Advertisements

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:

 

Advertisements

In Movement data in GIS #16, I presented a new way to deal with trajectory data using GeoPandas and how to load the trajectory GeoDataframes as a QGIS layer. Following up on this initial experiment, I’ve now implemented a first version of an algorithm that performs a spatial analysis on my GeoPandas trajectories.

The first spatial analysis algorithm I’ve implemented is Clip trajectories by extent. Implementing this algorithm revealed a couple of pitfalls:

  • To achieve correct results, we need to compute spatial intersections between linear trajectory segments and the extent. Therefore, we need to convert our point GeoDataframe to a line GeoDataframe.
  • Based on the spatial intersection, we need to take care of computing the corresponding timestamps of the events when trajectories enter or leave the extent.
  • A trajectory can intersect the extent multiple times. Therefore, we cannot simply use the global minimum and maximum timestamp of intersecting segments.
  • GeoPandas provides spatial intersection functionality but if the trajectory contains consecutive rows without location change, these will result in zero length lines and those cause an empty intersection result.

So far, the clip result only contains the trajectory id plus a suffix indicating the sequence of the intersection segments for a specific trajectory (because one trajectory can intersect the extent multiple times). The following screenshot shows one highlighted trajectory that intersects the extent three times and the resulting clipped trajectories:

This algorithm together with the basic trajectory from points algorithm is now available in a Processing algorithm provider plugin called Processing Trajectory.

Note: This plugin depends on GeoPandas.

Note for Windows users: GeoPandas is not a standard package that is available in OSGeo4W, so you’ll have to install it manually. (For the necessary steps, see this answer on gis.stackexchange.com)

The implemented tests show how to use the Trajectory class independently of QGIS. So far, I’m only testing the spatial properties though:

def test_two_intersections_with_same_polygon(self):
    polygon = Polygon([(5,-5),(7,-5),(7,12),(5,12),(5,-5)])
    data = [{'id':1, 'geometry':Point(0,0), 't':datetime(2018,1,1,12,0,0)},
        {'id':1, 'geometry':Point(6,0), 't':datetime(2018,1,1,12,10,0)},
        {'id':1, 'geometry':Point(10,0), 't':datetime(2018,1,1,12,15,0)},
        {'id':1, 'geometry':Point(10,10), 't':datetime(2018,1,1,12,30,0)},
        {'id':1, 'geometry':Point(0,10), 't':datetime(2018,1,1,13,0,0)}]
    df = pd.DataFrame(data).set_index('t')
    geo_df = GeoDataFrame(df, crs={'init': '31256'})
    traj = Trajectory(1, geo_df)
    intersections = traj.intersection(polygon)
    result = []
    for x in intersections:
        result.append(x.to_linestring())
    expected_result = [LineString([(5,0),(6,0),(7,0)]), LineString([(7,10),(5,10)])]
    self.assertEqual(result, expected_result) 

One issue with implementing the algorithms as QGIS Processing tools in this way is that the tools are independent of one another. That means that each tool has to repeat the expensive step of creating the trajectory objects in memory. I’m not sure this can be solved.

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.

If you follow me on Twitter, you have probably already heard that the ebook of “QGIS Map Design 2nd Edition” has now been published and we are expecting the print version to be up for sale later this month. Gretchen Peterson and I – together with our editor Gary Sherman (yes, that Gary Sherman!) – have been working hard to provide you with tons of new and improved map design workflows and many many completely new maps. By Gretchen’s count, this edition contains 23 new maps, so it’s very hard to pick a favorite!

Like the 1st edition, we provide increasingly advanced recipes in three chapters, each focusing on either layer styling, labeling, or creating print layouts. If I had to pick a favorite, I’d have to go with “Mastering Rotated Maps”, one of the advanced recipes in the print layouts chapter. It looks deceptively simple but it combines a variety of great QGIS features and clever ideas to design a map that provides information on multiple levels of detail. Besides the name inspiring rotated map items, this design combines

  • map overviews
  • map themes
  • graduated lines and polygons
  • a rotated north arrow
  • fancy leader lines

all in one:

“QGIS Map Design 2nd Edition” provides how-to instructions, as well as data and project files for each recipe. So you can jump right into it and work with the provided materials or apply the techniques to your own data.

The ebook is available at LocatePress.

Need to geocode some addresses? Here’s a five-lines-of-code solution based on “An A-Z of useful Python tricks” by Peter Gleeson:

from geopy import GoogleV3
place = "Krems an der Donau"
location = GoogleV3().geocode(place)
print(location.address)
print("POINT({},{})".format(location.latitude,location.longitude))

For more info, check out geopy:

geopy is a Python 2 and 3 client for several popular geocoding web services.
geopy includes geocoder classes for the OpenStreetMap Nominatim, ESRI ArcGIS, Google Geocoding API (V3), Baidu Maps, Bing Maps API, Yandex, IGN France, GeoNames, Pelias, geocode.earth, OpenMapQuest, PickPoint, What3Words, OpenCage, SmartyStreets, GeocodeFarm, and Here geocoder services.

This is a guest post by Time Manager collaborator and Python expert, Ariadni-Karolina Alexiou.

Today we’re going to look at how to visualize the error bounds of a GPS trace in time. The goal is to do an in-depth visual exploration using QGIS and Time Manager in order to learn more about the data we have.

The Data

We have a file that contains GPS locations of an object in time, which has been created by a GPS tracker. The tracker also keeps track of the error covariance matrix for each point in time, that is, what confidence it has in the measurements it gives. Here is what the file looks like:

data.png

Error Covariance Matrix

What are those sd* fields? According to the manual: The estimated standard deviations of the solution assuming a priori error model and error parameters by the positioning options. What it basically means is that the real GPS location will be located no further than three standard deviations across north and east from the measured location, most of (99.7%) the time. A way to represent this visually is to create an ellipse that maps this area of where the real location can be.ellipse_ab

An ellipse can be uniquely defined from the lengths of the segments a and b and its rotation angle. For more details on how to get those ellipse parameters from the covariance matrix, please see the footnote.

Ground truth data

We also happen to have a file with the actual locations (also in longitudes and latitudes) of the object for the same time frame as the GPS (also in seconds), provided through another tracking method which is more accurate in this case.

actual_data

This is because, the object was me running on a rooftop in Zürich wearing several tracking devices (not just GPS), and I knew exactly which floor tiles I was hitting.

The goal is to explore, visually, the relationship between the GPS data and the actual locations in time. I hope to get an idea of the accuracy, and what can influence it.

First look

Loading the GPS data into QGIS and Time Manager, we can indeed see the GPS locations vis-a-vis the actual locations in time.

actual_vs_gps

Let’s see if the actual locations that were measured independently fall inside the ellipse coverage area. To do this, we need to use the covariance data to render ellipses.

Creating the ellipses

I considered using the ellipses marker from QGIS.

ellipse_marker.png

It is possible to switch from Millimeter to Map Unit and edit a data defined override for symbol width, height and rotation. Symbol width would be the a parameter of the ellipse, symbol height the b parameter and rotation simply the angle. The thing is, we haven’t computed any of these values yet, we just have the error covariance values in our dataset.

Because of the re-projections and matrix calculations inherent into extracting the a, b and angle of the error ellipse at each point in time, I decided to do this calculation offline using Python and relevant libraries, and then simply add a WKT text field with a polygon representation of the ellipse to the file I had. That way, the augmented data could be re-used outside QGIS, for example, to visualize using Leaflet or similar. I could have done a hybrid solution, where I calculated a, b and the angle offline, and then used the dynamic rendering capabilities of QGIS, as well.

I also decided to dump the csv into an sqlite database with an index on the time column, to make time range queries (which Time Manager does) run faster.

Putting it all together

The code for transforming the initial GPS data csv file into an sqlite database can be found in my github along with a small sample of the file containing the GPS data.

I created three ellipses per timestamp, to represent the three standard deviations. Opening QGIS (I used version: 2.12, Las Palmas) and going to Layer>Add Layer>Add SpatialLite Layer, we see the following dialog:

add_spatialite2.png

After adding the layer (say, for the second standard deviation ellipse), we can add it to Time Manager like so:

add_to_tm

We do the process three times to add the three types of ellipses, taking care to style each ellipse differently. I used transparent fill for the second and third standard deviation ellipses.

I also added the data of my  actual positions.

Here is an exported video of the trace (at a place in time where I go forward, backwards and forward again and then stay still).

gps

Conclusions

Looking at the relationship between the actual data and the GPS data, we can see the following:

  • Although the actual position differs from the measured one, the actual position always lies within one or two standard deviations of the measured position (so, inside the purple and golden ellipses).
  • The direction of movement has greater uncertainty (the ellipse is elongated across the line I am running on).
  • When I am standing still, the GPS position is still moving, and unfortunately does not converge to my actual stationary position, but drifts. More research is needed regarding what happens with the GPS data when the tracker is actually still.
  • The GPS position doesn’t jump erratically, which can be good, however, it seems to have trouble ‘catching up’ with the actual position. This means if we’re looking to measure velocity in particular, the GPS tracker might underestimate that.

These findings are empirical, since they are extracted from a single visualization, but we have already learned some new things. We have some new ideas for what questions to ask on a large scale in the data, what additional experiments to run in the future and what limitations we may need to be aware of.

Thanks for reading!

Footnote: Error Covariance Matrix calculations

The error covariance matrix is (according to the definitions of the sd* columns in the manual):

sde * sde sign(sdne) * sdne * sdne
sign(sdne) * sdne * sdne sdn * sdn

It is not a diagonal matrix, which means that the errors across the ‘north’ dimension and the ‘east’ dimension, are not exactly independent.

An important detail is that, while the position is given in longitudes and latitudes, the sdn, sde and sdne fields are in meters. To address this in the code, we convert the longitude and latitudes using UTM projection, so that they are also in meters (northings and eastings).

For more details on the mathematics used to plot the ellipses check out this article by Robert Eisele and the implementation of the ellipse calculations on my github.

%d bloggers like this: