# Individual measures¶

 radius_of_gyration(traj[, show_progress]) Radius of gyration. k_radius_of_gyration(traj[, k, show_progress]) k-radius of gyration. random_entropy(traj[, show_progress]) Random entropy. uncorrelated_entropy(traj[, normalize, …]) Uncorrelated entropy. real_entropy(traj[, show_progress]) Real entropy. jump_lengths(traj[, show_progress, merge]) Jump lengths. maximum_distance(traj[, show_progress]) Maximum distance. distance_straight_line(traj[, show_progress]) Distance straight line. waiting_times(traj[, show_progress, merge]) Waiting times. number_of_locations(traj[, show_progress]) Number of distinct locations. home_location(traj[, start_night, …]) Home location. max_distance_from_home(traj[, start_night, …]) Maximum distance from home. number_of_visits(traj[, show_progress]) Number of visits. location_frequency(traj[, normalize, …]) Location frequency. individual_mobility_network(traj[, …]) Individual Mobility Network. recency_rank(traj[, show_progress]) Recency rank. frequency_rank(traj[, show_progress]) Frequency rank.
skmob.measures.individual.distance_straight_line(traj, show_progress=True)

Distance straight line.

Compute the distance (in kilometers) travelled straight line by a set of individuals in a TrajDataFrame. The distance straight line $$d_{SL}$$ travelled by an individual $$u$$ is computed as the sum of the distances travelled $$u$$:

$d_{SL} = \sum_{j=2}^{n_u} dist(r_{j-1}, r_j)$

where $$n_u$$ is the number of points recorded for $$u$$, $$r_{j-1}$$ and $$r_j$$ are two consecutive points, described as a $$(latitude, longitude)$$ pair, in $$u$$’s time-ordered trajectory, and $$dist$$ is the geographic distance between the two points [WTDED2015].

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the straight line distance traveled by the individuals. Note that $$NaN$$ indicates that an individual visited just one location and hence distance is not defined.

Return type

pandas DataFrame

Warning

The input TrajDataFrame must be sorted in ascending order by datetime.

Examples

>>> import skmob
>>> from skmob.measures.individual import distance_straight_line
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> dsl_df = distance_straight_line(tdf)
uid  distance_straight_line
0    0           374530.954882
1    1           774346.816009
2    2            88710.682464
3    3           470986.771764
4    4           214623.524252

skmob.measures.individual.frequency_rank(traj, show_progress=True)

Frequency rank.

Compute the frequency rank of the locations of a set of individuals in a TrajDataFrame. The frequency rank $$K_f(r_i)$$ of a location $$r_i$$ of an individual $$u$$ is $$K_f(r_i) = 1$$ if location $$r_i$$ is the most visited location, it is $$K_f(r_i) = 2$$ if $$r_i$$ is the second-most visited location, and so on [BDEM2015].

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the frequency rank for each location of the individuals.

Return type

pandas DataFrame

Examples

>>> import skmob
>>> from skmob.measures.individual import frequency_rank
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> fr_df = frequency_rank(tdf)
lat         lng  frequency_rank
uid
0   0  39.762146 -104.982480               1
1  39.891077 -105.068532               2
2  39.739154 -104.984703               3
3  39.891586 -105.068463               4
4  39.827022 -105.143191               5

skmob.measures.individual.home_location(traj, start_night='22:00', end_night='07:00', show_progress=True)

Home location.

Compute the home location of a set of individuals in a TrajDataFrame. The home location $$h(u)$$ of an individual $$u$$ is defined as the location $$u$$ visits the most during nighttime [CBTDHVSB2012] [PSO2012]:

$h(u) = \arg\max_{i} |\{r_i | t(r_i) \in [t_{startnight}, t_{endnight}] \}|$

where $$r_i$$ is a location visited by $$u$$, $$t(r_i)$$ is the time when $$u$$ visited $$r_i$$, and $$t_{startnight}$$ and $$t_{endnight}$$ indicates the times when nighttime starts and ends, respectively.

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• start_night (str, optional) – the starting time of the night (format HH:MM). The default is ‘22:00’.

• end_night (str, optional) – the ending time for the night (format HH:MM). The default is ‘07:00’.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the home location, as a $$(latitude, longitude)$$ pair, of the individuals.

Return type

pandas DataFrame

Examples

>>> import skmob
>>> from skmob.measures.individual import home_location
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> hl_df = home_location(tdf)
uid        lat         lng
0    0  39.891077 -105.068532
1    1  37.630490 -122.411084
2    2  39.739154 -104.984703
3    3  37.748170 -122.459192
4    4  60.180171   24.949728


References

CBTDHVSB2012

Csáji, B. C., Browet, A., Traag, V. A., Delvenne, J.-C., Huens, E., Van Dooren, P., Smoreda, Z. & Blondel, V. D. (2012) Exploring the Mobility of Mobile Phone Users. Physica A: Statistical Mechanics and its Applications 392(6), 1459-1473, https://www.sciencedirect.com/science/article/pii/S0378437112010059

PSO2012

Phithakkitnukoon, S., Smoreda, Z. & Olivier, P. (2012) Socio-geography of human mobility: A study using longitudinal mobile phone data. PLOS ONE 7(6): e39253. https://doi.org/10.1371/journal.pone.0039253

skmob.measures.individual.individual_mobility_network(traj, self_loops=False, show_progress=True)

Individual Mobility Network.

Compute the individual mobility network of a set of individuals in a TrajDataFrame. An Individual Mobility Network (aka IMN) of an individual $$u$$ is a directed graph $$G_u=(V,E)$$, where $$V$$ is the set of nodes and $$E$$ is the set of edges. Nodes indicate locations visisted by $$u$$, and edges indicate trips between two locations by $$u$$. On the edges the following function is defined:

$\omega: E \rightarrow \mathbb{N}$

which returns the weight of an edge, i.e., the number of travels performed by $$u$$ on that edge [RGNPPG2014] [BL2012] [SQBB2010].

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• self_loops (boolean, optional) – if True, adds self loops also. The default is False.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the individual mobility network of each individual.

Return type

pandas DataFrame

Warning

The input TrajDataFrame must be sorted in ascending order by datetime.

Examples

>>> import skmob
>>> from skmob.measures.individual import individual_mobility_network
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> imn_df = individual_mobility_network(tdf)
uid  lat_origin  lng_origin   lat_dest    lng_dest n_trips
0    0   37.774929 -122.419415  37.600747 -122.382376       1
1    0   37.600747 -122.382376  37.615223 -122.389979       1
2    0   37.600747 -122.382376  37.580304 -122.343679       1
3    0   37.615223 -122.389979  39.878664 -104.682105       1
4    0   37.615223 -122.389979  37.580304 -122.343679       1


References

RGNPPG2014

Rinzivillo, S., Gabrielli, L., Nanni, M., Pappalardo, L., Pedreschi, D. & Giannotti, F. (2012) The purpose of motion: Learning activities from Individual Mobility Networks. Proceedings of the 2014 IEEE International Conference on Data Science and Advanced Analytics, 312-318, https://ieeexplore.ieee.org/document/7058090

BL2012

Bagrow, J. P. & Lin, Y.-R. (2012) Mesoscopic Structure and Social Aspects of Human Mobility. PLOS ONE 7(5): e37676. https://doi.org/10.1371/journal.pone.0037676

skmob.measures.individual.jump_lengths(traj, show_progress=True, merge=False)

Jump lengths.

Compute the jump lengths (in kilometers) of a set of individuals in a TrajDataFrame. A jump length (or trip distance) $$\Delta r$$ is defined as the geographic distance between two consecutive points visited by $$u$$:

$\Delta r = dist(r_i, r_{i + 1})$

where $$r_i$$ and $$r_{i + 1}$$ are two consecutive points, described as a latitude, longitude pair, in the time-ordered trajectory of an individual, and $$dist$$ is the geographic distance between the two points [BHG2006] [GHB2008] [PRQPG2013].

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

• merge (boolean, optional) – if True, merge the individuals’ lists into one list. The default is False.

Returns

the jump lengths for each individual, where $$NaN$$ indicates that an individual visited just one location and hence distance is not defined; or a list with all jumps together if merge is True.

Return type

pandas DataFrame or list

Warning

The input TrajDataFrame must be sorted in ascending order by datetime.

Examples

>>> import skmob
>>> from skmob.measures.individual import jump_lengths
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> jl_df = jump_lengths(tdf)
uid                                       jump_lengths
0    0  [19.640467328877936, 0.0, 0.0, 1.7434311010381...
1    1  [6.505330424378251, 46.75436600375988, 53.9284...
2    2  [0.0, 0.0, 0.0, 0.0, 3.6410097195943507, 0.0, ...
3    3  [3861.2706300798827, 4.061631313492122, 5.9163...
4    4  [15511.92758595804, 0.0, 15511.92758595804, 1....
>>> jl_list = jump_lengths(tdf, merge=True)
>>> print(jl_list[:10]) # print the first ten elements in the list
[19.640467328877936, 0.0, 0.0, 1.743431101038163, 1553.5011134765616, 0.0, 30.14517724008101, 0.0, 2.563647571198179, 1.9309489380903868]


References

BHG2006

Brockmann, D., Hufnagel, L. & Geisel, T. (2006) The scaling laws of human travel. Nature 439, 462-465, https://www.nature.com/articles/nature04292

Compute the k-radii of gyration (in kilometers) of a set of individuals in a TrajDataFrame. The k-radius of gyration of an individual $$u$$ is defined as [PSRPGB2015]:

$r_g^{(k)}(u) = \sqrt{\frac{1}{n_u^{(k)}} \sum_{i=1}^k (r_i(u) - r_{cm}^{(k)}(u))^2}$

where $$r_i(u)$$ represents the $$n_u^{(k)}$$ positions recorded for $$u$$ on their k most frequent locations, and $$r_{cm}^{(k)}(u)$$ is the center of mass of $$u$$’s trajectory considering the visits to the k most frequent locations only. In mobility analysis, the k-radius of gyration indicates the characteristic distance travelled by that individual as induced by their k most frequent locations.

Parameters
• traj (TrajDataFrame) – the trajectories of the individual.

• k (int, optional) – the number of most frequent locations to consider. The default is 2. The possible range of values is $$[2, +inf]$$.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the k-radii of gyration of the individuals

Return type

pandas DataFrame

Examples

>>> import skmob
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
0    0               7.730516
1    1               3.620671
2    2               6.366549
3    3              10.543072
4    4            3910.808802


References

PSRPGB2015

Pappalardo, L., Simini, F. Rinzivillo, S., Pedreschi, D. Giannotti, F. & Barabasi, A. L. (2015) Returners and Explorers dichotomy in human mobility. Nature Communications 6, https://www.nature.com/articles/ncomms9166

skmob.measures.individual.location_frequency(traj, normalize=True, as_ranks=False, show_progress=True, location_columns=['lat', 'lng'])

Location frequency.

Compute the visitation frequency of each location, for a set of individuals in a TrajDataFrame. Given an individual $$u$$, the visitation frequency of a location $$r_i$$ is the number of visits to that location by $$u$$. The visitation frequency $$f(r_i)$$ of location $$r_i$$ is also defined in the literaure as the probability of visiting location $$r_i$$ by $$u$$ [SKWB2010] [PF2018]:

$f(r_i) = \frac{n(r_i)}{n_u}$

where $$n(r_i)$$ is the number of visits to location $$r_i$$ by $$u$$, and $$n_u$$ is the total number of data points in $$u$$’s trajectory.

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• normalize (boolean, optional) – if True, the number of visits to a location by an individual is computed as probability, i.e., divided by the individual’s total number of visits. The default is True.

• as_ranks (boolean, optional) – if True, return a list where element $$i$$ indicates the average visitation frequency of the $$i$$-th most frequent location. The default is False.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

• location_columns (list, optional) – the name of the column(s) indicating the location. The default is [constants.LATITUDE, constants.LONGITUDE].

Returns

the location frequency for each location for each individual, or the ranks list for each individual.

Return type

pandas DataFrame or list

Examples

>>> import skmob
>>> from skmob.measures.individual import location_frequency
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> lf_df = location_frequency(tdf, normalize=False).reset_index()
uid        lat         lng  location_frequency
0    0  39.762146 -104.982480                 214
1    0  39.891077 -105.068532                 137
2    0  39.739154 -104.984703                 126
3    0  39.891586 -105.068463                  72
4    0  39.827022 -105.143191                  53
>>> lf_df = location_frequency(tdf, normalize=True).reset_index() # frequencies ad probabilities
uid        lat         lng  location_frequency
0    0  39.762146 -104.982480            0.101953
1    0  39.891077 -105.068532            0.065269
2    0  39.739154 -104.984703            0.060029
3    0  39.891586 -105.068463            0.034302
4    0  39.827022 -105.143191            0.025250
>>> ranks = location_frequency(tdf, as_ranks=True) # as rank list
>>> print(ranks[:10])
[0.26774954912290716, 0.12699129836809203, 0.07090642778490935, 0.04627646190564675, 0.03657120208870922, 0.029353331229094993, 0.025050267239164755, 0.020284764933447663, 0.018437443393907686, 0.01656729815097415]


visits_per_location

skmob.measures.individual.max_distance_from_home(traj, start_night='22:00', end_night='07:00', show_progress=True)

Maximum distance from home.

Compute the maximum distance (in kilometers) traveled from their home location by a set of individuals in a TrajDataFrame. The maximum distance from home $$dh_{max}(u)$$ of an individual $$u$$ is defined as [CM2015]:

$dh_{max}(u) = \max\limits_{1 \leq i \lt j \lt n_u} dist(r_i, h(u))$

where $$n_u$$ is the number of points recorded for $$u$$, $$r_i$$ is a location visited by $$u$$ described as a $$(latitude, longitude)$$ pair, $$h(u)$$ is the home location of $$u$$, and $$dist$$ is the geographic distance between two points.

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• start_night (str, optional) – the starting time of the night (format HH:MM). The default is ‘22:00’.

• end_night (str, optional) – the ending time for the night (format HH:MM). The default is ‘07:00’.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the maximum distance from home of the individuals.

Return type

pandas DataFrame

Examples

>>> import skmob
>>> from skmob.measures.individual import max_distance_from_home
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> dh_max = max_distance_from_home(tdf)
uid  max_distance_from_home
0    0            11286.942949
1    1            12800.547682
2    2            11282.748348
3    3            12799.754644
4    4            15512.788707


References

CM2015

Canzian, L. & Musolesi, M. (2015) Trajectories of depression: unobtrusive monitoring of depressive states by means of smartphone mobility traces analysis. Proceedings of the 2015 ACM International Joint Conference on Pervasive and Ubiquitous Computing, 1293-1304, https://dl.acm.org/citation.cfm?id=2805845

skmob.measures.individual.maximum_distance(traj, show_progress=True)

Maximum distance.

Compute the maximum distance (in kilometers) traveled by a set of individuals in a TrajDataFrame. The maximum distance $$d_{max}$$ travelled by an individual $$u$$ is defined as:

$d_{max} = \max\limits_{1 \leq i \lt j \lt n_u} dist(r_i, r_j)$

where $$n_u$$ is the number of points recorded for $$u$$, $$r_i$$ and $$r_{i + 1}$$ are two consecutive points, described as a $$(latitude, longitude)$$ pair, in $$u$$’s time-ordered trajectory, and $$dist$$ is the geographic distance between the two points [WTDED2015] [LBH2012].

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the maximum traveled distance for each individual. Note that $$NaN$$ indicates that an individual visited just one location and so the maximum distance is not defined.

Return type

pandas DataFrame

Examples

>>> import skmob
>>> from skmob.measures.individual import maximum_distance
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> md_df = maximum_distance(tdf)
uid  maximum_distance
0    0      11294.436420
1    1      12804.895064
2    2      11286.745660
3    3      12803.259219
4    4      15511.927586


References

WTDED2015(1,2)

Williams, N. E., Thomas, T. A., Dunbar, M., Eagle, N. & Dobra, A. (2015) Measures of Human Mobility Using Mobile Phone Records Enhanced with GIS Data. PLOS ONE 10(7): e0133630. https://doi.org/10.1371/journal.pone.0133630

LBH2012

Lu, X., Bengtsson, L. & Holme, P. (2012) Predictability of population displacement after the 2010 haiti earthquake. Proceedings of the National Academy of Sciences 109 (29) 11576-11581; https://doi.org/10.1073/pnas.1203882109

skmob.measures.individual.number_of_locations(traj, show_progress=True)

Number of distinct locations.

Compute the number of distinct locations visited by a set of individuals in a TrajDataFrame [GHB2008].

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the number of distinct locations visited by the individuals.

Return type

pandas DataFrame

Examples

>>> import skmob
>>> from skmob.measures.individual import number_of_locations
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> nl_df = number_of_locations(tdf)
uid  number_of_locations
0    0                  542
1    1                   97
2    2                  460
3    3                  614
4    4                  216

skmob.measures.individual.number_of_visits(traj, show_progress=True)

Number of visits.

Compute the number of visits (i.e., data points) for each individual in a TrajDataFrame.

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the number of visits or points per each individual.

Return type

pandas DataFrame

Examples

>>> import skmob
>>> from skmob.measures.individual import number_of_visits
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> num_v_df = number_of_visits(tdf)
uid  number_of_visits
0    0              2099
1    1              1210
2    2              2100
3    3              1807
4    4               779


Compute the radii of gyration (in kilometers) of a set of individuals in a TrajDataFrame. The radius of gyration of an individual $$u$$ is defined as [GHB2008] [PRQPG2013]:

$r_g(u) = \sqrt{ \frac{1}{n_u} \sum_{i=1}^{n_u} dist(r_i(u) - r_{cm}(u))^2}$

where $$r_i(u)$$ represents the $$n_u$$ positions recorded for $$u$$, and $$r_{cm}(u)$$ is the center of mass of $$u$$’s trajectory. In mobility analysis, the radius of gyration indicates the characteristic distance travelled by $$u$$.

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the radius of gyration of each individual.

Return type

pandas DataFrame

Examples

>>> import skmob
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
0    0         1564.436792
1    1         2467.773523
2    2         1439.649774
3    3         1752.604191
4    4         5380.503250


References

GHB2008(1,2,3)

González, M. C., Hidalgo, C. A. & Barabási, A. L. (2008) Understanding individual human mobility patterns. Nature, 453, 779–782, https://www.nature.com/articles/nature06958.

PRQPG2013(1,2)

Pappalardo, L., Rinzivillo, S., Qu, Z., Pedreschi, D. & Giannotti, F. (2013) Understanding the patterns of car travel. European Physics Journal Special Topics 215(1), 61-73, https://link.springer.com/article/10.1140%2Fepjst%2Fe2013-01715-5

skmob.measures.individual.random_entropy(traj, show_progress=True)

Random entropy.

Compute the random entropy of a set of individuals in a TrajDataFrame. The random entropy of an individual $$u$$ is defined as [EP2009] [SQBB2010]:

$E_{rand}(u) = log_2(N_u)$

where $$N_u$$ is the number of distinct locations visited by $$u$$, capturing the degree of predictability of $$u$$’s whereabouts if each location is visited with equal probability.

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the random entropy of the individuals.

Return type

pandas DataFrame

Examples

>>> import skmob
>>> from skmob.measures.individual import random_entropy
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> re_df = random_entropy(tdf)
uid  random_entropy
0    0        9.082149
1    1        6.599913
2    2        8.845490
3    3        9.262095
4    4        7.754888


References

EP2009(1,2)

Eagle, N. & Pentland, A. S. (2009) Eigenbehaviors: identifying structure in routine. Behavioral Ecology and Sociobiology 63(7), 1057-1066, https://link.springer.com/article/10.1007/s00265-009-0830-6

SQBB2010(1,2,3,4)

Song, C., Qu, Z., Blumm, N. & Barabási, A. L. (2010) Limits of Predictability in Human Mobility. Science 327(5968), 1018-1021, https://science.sciencemag.org/content/327/5968/1018

skmob.measures.individual.real_entropy(traj, show_progress=True)

Real entropy.

Compute the real entropy of a set of individuals in a TrajDataFrame. The real entropy of an individual $$u$$ is defined as [SQBB2010]:

$E(u) = - \sum_{T'_u}P(T'_u)log_2[P(T_u^i)]$

where $$P(T'_u)$$ is the probability of finding a particular time-ordered subsequence $$T'_u$$ in the trajectory $$T_u$$. The real entropy hence depends not only on the frequency of visitation, but also the order in which the nodes were visited and the time spent at each location, thus capturing the full spatiotemporal order present in an $$u$$’s mobility patterns.

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the real entropy of the individuals

Return type

pandas DataFrame

Warning

The input TrajDataFrame must be sorted in ascending order by datetime. Note that the computation of this measure is, by construction, slow.

Examples

>>> import skmob
>>> from skmob.measures.individual import real_entropy
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> re_df = real_entropy(tdf[tdf.uid < 50]) # computed on a subset of individuals
uid  real_entropy
0    0      4.906479
1    1      2.207224
2    2      4.467225
3    3      4.782442
4    4      3.585371

skmob.measures.individual.recency_rank(traj, show_progress=True)

Recency rank.

Compute the recency rank of the locations of a set of individuals in a TrajDataFrame. The recency rank $$K_s(r_i)$$ of a location $$r_i$$ of an individual $$u$$ is $$K_s(r_i) = 1$$ if location $$r_i$$ is the last visited location, it is $$K_s(r_i) = 2$$ if $$r_i$$ is the second-last visited location, and so on [BDEM2015].

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the recency rank for each location of the individuals.

Return type

pandas DataFrame

Warning

The input TrajDataFrame must be sorted in ascending order by datetime.

Examples

>>> import skmob
>>> from skmob.measures.individual import recency_rank
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> rr_df = recency_rank(tdf)
lat         lng  recency_rank
uid
0   0  39.891383 -105.070814             1
1  39.891077 -105.068532             2
2  39.750469 -104.999073             3
3  39.752713 -104.996337             4
4  39.752508 -104.996637             5


References

BDEM2015(1,2)

Barbosa, H., de Lima-Neto, F. B., Evsukoff, A., Menezes, R. (2015) The effect of recency to human mobility, EPJ Data Science 4(21), https://epjdatascience.springeropen.com/articles/10.1140/epjds/s13688-015-0059-8

skmob.measures.individual.uncorrelated_entropy(traj, normalize=False, show_progress=True)

Uncorrelated entropy.

Compute the temporal-uncorrelated entropy of a set of individuals in a TrajDataFrame. The temporal-uncorrelated entropy of an individual $$u$$ is defined as [EP2009] [SQBB2010] [PVGSPG2016]:

$E_{unc}(u) = - \sum_{j=1}^{N_u} p_u(j) log_2 p_u(j)$

where $$N_u$$ is the number of distinct locations visited by $$u$$ and $$p_u(j)$$ is the historical probability that a location $$j$$ was visited by $$u$$. The temporal-uncorrelated entropy characterizes the heterogeneity of $$u$$’s visitation patterns.

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• normalize (boolean, optional) – if True, normalize the entropy in the range $$[0, 1]$$ by dividing by $$log_2(N_u)$$, where $$N$$ is the number of distinct locations visited by individual $$u$$. The default is False.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

Returns

the temporal-uncorrelated entropy of the individuals.

Return type

pandas DataFrame

Examples

>>> import skmob
>>> from skmob.measures.individual import uncorrelated_entropy
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> ue_df = uncorrelated_entropy(tdf, normalize=True)
uid  norm_uncorrelated_entropy
0    0                   0.819430
1    1                   0.552972
2    2                   0.764304
3    3                   0.794553
4    4                   0.756421


References

PVGSPG2016

Pappalardo, L., Vanhoof, M., Gabrielli, L., Smoreda, Z., Pedreschi, D. & Giannotti, F. (2016) An analytical framework to nowcast well-being using mobile phone data. International Journal of Data Science and Analytics 2(75), 75-92, https://link.springer.com/article/10.1007/s41060-016-0013-2

skmob.measures.individual.waiting_times(traj, show_progress=True, merge=False)

Waiting times.

Compute the waiting times (in seconds) between the movements of each individual in a TrajDataFrame. A waiting time (or inter-time) by an individual $$u$$ is defined as the time between two consecutive points in $$u$$’s trajectory:

$\Delta t = |t(r_i) - t(r_{i + 1})|$

where $$r_i$$ and $$r_{i + 1}$$ are two consecutive points, described as a $$(latitude, longitude)$$ pair, in the time-ordered trajectory of $$u$$, and $$t(r)$$ indicates the time when $$u$$ visits point $$r$$ [SKWB2010] [PF2018].

Parameters
• traj (TrajDataFrame) – the trajectories of the individuals.

• show_progress (boolean, optional) – if True, show a progress bar. The default is True.

• merge (boolean, optional) – if True, merge the individuals’ lists into one list. The default is False.

Returns

the list of waiting times for each individual, where $$NaN$$ indicates that an individual visited just one location and hence waiting time is not defined; or a list with all waiting times together if merge is True.

Return type

pandas DataFrame or list

Warning

The input TrajDataFrame must by sorted in ascending order by datetime.

Examples

>>> import skmob
>>> from skmob.measures.individual import waiting_times
>>> url = "https://snap.stanford.edu/data/loc-brightkite_totalCheckins.txt.gz"
names=['user', 'check-in_time', 'latitude', 'longitude', 'location id'])
>>> tdf = skmob.TrajDataFrame(df, latitude='latitude', longitude='longitude', datetime='check-in_time', user_id='user')
>>> wt_df = waiting_times(tdf)
uid                                      waiting_times
0    0  [2358.0, 136.0, 303.0, 1836.0, 14869.0, 517.0,...
1    1  [43460.0, 34353.0, 8347.0, 40694.0, 281.0, 16....
2    2  [293.0, 308.0, 228.0, 402.0, 16086.0, 665.0, 9...
3    3  [10200079.0, 30864.0, 54415.0, 2135.0, 63.0, 1...
4    4  [82845.0, 56.0, 415156.0, 1372.0, 23.0, 42679....
>>> wl_list = waiting_times(tdf, merge=True)
>>> print(wl_list[:10])
[2358.0, 136.0, 303.0, 1836.0, 14869.0, 517.0, 8995.0, 41306.0, 949.0, 11782.0]


References

SKWB2010(1,2)

Song, C., Koren, T., Wang, P. & Barabasi, A.L. (2010) Modelling the scaling properties of human mobility. Nature Physics 6, 818-823, https://www.nature.com/articles/nphys1760

PF2018(1,2)

Pappalardo, L. & Simini, F. (2018) Data-driven generation of spatio-temporal routines in human mobility. Data Mining and Knowledge Discovery 32, 787-829, https://link.springer.com/article/10.1007/s10618-017-0548-4