Normalization in Sequence Analysis

Raw sequence distances are measured on the scale created by your distance method, sequence length, alphabet, and cost settings. A distance of 10 can mean a large difference for short sequences and a modest difference for long sequences. Normalization rescales distances so that comparisons are easier to read across sequences, groups, or study contexts.

Normalization is not only a repair for unequal sequence lengths. Elzinga and Studer (2019) show that normalization also matters when the goal is to interpret distance as dissimilarity, compare cohorts or regions, or study sequences whose complexity differs across contexts. The practical question is therefore not whether normalization is good in general. The question is what comparison you want the distance to support.

When to Normalize

Use normalization when raw distances would be hard to compare directly.

Situation	Why normalization helps	Typical choice
Sequences have different lengths	Longer sequences can accumulate larger edit costs	norm="auto" or a method-specific choice
You compare cohorts, regions, or periods	Sequence complexity can differ across contexts	norm="auto"; consider norm="ElzingaStuder" for reference-based comparisons
You need distance to behave like dissimilarity	A bounded scale supports statements such as similarity = 1 - distance	norm="ElzingaStuder" when its assumptions fit
You work with DSS-style sequences	Removing repeated adjacent states changes sequence length and complexity	A reference-based normalization can be useful
The distance method already normalizes internally	Extra normalization may not add useful information	Check the function reference before overriding

You might keep raw distances when all sequences have the same length, come from the same context, and the analysis is exploratory. In that case, report that distances are unnormalized and explain what the raw scale means.

What Normalization Changes

A raw distance depends on at least three design choices.

1. Length. More positions can create more opportunities for mismatches, insertions, or deletions.
2. Alphabet. A sequence with three states has a different range of possible substitutions than a sequence with ten states.
3. Cost structure. Optimal Matching distances depend on substitution costs (sm) and insertion or deletion costs (indel).

Normalization does not remove these design choices. It rescales the resulting distances after those choices have been made. You still need to justify the distance measure and cost settings before interpreting a normalized matrix.

Common Normalization Methods

Sequenzo follows the normalization options exposed by get_distance_matrix(). The table below gives the practical reading of each option.

Option	Main idea	Typical use
norm="none"	Return raw distances	Diagnostics, exploratory checks, or when another method handles scaling
norm="maxlength"	Divide by the longer sequence length	OM, HAM, and DHD style comparisons
norm="gmean"	Normalize similarity by the geometric mean of self-similarities	LCS, LCP, and RLCP style measures
norm="maxdist"	Divide by a theoretical pairwise maximum	Spell and prefix variants where gmean can be unstable
norm="YujianBo"	Apply an edit-distance correction	OM variants where metric behavior matters
norm="ElzingaStuder"	Use the reference-based transformation from Elzinga and Studer (2019)	Comparisons where similarity = 1 - distance is part of the interpretation
norm="auto"	Let Sequenzo choose a method-specific option	Default recommendation when you do not need a specific normalization

norm="auto" is explicit. The function signature defaults to norm="none", so pass norm="auto" when you want Sequenzo to select a method-specific normalization.

LCP-Family Rules

The LCP family has method-specific normalization rules. These are enforced by the public API, so unsupported combinations now raise clear errors instead of silently producing questionable values.

Method family	norm="auto" selects	Supported choices
LCP, RLCP	gmean	none, maxlength, gmean, maxdist, YujianBo, auto, ElzingaStuder
LCPspell, RLCPspell	maxdist	none, maxdist, auto, ElzingaStuder
LCPmst, RLCPmst	maxdist	none, gmean, maxdist, YujianBo, auto, ElzingaStuder
LCPprod, RLCPprod	none	none, auto

For LCPspell and RLCPspell, duration_ref is a design-level reference scale. Use the observation window or another pre-specified upper bound. If norm="maxdist" and expcost>0, Sequenzo checks that duration_ref is at least the largest observed active spell duration.

LCPprod and RLCPprod return raw squared-duration dissimilarities. They are useful for the product-duration definition, but they are not normalized to a bounded scale.

Using Normalization in Sequenzo

The example below uses SequenceData(...), which is the constructor exposed by the current package source.

from sequenzo import SequenceData, get_distance_matrix

time_cols = ["T1", "T2", "T3", "T4"]
states = ["Education", "Employment", "Unemployment"]

seqdata = SequenceData(
    data=df,
    time=time_cols,
    id_col="id",
    states=states,
)

dist_auto = get_distance_matrix(
    seqdata=seqdata,
    method="OM",
    sm="CONSTANT",
    indel=1,
    norm="auto",
)

For a reference-based normalization, pass norm="ElzingaStuder" and specify the reference sequence by index. The reference should match the question. A medoid, a template sequence, or a deliberately chosen baseline can all be defensible choices, but they answer different questions.

dist_ref = get_distance_matrix(
    seqdata=seqdata,
    method="OM",
    sm="CONSTANT",
    indel=1,
    norm="ElzingaStuder",
    normalization_reference_index=0,
)

With norm="ElzingaStuder", the normalized distance from the reference object to any non-reference object is scaled through the reference-based formula. This can be useful when the analysis asks how far sequences depart from a selected baseline.

When using norm="ElzingaStuder", prefer computing the full pairwise matrix and setting normalization_reference_index. A single refseq is allowed, but all non-zero distances to that same reference collapse to 1 after the reference-based post-processing, so it is usually less informative.

How to Choose

Start with the research question, then choose the distance, costs, and normalization in that order.

Research need	Practical choice
Standard OM workflow with no special normalization argument	Use method="OM", specify sm and indel, and set norm="auto"
Direct comparison with TraMineR-style OM defaults	Consider norm="maxlength" and report the cost settings
LCS or common-prefix comparisons	Use norm="auto" or inspect the gmean choice explicitly
Spell-based variants	Check the method family: OM-style spell measures often use YujianBo; LCPspell and LCPmst use maxdist; LCPprod remains raw
A baseline or template is substantively meaningful	Consider norm="ElzingaStuder" with a named reference

Avoid using normalization to hide missing-data problems. If sequences are censored or contain structural gaps, address that in preprocessing or sensitivity analysis first. Normalization can make distances comparable, but it does not decide whether the observed data are comparable.

Reporting Normalized Distances

A short report should state:

1. the distance method;
2. substitution and indel costs, when relevant;
3. the normalization method;
4. the reference sequence, if norm="ElzingaStuder" is used;
5. whether sensitivity checks with another normalization changed the substantive conclusion.

For example:

We computed Optimal Matching distances with constant substitution cost 2, indel cost 1, and norm="auto". As a sensitivity check, we repeated the analysis with raw distances and with norm="ElzingaStuder" using the sample medoid as reference. The main cluster interpretation was unchanged.

See Also

• Dissimilarity Measures explains the distances being normalized.
• Matrices in Dissimilarity Measures separates the matrices involved.
• get_distance_matrix() documents the norm parameter.

References

Abbott, A., & Forrest, J. (1986). Optimal matching methods for historical sequences. Journal of Interdisciplinary History, 16(3), 471-494. https://doi.org/10.2307/204500

Elzinga, C. H., & Studer, M. (2015). Spell sequences, state proximities, and distance metrics. Sociological Methods & Research, 44(1), 3-47. https://doi.org/10.1177/0049124114540707

Elzinga, C. H., & Studer, M. (2019). Normalization of distance and similarity in sequence analysis. Sociological Methods & Research, 48(4), 877-904. https://doi.org/10.1177/0049124117701487

Gabadinho, A., Ritschard, G., Muller, N. S., & Studer, M. (2011). Analyzing and visualizing state sequences in R with TraMineR. Journal of Statistical Software, 40(4), 1-37. https://doi.org/10.18637/jss.v040.i04

Studer, M., & Ritschard, G. (2016). What matters in differences between life trajectories: A comparative review of sequence dissimilarity measures. Journal of the Royal Statistical Society: Series A, 179(2), 481-511. https://doi.org/10.1111/rssa.12125

Author: Yuqi Liang