Merge branch 'minor-fix' into 'master'

Avoiding weird error when using custom mad See merge request !8

Merge branch 'minor-fix' into 'master'
17c4578e · Paul McCarthy · e81c542c · 1fb57681 · 17c4578e · 17c4578e
Commit 17c4578e authored 4 years ago by Paul McCarthy
--- a/applications/pandas/pandas.ipynb
+++ b/applications/pandas/pandas.ipynb
@@ -963,7 +963,7 @@
   "metadata": {},
   "outputs": [],
   "source": [
-    "titanic.groupby(['class', 'survived']).aggregate((np.median, mad))"
+    "titanic.groupby(['class', 'survived']).aggregate((np.median, 'mad'))"
   ]
  },
  {
@@ -984,7 +984,7 @@
   "metadata": {},
   "outputs": [],
   "source": [
-    "df_full = titanic.groupby(['class', 'survived']).aggregate((np.median, mad))"
+    "df_full = titanic.groupby(['class', 'survived']).aggregate((np.median, 'mad'))"
   ]
  },
  {

 %% Cell type:markdown id: tags:

 # Pandas

 Pandas is a data analysis library focused on the cleaning and exploration of
 tabular data.

 Some useful links are:
 - [main website](https://pandas.pydata.org)
 - [documentation](http://pandas.pydata.org/pandas-docs/stable/)<sup>1</sup>
 - [Python Data Science Handbook](https://jakevdp.github.io/PythonDataScienceHandbook/)<sup>1</sup> by
  Jake van der Plas
 - [List of Pandas tutorials](https://pandas.pydata.org/pandas-docs/stable/getting_started/tutorials.html)

 <sup>1</sup> This tutorial borrows heavily from the pandas documentation and
 the Python Data Science Handbook

 %% Cell type:code id: tags:

 ``` 
 %pylab inline
 import pandas as pd  # pd is the usual abbreviation for pandas
 import matplotlib.pyplot as plt # matplotlib for plotting
 import seaborn as sns  # seaborn is the main plotting library for Pandas
 import statsmodels.api as sm  # statsmodels fits linear models to pandas data
 import statsmodels.formula.api as smf
 from IPython.display import Image
 sns.set()  # use the prettier seaborn plotting settings rather than the default matplotlib one
 ```

 %% Cell type:markdown id: tags:

 > We will mostly be using `seaborn` instead of `matplotlib` for
 > visualisation. But `seaborn` is actually an extension to `matplotlib`, so we
 > are still using the latter under the hood.

 ## Loading in data

 Pandas supports a wide range of I/O tools to load from text files, binary files,
 and SQL databases. You can find a table with all supported formats
 [here](http://pandas.pydata.org/pandas-docs/stable/io.html).

 %% Cell type:code id: tags:

 ``` 
 titanic = pd.read_csv('https://raw.githubusercontent.com/mwaskom/seaborn-data/master/titanic.csv')
 titanic
 ```

 %% Cell type:markdown id: tags:

 This loads the data into a
 [`DataFrame`](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.html)
 object, which is the main object we will be interacting with in pandas. It
 represents a table of data.  The input functions for other file formats all look like
 `pd.read_{format}`.  Note that we can provide the URL to the dataset, without having
 to download it beforehand.

 We can write out the dataset using `<dataframe>.to_{format}(<filename>)`:

 %% Cell type:code id: tags:

 ``` 
 titanic.to_csv('titanic_copy.csv', index=False)  # we set index to False to prevent pandas from storing the row indices (explained in more detail below)
 ```

 %% Cell type:markdown id: tags:

 If you can not connect to the internet, you can run the command below to load
 this locally stored titanic dataset

 %% Cell type:code id: tags:

 ``` 
 titanic = pd.read_csv('titanic.csv')
 titanic
 ```

 %% Cell type:markdown id: tags:

 Note that the titanic dataset was also available to us as one of the standard
 datasets included with seaborn. We could load it from there using

 %% Cell type:code id: tags:

 ``` 
 sns.load_dataset('titanic')
 ```

 %% Cell type:markdown id: tags:

 `Dataframes` can also be created from other python objects, using
 `pd.DataFrame.from_{other type}`. The most useful of these is `from_dict`,
 which converts a mapping of the columns to a pandas `DataFrame` (i.e., table).

 %% Cell type:code id: tags:

 ``` 
 pd.DataFrame.from_dict({
    'random numbers': np.random.rand(5),
    'sequence (int)': np.arange(5),
    'sequence (float)': np.linspace(0, 5, 5),
    'letters': list('abcde'),
    'constant_value': 'same_value'
 })
 ```

 %% Cell type:markdown id: tags:

 ## Getting your data out
 For many applications (e.g., ICA, machine learning) you might want to
 extract your data as a numpy array, even though more and more projects
 support pandas Dataframes directly. The underlying numpy array can be
 accessed using the `to_numpy` method

 %% Cell type:code id: tags:

 ``` 
 titanic.to_numpy()
 ```

 %% Cell type:markdown id: tags:

 Note that the type of the returned array is the most common type (in this case
 object). If you just want the numeric parts of the table you can use
 `select_dtypes`, which selects specific columns based on their dtype:

 %% Cell type:code id: tags:

 ``` 
 titanic.select_dtypes(include=np.number).to_numpy()
 ```

 %% Cell type:markdown id: tags:

 Note that the numpy array has no information on the column names or row indices.
 Alternatively, when you want to include the categorical variables in your later
 analysis (e.g., for machine learning), you can extract dummy variables using:

 %% Cell type:code id: tags:

 ``` 
 pd.get_dummies(titanic)
 ```

 %% Cell type:markdown id: tags:

 ## Accessing parts of the data

 [Documentation on indexing](http://pandas.pydata.org/pandas-docs/stable/indexing.html)

 ### Selecting columns by name

 Single columns can be selected using the normal python indexing:

 %% Cell type:code id: tags:

 ``` 
 titanic['embark_town']
 ```

 %% Cell type:markdown id: tags:

 If the column names is a valid python identifier (i.e., is a string that does not contain stuff like spaces)
 we can also access it as an attribute

 %% Cell type:code id: tags:

 ``` 
 titanic.embark_town
 ```

 %% Cell type:markdown id: tags:

 Note that this returns a single column is represented by a pandas
 [`Series`](https://pandas.pydata.org/pandas-docs/stable/generated/pandas.Series.html)
 rather than a `DataFrame` object. A `Series` is a 1-dimensional array
 representing a single column, while a `DataFrame` is a collection of `Series`
 representing a table.  Multiple columns can be returned by providing a
 list of columns names. This will return a `DataFrame`:

 %% Cell type:code id: tags:

 ``` 
 titanic[['class', 'alive']]
 ```

 %% Cell type:markdown id: tags:

 Note that you have to provide a list here (square brackets). If you provide a
 tuple (round brackets) pandas will think you are trying to access a single
 column that has that tuple as a name:

 %% Cell type:code id: tags:

 ``` 
 titanic[('class', 'alive')]
 ```

 %% Cell type:markdown id: tags:

 In this case there is no column called `('class', 'alive')` leading to an
 error.  Later on we will see some uses to having columns named like this.

 The same interface can be used to add a new column (or updating an existing one):

 %% Cell type:code id: tags:

 ``` 
 titanic["my_column"] = 10
 titanic
 ```

 %% Cell type:markdown id: tags:

 We can delete a column using:

 %% Cell type:code id: tags:

 ``` 
 del titanic["my_column"]
 ```

 %% Cell type:markdown id: tags:

 ### Indexing rows by name or integer

 Individual rows can be accessed based on their name (i.e., the index) or integer
 (i.e., which row it is in). In our current table this will give the same
 results. To ensure that these are different, let's sort our titanic dataset
 based on the passenger fare:

 %% Cell type:code id: tags:

 ``` 
 titanic_sorted = titanic.sort_values('fare')
 titanic_sorted
 ```

 %% Cell type:markdown id: tags:

 Note that the re-sorting did not change the values in the index (i.e., left-most
 column).

 We can select the first row of this newly sorted table using `iloc`

 %% Cell type:code id: tags:

 ``` 
 titanic_sorted.iloc[0]
 ```

 %% Cell type:markdown id: tags:

 We can select the row with the index 0 using

 %% Cell type:code id: tags:

 ``` 
 titanic_sorted.loc[0]
 ```

 %% Cell type:markdown id: tags:

 Note that this gives the same passenger as the first row of the initial table
 before sorting

 %% Cell type:code id: tags:

 ``` 
 titanic.iloc[0]
 ```

 %% Cell type:markdown id: tags:

 Another common way to access the first or last N rows of a table is using the
 head/tail methods

 %% Cell type:code id: tags:

 ``` 
 titanic_sorted.head(3)
 ```

 %% Cell type:code id: tags:

 ``` 
 titanic_sorted.tail(3)
 ```

 %% Cell type:markdown id: tags:

 Note that nearly all methods in pandas return a new `DataFrame`, which means
 that we can "chain" methods. What we mean by "chaining" is that rather than
 storing a returned `DataFrame`, we instead call a different method on it.
 For example, below the `tail` method call returns a new `DataFrame`
 on which we then call the `head` method:

 %% Cell type:code id: tags:

 ``` 
 titanic_sorted.tail(10).head(5)  # select the first 5 of the last 10 passengers in the database
 ```

 %% Cell type:markdown id: tags:

 > This chaining is usually very efficient, because when creating a new `DataFrame`
 > pandas minimizes the copying of the underlying data (similar to what `numpy` arrays do).
 > While this allows for fast manipulation of the data, it does mean that if you update the
 > data in one of your `DataFrame` the data *might* also be changed in related `DataFrame`s

 %% Cell type:code id: tags:

 ``` 
 titanic_sorted.iloc[-10:-5]  # alternative way to get the same passengers
 ```

 %% Cell type:markdown id: tags:

 **Exercise**: use sorting and tail/head or indexing to find the 10 youngest
 passengers on the titanic. Try to do this on a single line by chaining calls
 to the titanic `DataFrame` object

 %% Cell type:code id: tags:

 ``` 
 titanic.sort_values...
 ```

 %% Cell type:markdown id: tags:

 ### Indexing rows by value

 One very common way to select specific columns is by their value

 %% Cell type:code id: tags:

 ``` 
 titanic[titanic.sex == 'female']  # selects all females
 ```

 %% Cell type:code id: tags:

 ``` 
 # select all passengers older than 60 who departed from Southampton
 titanic[(titanic.age > 60) & (titanic['embark_town'] == 'Southampton')]
 ```

 %% Cell type:markdown id: tags:

 Note that this required typing `titanic` quite often.
 The `query` method allows you to get the same result with much less
 typing (see [documentation](http://pandas.pydata.org/pandas-docs/stable/indexing.html#the-query-method))
 (note that using the `query` method is also faster and uses a lot less
 memory).

 > You may have trouble using the `query` method with columns which have
 > a name that cannot be used as a Python identifier.

 %% Cell type:code id: tags:

 ``` 
 titanic.query('(age > 60) & (embark_town == "Southampton")')
 ```

 %% Cell type:markdown id: tags:

 When selecting a categorical multiple options from a categorical values you
 might want to use `isin`:

 %% Cell type:code id: tags:

 ``` 
 titanic[titanic['class'].isin(['First','Second'])]  # select all passangers not in first or second class
 ```

 %% Cell type:markdown id: tags:

 Particularly useful when selecting data like this is the `isna` method which
 finds all missing data

 %% Cell type:code id: tags:

 ``` 
 titanic[~titanic.age.isna()]  # select all passengers whose age is not N/A
 ```

 %% Cell type:markdown id: tags:

 This removing of missing numbers is so common that it has is own method

 %% Cell type:code id: tags:

 ``` 
 titanic.dropna()  # drops all passengers that have some datapoint missing
 ```

 %% Cell type:code id: tags:

 ``` 
 titanic.dropna(subset=['age', 'fare'])  # Only drop passengers with missing ages or fares
 ```

 %% Cell type:markdown id: tags:

 **Exercise**: use sorting, indexing by value, `dropna` and `tail`/`head` or
 indexing to find the 10 oldest female passengers on the titanic. Try to do
 this on a single line by chaining calls to the titanic `DataFrame` object

 %% Cell type:code id: tags:

 ``` 
 titanic...
 ```

 %% Cell type:markdown id: tags:

 ## Plotting the data

 Before we start analyzing the data, let's play around with visualizing it.
 Pandas does have some basic built-in plotting options:

 %% Cell type:code id: tags:

 ``` 
 titanic.fare.hist(bins=20, log=True)
 ```

 %% Cell type:code id: tags:

 ``` 
 titanic.age.plot()
 ```

 %% Cell type:markdown id: tags:

 To plot all variables simply call `plot` or `hist` on the full `DataFrame`
 rather than a single `Series` (i.e., column). You might want to set `subplots=True`
 to plot each variable in a different subplot.

 %% Cell type:code id: tags:

 ``` 
 titanic.plot(subplots=True)
 ```

 %% Cell type:markdown id: tags:

 Individual `Series` are essentially 1D arrays, so we can use them as such in
 `matplotlib`

 %% Cell type:code id: tags:

 ``` 
 plt.scatter(titanic.age, titanic.fare)
 ```

 %% Cell type:markdown id: tags:

 However, for most purposes much nicer plots can be obtained using
 [Seaborn](https://seaborn.pydata.org). Seaborn has support to produce plots
 showing the
 [univariate](https://seaborn.pydata.org/tutorial/distributions.html#plotting-univariate-distributions)
 or
 [bivariate](https://seaborn.pydata.org/tutorial/distributions.html#plotting-bivariate-distributions)
 distribution of data in a single or a grid of plots.  Most of the seaborn
 plotting functions expect to get a pandas `DataFrame` (although they will work
 with Numpy arrays as well). So we can plot age vs. fare like:

 %% Cell type:code id: tags:

 ``` 
 sns.jointplot('age', 'fare', data=titanic)
 ```

 %% Cell type:markdown id: tags:

 **Exercise**: check the documentation from `sns.jointplot` (hover the mouse
 over the text `jointplot` and press shift-tab) to find out how to turn the
 scatter plot into a density (kde) map

 %% Cell type:code id: tags:

 ``` 
 sns.jointplot('age', 'fare', data=titanic, ...)
 ```

 %% Cell type:markdown id: tags:

 Here is just a brief example of how we can use multiple columns to illustrate
 the data in more detail

 %% Cell type:code id: tags:

 ``` 
 sns.relplot(x='age', y='fare', col='class', hue='sex', data=titanic,
           col_order=('First', 'Second', 'Third'))
 ```

 %% Cell type:markdown id: tags:

 **Exercise**: Split the plot above into two rows with the first row including
 the passengers who survived and the second row those who did not (you might
 have to check the documentation again by using shift-tab while hovering the
 mouse over `relplot`)

 %% Cell type:code id: tags:

 ``` 
 sns.relplot(x='age', y='fare', col='class', hue='sex', data=titanic,
           col_order=('First', 'Second', 'Third')...)
 ```

 %% Cell type:markdown id: tags:

 One of the nice thing of Seaborn is how easy it is to update how these plots
 look. You can read more about that
 [here](https://seaborn.pydata.org/tutorial/aesthetics.html). For example, to
 increase the font size to get a plot more approriate for a talk, you can use:

 %% Cell type:code id: tags:

 ``` 
 sns.set_context('talk')
 sns.violinplot(x='class', y='age', hue='sex', data=titanic, split=True,
               order=('First', 'Second', 'Third'))
 ```

 %% Cell type:markdown id: tags:

 ## Summarizing the data (mean, std, etc.)

 There are a large number of built-in methods to summarize the observations in
 a Pandas `DataFrame`. Most of these will return a `Series` with the columns
 names as index:

 %% Cell type:code id: tags:

 ``` 
 titanic.mean()
 ```

 %% Cell type:code id: tags:

 ``` 
 titanic.quantile(0.75)
 ```

 %% Cell type:markdown id: tags:

 One very useful one is `describe`, which gives an overview of many common
 summary measures

 %% Cell type:code id: tags:

 ``` 
 titanic.describe()
 ```

 %% Cell type:markdown id: tags:

 Note that non-numeric columns are ignored when summarizing data in this way.

 For a more detailed exploration of the data, you might want to check
 [pandas_profiling](https://pandas-profiling.github.io/pandas-profiling/docs/)
 (not installed in fslpython, so the following will not run in fslpython):

 %% Cell type:code id: tags:

 ``` 
 from pandas_profiling import ProfileReport
 profile = ProfileReport(titanic, title='Titanic Report', html={'style':{'full_width':True}})
 profile.to_widgets()
 ```

 %% Cell type:markdown id: tags:

 We can also define our own functions to apply to the columns (in this case we
 have to explicitly select the numeric columns).

 %% Cell type:code id: tags:

 ``` 
 def mad(series):
    """
    Computes the median absolute deviatation (MAD)

    This is a outlier-resistant measure of the standard deviation
    """
    no_nan = series.dropna()
    return np.median(abs(no_nan - np.nanmedian(no_nan)))

 titanic.select_dtypes(np.number).apply(mad)
 ```

 %% Cell type:markdown id: tags:

 We can also provide multiple functions to the `apply` method (note that
 functions can be provided as strings)

 %% Cell type:code id: tags:

 ``` 
 titanic.select_dtypes(np.number).apply(['mean', np.median, np.std, mad])
 ```

 %% Cell type:markdown id: tags:

 ### Grouping by

 One of the more powerful features in pandas is `groupby`, which splits the dataset based
 on a categorical variable. The book contains a clear tutorial on that feature
 [here](https://jakevdp.github.io/PythonDataScienceHandbook/03.08-aggregation-and-grouping.html). You
 can check the pandas documentation
 [here](http://pandas.pydata.org/pandas-docs/stable/groupby.html) for a more
 formal introduction. One simple use is just to put it into a loop

 %% Cell type:code id: tags:

 ``` 
 for cls, part_table in titanic.groupby('class'):
    print(f'Mean fare in {cls.lower()} class: {part_table.fare.mean()}')
 ```

 %% Cell type:markdown id: tags:

 However, it is more often combined with one of the aggregation functions
 discussed above as illustrated in this figure from the [Python data science
 handbook](https://jakevdp.github.io/PythonDataScienceHandbook/06.00-figure-code.html#Split-Apply-Combine)

 ![group by image](group_by.png)

 %% Cell type:code id: tags:

 ``` 
 titanic.groupby('class').mean()
 ```

 %% Cell type:markdown id: tags:

 We can also group by multiple variables at once

 %% Cell type:code id: tags:

 ``` 
 titanic.groupby(['class', 'survived']).mean()  # as always in pandas supply multiple column names as lists, not tuples
 ```

 %% Cell type:markdown id: tags:

 When grouping it can help to use the `cut` method to split a continuous variable
 into a categorical one

 %% Cell type:code id: tags:

 ``` 
 titanic.groupby(['class', pd.cut(titanic.age, bins=(0, 18, 50, np.inf))]).mean()
 ```

 %% Cell type:markdown id: tags:

 We can use the `aggregate` method to apply a different function to each series

 %% Cell type:code id: tags:

 ``` 
-titanic.groupby(['class', 'survived']).aggregate((np.median, mad))
+titanic.groupby(['class', 'survived']).aggregate((np.median, 'mad'))
 ```

 %% Cell type:markdown id: tags:

 Note that both the index (on the left) and the column names (on the top) now
 have multiple levels. Such an index is referred to as `MultiIndex`.
 This does complicate selecting specific columns/rows. You can read more of using
 `MultiIndex` [here](http://pandas.pydata.org/pandas-docs/stable/advanced.html).
 The short version is that columns can be selected using direct indexing (as
 discussed above)

 %% Cell type:code id: tags:

 ``` 
-df_full = titanic.groupby(['class', 'survived']).aggregate((np.median, mad))
+df_full = titanic.groupby(['class', 'survived']).aggregate((np.median, 'mad'))
 ```

 %% Cell type:code id: tags:

 ``` 
 df_full[('age', 'median')]  # selects median age column; note that the round brackets are optional
 ```

 %% Cell type:code id: tags:

 ``` 
 df_full['age']  # selects both age columns
 ```

 %% Cell type:markdown id: tags:

 Remember that indexing based on the index was done through `loc`. The rest is
 the same as for the columns above

 %% Cell type:code id: tags:

 ``` 
 df_full.loc[('First', 0)]
 ```

 %% Cell type:code id: tags:

 ``` 
 df_full.loc['First']
 ```

 %% Cell type:markdown id: tags:

 More advanced use of the `MultiIndex` is possible through `xs`:

 %% Cell type:code id: tags:

 ``` 
 df_full.xs(0, level='survived') # selects all the zero's from the survived index
 ```

 %% Cell type:code id: tags:

 ``` 
 df_full.xs('mad', axis=1, level=1) # selects mad from the second level in the columns (i.e., axis=1)
 ```

 %% Cell type:markdown id: tags:

 ## Reshaping tables

 If we were interested in how the survival rate depends on the class and sex of
 the passengers we could simply use a groupby:

 %% Cell type:code id: tags:

 ``` 
 titanic.groupby(['class', 'sex']).survived.mean()
 ```

 %% Cell type:markdown id: tags:

 However, this single-column table is difficult to read. The reason for this is
 that the indexing is multi-leveled (called `MultiIndex` in pandas), while there
 is only a single column. We would like to move one of the levels in the index to
 the columns. This can be done using `stack`/`unstack`:

 - `unstack`: Moves one level in the index to the columns
 - `stack`: Moves one level in the columns to the index

 %% Cell type:code id: tags:

 ``` 
 titanic.groupby(['class', 'sex']).survived.mean().unstack('sex')
 ```

 %% Cell type:markdown id: tags:

 The former table, where the different groups are defined in different rows, is
 often referred to as long-form. After unstacking the table is often referred to
 as wide-form as the different groups (sex in this case) is now represented as
 different columns. In pandas some operations are easier on long-form tables
 (e.g., `groupby`) while others require wide_form tables (e.g., making scatter
 plots of two variables). You can go back and forth using `unstack` or `stack` as
 illustrated above, but as this is a crucial part of pandas there are many
 alternatives, such as `pivot_table`, `melt`, and `wide_to_long`, which we will
 discuss below.

 We can prettify the table further using seaborn

 %% Cell type:code id: tags:

 ``` 
 ax = sns.heatmap(titanic.groupby(['class', 'sex']).survived.mean().unstack('sex'),
                 annot=True)
 ax.set_title('survival rate')
 ```

 %% Cell type:markdown id: tags:

 > There are also many ways to produce prettier tables in pandas.
 > This is documented [here](http://pandas.pydata.org/pandas-docs/stable/style.html).

 Because this stacking/unstacking is fairly common after a groupby operation,
 there is a shortcut for it: `pivot_table`

 %% Cell type:code id: tags:

 ``` 
 titanic.pivot_table('survived', 'class', 'sex')
 ```

 %% Cell type:markdown id: tags:

 The first argument is the numeric variable that will be summarised.
 The next arguments indicates which categorical variable(s) should be
 used as respectively index or column.

 As usual in pandas we can also provide multiple column names

 %% Cell type:code id: tags:

 ``` 
 sns.heatmap(titanic.pivot_table('survived', ['class', 'embark_town'], ['sex', pd.cut(titanic.age, (0, 18, np.inf))]), annot=True)
 ```

 %% Cell type:markdown id: tags:

 We can also change the function to be used to aggregate the data (by default the mean is computed)

 %% Cell type:code id: tags:

 ``` 
 sns.heatmap(titanic.pivot_table('survived', ['class', 'embark_town'], ['sex', pd.cut(titanic.age, (0, 18, np.inf))],
                                aggfunc='count'), annot=True)
 ```

 %% Cell type:markdown id: tags:

 As in `groupby` the aggregation function can be a string of a common aggregation
 function, or any function that should be applied.

 We can even apply different aggregate functions to different columns

 %% Cell type:code id: tags:

 ``` 
 titanic.pivot_table(index='class', columns='sex',
                    aggfunc={'survived': 'count', 'fare': np.mean}) # compute number of survivors and mean fare
 ```

 %% Cell type:markdown id: tags:

 The opposite of `pivot_table` is `melt`. This can be used to change a wide-form
 table into a long-form table. This is not particularly useful on the titanic
 dataset, so let's create a new table where this might be useful. Let's say we
 have a dataset listing the FA and MD values in various WM tracts:

 %% Cell type:code id: tags:

 ``` 
 tracts = ('Corpus callosum', 'Internal capsule', 'SLF', 'Arcuate fasciculus')
 df_wide = pd.DataFrame.from_dict(dict({'subject': list('ABCDEFGHIJ')}, **{
    f'FA({tract})': np.random.rand(10) for tract in tracts }, **{
    f'MD({tract})': np.random.rand(10) * 1e-3 for tract in tracts
 }))
 df_wide
 ```

 %% Cell type:markdown id: tags:

 This wide-form table (i.e., all the information is in different columns) makes
 it hard to select just all the FA values or only the values associated with the
 SLF. For this it would be easier to list all the values in a single column.
 Most of the tools discussed above (e.g., `group_by` or `seaborn` plotting) work
 better with long-form data, which we can obtain from `melt`:

 %% Cell type:code id: tags:

 ``` 
 df_long = df_wide.melt('subject', var_name='measurement', value_name='dti_value')
 df_long.head(12)
 ```

 %% Cell type:markdown id: tags:

 We can see that `melt` took all the columns (we could also have specified a
 specific sub-set) and returned each measurement as a seperate row. We probably
 want to seperate the measurement column into the measurement type (FA or MD) and
 the tract name. Many string manipulation function are available in the
 `DataFrame` object under `DataFrame.str`
 ([tutorial](http://pandas.pydata.org/pandas-docs/stable/text.html))

 %% Cell type:code id: tags:

 ``` 
 df_long['variable'] = df_long.measurement.str.slice(0, 2)  # first two letters correspond to FA or MD
 df_long['tract'] = df_long.measurement.str.slice(3, -1)  # fourth till the second-to-last letter correspond to the tract
 df_long.head(12)
 ```

 %% Cell type:markdown id: tags:

 Finally we probably do want the FA and MD variables as different columns.

 **Exercise**: Use `pivot_table` or `stack`/`unstack` to create a column for MD
 and FA.

 %% Cell type:code id: tags:

 ``` 
 df_unstacked = df_long.
 ```

 %% Cell type:markdown id: tags:

 We can now use the tools discussed above to visualize the table (`seaborn`) or
 to group the table based on tract (`groupby` or `pivot_table`).

 %% Cell type:code id: tags:

 ``` 
 # feel free to analyze this random data in more detail
 ```

 %% Cell type:markdown id: tags:

 In general pandas is better at handling long-form than wide-form data, because
 you can use `groupby` on long-form data.
 For better visualization of the data an intermediate format is often preferred. One
 exception is calculating a covariance (`DataFrame.cov`) or correlation
 (`DataFrame.corr`) matrices which require the variables that will be compared
 to be columns:

 %% Cell type:code id: tags:

 ``` 
 sns.heatmap(df_wide.corr(), cmap=sns.diverging_palette(240, 10, s=99, n=300), )
 ```

 %% Cell type:markdown id: tags:

 ## Linear fitting (`statsmodels`)

 Linear fitting between the different columns is available through the
 [`statsmodels`](https://www.statsmodels.org/stable/index.html) library. A nice
 way to play around with a wide variety of possible models is to use R-style
 formulas. The usage of the formulas in `statsmodels` is described
 [here](https://www.statsmodels.org/dev/example_formulas.html). You can find a
 more detailed description of the R-style formulas
 [here](https://patsy.readthedocs.io/en/latest/formulas.html#the-formula-language).

 In short these functions describe the linear model as a string. For example,
 `"y ~ x + a + x * a"` fits the variable `y` as a function of `x`, `a`, and the
 interaction between `x` and `a`. The intercept parameter is included by default
 (you can add `"+ 0"` to remove it).

 %% Cell type:code id: tags:

 ``` 
 result = smf.logit('survived ~ age + sex + age * sex', data=titanic).fit()
 print(result.summary())
 ```

 %% Cell type:markdown id: tags:

 Note that `statsmodels` understands categorical variables and automatically
 replaces them with dummy variables.

 Above we used logistic regression, which is appropriate for the binary
 survival rate. A wide variety of linear models are available. Let's try a GLM,
 but assume that the fare is drawn from a Gamma distribution:

 %% Cell type:code id: tags:

 ``` 
 age_dmean = titanic.age - titanic.age.mean()
 result = smf.glm('fare ~ age_dmean + embark_town', data=titanic).fit()
 print(result.summary())
 ```

 %% Cell type:markdown id: tags:

 Cherbourg passengers clearly paid a lot more...


 Note that we did not actually add the "age_dmean" `Series` to the
 `DataFrame`. `statsmodels` (or more precisely the underlying
 [patsy](https://patsy.readthedocs.io/en/latest/) library) automatically
 extracted this from our environment. This can lead to confusing behaviour...

 # More reading
 If you regularly keep running out of memory when handling large tables (e.g., Biobank), you should consider using [dask DataFrames](https://docs.dask.org/en/latest/dataframe.html) instead. They have the same interface as `pandas` `DataFrame`, but will split up your instructions into smaller jobs that can run without the full table being loaded into memory. Many other options for out-of-core computation are listed [here](https://pandas.pydata.org/docs/ecosystem.html#out-of-core).

 Not all data is well represented by a 2D table. If you want more dimensions to for example represent MRI data, have a look at [xarray](https://github.com/pydata/xarray), which has a pandas-like interface for higher-dimensional objects.

 Other useful features:
 - [Concatenating and merging tables](https://pandas.pydata.org/pandas-docs/stable/getting_started/intro_tutorials/08_combine_dataframes.html)
 - [Lots of time series support](https://pandas.pydata.org/pandas-docs/stable/getting_started/intro_tutorials/09_timeseries.html)
 - [Rolling Window
  functions](http://pandas.pydata.org/pandas-docs/stable/computation.html#window-
  functions) for after you have meaningfully sorted your data
 - and much, much more

--- a/applications/pandas/pandas.md
+++ b/applications/pandas/pandas.md
@@ -462,7 +462,7 @@ titanic.groupby(['class', pd.cut(titanic.age, bins=(0, 18, 50, np.inf))]).mean()
 We can use the `aggregate` method to apply a different function to each series

 ```
-titanic.groupby(['class', 'survived']).aggregate((np.median, mad))
+titanic.groupby(['class', 'survived']).aggregate((np.median, 'mad'))
 ```

 Note that both the index (on the left) and the column names (on the top) now
@@ -473,7 +473,7 @@ The short version is that columns can be selected using direct indexing (as
 discussed above)

 ```
-df_full = titanic.groupby(['class', 'survived']).aggregate((np.median, mad))
+df_full = titanic.groupby(['class', 'survived']).aggregate((np.median, 'mad'))
 ```

 ```