Measuring the Distance Between Activity Populations

Created in March 29, 2026

2026   ·   acteval,   transport,   evaluation     ·   Dev  

Acteval

This is a post to introduce key features in acteval - a project for measuring the difference between populations of activity schedules using density estimation.

Activity schedules are the things we do and when. Like leave the house at 7, work from 8.30 till 5.30, then nip to the shops for 20 minutes on the way home.

A tyical use case for acteval is to check how well you modelled or synthesised schedules match some target distribution.

In this post we walk through a full pipeline on a pair of constructed example populations, from raw DataFrames to domain-level rankings.

What are activity sequences?

A daily activity schedule is a sequence of non-overlapping episodes that typically span a 24-hour day. Each episode records who did it (we use person ID - pid), what activity they performed (act), and when (start, end, duration):

import pandas as pd
from acteval.describe import plot

example = pd.DataFrame([
    {"pid": 0, "act": "home",    "start":    0, "end":  450, "duration": 450},
    {"pid": 0, "act": "work",    "start":  450, "end":  810, "duration": 360},
    {"pid": 0, "act": "eat_out", "start":  810, "end":  870, "duration":  60},
    {"pid": 0, "act": "work",    "start":  870, "end": 1080, "duration": 210},
    {"pid": 0, "act": "home",    "start": 1080, "end": 1440, "duration": 360},
])

ACTS = {
    "home": "#5b8dd9",
    "work": "#d95b5b",
    "eat_out": "#e8a838",
    "shop": "#5db85d",
    "leisure": "#a05bb8",
    "education": "#5bbcbc",
}

plot.gantt(
    populations={"Sample": example},
    act_colors=ACTS,
    acts=ACTS.keys(),
)
Example activity schedule.

Times are typically minutes from midnight but for now any consistent unit works; the library normalises internally.

What sort of distances?

On it’s own a schedule is a complex object. As a though experiment we can image representing a schedule as 1440 activity choices (each representing a minute of the day). If we were to represent a schedule with just two possible activity types (say home and work) then there are $2^{1440}$ possible schedules. A population is then comprised of potentially millions of person’s schedules.

import pandas as pd
from acteval.describe import plot
from acteval.scripts.generate_blog_plots import generate_suburban_workers, generate_lifestyle

A = generate_suburban_workers(1000)

plot.gantt(populations={"A": A}, act_colors=ACTS, acts=ACTS.keys())
Example activity schedule.

We can think of these populations as really big complex distributions, with mixtures of discrete and continuous dimensions. Vast swathes of this theorized disribution are empty (no one should flip-flop hundreds of times between work and home in a day) and some are quite dense (like the classic home-work-home sequence).

Our focus is to pick out subtle changes, including changes to pattens in activity particapations, changes to the order of activities, their durations and when they happen.

To do this we try to measure differences in a meaningful way. Specifically, we approximate the nasty complex distribution of the population with loads and loads of meaningful marginal distributions. For example:

  • how often do people participate in home, work, shop, and so on?
  • how often do people travel from work to shop, shop to work and so on?
  • when do people tend to start work and how long do they work for?

We tend to refer to these as population features, rather than marginal distributions.

Features

We expose population features as pre-computed numpy arrays via the eval.Population class:

from acteval import Population
from acteval.describe import plot

A = Population(generate_urban_workers(1000))
print(A.count_matrix[-3:])
# [
#  [0 1 2 1 0 0]
#  [0 0 2 0 1 1]
#  [0 0 2 0 0 1]
# ]
print(A.int_to_act)
# ['eat_out' 'education' 'home' 'leisure' 'shop' 'work']

Population integer-encodes activities and person IDs on construction, and lazily caches expensive derived quantities (n-gram keys, count matrices) on first access. This pays off when evaluating many synthetic models against the same observed data.

Feature distributions

These features form distributions. Such as the numbers of activities in each plan:

from acteval.features.participation import sequence_lengths
from acteval.describe import plot

A = urban_workers(1000)

print(sequence_lengths(Population(A)).aggregate())
# {'sequence lengths': (array([3., 4., 5.]), array([230, 395, 375]))}

_ = plot.sequence_lengths({"A": A})
Example population feature distributions.

Note that sequence_lengths_per_pid returns a PidFeatures object. We then use aggregate to extract the distribution as a tuple of counts and their frequncies. PidFeatures can be subset based on some sub-population of person ids to get more refined distributions. But more on this later.

Feature distances

Consider two distribution of sequence lengths, one from population A, the other from population B:

from acteval.features.participation import sequence_lengths
from acteval.describe import plot

A = urban_workers(1000)
B = leisure_dominant(1000)

print(sequence_lengths(Population(A)).aggregate())
# {'sequence lengths': (array([3., 4., 5.]), array([230, 395, 375]))}

print(sequence_lengths(Population(B)).aggregate())
# {'sequence lengths': (array([3., 4., 5.]), array([407, 520,  73]))}

_ = plot.sequence_lengths({"A": A, "B": B})
Example population feature distributions.

We measure the distance between these two distributions using Earth Mover’s Distance (EMD), also known as the Wasserstein distance. Informally: imagine each distribution as a pile of soil spread across a number line. The EMD is the minimum amount of work needed to rearrange one pile into the shape of the other, where work = area × distance moved.


from acteval.distance.wasserstein import emd
from acteval.features.participation import sequence_lengths

A = urban_workers(1000)
B = leisure_dominant(1000)

features_A = sequence_lengths(Population(A)).aggregate()
features_B = sequence_lengths(Population(B)).aggregate()

print(emd(features_A["sequence lengths"], features_B["sequence lengths"]))
# 0.47899999999999987

We like EMD because the unit of distance is often quite meaningful. For example, a distance of 1, between sequence length distributions, is equivalent to saying that a population’s schedules are typically one activitiy longer or shorter than another. But keep in mind they could also have the same expected value, but be distributed more and less flatly. To find out, a user has to plot the distribution or calculate descriptive metrics.

from acteval.distance.wasserstein import emd

A = urban_workers(1000)
B = leisure_dominant(1000)

features_A = sequence_lengths(Population(A)).aggregate()
features_B = sequence_lengths(Population(B)).aggregate()

vals_A, counts_A = features_A["sequence lengths"]
vals_B, counts_B = features_B["sequence lengths"]

mean_A = (vals_A * counts_A).sum() / counts_A.sum()
mean_B = (vals_B * counts_B).sum() / counts_B.sum()

print("Expected number of actvities per sequence:")
print(f"  Population A: {mean_A:.2f}")
print(f"  Population B: {mean_B:.2f}")
# Expected number of actvities per sequence:
#   Population A: 4.14
#   Population B: 3.67

In practice, acteval represents distributions as weighted histograms — (values, weights) tuples — and computes EMD via the POT library.

Note that we are measuring the distance between populations of schedules. Compared population don’t need to be comprised of the same persons or be the same size. Comparison is not therefore pairwise. individual schedules, is being added to acteval (see the pair-wise module).

Bringing it all togther

The sequence_length feature is a useful comparison, but obviously there’s a lot more going on in activity schedules. The acteval strategy is to simply consider loads and loads of features.

For example, we consider the number of times each activity type occurs (participation rates), the number of times each transition from one activity type to another occurs (2-grams), the durations of each activity type (durations), and so on. The full catelogue, and which of these are used in the default evaluation configuration are available from acteval.features.catalogue:

from acteval.features import catalogue

print(catalogue.list_features().to_markdown())
  domain group config_key description in_default_config
0 participations sequence lengths lengths Distribution of number of episodes per person. True
1 participations participation rate rates How many times each person participates in each activity. True
2 participations pair participation rate pair_rates Co-participation counts for all activity pairs. True
3 participations seq participation rate seq_rates Participation rates keyed by sequence position (e.g. ‘0home’, ‘1work’). False
4 participations enum participation rate enum_rates Participation rates keyed by n-th occurrence of each activity (e.g. ‘home0’, ‘home1’). False
5 timing start times start_times Start-time distribution per activity × occurrence index. True
6 timing durations durations Duration distribution per activity × occurrence index. True
7 timing start-durations start_durations Joint (start, duration) 2-D distribution per activity. True
8 timing joint-durations joint_durations Joint (duration_i, duration_{i+1}) distribution for consecutive activity pairs. True
9 timing start times by act start_times_by_act Start-time distribution per activity (no occurrence index). False
10 timing end times by act end_times_by_act End-time distribution per activity (no occurrence index). False
11 timing durations by act durations_by_act Duration distribution per activity (no occurrence index). False
12 timing time consistency time_consistency Per-person flags: starts at 0, ends at 1440, total duration equals 1440. False
13 transitions 2-gram 2-gram Consecutive activity pair (bigram) counts per person. True
14 transitions 3-gram 3-gram Consecutive activity triple (trigram) counts per person. True
15 transitions 4-gram 4-gram Consecutive activity quad (4-gram) counts per person. True
16 transitions full sequences full_sequences Per-person indicator for each unique full abbreviated tour string (e.g. ‘h>w>h’). False

…Which is a lot, so to be more useful for quick comparisons, we encourage aggregations to group and domain levels. The highest level, domain, consists of the following:

  • participations: people taking part in activities
  • transitions: people moving between activities
  • timing: when and for how long people do things

The acteval.Evaluator orchestrated all these comparisons in an efficient way. It also looks after descriptive metrics and non-density estimation features, such as for measuring correctness and creativity.

from acteval import Evaluator

A = urban_workers(1000)
B = leisure_dominant(1000)
C = education_leaning(1000)

evaluator = Evaluator(target=A)
print(evaluator.compare({"B": B, "C": C}))

# EvalResult — 2 model(s): B, C
#                        B         C
# domain                            
# creativity      0.006521  0.016128
# feasibility     0.000000  0.000000
# participations  0.263421  0.205167
# timing          0.061743  0.031365
# transitions     0.243480  0.184492

More to come

  • Creativity and diversity
  • Attributes
  • Reporting
  • Pair-wise