In this post I’ll demonstrate how I created annotated heatmaps (or what I’m calling a rank change grid) showing change in rank over time using Python and Matplotlib’s imshow plots. I was writing a report on population trends and internal migration using the IRS county to county migration dataset, and wanted to depict the top origins and destinations of migrants for New York City and the New York Metropolitan Area and how they changed from year to year.

I hit upon this idea based on an example in the Matplotlib documentation using the imshow plot. Imshow was designed for manipulating and creating images, but since images are composed of rows and columns of pixels you can use this function to create grids (for GIS folks, think of a raster). The rows can indicate rank from 1 to N, while the columns could represent time, which in my case is years. I could label each grid cell with the name of a place (i.e. origin or destination), and if a place changes ranks over time I could assign the cell a color indicating increase or decrease; otherwise I’d assign a neutral color indicating no change. The idea is that you could look at place at a given rank in year 1 and follow it across the chart by looking at the label. If a new place appears in a given position, the color change clues you in, and you can quickly scan to see whether a given place went up or down.

The image below shows change in rank for the top metro area destinations for migrants leaving the NYC metro from 2011 to 2018. You can see that metro Miami was the top destination for several years, up until 2016-17 when it flips positions with metro Philadelphia, which had been the number 2 destination. The sudden switch from a neutral color indicates that the place occupying this rank is new. You can also follow how 3rd ranked Bridgeport falls to 4th place in the 2nd year (displaced by Los Angeles), remains in 4th place for a few years, and then falls to 5th place (again bumped by Los Angeles, which falls from 3rd to 4th as it’s bumped by Poughkeepsie).

I opted for this over a more traditional approach called a bump chart (also referred to a slope chart or graph), with time on the x-axis and ranks on the y-axis, and observations labeled at either the first or last point in time. Each observation is assigned a specific color or symbol, and lines connect each observation to its changing position in rank so you can follow it along the chart. Interpreting these charts can be challenging; if there are frequent changes in rank the whole thing begins to look like spaghetti, and the more observations you have the tougher it gets to interpret. Most of the examples I found depicted a small and finite number of observations. I have hundreds of observations and only want to see the top ten, and if observations fall in and out of the top N ranks you get several discontinuous lines which look odd. Lastly, neither Matplotlib or Pandas have a default function for creating bump charts, although I found a few examples where you could create your own.

Creating the rank change grids was a three-part process that required: taking the existing data and transforming it into an array of the top or bottom N values that you want to show, using that array to generate an array that shows change in ranks over time, and generating a plot using both arrays, one for the value and the other for the labels. I’ll tackle each piece in this post. I’ve embedded the functions at the end of each explanation; you can also look at my GitHub repo that has the Jupyter Notebook I used for the analysis for the paper (to be published in Sept 2020).

Create the Initial Arrays

In the paper I was studying flows between NYC and other counties, and the NYC metro area and other metropolitan statisical areas. I’ll refer just to the metro areas as my example in this post, but my functions were written to handle both types of places, stored in separate dataframes. I began with a large dataframe with every metro that exchanged migrants with the NYC metro. There is a row for each metro where the index is the Census Bureau’s unique FIPS code, and columns that show inflows, outflows, and net flows year by year (see image below). There are some rows that represent aggregates, such as flows to all non-metro areas and the sum of individual metro flows that could not be disclosed due to privacy regulations.

The first step is to create an array that has just the top or bottom N places that I want to depict, just for one flow variable (in, out, or net). Why an array? Arrays are pretty solid structures that allow you to select specific rows and columns, and they mesh nicely with imshow charts as each location in the matrix can correspond with the same location in the chart. Most of the examples I looked at used arrays. It’s possible to use other structures but it’s more tedious; nested Python lists don’t have explicit rows and columns so a lot of looping and slicing is required, and with dataframes there always seems to be some catch with data types, messing with the index versus the values, or something else. I went with NumPy’s array type.

I wrote a function where I pass in the dataframe, the type of variable (in, out, or net flow), the number of places I want, whether they are counties or metro areas, and whether I want the top or bottom N records (true or false). Two arrays are returned: the first shows the FIPS unique ID numbers of each place, while the second returns the labels. You don’t have to do anything to calculate actual ranks, because once the data is sorted the ranks become implicit; each row represents ranks 1 through 10, each column represents a year, and the ID or label for a place that occupies each position indicates its rank for that year.

In my dataframe, the names of the columns are prefixed based on the type of variable (inflow, outflow, or net flow), followed by the year, i.e. inflows_2011_12. In the function, I subset the dataframe by selecting columns that start with the variable I want. I have to deal with different issues based on whether I’m looking at counties or metro areas, and I need to get rid of any IDs that are for summary values like the non-metro areas; these IDS are stored in a list called suppressed, and the ~df.indexisin(suppressed) is pandaesque for taking anything that’s not in this list (the tilde acts as not). Then, I select the top or bottom values for each year, and append them to lists in a nested list (each sub-list represents the top / bottom N places in order for a given year). Next, I get the labels I want by creating a dictionary that relates all ID codes to label names, pull out the labels for the actual N values that I have, and format them before appending them to lists in a nested list. For example, the metro labels are really long and won’t fit in the chart, so I split them and grab just the first piece: Albany-Schenectady-Troy, NY becomes Albany (split using the dash) while Akron, OH becomes Akron (if no dash is present, split at comma). At the end, I use np.array to turn the nested lists into arrays, and transpose (T) them so rows become ranks and years become values. The result is below:

# Create array of top N geographies by flow type, with rows as ranks and columns as years
# Returns 2 arrays with values for geographies (id codes) and place names
# Must specify: number of places to rank, counties or metros, or sort by largest or smallest (True or False)
def create_arrays(df,flowtype,nsize,gtype,largest):
    geogs=[]
    cols=[c for c in df if c.startswith(flowtype)]
    for c in cols:
        if gtype=='counties':
            row=df.loc[~df.index.isin(suppressed),[c]]
        elif gtype=='metros':
            row=df.loc[~df.index.isin(msuppressed),[c]]
        if largest is True:
            row=row[c].nlargest(nsize)
        elif largest is False:
            row=row[c].nsmallest(nsize)
        idxs=list(row.index)
        geogs.append(idxs)

    if gtype=='counties':
        fips=df.to_dict()['co_name']
    elif gtype=='metros':
        fips=df.to_dict()['mname']
    labels=[]
    for row in geogs:
        line=[]
        for uid in row:
            if gtype=='counties':
                if fips[uid]=='District of Columbia, DC':
                    line.append('Washington\n DC')
                else:
                    line.append(fips[uid].replace('County, ','\n')) #creates short labels
            elif gtype=='metros':
                if '-' in fips[uid]:
                    line.append(fips[uid].split('-')[0]) #creates short labels
                else:
                    line.append(fips[uid].split(',')[0])
        labels.append(line)

    a_geogs=np.array(geogs).T
    a_labels=np.array(labels).T

    return a_geogs, a_labels

Change in Rank Array

Using the array of geographic ID codes, I can feed this into function number two to create a new array that indicates change in rank over time. It’s better to use the ID code array as we guarantee that the IDs are unique; labels (place names) may not be unique and pose all kinds of formatting issues. All places are assigned a value of 0 for the first year, as there is no previous year to compare them to. Then, for each subsequent year, we look at each value (ID code) and compare it to the value in the same position (rank) in the previous column (year). If the value is the same, that place holds the same rank and is assigned a 0. Otherwise, if it’s different we look at the new value and see what position it was in in the previous year. If it was in a higher position last year, then it has declined and we assign -1. If it was in a lower position last year or was not in the array in that column (i.e. below the top 10 in that year) it has increased and we assign it a value of 1. This result is shown below:

# Create array showing how top N geographies have changed ranks over time, with rows as rank changes and
# columns as years. Returns 1 array with values: 0 (no change), 1 (increased rank), and -1 (descreased rank)
def rank_change(geoarray):
    rowcount=geoarray.shape[0]
    colcount=geoarray.shape[1]

    # Create a number of blank lists
    changelist = [[] for _ in range(rowcount)]

    for i in range(colcount):
        if i==0:
            # Rank change for 1st year is 0, as there is no previous year
            for j in range(rowcount):
                changelist[j].append(0)
        else:
            col=geoarray[:,i] #Get all values in this col
            prevcol=geoarray[:,i-1] #Get all values in previous col
            for v in col:
                array_pos=np.where(col == v) #returns array
                current_pos=int(array_pos[0]) #get first array value
                array_pos2=np.where(prevcol == v) #returns array
                if len(array_pos2[0])==0: #if array is empty, because place was not in previous year
                    previous_pos=current_pos+1
                else:
                    previous_pos=int(array_pos2[0]) #get first array value
                if current_pos==previous_pos:
                    changelist[current_pos].append(0)
                    #No change in rank
                elif current_posprevious_pos: #Larger value = smaller rank
                    changelist[current_pos].append(-1)
                    #Rank has decreased
                else:
                    pass

    rankchange=np.array(changelist)
    return rankchange 

Create the Plot

Now we can create the actual chart! The rank change array is what will actually be charted, but we will use the labels array to display the names of each place. The values that occupy the positions in each array pertain to the same place. The chart function takes the names of both these arrays as input. I do some fiddling around at the beginning to get the labels for the x and y axis the way I want them. Matplotlib allows you to modify every iota of your plot, which is in equal measures flexible and overwhelming. I wanted to make sure that I showed all the tick labels, and changed the default grid lines to make them thicker and lighter. It took a great deal of fiddling to get these details right, but there were plenty of examples to look at (Matplotlib docs, cookbook, Stack Overflow, and this example in particular). For the legend, shrinking the colorbar was a nice option so it’s not ridiculously huge, and I assign -1, 0, and 1 to specific colors denoting decrease, no change, and increase. I loop over the data values to get their corresponding labels, and depending on the color that’s assigned I can modify whether the text is dark or light (so you can see it against the background of the cell). The result is what you saw at the beginning of this post for outflows (top destinations for migrants leaving the NY metro). The function call is below:

# Create grid plot based on an array that shows change in ranks and an array of cell labels
def rank_grid(rank_change,labels):
    alabels=labels
    xlabels=[yr.replace('_','-') for yr in years]
    ranklabels=['1st','2nd','3rd','4th','5th','6th','7th','8th','9th','10th',
               '11th','12th','13th','14th','15th','16th','17th','18th','19th','20th']
    nsize=rank_change.shape[0]
    ylabels=ranklabels[:nsize]

    mycolors = colors.ListedColormap(['#de425b','#f7f7f7','#67a9cf'])
    fig, ax = plt.subplots(figsize=(10,10))
    im = ax.imshow(rank_change, cmap=mycolors)

    # Show all ticks...
    ax.set_xticks(np.arange(len(xlabels)))
    ax.set_yticks(np.arange(len(ylabels)))
    # ... and label them with the respective list entries
    ax.set_xticklabels(xlabels)
    ax.set_yticklabels(ylabels)

    # Create white grid.
    ax.set_xticks(np.arange(rank_change.shape[1]+1)-.5, minor=True)
    ax.set_yticks(np.arange(rank_change.shape[0]+1)-.5, minor=True)
    ax.grid(which="minor", color="w", linestyle='-', linewidth=3)
    ax.grid(which="major",visible=False)

    cbar = ax.figure.colorbar(im, ax=ax, ticks=[1,0,-1], shrink=0.5)
    cbar.ax.set_yticklabels(['Increased','No Change','Decreased'])

    # Loop over data dimensions and create text annotations.
    for i in range(len(ylabels)):
        for j in range(len(xlabels)):
            if rank_change[i,j] < 0:
                text = ax.text(j, i, alabels[i, j],
                           ha="center", va="center", color="w", fontsize=10)
            else:
                text = ax.text(j, i, alabels[i, j],
                           ha="center", va="center", color="k", fontsize=10)

    #ax.set_title("Change in Rank Over Time")
    plt.xticks(fontsize=12)
    plt.yticks(fontsize=12)
    fig.tight_layout()
    plt.show()
    return ax 

Conclusions and Alternatives

I found that this approach worked well for my particular circumstances, where I had a limited number of data points to show and the ranks didn’t fluctuate much from year to year. The charts for ten observations displayed over seven points in time fit easily onto standard letter-sized paper; I could even get away with adding two additional observations and an eighth point in time if I modified the size and placement of the legend. However, beyond that you can begin to run into trouble. I generated charts for the top twenty places so I could see the results for my own analysis, but it was much too large to create a publishable graphic (at least in print). If you decrease the dimensions for the chart or reduce the size of the grid cells, the labels start to become unreadable (print that’s too small or overlapping labels).

There are a number of possibilities for circumventing this. One would be to use shorter labels; if we were working with states or provinces we can use the two-letter postal codes, or ISO country codes in the case of countries. Not an option in my example. Alternatively, we could move the place names to the y-axis (sorted alphabetically or by first or final year rank) and then use the rank as the annotation label. This would be a fundamentally different chart; you could see how one place changes in rank over time, but it would be tougher to discern which places were the most important source / destination for the area you’re studying (you’d have to skim through the whole chart). Or, you could keep ranks on the y-axis and assign each place a unique color in the legend, shade the grid cells using that color, and thus follow the changing colors with your eye. But this flops is you have too many places / colors.

A different caveat is this approach doesn’t work so well if there is a lot of fluctuation in ranks from year to year. In this example, the top inflows and outflows were relatively stable from year to year. There were enough places that held the same rank that you could follow the places that changed positions. We saw the example above for outflows, below is an example for inflows (i.e. the top origins or sources of migrants moving to the NY metro):

In contrast, the ranks for net flows were highly variable. There was so much change that the chart appears as a solid block of colors with few neutral (unchanged) values, making it difficult to see what’s going on. An example of this is below, representing net flows for the NYC metro area. This is the difference between inflows and outflows, and the chart represents metros that receive more migrants from New York than they send (i.e. net receivers of NY migrants). While I didn’t use the net flow charts in my paper, it was still worth generating as it made it clear to me that net flow ranks fluctuate quite a bit, which was a fact I could state in the text.

There are also a few alternatives to using imshow. Matplotlib’s pcolor plot can produce similar effects but with rectangles instead of square grid cells. That could allow for more observations and longer labels. I thought it was less visually pleasing than the equal grid, and early on I found that implementing it was clunkier so I went no further. My other idea was to create a table instead of a chart. Pandas has functions for formatting dataframes in a Jupyter Notebook, and there are options for exporting the results out to HTML. Formatting is the downside – if you create a plot as an image, you export it out and can then embed it into any document format you like. When you’re exporting tables out of a notebook, you’re only exporting the content and not the format. With a table, the content and formatting is separate, and the latter is often tightly bound to the publication format (Word, LaTeX, HTML, etc.) You can design with this in mind if you’re self-publishing a blog post or report, but this is not feasible when you’re submitting something for publication where an editor or designer will be doing the layout.

I really wanted to produce something that I could code and run automatically in many different iterations, and was happy with this solution. It was an interesting experiment, as I grappled with taking something that seemed intuitive to do the old-fashioned way (see below) and reproducing it in a digital, repeatable format.