Plotting and Tabular Data
OverviewTeaching: 50 min
Exercises: 5 minQuestions
How do we make scatter plots in Matplotlib?
How do we store data in a Pandas
Select rows and columns from an Astropy
Use Matplotlib to make a scatter plot.
Use Gala to transform coordinates.
Make a Pandas
DataFrameand use a Boolean
Seriesto select rows.
DataFramein an HDF5 file.
In the previous episode, we wrote a query to select stars from the region of the sky where we expect GD-1 to be, and saved the results in a FITS file.
Now we will read that data back in and implement the next step in the analysis, identifying stars with the proper motion we expect for GD-1.
We will read back the results from the previous lesson, which we saved in a FITS file.
Then we will transform the coordinates and proper motion data from ICRS back to the coordinate frame of GD-1.
We will put those results into a Pandas
Starting from this episode
If you are starting a new notebook for this episode, expand this section for information you will need to get started.
In the previous episode, we ran a query on the Gaia server, downloaded data for roughly 140,000 stars, and saved the data in a FITS file. We will use that data for this episode. Whether you are working from a new notebook or coming back from a checkpoint, reloading the data will save you from having to run the query again.
If you are starting this episode here or starting this episode in a new notebook, you will need to run the following lines of code.
This imports previously imported functions:
import astropy.units as u from astropy.coordinates import SkyCoord from gala.coordinates import GD1Koposov10 from astropy.table import Table from episode_functions import *
The following code loads in the data (instructions for downloading data can be found in the setup instructions). You may need to add a the path to the filename variable below (e.g.
filename = 'student_download/data/gd1_results.fits')
filename = 'gd1_results.fits' polygon_results = Table.read(filename) gd1_frame = GD1Koposov10()
Selecting rows and columns
In the previous episode, we selected spatial and proper motion information from the Gaia catalog for stars around a small part of GD-1. The output was returned as an Astropy
We can use
info to check the contents.
<Table length=140339> name dtype unit description --------- ------- -------- ------------------------------------------------------------------ source_id int64 Unique source identifier (unique within a particular Data Release) ra float64 deg Right ascension dec float64 deg Declination pmra float64 mas / yr Proper motion in right ascension direction pmdec float64 mas / yr Proper motion in declination direction parallax float64 mas Parallax
In this episode, we will see operations for selecting columns and rows
from an Astropy
Table. You can find more information about these
operations in the Astropy
We can get the names of the columns like this:
['source_id', 'ra', 'dec', 'pmra', 'pmdec', 'parallax']
And select an individual column like this:
<Column name='ra' dtype='float64' unit='deg' description='Right ascension' length=140339> 142.48301935991023 142.25452941346344 142.64528557468074 142.57739430926034 142.58913564478618 141.81762228999614 143.18339801317677 142.9347319464589 142.26769745823267 142.89551292869012 [Output truncated]
The result is a
Column object that contains the data, and also the
data type, units, and name of the column.
The rows in the
Table are numbered from 0 to
n is the
number of rows. We can select the first row like this:
<Row index=0> source_id ra dec pmra pmdec parallax deg deg mas / yr mas / yr mas int64 float64 float64 float64 float64 float64 ------------------ ------------------ ----------------- ------------------- ----------------- ------------------- 637987125186749568 142.48301935991023 21.75771616932985 -2.5168384683875766 2.941813096629439 -0.2573448962333354
The result is a
Notice that the bracket operator can be used to select both columns and rows. You might wonder how it knows which to select. If the expression in brackets is a string, it selects a column; if the expression is an integer, it selects a row.
If you apply the bracket operator twice, you can select a column and then an element from the column.
Or you can select a row and then an element from the row.
You get the same result either way.
To see what the results look like, we will use a scatter plot. The library we will use is Matplotlib, which is the most widely-used plotting library for Python. The Matplotlib interface is based on MATLAB (hence the name), so if you know MATLAB, some of it will be familiar.
We will import like this:
import matplotlib.pyplot as plt
Pyplot is part of the Matplotlib library. It is conventional to
import it using the shortened name
Keeping plots in the notebook
In recent versions of Jupyter, plots appear “inline”; that is, they are part of the notebook. In some older versions, plots appear in a new window. If your plots appear in a new window, you might want to run the following Jupyter magic command in a notebook cell:
scatteris more versatile; for example, you can make every point in a scatter plot a different color.
plotis more limited, but for simple cases, it can be substantially faster.
Jake Vanderplas explains these differences in The Python Data Science Handbook.
Since we are plotting more than 100,000 points and they are all the
same size and color, we will use
Here is a scatter plot of the stars we selected in the GD-1 region with right ascension on the x-axis and declination on the y-axis, both ICRS coordinates in degrees.
x = polygon_results['ra'] y = polygon_results['dec'] plt.plot(x, y, 'ko') plt.xlabel('ra (degree ICRS)') plt.ylabel('dec (degree ICRS)')
<Figure size 432x288 with 1 Axes>
The arguments to
y, and a string that specifies
the style. In this case, the letters
ko indicate that we want a
black, round marker (
k is for black because
b is for blue).
ylabel put labels on the axes.
Looking at this plot, we can see that the region we selected, which is a rectangle in GD-1 coordinates, is a non-rectanglar region in ICRS coordinates.
However, this scatter plot has a problem. It is “overplotted”, which means that there are so many overlapping points, we cannot distinguish between high and low density areas.
To fix this, we can provide optional arguments to control the size and transparency of the points.
Exercise (5 minutes)
In the call to
plt.plot, use the keyword argument
markersizeto make the markers smaller.
Then add the keyword argument
alphato make the markers partly transparent.
Adjust these arguments until you think the figure shows the data most clearly.
Note: Once you have made these changes, you might notice that the figure shows stripes with lower density of stars. These stripes are caused by the way Gaia scans the sky, which you can read about here. The dataset we are using, Gaia Data Release 2, covers 22 months of observations; during this time, some parts of the sky were scanned more than others.
x = polygon_results['ra'] y = polygon_results['dec'] plt.plot(x, y, 'ko', markersize=0.1, alpha=0.1) plt.xlabel('ra (degree ICRS)') plt.ylabel('dec (degree ICRS)')
Remember that we selected data from a rectangle of coordinates in the GD-1 frame, then transformed them to ICRS when we constructed the query. The coordinates in the query results are in ICRS.
To plot them, we will transform them back to the GD-1 frame; that way, the axes of the figure are aligned with the orbit of GD-1, which is useful for two reasons:
By transforming the coordinates, we can identify stars that are likely to be in GD-1 by selecting stars near the centerline of the stream, where φ2 is close to 0.
By transforming the proper motions, we can identify stars with non-zero proper motion along the φ1 axis, which are likely to be part of GD-1.
To do the transformation, we will put the results into a
object. In a previous episode, we created a
SkyCoord object like
skycoord = SkyCoord(ra=polygon_results['ra'], dec=polygon_results['dec'])
Notice that we did not specify the reference frame. That is because when
ICRS frame is assumed by default.
SkyCoord object can keep track not just of location, but also proper motions.
This means that we can initialize a
SkyCoord object with location and proper motions,
then use all of these quantities together to transform into the GD-1 frame.
Now we are going to do something similar, but now we will take advantage of the
capacity to include and track space motion information in addition to
dec. We will now also include:
pmdec, which are proper motion in the
radial_velocity, which are important for the reflex correction and will be discussed in that section.
distance = 8 * u.kpc radial_velocity= 0 * u.km/u.s skycoord = SkyCoord(ra=polygon_results['ra'], dec=polygon_results['dec'], pm_ra_cosdec=polygon_results['pmra'], pm_dec=polygon_results['pmdec'], distance=distance, radial_velocity=radial_velocity)
For the first four arguments, we use columns from
radial_velocity we use constants, which we explain in the section on reflex correction.
The result is an Astropy
SkyCoord object, which we can transform to
the GD-1 frame.
transformed = skycoord.transform_to(gd1_frame)
The result is another
SkyCoord object, now in the GD-1 frame.
The next step is to correct the proper motion measurements for the effect of the motion of our solar system around the Galactic center.
When we created
skycoord, we provided constant values for
radial_velocity rather than measurements from Gaia.
That might seem like a strange thing to do, but here is the motivation:
Because the stars in GD-1 are so far away, parallaxes measured by Gaia are negligible, making the distance estimates unreliable.
So we replace them with our current best estimate of the mean distance to GD-1, about 8 kpc. See Koposov, Rix, and Hogg, 2010.
For the other stars in the table, this distance estimate will be inaccurate, so reflex correction will not be correct. But that should have only a small effect on our ability to identify stars with the proper motion we expect for GD-1.
The measurement of radial velocity has no effect on the correction for proper motion, but we have to provide a value to avoid errors in the reflex correction calculation. So we provide
0as an arbitrary place-keeper.
With this preparation, we can use
reflex_correct from Gala
to correct for the motion of the solar system.
from gala.coordinates import reflex_correct skycoord_gd1 = reflex_correct(transformed)
The result is a
SkyCoord object that contains
phi2, which represent the transformed coordinates in the GD-1 frame.
pm_phi2, which represent the transformed proper motions that have been corrected for the motion of the solar system around the Galactic center.
We can select the coordinates and plot them like this:
x = skycoord_gd1.phi1 y = skycoord_gd1.phi2 plt.plot(x, y, 'ko', markersize=0.1, alpha=0.1) plt.xlabel('phi1 (degree GD1)') plt.ylabel('phi2 (degree GD1)')
<Figure size 432x288 with 1 Axes>
We started with a rectangle in the GD-1 frame. When transformed to the ICRS frame, it is a non-rectangular region. Now, transformed back to the GD-1 frame, it is a rectangle again.
At this point we have two objects containing different sets of the
data relating to identifying stars in GD-1.
polygon_results is the Astropy
Table we downloaded from Gaia.
skycoord_gd1 is a
SkyCoord object that contains the
transformed coordinates and proper motions.
On one hand, this division of labor makes sense because each object provides different capabilities. But working with multiple object types can be awkward. It will be more convenient to choose one object and get all of the data into it.
Now we can extract the columns we want from
skycoord_gd1 and add
them as columns in the Astropy
phi2 contain the
polygon_results['phi1'] = skycoord_gd1.phi1 polygon_results['phi2'] = skycoord_gd1.phi2 polygon_results.info()
<Table length=140339> name dtype unit description --------- ------- -------- ------------------------------------------------------------------ source_id int64 Unique source identifier (unique within a particular Data Release) ra float64 deg Right ascension dec float64 deg Declination pmra float64 mas / yr Proper motion in right ascension direction pmdec float64 mas / yr Proper motion in declination direction parallax float64 mas Parallax phi1 float64 deg phi2 float64 deg
pm_phi2 contain the components of proper
motion in the transformed frame.
polygon_results['pm_phi1'] = skycoord_gd1.pm_phi1_cosphi2 polygon_results['pm_phi2'] = skycoord_gd1.pm_phi2 polygon_results.info()
<Table length=140339> name dtype unit description --------- ------- -------- ------------------------------------------------------------------ source_id int64 Unique source identifier (unique within a particular Data Release) ra float64 deg Right ascension dec float64 deg Declination pmra float64 mas / yr Proper motion in right ascension direction pmdec float64 mas / yr Proper motion in declination direction parallax float64 mas Parallax phi1 float64 deg phi2 float64 deg pm_phi1 float64 mas / yr pm_phi2 float64 mas / yr
Detail If you notice that
SkyCoordhas an attribute called
proper_motion, you might wonder why we are not using it.
We could have:
proper_motioncontains the same data as
pm_phi2, but in a different format.
DataFrames versus Astropy
Two common choices are the Pandas
DataFrames and Astropy
Tables share many of the same characteristics and most of the manipulations that we do can be done with either. As you become more familiar with each, you will develop a sense of which one you prefer for different tasks. For instance you may choose to use Astropy
Tables to read in data, especially astronomy specific data formats, but Pandas
DataFrames to inspect the data. Fortunately, Astropy makes it easy to convert between the two data types. We will choose to use Pandas
DataFrame, for two reasons:
It provides capabilities that are (almost) a superset of the other data structures, so it is the all-in-one solution.
Pandas is a general-purpose tool that is useful in many domains, especially data science. If you are going to develop expertise in one tool, Pandas is a good choice.
However, compared to an Astropy
Table, Pandas has one big drawback: it does not keep the metadata associated with the table, including the units for the columns. Nevertheless, we think its a useful data type to be familiar with.
It is straightforward to convert an Astropy
Table to a Pandas
import pandas as pd results_df = polygon_results.to_pandas()
shape, which shows the number of rows and columns.
It also provides
head, which displays the first few rows.
useful for spot-checking large results as you go along.
source_id ra dec pmra pmdec parallax phi1 phi2 pm_phi1 pm_phi2 0 637987125186749568 142.483019 21.757716 -2.516838 2.941813 -0.257345 -54.975623 -3.659349 6.429945 6.518157 1 638285195917112960 142.254529 22.476168 2.662702 -12.165984 0.422728 -54.498247 -3.081524 -3.168637 -6.206795 2 638073505568978688 142.645286 22.166932 18.306747 -7.950660 0.103640 -54.551634 -3.554229 9.129447 -16.819570 3 638086386175786752 142.577394 22.227920 0.987786 -2.584105 -0.857327 -54.536457 -3.467966 3.837120 0.526461 4 638049655615392384 142.589136 22.110783 0.244439 -4.941079 0.099625 -54.627448 -3.542738 1.466103 -0.185292
Attributes vs functions
shapeis an attribute, so we display its value without calling it as a function.
headis a function, so we need the parentheses.
Before we go any further, we will take all of the steps that we have done
and consolidate them into a single function that we can use to take the
coordinates and proper motion that we get as an Astropy
Table from our
Gaia query, add columns representing the reflex corrected
GD-1 coordinates and proper motions, and transform it into a
This is a general function that we will use multiple times as we build different
queries so we want to write it once and then call the function rather than having
to copy and paste the code over and over again.
def make_dataframe(table): """Transform coordinates from ICRS to GD-1 frame. table: Astropy Table returns: Pandas DataFrame """ #Create a SkyCoord object with the coordinates and proper motions # in the input table skycoord = SkyCoord( ra=table['ra'], dec=table['dec'], pm_ra_cosdec=table['pmra'], pm_dec=table['pmdec'], distance=8*u.kpc, radial_velocity=0*u.km/u.s) # Define the GD-1 reference frame gd1_frame = GD1Koposov10() # Transform input coordinates to the GD-1 reference frame transformed = skycoord.transform_to(gd1_frame) # Correct GD-1 coordinates for solar system motion around galactic center skycoord_gd1 = reflex_correct(transformed) #Add GD-1 reference frame columns for coordinates and proper motions table['phi1'] = skycoord_gd1.phi1 table['phi2'] = skycoord_gd1.phi2 table['pm_phi1'] = skycoord_gd1.pm_phi1_cosphi2 table['pm_phi2'] = skycoord_gd1.pm_phi2 # Create DataFrame df = table.to_pandas() return df
Here is how we use the function:
results_df = make_dataframe(polygon_results)
Saving the DataFrame
At this point we have run a successful query and combined the results into a single
DataFrame. This is a good time to save the data.
To save a Pandas
DataFrame, one option is to convert it to an
Table, like this:
from astropy.table import Table results_table = Table.from_pandas(results_df) type(results_table)
Then we could write the
Table to a FITS file, as we did in the
But, like Astropy, Pandas provides functions to write DataFrames in other formats; to
see what they are find the functions here that begin with
One of the best options is HDF5, which is Version 5 of Hierarchical Data Format.
HDF5 is a binary format, so files are small and fast to read and write (like FITS, but unlike XML).
An HDF5 file is similar to an SQL database in the sense that it can contain more than one table, although in HDF5 vocabulary, a table is called a Dataset. (Multi-extension FITS files can also contain more than one table.)
And HDF5 stores the metadata associated with the table, including column names, row labels, and data types (like FITS).
Finally, HDF5 is a cross-language standard, so if you write an HDF5 file with Pandas, you can read it back with many other software tools (more than FITS).
We can write a Pandas
DataFrame to an HDF5 file like this:
filename = 'gd1_data.hdf' results_df.to_hdf(filename, 'results_df', mode='w')
Because an HDF5 file can contain more than one Dataset, we have to provide a name, or “key”, that identifies the Dataset in the file.
We could use any string as the key, but it is generally a good practice
to use a descriptive name (just like your
DataFrame variable name) so
we will give the Dataset in the file the same name (key) as the
By default, writing a
DataFrame appends a new dataset to an existing HDF5 file.
We will use the argument
mode='w' to overwrite the
file if it already exists rather than append another dataset to it.
In this episode, we re-loaded the Gaia data we saved from a previous query.
We transformed the coordinates and proper motion from ICRS to a frame
aligned with the orbit of GD-1, stored the results in a Pandas
DataFrame, and visualized them.
We combined all of these steps into a single function that we can reuse in the future to go straight from the output of a query with object coordinates in the ICRS reference frame directly to a Pandas DataFrame that includes object coordinates in the GD-1 reference frame.
We saved our results to an HDF5 file which we can use to restart the analysis from this stage or verify our results at some future time.
When you make a scatter plot, adjust the size of the markers and their transparency so the figure is not overplotted; otherwise it can misrepresent the data badly.
For simple scatter plots in Matplotlib,
plotis faster than
Tableand a Pandas
DataFrameare similar in many ways and they provide many of the same functions. They have pros and cons, but for many projects, either one would be a reasonable choice.
To store data from a Pandas
DataFrame, a good option is an HDF5 file, which can contain multiple Datasets (we’ll dig in more in the Join lesson).