Merge branch 'pandas-update' into 'master'

Updates to the pandas practical/talk See merge request fsl/win-pytreat!25

Merge branch 'pandas-update' into 'master'
74bd46ee · Michiel Cottaar · 8f23ab03 · 5fc3b2ce · 74bd46ee · 74bd46ee
Commit 74bd46ee authored 3 years ago by Michiel Cottaar
--- a/applications/pandas/pandas.ipynb
+++ b/applications/pandas/pandas.ipynb
@@ -2,6 +2,7 @@
 "cells": [
  {
   "cell_type": "markdown",
+   "id": "9803940b",
   "metadata": {},
   "source": [
    "# Pandas\n",
@@ -23,6 +24,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "eb7f5417",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -38,6 +40,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "9f998231",
   "metadata": {},
   "source": [
    "> We will mostly be using `seaborn` instead of `matplotlib` for\n",
@@ -54,6 +57,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "7020257e",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -63,6 +67,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "6a17483f",
   "metadata": {},
   "source": [
    "This loads the data into a\n",
@@ -78,6 +83,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "4d0436b1",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -86,6 +92,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "173c767f",
   "metadata": {},
   "source": [
    "If you can not connect to the internet, you can run the command below to load\n",
@@ -95,6 +102,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "9a047455",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -104,6 +112,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "6b801a28",
   "metadata": {},
   "source": [
    "Note that the titanic dataset was also available to us as one of the standard\n",
@@ -113,6 +122,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "7be7954c",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -121,6 +131,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "112b9665",
   "metadata": {},
   "source": [
    "`Dataframes` can also be created from other python objects, using\n",
@@ -131,6 +142,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "de3236a1",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -145,18 +157,50 @@
  },
  {
   "cell_type": "markdown",
+   "id": "0761c3c8",
   "metadata": {},
   "source": [
+    "## A note on types\n",
+    "Each column in the pandas dataframe has its own data type, which can be:\n",
+    "- integer or float for numbers\n",
+    "- boolean for True/False\n",
+    "- datetime for defining specific times (and timedelta for durations)\n",
+    "- categorical, where each element is selected from a finite list of text values\n",
+    "- objects for anything else used for strings or columns with mixed elements\n",
+    "Each element in the column must match the type of the whole column. \n",
+    "When reading in a dataset pandas will try to assign the most specific type to each column.\n",
+    "Every pandas datatype also has support for missing data (which we will look more at below).\n",
+    "\n",
+    "One can check the type of each column using:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "398a2240",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "titanic.dtypes"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "839cfc99",
+   "metadata": {},
+   "source": [
+    "Note that in much of python data types are referred to as dtypes.\n",
    "## Getting your data out\n",
-    "For many applications (e.g., ICA, machine learning) you might want to\n",
+    "For some applications you might want to\n",
    "extract your data as a numpy array, even though more and more projects \n",
-    "support pandas Dataframes directly. The underlying numpy array can be \n",
-    "accessed using the `to_numpy` method"
+    "support pandas Dataframes directly (including \"scikit-learn\"). \n",
+    "The underlying numpy array can be accessed using the `to_numpy` method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "987ee604",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -165,16 +209,25 @@
  },
  {
   "cell_type": "markdown",
+   "id": "1d210770",
   "metadata": {},
   "source": [
-    "Note that the type of the returned array is the most common type (in this case\n",
-    "object). If you just want the numeric parts of the table you can use\n",
-    "`select_dtypes`, which selects specific columns based on their dtype:"
+    "Similarly to the `pandas` types discussed above,\n",
+    "`numpy` also requires all elements to have the same type.\n",
+    "However, `numpy` requires all elements in the whole array,\n",
+    "not just a single column to be the same type.\n",
+    "In this case this means that all data had to be converted\n",
+    "to the generic \"object\" type, which is not particularly useful.\n",
+    "\n",
+    "For most analyses, we would only be interested in the numeric columns.\n",
+    "Thise can be extracted using `select_dtypes`, which selects specific columns \n",
+    "based on their data type (dtype):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "7836cb90",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -183,16 +236,26 @@
  },
  {
   "cell_type": "markdown",
+   "id": "85bbba82",
   "metadata": {},
   "source": [
-    "Note that the numpy array has no information on the column names or row indices.\n",
-    "Alternatively, when you want to include the categorical variables in your later\n",
-    "analysis (e.g., for machine learning), you can extract dummy variables using:"
+    "Now we get an array with a numeric type rather than the generic \"object\",\n",
+    "which is a lot more useful as we can now run math operations on the\n",
+    "resulting array (e.g., PCA).\n",
+    "\n",
+    "Finally, let's have a look at extracting categorical variables.\n",
+    "These are columns where each element has one of a finite list of possible values\n",
+    "(e.g., the \"embark_town\" column being \"Southampton\", \"Cherbourg\", or, \"Queenstown,\n",
+    "which are the three towns the Titanic docked to let on passengers).\n",
+    "As we will see below, `pandas` has extensive support for categorical values,\n",
+    "but many other tools do not. To support those tools, `pandas` allows you to \n",
+    "replace such columns with dummy variables:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "60c7e8fc",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -201,8 +264,14 @@
  },
  {
   "cell_type": "markdown",
+   "id": "1defde14",
   "metadata": {},
   "source": [
+    "Note that rather than having a single \"embark_town\" column with a categorical type,\n",
+    "we now have three columns named \"embark_town_<name>\" with a 1 for every passenger\n",
+    "who embarked in that town. These numeric columns can then be fed into a GLM or\n",
+    "a machine learning algorithm.\n",
+    "\n",
    "## Accessing parts of the data\n",
    "\n",
    "[Documentation on indexing](http://pandas.pydata.org/pandas-docs/stable/indexing.html)\n",
@@ -215,6 +284,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "fa00ea38",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -223,6 +293,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "2bb923fa",
   "metadata": {},
   "source": [
    "If the column names is a valid python identifier (i.e., is a string that does not contain stuff like spaces)\n",
@@ -232,6 +303,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "acf0cfc6",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -240,6 +312,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "1bfb79b1",
   "metadata": {},
   "source": [
    "Note that this returns a single column is represented by a pandas\n",
@@ -253,6 +326,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "7c097dc4",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -261,6 +335,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "027848a7",
   "metadata": {},
   "source": [
    "Note that you have to provide a list here (square brackets). If you provide a\n",
@@ -271,6 +346,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "5be86a4a",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -279,6 +355,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "bbf62da6",
   "metadata": {},
   "source": [
    "In this case there is no column called `('class', 'alive')` leading to an\n",
@@ -290,6 +367,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "8fc35ce3",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -299,6 +377,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "766e1a41",
   "metadata": {},
   "source": [
    "We can delete a column using:"
@@ -307,6 +386,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "61d91bdf",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -315,6 +395,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "9ea81208",
   "metadata": {},
   "source": [
    "### Indexing rows by name or integer\n",
@@ -328,6 +409,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "e6263074",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -337,6 +419,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "0a6d1311",
   "metadata": {},
   "source": [
    "Note that the re-sorting did not change the values in the index (i.e., left-most\n",
@@ -348,6 +431,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "1016cb0b",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -356,6 +440,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "00bfb183",
   "metadata": {},
   "source": [
    "We can select the row with the index 0 using"
@@ -364,6 +449,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "63cb04fc",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -372,6 +458,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "b0bf4c58",
   "metadata": {},
   "source": [
    "Note that this gives the same passenger as the first row of the initial table\n",
@@ -381,6 +468,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "ece87592",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -389,6 +477,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "738c3b6c",
   "metadata": {},
   "source": [
    "Another common way to access the first or last N rows of a table is using the\n",
@@ -398,6 +487,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "8937c751",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -407,6 +497,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "0333e9cc",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -415,6 +506,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "e846457f",
   "metadata": {},
   "source": [
    "Note that nearly all methods in pandas return a new `DataFrame`, which means\n",
@@ -427,6 +519,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "84cc7089",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -435,6 +528,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "0611a8bf",
   "metadata": {},
   "source": [
    "> This chaining is usually very efficient, because when creating a new `DataFrame`\n",
@@ -446,6 +540,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "dbd982e7",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -454,6 +549,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "71a1e3a3",
   "metadata": {},
   "source": [
    "**Exercise**: use sorting and tail/head or indexing to find the 10 youngest\n",
@@ -464,6 +560,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "7272cae5",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -472,6 +569,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "c836a51b",
   "metadata": {},
   "source": [
    "### Indexing rows by value\n",
@@ -482,6 +580,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "7dbabecf",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -491,6 +590,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "e9503367",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -500,6 +600,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "edefb548",
   "metadata": {},
   "source": [
    "Note that this required typing `titanic` quite often.\n",
@@ -515,6 +616,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "0c710cfa",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -523,6 +625,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "d8545d1a",
   "metadata": {},
   "source": [
    "When selecting a categorical multiple options from a categorical values you \n",
@@ -532,6 +635,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "93a73be5",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -540,6 +644,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "129a27a3",
   "metadata": {},
   "source": [
    "Particularly useful when selecting data like this is the `isna` method which\n",
@@ -549,6 +654,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "66af0870",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -557,6 +663,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "ce7d05c3",
   "metadata": {},
   "source": [
    "This removing of missing numbers is so common that it has is own method"
@@ -565,6 +672,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "79d28611",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -574,6 +682,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "34ebdb36",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -582,6 +691,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "82ddcb59",
   "metadata": {},
   "source": [
    "**Exercise**: use sorting, indexing by value, `dropna` and `tail`/`head` or\n",
@@ -592,6 +702,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "1fe9d398",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -600,6 +711,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "c394e5ac",
   "metadata": {},
   "source": [
    "## Plotting the data\n",
@@ -611,6 +723,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "1d443d44",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -620,6 +733,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "8bd6d770",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -628,6 +742,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "dc8e64a9",
   "metadata": {},
   "source": [
    "To plot all variables simply call `plot` or `hist` on the full `DataFrame`\n",
@@ -638,6 +753,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "ab6c3514",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -646,6 +762,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "8140217e",
   "metadata": {},
   "source": [
    "Individual `Series` are essentially 1D arrays, so we can use them as such in\n",
@@ -655,6 +772,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "48a59b56",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -663,6 +781,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "07cf3584",
   "metadata": {},
   "source": [
    "However, for most purposes much nicer plots can be obtained using\n",
@@ -679,6 +798,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "8563a7a1",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -687,6 +807,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "0e752be0",
   "metadata": {},
   "source": [
    "**Exercise**: check the documentation from `sns.jointplot` (hover the mouse\n",
@@ -697,6 +818,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "4ccd4177",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -705,6 +827,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "4e513fbc",
   "metadata": {},
   "source": [
    "Here is just a brief example of how we can use multiple columns to illustrate\n",
@@ -714,6 +837,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "f5cb00af",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -723,6 +847,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "6ec94eac",
   "metadata": {},
   "source": [
    "**Exercise**: Split the plot above into two rows with the first row including\n",
@@ -734,6 +859,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "c6f6e763",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -743,6 +869,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "1d54e3ec",
   "metadata": {},
   "source": [
    "One of the nice thing of Seaborn is how easy it is to update how these plots\n",
@@ -754,6 +881,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "52183332",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -764,6 +892,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "ac35e133",
   "metadata": {},
   "source": [
    "## Summarizing the data (mean, std, etc.)\n",
@@ -776,6 +905,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "404be564",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -785,6 +915,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "bd6dd429",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -793,6 +924,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "3f0eaeb2",
   "metadata": {},
   "source": [
    "One very useful one is `describe`, which gives an overview of many common\n",
@@ -802,6 +934,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "e52493af",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -810,6 +943,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "7fd8fba3",
   "metadata": {},
   "source": [
    "Note that non-numeric columns are ignored when summarizing data in this way.\n",
@@ -822,6 +956,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "3fffbcb9",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -832,6 +967,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "d4c09639",
   "metadata": {},
   "source": [
    "We can also define our own functions to apply to the columns (in this case we\n",
@@ -841,6 +977,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "e1b90c3f",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -858,6 +995,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "f869c17f",
   "metadata": {},
   "source": [
    "We can also provide multiple functions to the `apply` method (note that\n",
@@ -867,6 +1005,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "2dd3d814",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -875,6 +1014,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "78e7e950",
   "metadata": {},
   "source": [
    "### Grouping by\n",
@@ -890,6 +1030,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "c271697e",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -899,6 +1040,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "5537b1e4",
   "metadata": {},
   "source": [
    "However, it is more often combined with one of the aggregation functions\n",
@@ -911,6 +1053,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "580a68d4",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -919,6 +1062,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "9a94b1c0",
   "metadata": {},
   "source": [
    "We can also group by multiple variables at once"
@@ -927,6 +1071,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "4d119923",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -935,6 +1080,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "9c5c1119",
   "metadata": {},
   "source": [
    "When grouping it can help to use the `cut` method to split a continuous variable\n",
@@ -944,6 +1090,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "e18ac0a4",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -952,6 +1099,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "0c3e2145",
   "metadata": {},
   "source": [
    "We can use the `aggregate` method to apply a different function to each series"
@@ -960,6 +1108,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "cf6abd30",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -968,6 +1117,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "eaca0a93",
   "metadata": {},
   "source": [
    "Note that both the index (on the left) and the column names (on the top) now\n",
@@ -981,6 +1131,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "79780e3b",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -990,6 +1141,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "fb15d602",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -999,6 +1151,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "e7ba4b48",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1007,6 +1160,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "5414e4c5",
   "metadata": {},
   "source": [
    "Remember that indexing based on the index was done through `loc`. The rest is\n",
@@ -1016,6 +1170,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "ed55f8ee",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1025,6 +1180,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "1376c35c",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1033,6 +1189,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "1289b2db",
   "metadata": {},
   "source": [
    "More advanced use of the `MultiIndex` is possible through `xs`:"
@@ -1041,6 +1198,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "472127b8",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1050,6 +1208,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "61e73d0b",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1058,6 +1217,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "7fc120ae",
   "metadata": {},
   "source": [
    "## Reshaping tables\n",
@@ -1069,6 +1229,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "6f3d1ccd",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1077,6 +1238,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "b9a13425",
   "metadata": {},
   "source": [
    "However, this single-column table is difficult to read. The reason for this is\n",
@@ -1091,6 +1253,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "c5cf521a",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1099,6 +1262,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "55d2c5a4",
   "metadata": {},
   "source": [
    "The former table, where the different groups are defined in different rows, is\n",
@@ -1117,6 +1281,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "69301d4e",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1127,6 +1292,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "d81eb236",
   "metadata": {},
   "source": [
    "> There are also many ways to produce prettier tables in pandas. \n",
@@ -1139,6 +1305,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "2ddece59",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1147,6 +1314,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "e730a134",
   "metadata": {},
   "source": [
    "The first argument is the numeric variable that will be summarised. \n",
@@ -1159,6 +1327,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "641a14bf",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1167,6 +1336,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "ee37ad6b",
   "metadata": {},
   "source": [
    "We can also change the function to be used to aggregate the data (by default the mean is computed)"
@@ -1175,6 +1345,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "a2ebc6c2",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1184,6 +1355,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "b43f10b8",
   "metadata": {},
   "source": [
    "As in `groupby` the aggregation function can be a string of a common aggregation\n",
@@ -1195,6 +1367,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "dc66e7c0",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1204,6 +1377,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "25a8f2f1",
   "metadata": {},
   "source": [
    "The opposite of `pivot_table` is `melt`. This can be used to change a wide-form\n",
@@ -1215,6 +1389,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "2d509083",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1228,6 +1403,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "59b81e11",
   "metadata": {},
   "source": [
    "This wide-form table (i.e., all the information is in different columns) makes\n",
@@ -1240,6 +1416,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "70e01ab3",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1249,6 +1426,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "4c942e3b",
   "metadata": {},
   "source": [
    "We can see that `melt` took all the columns (we could also have specified a\n",
@@ -1262,6 +1440,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "ee385ba6",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1272,6 +1451,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "6145528a",
   "metadata": {},
   "source": [
    "Finally we probably do want the FA and MD variables as different columns.\n",
@@ -1283,6 +1463,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "2fb60939",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1291,6 +1472,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "2932f74b",
   "metadata": {},
   "source": [
    "We can now use the tools discussed above to visualize the table (`seaborn`) or\n",
@@ -1300,6 +1482,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "621dfde3",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1308,6 +1491,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "12aa9cb8",
   "metadata": {},
   "source": [
    "In general pandas is better at handling long-form than wide-form data, because\n",
@@ -1321,6 +1505,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "acd7334f",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1329,6 +1514,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "146c04ca",
   "metadata": {},
   "source": [
    "## Linear fitting (`statsmodels`)\n",
@@ -1350,6 +1536,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "d9b6fae7",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1359,6 +1546,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "ef53b1cd",
   "metadata": {},
   "source": [
    "Note that `statsmodels` understands categorical variables and automatically\n",
@@ -1372,6 +1560,7 @@
  {
   "cell_type": "code",
   "execution_count": null,
+   "id": "aa96dedf",
   "metadata": {},
   "outputs": [],
   "source": [
@@ -1382,6 +1571,7 @@
  },
  {
   "cell_type": "markdown",
+   "id": "74d2b523",
   "metadata": {},
   "source": [
    "Cherbourg passengers clearly paid a lot more...\n",
@@ -1400,14 +1590,13 @@
    "Other useful features:\n",
    "- [Concatenating and merging tables](https://pandas.pydata.org/pandas-docs/stable/getting_started/intro_tutorials/08_combine_dataframes.html)\n",
    "- [Lots of time series support](https://pandas.pydata.org/pandas-docs/stable/getting_started/intro_tutorials/09_timeseries.html)\n",
-    "- [Rolling Window\n",
-    "  functions](http://pandas.pydata.org/pandas-docs/stable/computation.html#window-\n",
-    "  functions) for after you have meaningfully sorted your data\n",
+    "- [Rolling Window functions](http://pandas.pydata.org/pandas-docs/stable/computation.html#window-functions) \n",
+    "  for after you have meaningfully sorted your data\n",
    "- and much, much more"
   ]
  }
 ],
 "metadata": {},
 "nbformat": 4,
- "nbformat_minor": 4
+ "nbformat_minor": 5
 }
-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:9803940b 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:
+%% Cell type:code id:eb7f5417 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:
+%% Cell type:markdown id:9f998231 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:
+%% Cell type:code id:7020257e tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:6a17483f 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:
+%% Cell type:code id:4d0436b1 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:
+%% Cell type:markdown id:173c767f 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:
+%% Cell type:code id:9a047455 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:6b801a28 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:
+%% Cell type:code id:7be7954c tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:112b9665 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:
+%% Cell type:code id:de3236a1 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:
+%% Cell type:markdown id:0761c3c8 tags:

+## A note on types
+Each column in the pandas dataframe has its own data type, which can be:
+- integer or float for numbers
+- boolean for True/False
+- datetime for defining specific times (and timedelta for durations)
+- categorical, where each element is selected from a finite list of text values
+- objects for anything else used for strings or columns with mixed elements
+Each element in the column must match the type of the whole column.
+When reading in a dataset pandas will try to assign the most specific type to each column.
+Every pandas datatype also has support for missing data (which we will look more at below).
+
+One can check the type of each column using:
+
+%% Cell type:code id:398a2240 tags:
+
+``` 
+titanic.dtypes
+```
+
+%% Cell type:markdown id:839cfc99 tags:
+
+Note that in much of python data types are referred to as dtypes.
 ## Getting your data out
-For many applications (e.g., ICA, machine learning) you might want to
+For some applications 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
+support pandas Dataframes directly (including "scikit-learn").
+The underlying numpy array can be accessed using the `to_numpy` method

-%% Cell type:code id: tags:
+%% Cell type:code id:987ee604 tags:

 ``` 
 titanic.to_numpy()
 ```

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:1d210770 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:
+Similarly to the `pandas` types discussed above,
+`numpy` also requires all elements to have the same type.
+However, `numpy` requires all elements in the whole array,
+not just a single column to be the same type.
+In this case this means that all data had to be converted
+to the generic "object" type, which is not particularly useful.
+
+For most analyses, we would only be interested in the numeric columns.
+Thise can be extracted using `select_dtypes`, which selects specific columns
+based on their data type (dtype):

-%% Cell type:code id: tags:
+%% Cell type:code id:7836cb90 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:85bbba82 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:
+Now we get an array with a numeric type rather than the generic "object",
+which is a lot more useful as we can now run math operations on the
+resulting array (e.g., PCA).
+
+Finally, let's have a look at extracting categorical variables.
+These are columns where each element has one of a finite list of possible values
+(e.g., the "embark_town" column being "Southampton", "Cherbourg", or, "Queenstown,
+which are the three towns the Titanic docked to let on passengers).
+As we will see below, `pandas` has extensive support for categorical values,
+but many other tools do not. To support those tools, `pandas` allows you to
+replace such columns with dummy variables:

-%% Cell type:code id: tags:
+%% Cell type:code id:60c7e8fc tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:1defde14 tags:
+
+Note that rather than having a single "embark_town" column with a categorical type,
+we now have three columns named "embark_town_<name>" with a 1 for every passenger
+who embarked in that town. These numeric columns can then be fed into a GLM or
+a machine learning algorithm.

 ## 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:
+%% Cell type:code id:fa00ea38 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:2bb923fa 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:
+%% Cell type:code id:acf0cfc6 tags:

 ``` 
 titanic.embark_town
 ```

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:1bfb79b1 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:
+%% Cell type:code id:7c097dc4 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:027848a7 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:
+%% Cell type:code id:5be86a4a tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:bbf62da6 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:
+%% Cell type:code id:8fc35ce3 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:766e1a41 tags:

 We can delete a column using:

-%% Cell type:code id: tags:
+%% Cell type:code id:61d91bdf tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:9ea81208 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:
+%% Cell type:code id:e6263074 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:0a6d1311 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:
+%% Cell type:code id:1016cb0b tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:00bfb183 tags:

 We can select the row with the index 0 using

-%% Cell type:code id: tags:
+%% Cell type:code id:63cb04fc tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:b0bf4c58 tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:ece87592 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:738c3b6c 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:
+%% Cell type:code id:8937c751 tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:0333e9cc tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:e846457f 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:
+%% Cell type:code id:84cc7089 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:0611a8bf 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:
+%% Cell type:code id:dbd982e7 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:71a1e3a3 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:
+%% Cell type:code id:7272cae5 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:c836a51b tags:

 ### Indexing rows by value

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

-%% Cell type:code id: tags:
+%% Cell type:code id:7dbabecf tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:e9503367 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:edefb548 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:
+%% Cell type:code id:0c710cfa tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:d8545d1a tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:93a73be5 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:129a27a3 tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:66af0870 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:ce7d05c3 tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:79d28611 tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:34ebdb36 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:82ddcb59 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:
+%% Cell type:code id:1fe9d398 tags:

 ``` 
 titanic...
 ```

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:c394e5ac 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:
+%% Cell type:code id:1d443d44 tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:8bd6d770 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:dc8e64a9 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:
+%% Cell type:code id:ab6c3514 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:8140217e tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:48a59b56 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:07cf3584 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:
+%% Cell type:code id:8563a7a1 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:0e752be0 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:
+%% Cell type:code id:4ccd4177 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:4e513fbc 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:
+%% Cell type:code id:f5cb00af tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:6ec94eac 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:
+%% Cell type:code id:c6f6e763 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:1d54e3ec 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:
+%% Cell type:code id:52183332 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:
+%% Cell type:markdown id:ac35e133 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:
+%% Cell type:code id:404be564 tags:

 ``` 
 titanic.mean()
 ```

-%% Cell type:code id: tags:
+%% Cell type:code id:bd6dd429 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:3f0eaeb2 tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:e52493af tags:

 ``` 
 titanic.describe()
 ```

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:7fd8fba3 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:
+%% Cell type:code id:3fffbcb9 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:
+%% Cell type:markdown id:d4c09639 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:
+%% Cell type:code id:e1b90c3f 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:
+%% Cell type:markdown id:f869c17f tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:2dd3d814 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:78e7e950 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:
+%% Cell type:code id:c271697e 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:
+%% Cell type:markdown id:5537b1e4 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:
+%% Cell type:code id:580a68d4 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:9a94b1c0 tags:

 We can also group by multiple variables at once

-%% Cell type:code id: tags:
+%% Cell type:code id:4d119923 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:9c5c1119 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:
+%% Cell type:code id:e18ac0a4 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:0c3e2145 tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:cf6abd30 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:eaca0a93 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:
+%% Cell type:code id:79780e3b tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:fb15d602 tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:e7ba4b48 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:5414e4c5 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:
+%% Cell type:code id:ed55f8ee tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:1376c35c tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:1289b2db tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:472127b8 tags:

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

-%% Cell type:code id: tags:
+%% Cell type:code id:61e73d0b 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:
+%% Cell type:markdown id:7fc120ae 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:
+%% Cell type:code id:6f3d1ccd tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:b9a13425 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:
+%% Cell type:code id:c5cf521a tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:55d2c5a4 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:
+%% Cell type:code id:69301d4e tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:d81eb236 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:
+%% Cell type:code id:2ddece59 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:e730a134 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:
+%% Cell type:code id:641a14bf 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:
+%% Cell type:markdown id:ee37ad6b 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:
+%% Cell type:code id:a2ebc6c2 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:
+%% Cell type:markdown id:b43f10b8 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:
+%% Cell type:code id:dc66e7c0 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:
+%% Cell type:markdown id:25a8f2f1 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:
+%% Cell type:code id:2d509083 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:
+%% Cell type:markdown id:59b81e11 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:
+%% Cell type:code id:70e01ab3 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:4c942e3b 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:
+%% Cell type:code id:ee385ba6 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:
+%% Cell type:markdown id:6145528a 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:
+%% Cell type:code id:2fb60939 tags:

 ``` 
 df_unstacked = df_long.
 ```

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:2932f74b 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:
+%% Cell type:code id:621dfde3 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:12aa9cb8 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:
+%% Cell type:code id:acd7334f tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:146c04ca 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:
+%% Cell type:code id:d9b6fae7 tags:

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

-%% Cell type:markdown id: tags:
+%% Cell type:markdown id:ef53b1cd 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:
+%% Cell type:code id:aa96dedf 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:
+%% Cell type:markdown id:74d2b523 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
+- [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
@@ -80,31 +80,63 @@ pd.DataFrame.from_dict({
    'constant_value': 'same_value'
 })
 ```
+
+## A note on types
+Each column in the pandas dataframe has its own data type, which can be:
+- integer or float for numbers
+- boolean for True/False
+- datetime for defining specific times (and timedelta for durations)
+- categorical, where each element is selected from a finite list of text values
+- objects for anything else used for strings or columns with mixed elements
+Each element in the column must match the type of the whole column. 
+When reading in a dataset pandas will try to assign the most specific type to each column.
+Every pandas datatype also has support for missing data (which we will look more at below).
+
+One can check the type of each column using:
+```
+titanic.dtypes
+```
+Note that in much of python data types are referred to as dtypes.
 ## Getting your data out
-For many applications (e.g., ICA, machine learning) you might want to
+For some applications 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
-
+support pandas Dataframes directly (including "scikit-learn"). 
+The underlying numpy array can be accessed using the `to_numpy` method
 ```
 titanic.to_numpy()
 ```

-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:
+Similarly to the `pandas` types discussed above,
+`numpy` also requires all elements to have the same type.
+However, `numpy` requires all elements in the whole array,
+not just a single column to be the same type.
+In this case this means that all data had to be converted
+to the generic "object" type, which is not particularly useful.

+For most analyses, we would only be interested in the numeric columns.
+Thise can be extracted using `select_dtypes`, which selects specific columns 
+based on their data type (dtype):
 ```
 titanic.select_dtypes(include=np.number).to_numpy()
 ```
+Now we get an array with a numeric type rather than the generic "object",
+which is a lot more useful as we can now run math operations on the
+resulting array (e.g., PCA).

-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:
-
+Finally, let's have a look at extracting categorical variables.
+These are columns where each element has one of a finite list of possible values
+(e.g., the "embark_town" column being "Southampton", "Cherbourg", or, "Queenstown,
+which are the three towns the Titanic docked to let on passengers).
+As we will see below, `pandas` has extensive support for categorical values,
+but many other tools do not. To support those tools, `pandas` allows you to 
+replace such columns with dummy variables:
 ```
 pd.get_dummies(titanic)
 ```
+Note that rather than having a single "embark_town" column with a categorical type,
+we now have three columns named "embark_town_<name>" with a 1 for every passenger
+who embarked in that town. These numeric columns can then be fed into a GLM or
+a machine learning algorithm.

 ## Accessing parts of the data

@@ -695,7 +727,6 @@ Not all data is well represented by a 2D table. If you want more dimensions to f
 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
+- [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
\ No newline at end of file