is a directional operation. Support can be broad for the query and sparse for the reference, or the reverse. A mapping result should be interpreted in light of these coverage patterns, not just the number of anchors or an embedding plot.

The diagnostics below assume that anchors were identified in a shared representation using the selected features and neighborhood settings, and then filtered and scored according to the procedure being used. They describe support conditional on those choices. They do not establish that a correspondence is biologically correct, and a high coverage value does not by itself guarantee good alignment.

Counts, Endpoints, and Coverage

Definition: Let \(R\) denote the reference cells, \(Q\) the query cells, and \(A\subseteq R\times Q\) the retained anchor pairs. The anchor count is \(|A|\). For a reference cell \(r\), its anchor degree \(d_R(r)\) is the number of query cells paired with it; for a query cell \(q\), \(d_Q(q)\) is the number of reference cells paired with it. A cell is anchor-covered if its degree is greater than zero.

The number of covered cells is not the number of pairs. One cell can participate in several pairs, so pair counts can increase without adding new covered cells. Report endpoint counts alongside the pair count, and report coverage as a proportion of the cells available on that side.

C_R=\frac{\sum_{r\in R} I\{d_R(r)>0\}}{|R|}, \qquad C_Q=\frac{\sum_{q\in Q} I\{d_Q(q)>0\}}{|Q|}.

Here \(I\{\cdot\}\) equals 1 when the condition is true and 0 otherwise. Overall coverage can hide differences among cell states. If state labels or provisional annotations are available for diagnosis, calculate the same quantities within each state, using the cells of that state in the relevant dataset as the denominator. State labels are diagnostic strata; they need not be used to construct the anchors.

C_R(g)=\frac{\#\{r\in R_g:d_R(r)>0\}}{|R_g|}, \qquad C_Q(g)=\frac{\#\{q\in Q_g:d_Q(q)>0\}}{|Q_g|}.

The pair count and endpoint counts together also reveal concentration. For example, the mean degree among covered reference cells is \(|A|/\#\{r:d_R(r)>0\}\). A high mean degree can indicate repeated support on a limited set of reference endpoints; it is not a measure of the average quality of an anchor.

Worked Example: Many Pairs, Limited Cell Coverage

Suppose a hypothetical reference contains 100 cells and a query contains 80 cells. The retained anchor set has 40 pairs, involving 20 distinct reference cells and 32 distinct query cells. Calculate coverage and mean degree among covered cells.

Reference coverage is \(20/100=0.20\), or 20%. Query coverage is \(32/80=0.40\), or 40%. Thus the 40 pairs do not imply that 40% of each dataset is supported.

Every anchor pair contributes one to the degree of one reference cell and one to the degree of one query cell. The mean degree among covered reference cells is \(40/20=2.00\); among covered query cells it is \(40/32=1.25\). These summaries indicate that the reference-side pairs are concentrated among fewer covered cells. They do not show whether any particular pair is reliable; anchor scores and local consistency must be examined separately.

Check Coverage Within States and Neighborhoods

State-stratified coverage helps identify where anchor evidence is missing or uneven. A low value in one state can be obscured by high coverage in a more abundant state. Compare both sides of the proposed match: a state may be well represented among covered reference cells but poorly covered in the query, or vice versa. Such a pattern is a warning that conclusions about that state may depend strongly on mapping direction.

Coverage can also be spatially uneven within a state. To assess this, define for each cell \(i\) a within-dataset neighborhood \(N_k(i)\) of \(k\) nearest cells in the same dataset and representation. Let \(a_j=I\{d(j)>0\}\) indicate whether cell \(j\) has at least one anchor. A simple local support fraction is:

L_k(i)=\frac{1}{k}\sum_{j\in N_k(i)}a_j.

Summarize these fractions by state using, for example, the median and lower quartile. A state with high overall coverage but low local support in one region may have a poorly supported subpopulation. The neighborhood size \(k\) is a diagnostic scale, not a universal constant: a very small \(k\) is sensitive to individual cells, whereas a very large \(k\) can average over distinct local structures. Compare a modest range of \(k\) values and use the same representation and neighborhood definition when comparing the two datasets.

Worked Example: Global Coverage Conceals a Local Gap

In a hypothetical query state, 18 of 20 cells have at least one anchor, giving \(18/20=0.90\), or 90% overall coverage. Consider a local neighborhood of 10 cells within that state. Only 2 of those 10 cells are anchor-covered, so its local support fraction is \(2/10=0.20\), or 20%.

The state-level percentage alone suggests broad support, but the neighborhood summary shows that the covered cells may be clustered elsewhere in the state. If several nearby query cells have low \(L_k(i)\), the mapping is weakly supported in that local region even though most cells in the full state have some anchor. This is a diagnostic signal to inspect the embedding, anchor endpoints, and relevant anchor scores; it is not proof that the region is a distinct biological state.

Mapping Direction Is a Separate Diagnostic

An anchor pair itself is not inherently directional: if \((r,q)\in A\), the relation links reference cell \(r\) and query cell \(q\). Direction enters when a mapping procedure uses anchors to transfer query cells into reference coordinates, or when the roles are reversed. The number of cells with usable support can differ by direction because the datasets have different sizes, densities, state composition, or endpoint coverage.

For a simple support audit, report the fraction of query cells with at least one anchor when mapping query to reference, and the corresponding fraction of reference cells when reversing the roles. If a mapping method requires more than one supporting anchor, use its actual eligibility rule rather than the at-least-one rule, and state that rule. Do not describe an unsupported cell as having a confident mapped position merely because a nearby cell in the other dataset has anchors.

A directional gap is descriptive. It can arise from unequal cell numbers or sampling density even when the shared states are genuine. It can also reflect poor matching in a portion of one dataset. The diagnostic identifies where interpretation needs caution; it does not determine which explanation is correct.

Worked Example: Comparing Mapping Support in Both Directions

Suppose a hypothetical reference has 100 cells and a query has 60. Under the mapping procedure's eligibility rule, 48 query cells and 35 reference cells have usable anchor support. Calculate the supported fractions for mapping query to reference and for mapping reference to query.

For query-to-reference mapping, the supported query fraction is \(48/60=0.80\), or 80%. For the reverse direction, the supported reference fraction is \(35/100=0.35\), or 35%. The same underlying anchor relation can therefore provide much broader eligibility on the query side than on the reference side.

This comparison does not mean that the anchor pairs change direction or that 80% of query mappings are correct. It says only that, under the stated eligibility rule, 80% of query cells and 35% of reference cells have usable support. State-level and local summaries are needed to see whether the gap is concentrated in particular regions.

A Short Justification for Reporting Both Sides

The reason pair counts cannot stand in for coverage follows directly from the degree definitions. Each pair contains exactly one reference endpoint and one query endpoint. Counting pairs by their reference endpoint counts every pair once, and counting them by their query endpoint also counts every pair once. Therefore:

\sum_{r\in R}d_R(r)=|A|=\sum_{q\in Q}d_Q(q).

But these equal totals do not imply equal numbers or proportions of covered cells. In the first worked example, the shared degree total is 40, yet there are 20 covered reference cells and 32 covered query cells. Dividing the same total by different numbers of covered endpoints gives different mean degrees; dividing those endpoint counts by different dataset sizes gives different coverage. Thus a pair count alone cannot identify either coverage or its distribution.

Diagnostics, Comparisons, and Failure Modes

A practical report should include the retained pair count, unique covered-cell counts and proportions on both sides, state-stratified coverage, and a local-support summary at stated neighborhood sizes. When anchor scores are available, inspect their distribution alongside these summaries. High coverage with consistently weak scores is not strong evidence, and a high average score can conceal a low-support region.

Diagnostic view	Question answered	Important limitation
Pair count	How many retained correspondences are present?	Repeated endpoints can inflate the total without broad cell coverage.
Endpoint coverage	What fraction of cells has any anchor support on each side?	Does not show local concentration or anchor correctness.
State-stratified coverage	Which annotated or provisional states have less support?	Depends on the state labels and their resolution.
Neighborhood support	Is support distributed across local regions?	Depends on representation and neighborhood size.
Directional eligibility	How many cells can be mapped under each role assignment?	Differences can reflect sampling and density as well as match quality.

These summaries complement, rather than replace, anchor-score inspection and neighborhood-consistency checks. A visualization can help locate sparse regions, but visual overlap is not a substitute for counting supported cells. Similarly, comparison with mutual-nearest-neighbor matches can provide context, but different methods can produce different match sets; agreement is informative, not a formal validation guarantee.

Common failure modes include interpreting the pair count as the number of matched cells, reporting only the larger coverage proportion, and treating asymmetric support as evidence that one dataset is the correct biological reference. Another mistake is to compare local coverage calculated with different representations or neighborhood sizes and attribute the difference to the data. Keep the diagnostic settings comparable, show denominators, and describe findings as support patterns rather than as proof of shared or absent biology.

Key takeaway: Diagnose anchor support on both datasets, within relevant states, and across local neighborhoods. Pair totals describe edges in the anchor relation; coverage describes participating cells; directionality describes which cells a particular mapping operation can use. None alone certifies a biological match.

Check Your Understanding

Answer each question using the definitions above. Short solutions follow the questions.

A reference has 50 cells, a query has 40, and 12 anchor pairs involve 8 distinct reference cells and 10 distinct query cells. What are the two coverage proportions?
Why can two anchor sets with the same pair count have different endpoint coverage?
A state has 70% overall anchor coverage, but many neighborhoods have local support fractions below 0.25. What should be reported, and what conclusion is not justified?
If query-to-reference mapping has higher eligibility than reference-to-query mapping, does that mean the anchor relation is directional? Explain.
Name two settings that should be held comparable when comparing neighborhood support across datasets.

Solutions. (1) Reference coverage is \(8/50=0.16\), or 16%; query coverage is \(10/40=0.25\), or 25%. (2) Pairs may reuse endpoints to different degrees, so the same number of edges can involve different numbers of unique cells. (3) Report both the 70% state coverage and the low local-support pattern, including the neighborhood definition. Do not conclude from these diagnostics alone that the low-support region is biologically absent or invalid. (4) No. An anchor pair remains a relation between two cells; eligibility can differ when the roles are reversed because of dataset sizes, densities, or support patterns. (5) Examples include the shared representation and the neighborhood size \(k\); the neighborhood construction should also be consistent.