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Today’s post continues where “Why you should be using PostGIS trajectories” leaves off. It’s the result of a collaboration with Eva Westermeier. I had the pleasure to supervise her internship at AIT last year and also co-supervised her Master’s thesis [0] on the topic of enriching trajectories with information about their geographic context.

Context-aware analysis of movement data is crucial for different domains and applications, from transport to ecology. While there is a wealth of data, efficient and user-friendly contextual trajectory analysis is still hampered by a lack of appropriate conceptual approaches and practical methods. (Westermeier, 2018)

Part of the work was focused on evaluating different approaches to adding context information from vector datasets to trajectories in PostGIS. For example, adding land cover context to animal movement data or adding information on anchoring and harbor areas to vessel movement data.

Classic point-based model vs. line-based model

The obvious approach is to intersect the trajectory points with context data. This is the classic point data model of contextual trajectories. It’s straightforward to add context information in the point-based model but it also generates large numbers of repeating annotations. In contrast, the line data model using, for example, PostGIS trajectories (LinestringM) is more compact since trajectories can be split into segments at context borders. This creates one annotation per segment and the individual segments are convenient to analyze (as described in part #12).

Spatio-temporal interpolation as provided by the line data model offers additional advantages for the analysis of annotated segments. Contextual segments start and end at the intersection of the trajectory linestring with context polygon borders. This means that there are no gaps like in the point-based model. Consequently, while the point-based model systematically underestimates segment length and duration, the line-based approach offers more meaningful segment length and duration measurements.

Schematic illustration of a subset of an annotated trajectory in two context classes, a) systematic underestimation of length or duration in the point data model, b) full length or duration between context polygon borders in the line data model (source: Westermeier (2018))

Another issue of the point data model is that brief context changes may be missed or represented by just one point location. This makes it impossible to compute the length or duration of the respective context segment. (Of course, depending on the application, it can be desirable to ignore brief context changes and make the annotation process robust towards irrelevant changes.)

Schematic illustration of context annotation for brief context changes, a) and b)
two variants for the point data model, c) gapless annotation in the line data model (source: Westermeier (2018) based on Buchin et al. (2014))

Beyond annotations, context can also be considered directly in an analysis, for example, when computing distances between trajectories and contextual point objects. In this case, the point-based approach systematically overestimates the distances.

Schematic illustration of distance measurement from a trajectory to an external
object, a) point data model, b) line data model (source: Westermeier (2018))

The above examples show that there are some good reasons to dump the classic point-based model. However, the line-based model is not without its own issues.

Issues

Computing the context annotations for trajectory segments is tricky. The main issue is that ST_Intersection drops the M values. This effectively destroys our trajectories! There are ways to deal with this issue – and the corresponding SQL queries are published in the thesis (p. 38-40) – but it’s a real bummer. Basically, ST_Intersection only provides geometric output. Therefore, we need to reconstruct the temporal information in order to create usable trajectory segments.

Finally, while the line-based model is well suited to add context from other vector data, it is less useful for context data from continuous rasters but that was beyond the scope of this work.

Conclusion

After the promising results of my initial investigations into PostGIS trajectories, I was optimistic that context annotations would be a straightforward add-on. The line-based approach has multiple advantages when it comes to analyzing contextual segments. Unfortunately, generating these contextual segments is much less convenient and also slower than I had hoped. Originally, I had planned to turn this work into a plugin for the Processing toolbox but the results of this work motivated me to look into other solutions. You’ve already seen some of the outcomes in part #20 “Trajectools v1 released!”.

References

[0] Westermeier, E.M. (2018). Contextual Trajectory Modeling and Analysis. Master Thesis, Interfaculty Department of Geoinformatics, University of Salzburg.


This post is part of a series. Read more about movement data in GIS.

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This post looks into the current AI hype and how it relates to geoinformatics in general and movement data analysis in GIS in particular. This is not an exhaustive review but aims to highlight some of the development within these fields. There are a lot of references in this post, including some to previous work of mine, so you can dive deeper into this topic on your own.

I’m looking forward to reading your take on this topic in the comments!

Introduction to AI

The dream of artificial intelligence (AI) that can think like a human (or even outsmart one) reaches back to the 1950s (Fig. 1, Tandon 2016). Machine learning aims to enable AI. However, classic machine learning approaches that have been developed over the last decades (such as: decision trees, inductive logic programming, clustering, reinforcement learning, neural networks, and Bayesian networks) have failed to achieve the goal of a general AI that would rival humans. Indeed, even narrow AI (technology that can only perform specific tasks) was mostly out of reach (Copeland 2018).

However, recent increases in computing power (be it GPUs, TPUs or CPUs) and algorithmic advances, particularly those based on neural networks, have made this dream (or nightmare) come closer (Rao 2017) and are fueling the current AI hype. It should be noted that artificial neural networks (ANN) are not a new technology. In fact, they used to be not very popular because they require large amounts of input data and computational power. However, in 2012, Andrew Ng at Google managed to create large enough neural networks and train them with massive amounts of data, an approach now know as deep learning (Copeland 2018).

Fig. 1: The evolution of artificial intelligence, machine learning, and deep learning. (Image source: Tandon 2016)

Machine learning & GIS

GIScience or geoinformatics is not new to machine learning. The most well-known application is probably supervised image classification, as implemented in countless commercial and open tools. This approach requires labeled training and test data (Fig. 2) to learn a prediction model that can, for example, classify land cover in remote sensing imagery. Many classification algorithms have been introduced, ranging from maximum likelihood classification to clustering (Congedo 2016) and neural networks.

Fig. 2: With supervised machine learning, the algorithm learns from labeled data. (Image source: Salian 2018)

Like in other fields, neural networks have intrigued geographers and GIScientists for a long time. For example, Hewitson & Crane (1994) state that “Neural nets offer a fascinating new strategy for spatial analysis, and their application holds enormous potential for the geographic sciences.” Early uses of neural network in GIScience include, for example: spatial interaction modeling (Openshaw 1998) and hydrological modeling of rainfall runoff (Dawson & Wilby 2001). More recently, neural networks and deep learning have enabled object recognition in georeferenced images. Most prominently, the research team at Mapillary (2016-2019) works on object recognition in street-level imagery (including fusion with other spatial data sources). Even Generative adversarial networks (GANs) (Fig. 3) have found their application in GIScience: for example, Zhu et al. (2017) (at the Berkeley AI Research (BAIR) laboratory) demonstrate how GANs can generate road maps from aerial images and vice versa, and Zhu et al. (2019) generate artificial digital elevation models.

Fig. 3: In a GAN, the discriminator is shown images from both the generator and from the training dataset. The discriminator is tasked with determining which images are real, and which are fakes from the generator. (Image source: Salian 2018)

However, besides general excitement about new machine learning approaches, researchers working on spatial analysis (Openshaw & Turton 1996) caution that “conventional classifiers, as provided in statistical packages, completely ignore most of the challenges of spatial data classification and handle a few inappropriately from a geographical perspective”. For example, data transformation using principal component or factor scores is sensitive to non-normal data distribution common in geographic data and many methods ignore spatial autocorrelation completely (Openshaw & Turton 1996). And neural networks are no exception: Convolutional neural networks (CNNs) are generally regarded appropriate for any problem involving pixels or spatial representations. However, Liu et al. (2018) demonstrate that they fail even for the seemingly trivial coordinate transform problem, which requires learning a mapping between coordinates in (x, y) Cartesian space and coordinates in one-hot pixel space.

The integration of spatial data challenges into machine learning is an ongoing area of research, for example in geostatistics (Hengl & Heuvelink 2019).

Machine learning and movement data

More and more movement data of people, vehicles, goods, and animals is becoming available. Developments in intelligent transportation systems specifically have been sparked by the availability of cheap GPS receivers and many models have been built that leverage floating car data (FCD) to classify traffic situations (for example, using visual analysis (Graser et al. 2012)), predict traffic speeds (for example, using linear regression models (Graser et al. 2016)), or detect movement anomalies (for example, using Gaussian mixture models (Graser & Widhalm 2018)). Beyond transportation, Valletta et al. (2017) describe applications of machine learning in animal movement and behavior.

Of course deep learning is making its way into movement data analysis as well. For example, Wang et al. (2018) and Kudinov (2018) trained neural networks to predict travel times in a transport networks. In contrast to conventional travel time prediction models (based on street graphs with associated speeds or travel times), these are considerably more computationally intensive. Kudinov (2018) for example, used 300 million simulated trips (start and end location, start time, and trip duration) as input and “spent about eight months of running one of the GP100 cards 24-7 in a search for an efficient architecture, spatial and statistical distributions of the training set, good values for multiple hyperparameters”.  More recently, Zhang et al. (2019) (at Microsoft Research Asia) used deep learning to predict flows in spatio-temporal networks. It remains to be seen if deep learning will manage to out-perform classical machine learning approaches for predictions in the transportation sector.

What would a transportation AI look like? Would it be able to drive a car and follow data-driven route recommendations (e.g. from waze.com) or would it purposefully ignore them because other – more basic systems – blindly follow it? Logistics AI might build on these kind of systems while simultaneously optimizing large fleets of vehicles. Transport planning AI might replace transport planners by providing reliable mobility demand predictions as well as resulting traffic models for varying infrastructure and policy scenarios.

Conclusions

The opportunities for using ML in geoinformatics are extensive and have been continuously explored for a multitude of different research problems and applications (from land use classification to travel time prediction). Geoinformatics is largely playing catch-up with the quick development in machine learning (including deep learning) that promise new and previously unseen possibilities. At the same time, it is necessary that geoinformatics researchers are aware of the particularities of spatial data, for example, by developing models that take spatial autocorrelation into account. Future research in geoinformatics should incorporate learnings from geostatistics to ensure that resulting machine learning models incorporate the geographical perspective.

References

  • Congedo, L. (2016). Semi-Automatic Classification Plugin Documentation. DOI: http://dx.doi.org/10.13140/RG.2.2.29474.02242/1
  • Copeland, M. (2016) What’s the Difference Between Artificial Intelligence, Machine Learning, and Deep Learning? https://blogs.nvidia.com/blog/2016/07/29/whats-difference-artificial-intelligence-machine-learning-deep-learning-ai/
  • Dawson, C. W., & Wilby, R. L. (2001). Hydrological modelling using artificial neural networks. Progress in physical Geography, 25(1), 80-108.
  • Graser, A., Ponweiser, W., Dragaschnig, M., Brandle, N., & Widhalm, P. (2012). Assessing traffic performance using position density of sparse FCD. In Intelligent Transportation Systems (ITSC), 2012 15th International IEEE Conference on (pp. 1001-1005). IEEE.
  • Graser, A., Leodolter, M., Koller, H., & Brändle, N. (2016) Improving vehicle speed estimates using street network centrality. International Journal of Cartography. doi:10.1080/23729333.2016.1189298.
  • Graser, A., & Widhalm, P. (2018). Modelling Massive AIS Streams with Quad Trees and Gaussian Mixtures. In: Mansourian, A., Pilesjö, P., Harrie, L., & von Lammeren, R. (Eds.), 2018. Geospatial Technologies for All : short papers, posters and poster abstracts of the 21th AGILE Conference on Geographic Information Science. Lund University 12-15 June 2018, Lund, Sweden. ISBN 978-3-319-78208-9. Accessible through https://agile-online.org/index.php/conference/proceedings/proceedings-2018
  • Hengl, T. Heuvelink, G.B.M. (2019) Workshop on Machine learning as a framework for predictive soil mapping https://www.cvent.com/events/pedometrics-2019/custom-116-81b34052775a43fcb6616a3f6740accd.aspx?dvce=1
  • Hewitson, B., Crane, R. G. (Eds.) (1994) Neural Nets: Applications in Geography. Springer.
  • Kudinov, D. (2018) Predicting travel times with artificial neural network and historical routes. https://community.esri.com/community/gis/applications/arcgis-pro/blog/2018/03/27/predicting-travel-times-with-artificial-neural-network-and-historical-routes
  • Liu, R., Lehman, J., Molino, P., Such, F. P., Frank, E., Sergeev, A., & Yosinski, J. (2018). An intriguing failing of convolutional neural networks and the coordconv solution. In Advances in Neural Information Processing Systems (pp. 9605-9616).
  • Mapillary Research (2016-2019) publications listed on https://research.mapillary.com/
  • Openshaw, S., & Turton, I. (1996). A parallel Kohonen algorithm for the classification of large spatial datasets. Computers & Geosciences, 22(9), 1019-1026.
  • Openshaw, S. (1998). Neural network, genetic, and fuzzy logic models of spatial interaction. Environment and Planning A, 30(10), 1857-1872.
  • Rao, R. C.S. (2017) New Product breakthroughs with recent advances in deep learning and future business opportunities. https://mse238blog.stanford.edu/2017/07/ramdev10/new-product-breakthroughs-with-recent-advances-in-deep-learning-and-future-business-opportunities/
  • Salian, I. (2018) SuperVize Me: What’s the Difference Between Supervised, Unsupervised, Semi-Supervised and Reinforcement Learning? https://blogs.nvidia.com/blog/2018/08/02/supervised-unsupervised-learning/
  • Tandon, K. (2016) AI & Machine Learning: The evolution, differences and connections https://www.linkedin.com/pulse/ai-machine-learning-evolution-differences-connections-kapil-tandon/
  • Valletta, J. J., Torney, C., Kings, M., Thornton, A., & Madden, J. (2017). Applications of machine learning in animal behaviour studies. Animal Behaviour, 124, 203-220.
  • Wang, D., Zhang, J., Cao, W., Li, J., & Zheng, Y. (2018). When will you arrive? estimating travel time based on deep neural networks. In Thirty-Second AAAI Conference on Artificial Intelligence.
  • Zhang, J., Zheng, Y., Sun, J., & Qi, D. (2019). Flow Prediction in Spatio-Temporal Networks Based on Multitask Deep Learning. IEEE Transactions on Knowledge and Data Engineering.
  • Zhu, J. Y., Park, T., Isola, P., & Efros, A. A. (2017). Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision (pp. 2223-2232).
  • Zhu, D., Cheng, X., Zhang, F., Yao, X., Gao, Y., & Liu, Y. (2019). Spatial interpolation using conditional generative adversarial neural networks. International Journal of Geographical Information Science, 1-24.

This post is part of a series. Read more about movement data in GIS.

MovingPandas is my attempt to provide a pure Python solution for trajectory data handling in GIS. MovingPandas provides trajectory classes and functions built on top of GeoPandas. 

To lower the entry barrier to getting started with MovingPandas, there’s now an interactive iPython notebook hosted on MyBinder. This notebook provides all the necessary imports and demonstrates how to create a Trajectory object.

Launch MyBinder for MovingPandas to get started!

In previous posts, I already wrote about Trajectools and some of the functionality it provides to QGIS Processing including:

There are also tools to compute heading and speed which I only talked about on Twitter.

Trajectools is now available from the QGIS plugin repository.

The plugin includes sample data from MarineCadastre downloads and the Geolife project.

Under the hood, Trajectools depends on GeoPandas!

If you are on Windows, here’s how to install GeoPandas for OSGeo4W:

  1. OSGeo4W installer: install python3-pip
  2. Environment variables: add GDAL_VERSION = 2.3.2 (or whichever version your OSGeo4W installation currently includes)
  3. OSGeo4W shell: call C:\OSGeo4W64\bin\py3_env.bat
  4. OSGeo4W shell: pip3 install geopandas (this will error at fiona)
  5. From https://www.lfd.uci.edu/~gohlke/pythonlibs/#fiona: download Fiona-1.7.13-cp37-cp37m-win_amd64.whl
  6. OSGeo4W shell: pip3 install path-to-download\Fiona-1.7.13-cp37-cp37m-win_amd64.whl
  7. OSGeo4W shell: pip3 install geopandas
  8. (optionally) From https://www.lfd.uci.edu/~gohlke/pythonlibs/#rtree: download Rtree-0.8.3-cp37-cp37m-win_amd64.whl and pip3 install it

If you want to use this functionality outside of QGIS, head over to my movingpandas project!

Many current movement data sources provide more or less continuous streams of object locations. For example, the AIS system provides continuous locations of vessels (mostly ships). This continuous stream of locations – let’s call it track – starts when we first record the vessel and ends with the last record. This start and end does not necessarily coincide with the start or end of a vessel voyage from one port to another. The stream start and end do not have any particular meaning. Instead, if we want to see what’s going on, we need to split the track into meaningful segments. One such segmentation – albeit a simple one – is to split tracks by day. This segmentation assumes that day/night changes affect the movement of our observed object. For many types of objects – those who mostly stay still during the night – this will work reasonably well.

For example, the following screenshot shows raw data of one particular vessel in the Boston region. By default, QGIS provides a Points to Path to convert points to lines. This tool takes one “group by” and one “order by” field. Therefore, if we want one trajectory per ship per day, we’d first have to create a new field that combines ship ID and day so that we can use this combination as a “group by” field. Additionally, the resulting lines loose all temporal information.

To simplify this workflow, Trajectools now provides a new algorithm that creates day trajectories and outputs LinestringM features. Using the Day trajectories from point layer tool, we can immediately see that our vessel of interest has been active for three consecutive days: entering our observation area on Nov 5th, moving to Boston where it stayed over night, then moving south to Weymouth on the next day, and leaving on the 7th.

Since the resulting trajectories are LinestringM features with time information stored in the M value, we can also visualize the speed of movement (as discussed in part #2 of this series):

We’ve seen a lot of explorative movement data analysis in the Movement data in GIS series so far. Beyond exploration, predictive analysis is another major topic in movement data analysis. One of the most obvious movement prediction use cases is trajectory prediction, i.e. trying to predict where a moving object will be in the future. The two main categories of trajectory prediction methods I see are those that try to predict the actual path that a moving object will take versus those that only try to predict the next destination.

Today, I want to focus on prediction methods that predict the path that a moving object is going to take. There are many different approaches from simple linear prediction to very sophisticated application-dependent methods. Regardless of the prediction method though, there is the question of how to evaluate the prediction results when these methods are applied to real-life data.

As long as we work with nice, densely, and regularly updated movement data, extracting evaluation samples is rather straightforward. To predict future movement, we need some information about past movement. Based on that past movement, we can then try to predict future positions. For example, given a trajectory that is twenty minutes long, we can extract a sample that provides five minutes of past movement, as well as the actually observed position five minutes into the future:

But what if the trajectory is irregularly updated? Do we interpolate the positions at the desired five minute timestamps? Do we try to shift the sample until – by chance – we find a section along the trajectory where the updates match our desired pattern? What if location timestamps include seconds or milliseconds and we therefore cannot find exact matches? Should we introduce a tolerance parameter that would allow us to match locations with approximately the same timestamp?

Depending on the duration of observation gaps in our trajectory, it might not be a good idea to simply interpolate locations since these interpolated locations could systematically bias our evaluation. Therefore, the safest approach may be to shift the sample pattern along the trajectory until a close match (within the specified tolerance) is found. This approach is now implemented in MovingPandas’ TrajectorySampler.

def test_sample_irregular_updates(self):
    df = pd.DataFrame([
        {'geometry':Point(0,0), 't':datetime(2018,1,1,12,0,1)},
        {'geometry':Point(0,3), 't':datetime(2018,1,1,12,3,2)},
        {'geometry':Point(0,6), 't':datetime(2018,1,1,12,6,1)},
        {'geometry':Point(0,9), 't':datetime(2018,1,1,12,9,2)},
        {'geometry':Point(0,10), 't':datetime(2018,1,1,12,10,2)},
        {'geometry':Point(0,14), 't':datetime(2018,1,1,12,14,3)},
        {'geometry':Point(0,19), 't':datetime(2018,1,1,12,19,4)},
        {'geometry':Point(0,20), 't':datetime(2018,1,1,12,20,0)}
        ]).set_index('t')
    geo_df = GeoDataFrame(df, crs={'init': '4326'})
    traj = Trajectory(1,geo_df)
    sampler = TrajectorySampler(traj, timedelta(seconds=5))
    past_timedelta = timedelta(minutes=5)
    future_timedelta = timedelta(minutes=5)
    sample = sampler.get_sample(past_timedelta, future_timedelta)
    result = sample.future_pos.wkt
    expected_result = "POINT (0 19)"
    self.assertEqual(result, expected_result)
    result = sample.past_traj.to_linestring().wkt
    expected_result = "LINESTRING (0 9, 0 10, 0 14)"
    self.assertEqual(result, expected_result)

The repository also includes a demo that illustrates how to split trajectories using a grid and finally extract samples:

 

Many of my previous posts in this series [1][2][3] have relied on PostGIS for trajectory data handling. While I love PostGIS, it feels like overkill to require a database to analyze smaller movement datasets. Wouldn’t it be great to have a pure Python solution?

If we look into moving object data literature, beyond the “trajectories are points with timestamps” perspective, which is common in GIS, we also encounter the “trajectories are time series with coordinates” perspective. I don’t know about you, but if I hear “time series” and Python, I think Pandas! In the Python Data Science Handbook, Jake VanderPlas writes:

Pandas was developed in the context of financial modeling, so as you might expect, it contains a fairly extensive set of tools for working with dates, times, and time-indexed data.

Of course, time series are one thing, but spatial data handling is another. Lucky for us, this is where GeoPandas comes in. GeoPandas has been around for a while and version 0.4 has been released in June 2018. So far, I haven’t found examples that use GeoPandas to manage movement data, so I’ve set out to give it a shot. My trajectory class uses a GeoDataFrame df for data storage. For visualization purposes, it can be converted to a LineString:

import pandas as pd 
from geopandas import GeoDataFrame
from shapely.geometry import Point, LineString

class Trajectory():
    def __init__(self, id, df, id_col):
        self.id = id
        self.df = df    
        self.id_col = id_col
        
    def __str__(self):
        return "Trajectory {1} ({2} to {3}) | Size: {0}".format(
            self.df.geometry.count(), self.id, self.get_start_time(), 
            self.get_end_time())
        
    def get_start_time(self):
        return self.df.index.min()
        
    def get_end_time(self):
        return self.df.index.max()
        
    def to_linestring(self):
        return self.make_line(self.df)
        
    def make_line(self, df):
        if df.size > 1:
            return df.groupby(self.id_col)['geometry'].apply(
                lambda x: LineString(x.tolist())).values[0]
        else:
            raise RuntimeError('Dataframe needs at least two points to make line!')

    def get_position_at(self, t):
        try:
            return self.df.loc[t]['geometry'][0]
        except:
            return self.df.iloc[self.df.index.drop_duplicates().get_loc(
                t, method='nearest')]['geometry']

Of course, this class can be used in stand-alone Python scripts, but it can also be used in QGIS. The following script takes data from a QGIS point layer, creates a GeoDataFrame, and finally generates trajectories. These trajectories can then be added to the map as a line layer.

All we need to do to ensure that our data is ordered by time is to set the GeoDataFrame’s index to the time field. From then on, Pandas takes care of the time series aspects and we can access the index as shown in the Trajectory.get_position_at() function above.

# Get data from a point layer
l = iface.activeLayer()
time_field_name = 't'
trajectory_id_field = 'trajectory_id' 
names = [field.name() for field in l.fields()]
data = []
for feature in l.getFeatures():
    my_dict = {}
    for i, a in enumerate(feature.attributes()):
        my_dict[names[i]] = a
    x = feature.geometry().asPoint().x()
    y = feature.geometry().asPoint().y()
    my_dict['geometry']=Point((x,y))
    data.append(my_dict)

# Create a GeoDataFrame
df = pd.DataFrame(data).set_index(time_field_name)
crs = {'init': l.crs().geographicCrsAuthId()} 
geo_df = GeoDataFrame(df, crs=crs)
print(geo_df)

# Test if spatial functions work
print(geo_df.dissolve([True]*len(geo_df)).centroid)

# Create a QGIS layer for trajectory lines
vl = QgsVectorLayer("LineString", "trajectories", "memory")
vl.setCrs(l.crs()) # doesn't stop popup :(
pr = vl.dataProvider()
pr.addAttributes([QgsField("id", QVariant.String)])
vl.updateFields() 

df_by_id = dict(tuple(geo_df.groupby(trajectory_id_field)))
trajectories = {}
for key, value in df_by_id.items():
    traj = Trajectory(key, value, trajectory_id_field)
    trajectories[key] = traj
    line = QgsGeometry.fromWkt(traj.to_linestring().wkt)
    f = QgsFeature()
    f.setGeometry(line)
    f.setAttributes([key])
    pr.addFeature(f) 
print(trajectories[1])

vl.updateExtents() 
QgsProject.instance().addMapLayer(vl)

The following screenshot shows this script applied to a sample of the Geolife datasets containing 100 trajectories with a total of 236,776 points. On my notebook, the runtime is approx. 20 seconds.

So far, GeoPandas has proven to be a convenient way to handle time series with coordinates. Trying to implement some trajectory analysis tools will show if it is indeed a promising data structure for trajectories.

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