Eight to Late

Sensemaking and Analytics for Organizations

A gentle introduction to data visualisation using R

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Data science students often focus on machine learning algorithms, overlooking some of the more routine but important skills of the profession.  I’ve lost count of the number of times I have advised students working on projects for industry clients to curb their keenness to code and work on understanding the data first.  This is important because, as people (ought to) know, data doesn’t speak for itself, it has to be given a voice; and as data-scarred professionals know from hard-earned experience, one of the best ways to do this is through visualisation.

Data visualisation is sometimes (often?) approached as a bag of tricks to be learnt individually, with no little or no reference to any underlying principles. Reading Hadley Wickham’s paper on the grammar of graphics was an epiphany; it showed me how different types of graphics can be constructed in a consistent way using common elements. Among other things, the grammar makes visualisation a logical affair rather than a set of tricks. This post is a brief – and hopefully logical – introduction to visualisation using ggplot2, Wickham’s implementation of a grammar of graphics.

In keeping with the practical bent of this series we’ll  focus on worked examples, illustrating elements of the grammar as we go along. We’ll first briefly describe the elements of the grammar and then show how these are used to build different types of visualisations.

A grammar of graphics

Most visualisations are constructed from common elements that are pieced together in prescribed ways.  The elements can be grouped into the following categories:

  • Data – this is obvious, without data there is no story to tell and definitely no plot!
  • Mappings – these are correspondences between data and display elements such as spatial location, shape or colour. Mappings are referred to as aesthetics in Wickham’s grammar.
  • Scales – these are transformations (conversions) of data values to numbers that can be displayed on-screen. There should be one scale per mapping. ggplot typically does the scaling transparently, without users having to worry about it. One situation in which you might need to mess with default scales is when you want to zoom in on a particular range of values. We’ll see an example or two of this later in this article.
  • Geometric objects – these specify the geometry of the visualisation. For example, in ggplot2 a scatter plot is specified via a point geometry whereas a fitting curve is represented by a smooth geometry. ggplot2 has a range of geometries available of which we will illustrate just a few.
  • Coordinate system – this specifies the system used to position data points on the graphic. Examples of coordinate systems are Cartesian and polar. We’ll deal with Cartesian systems in this tutorial. See this post for a nice illustration of how one can use polar plots creatively.
  • Facets – a facet specifies how data can be split over multiple plots to improve clarity. We’ll look at this briefly towards the end of this article.

The basic idea of a layered grammar of graphics is that each of these elements can be combined – literally added layer by layer – to achieve a desired visual result. Exactly how this is done will become clear as we work through some examples. So without further ado, let’s get to it.

Hatching (gg)plots

In what follows we’ll use the NSW Government Schools dataset,  made available via the state government’s open data initiative. The data is in csv format. If you cannot access the original dataset from the aforementioned link, you can download an Excel file with the data here (remember to save it as a csv before running the code!).

The first task – assuming that you have a working R/RStudio environment –  is to load the data into R. To keep things simple we’ll delete a number of columns (as shown in the code) and keep only  rows that are complete, i.e. those that have no missing values. Here’s the code:

#set working directory if needed (modify path as needed)
#setwd(“C:/Users/Kailash/Documents/ggplot”)
#load required library
library(ggplot2)
#load dataset (ensure datafile is in directory!)
nsw_schools <- read.csv(“NSW-Public-Schools-Master-Dataset-07032017.csv”)
#build expression for columns to delete
colnames_for_deletion <- paste0(“AgeID|”,”street|”,”indigenous_pct|”,
“lbote_pct|”,”opportunity_class|”,”school_specialty_type|”,
“school_subtype|”,”support_classes|”,”preschool_ind|”,
“distance_education|”,”intensive_english_centre|”,”phone|”,
“school_email|”,”fax|”,”late_opening_school|”,
“date_1st_teacher|”,”lga|”,”electorate|”,”fed_electorate|”,
“operational_directorate|”,”principal_network|”,
“facs_district|”,”local_health_district|”,”date_extracted”)
#get indexes of cols for deletion
cols_for_deletion <- grep(colnames_for_deletion,colnames(nsw_schools))
#delete them
nsw_schools <- nsw_schools[,-cols_for_deletion]
#structure and number of rows
str(nsw_schools)
nrow(nsw_schools)
#remove rows with NAs
nsw_schools <- nsw_schools[complete.cases(nsw_schools),]
#rowcount
nrow(nsw_schools)
#convert student number to numeric datatype.
#Need to convert factor to character first…
#…alternately, load data with StringsAsFactors set to FALSE
nsw_schools$student_number <- as.numeric(as.character(nsw_schools$student_number))
#a couple of character strings have been coerced to NA. Remove these
nsw_schools <- nsw_schools[complete.cases(nsw_schools),]

A note regarding the last line of code above, a couple of schools have “np” entered for the student_number variable. These are coerced to NA in the numeric conversion. The last line removes these two schools from the dataset.

Apart from student numbers and location data, we have retained level of schooling (primary, secondary etc.) and ICSEA ranking. The location information includes attributes such as suburb, postcode, region, remoteness as well as latitude and longitude. We’ll use only remoteness in this post.

The first thing that caught my eye in the data was was the ICSEA ranking.  Before going any further, I should mention that the  Australian Curriculum Assessment and Reporting Authority   (the  organisation responsible for developing the ICSEA system) emphasises that the score  is not a school ranking, but a measure of socio-educational advantage  of the student population in a school. Among other things, this is related to family background and geographic location.  The average ICSEA score is set at an average of 1000, which can be used as a reference level.

I thought a natural first step would be to see how ICSEA varies as a function of the other variables in the dataset such as student numbers and location (remoteness, for example). To begin with, let’s plot ICSEA rank as a function of student number. As it is our first plot, let’s take it step by step to understand how the layered grammar works. Here we go:

#specify data layer
p <- ggplot(data=nsw_schools)
#display plot
p

This displays a blank plot because we have not specified a mapping and geometry to go with the data. To get a plot we need to specify both. Let’s start with a scatterplot, which is specified via a point geometry.  Within the geometry function, variables are mapped to visual properties of the  using  aesthetic mappings. Here’s the code:

#specify a point geometry (geom_point)
p <- p + geom_point(mapping = aes(x=student_number,y=ICSEA_Value))
#…lo and behold our first plot
p

The resulting plot is shown in Figure 1.

Figure 1: Scatterplot of ICSEA score versus student numbers

At first sight there are two points that stand out: 1) there are fewer number of large schools, which we’ll look into in more detail later and 2) larger schools seem to have a higher ICSEA score on average.   To dig a little deeper into the latter, let’s add a linear trend line. We do that by adding another layer (geometry) to the scatterplot like so:

#add a trendline using geom_smooth
p <- p + geom_smooth(mapping= aes(x=student_number,y=ICSEA_Value),method=”lm”)
#scatter plot with trendline
p

The result is shown in Figure 2.

Figure 2: scatterplot of ICSEA vs student number with linear trendline

The lm method does a linear regression on the data.  The shaded area around the line is the 95% confidence level of the regression line (i.e that it is 95% certain that the true regression line lies in the shaded region). Note that geom_smooth   provides a range of smoothing functions including generalised linear and local regression (loess) models.

You may have noted that we’ve specified the aesthetic mappings in both geom_point and geom_smooth. To avoid this duplication, we can simply specify the mapping, once in the top level ggplot call (the first layer) like so:

#rewrite the above, specifying the mapping in the ggplot call instead of geom
p <- ggplot(data=nsw_schools,mapping= aes(x=student_number,y=ICSEA_Value)) +
geom_point()+
geom_smooth(method=”lm”)
#display plot, same as Fig 2
p

From Figure 2, one can see a clear positive correlation between student numbers and ICSEA scores, let’s zoom in around the average value (1000) to see this more clearly…

#set display to 900 < y < 1100
p <- p + coord_cartesian(ylim =c(900,1100))
#display plot
p

The coord_cartesian function is used to zoom the plot to without changing any other settings. The result is shown in Figure 3.

Figure 3: Zoomed view of Figure 2 for 900 < ICSEA <1100

To  make things clearer, let’s add a reference line at the average:

#add horizontal reference line at the avg ICSEA score
p <- p + geom_hline(yintercept=1000)
#display plot
p

The result, shown in Figure 4, indicates quite clearly that larger schools tend to have higher ICSEA scores. That said, there is a twist in the tale which we’ll come to a bit later.

Figure 4: Zoomed view with reference line at average value of ICSEA

As a side note, you would use geom_vline to zoom in on a specific range of x values and geom_abline to add a reference line with a specified slope and intercept. See this article on ggplot reference lines for more.

OK, now that we have seen how ICSEA scores vary with student numbers let’s switch tack and incorporate another variable in the mix.  An obvious one is remoteness. Let’s do a scatterplot as in Figure 1, but now colouring each point according to its remoteness value. This is done using the colour aesthetic as shown below:

#Map aecg_remoteness to colour aesthetic
p <- ggplot(data=nsw_schools, aes(x=student_number,y=ICSEA_Value,  colour=ASGS_remoteness)) +
geom_point()
#display plot
p

The resulting plot is shown in Figure 5.

Figure 5: ICSEA score as a function of student number and remoteness category

Aha, a couple of things become apparent. First up, large schools tend to be in metro areas, which makes good sense. Secondly, it appears that metro area schools have a distinct socio-educational advantage over regional and remote area schools. Let’s add trendlines by remoteness category as well to confirm that this is indeed so:

#add reference line at avg + trendlines for each remoteness category
p <- p + geom_hline(yintercept=1000) + geom_smooth(method=”lm”)
#display plot
p

The plot, which is shown in Figure 6, indicates clearly that  ICSEA scores decrease on the average as we move away from metro areas.

Figure 6: ICSEA scores vs student numbers and remoteness, with trendlines for each remoteness category

Moreover, larger schools metropolitan areas tend to have higher than average scores (above 1000),  regional areas tend to have lower than average scores overall, with remote areas being markedly more disadvantaged than both metro and regional areas.  This is no surprise, but the visualisations show just how stark the  differences are.

It is also interesting that, in contrast to metro and (to some extent) regional areas, there negative correlation between student numbers and scores for remote schools. One can also use  local regression to get a better picture of how ICSEA varies with student numbers and remoteness. To do this, we simply use the loess method instead of lm:

#redo plot using loess smoothing instead of lm
p <- ggplot(data=nsw_schools, aes(x=student_number,y=ICSEA_Value, colour=ASGS_remoteness)) +
geom_point() + geom_hline(yintercept=1000) + geom_smooth(method=”loess”)
#display plot
p

The result, shown in Figure 7,  has  a number  of interesting features that would have been worth pursuing further were we analysing this dataset in a real life project.  For example, why do small schools tend to have lower than benchmark scores?

Figure 7: ICSEA scores vs student numbers and remoteness with loess regression curves.

From even a casual look at figures 6 and 7, it is clear that the confidence intervals for remote areas are huge. This suggests that the number of datapoints for these regions are a) small and b) very scattered.  Let’s quantify the number by getting counts using the table function (I know, we could plot this too…and we will do so a little later). We’ll also transpose the results using data.frame to make them more readable:

#get school counts per remoteness category
data.frame(table(nsw_schools$ASGS_remoteness))
Var1 Freq
1 0
2 Inner Regional Australia 561
3 Major Cities of Australia 1077
4 Outer Regional Australia 337
5 Remote Australia 33
6 Very Remote Australia 14

The number of datapoints for remote regions is much less than those for metro and regional areas. Let’s repeat the loess plot with only the two remote regions. Here’s the code:

#create vector containing desired categories
remote_regions <- c(‘Remote Australia’,’Very Remote Australia’)
#redo loess plot with only remote regions included
p <- ggplot(data=nsw_schools[nsw_schools$ASGS_remoteness %in% remote_regions,], aes(x=student_number,y=ICSEA_Value, colour=ASGS_remoteness)) +
geom_point() + geom_hline(yintercept=1000) + geom_smooth(method=”loess”)
#display plot
p

The plot, shown in Figure 8, shows that there is indeed a huge variation in the (small number) of datapoints, and the confidence intervals reflect that. An interesting feature is that some small remote schools have above average scores. If we were doing a project on this data, this would be a feature worth pursuing further as it would likely be of interest to education policymakers.

Figure 8: Loess plots as in Figure 7 for remote region schools

Note that there is a difference in the x axis scale between Figures 7 and 8 – the former goes from 0 to 2000 whereas the  latter goes up to 400 only. So for a fair comparison, between remote and other areas, you may want to re-plot Figure 7, zooming in on student numbers between 0 and 400 (or even less). This will also enable you to see the complicated dependence of scores on student numbers more clearly across all regions.

We’ll leave the scores vs student numbers story there and move on to another  geometry – the well-loved bar chart. The first one is a visualisation of the remoteness category count that we did earlier. The relevant geometry function is geom_bar, and the code is as easy as:

#frequency plot
p <- ggplot(data=nsw_schools, aes(x=ASGS_remoteness)) + geom_bar()
#display plot
p

The plot is shown in Figure 9.

Figure 9: School count by remoteness categories

The category labels on the x axis are too long and look messy. This can be fixed by tilting them to a 45 degree angle so that they don’t run into each other as they most likely did when you ran the code on your computer. This is done by modifying the axis.text element of the plot theme. Additionally, it would be nice to get counts on top of each category bar. The way to do that is using another geometry function, geom_text. Here’s the code incorporating the two modifications:

#frequency plot
p <- p + geom_text(stat=’count’,aes(label= ..count..),vjust=-1)+
theme(axis.text.x=element_text(angle=45, hjust=1))
#display plot
p

The result is shown in Figure 10.

Figure 10: Bar plot of remoteness with counts and angled x labels

Some things to note: : stat=count tells ggplot to compute counts by category and the aesthetic label = ..count.. tells ggplot to access the internal variable that stores those counts. The the vertical justification setting, vjust=-1, tells ggplot to display the counts on top of the bars. Play around with different values of vjust to see how it works. The code to adjust label angles is self explanatory.

It would be nice to reorder the bars by frequency. This is easily done via fct_infreq function in the forcats package like so:

#use factor tools
library(forcats)
#descending
p <- ggplot(data=nsw_schools) +
geom_bar(mapping = aes(x=fct_infreq(ASGS_remoteness)))+
theme(axis.text.x=element_text(angle=45, hjust=1))
#display plot
p

The result is shown in Figure 11.

Figure 11: Barplot of Figure 10 sorted by descending count

To reverse the order, invoke fct_rev, which reverses the sort order:

#reverse sort order to ascending
p <- ggplot(data=nsw_schools) +
geom_bar(mapping = aes(x=fct_rev(fct_infreq(ASGS_remoteness))))+
theme(axis.text.x=element_text(angle=45, hjust=1))
#display plot
p

The resulting plot is shown in Figure 12.

Figure 12: Bar plot of Figure 10 sorted by ascending count

If this is all too grey for us, we can always add some colour. This is done using the fill aesthetic as follows:

#add colour using the fill aesthetic
p <- ggplot(data=nsw_schools) +
geom_bar(mapping = aes(x=ASGS_remoteness, fill=ASGS_remoteness))+
theme(axis.text.x=element_text(angle=45, hjust=1))
#display plot
p

The resulting plot is shown in Figure 13.

Figure 13: Coloured bar plot of school count by remoteness

Note that, in the above, that we have mapped fill and x to the same variable, remoteness which makes the legend superfluous. I will leave it to you to figure out how to suppress the legend – Google is your friend.

We could also map fill to another variable, which effectively adds another dimension to the plot. Here’s how:

#map fill to another variable
p <- ggplot(data=nsw_schools) +
geom_bar(mapping = aes(x=ASGS_remoteness, fill=level_of_schooling))+
theme(axis.text.x=element_text(angle=45, hjust=1))
#display plot
p

The plot is shown in Figure 14. The new variable, level of schooling, is displayed via proportionate coloured segments stacked up in each bar. The default stacking is one on top of the other.

Figure 14: Bar plot of school counts as a function of remoteness and school level

Alternately, one can stack them up side by side by setting the position argument to dodge as follows:

#stack side by side
p <- ggplot(data=nsw_schools) +
geom_bar(mapping = aes(x=ASGS_remoteness,fill=level_of_schooling),position =”dodge”)+
theme(axis.text.x=element_text(angle=45, hjust=1))
#display plot
p

The plot is shown in Figure 15.

Figure 15: Same data as in Figure 14 stacked side-by-side

Finally, setting the  position argument to fill  normalises the bar heights and gives us the proportions of level of schooling for each remoteness category. That sentence will  make more sense when you see Figure 16 below. Here’s the code, followed by the figure:

#proportion plot
p <- ggplot(data=nsw_schools) +
geom_bar(mapping = aes(x=ASGS_remoteness,fill=level_of_schooling),position = “fill”)+
theme(axis.text.x=element_text(angle=45, hjust=1))
#display plot
p

Obviously,  we lose frequency information since the bar heights are normalised.

Figure 16: Proportions of school levels for remoteness categories

An  interesting feature here is that  the proportion of central and community schools increases with remoteness. Unlike primary and secondary schools, central / community schools provide education from Kindergarten through Year 12. As remote areas have smaller numbers of students, it makes sense to consolidate educational resources in institutions that provide schooling at all levels .

Finally, to close the loop so to speak,  let’s revisit our very first plot in Figure 1 and try to simplify it in another way. We’ll use faceting to  split it out into separate plots, one per remoteness category. First, we’ll organise the subplots horizontally using facet_grid:

#faceting – subplots laid out horizontally (faceted variable on right of formula)
p <- ggplot(data=nsw_schools) + geom_point(mapping = aes(x=student_number,y=ICSEA_Value))+
facet_grid(~ASGS_remoteness)
#display plot
p

The plot is shown in Figure 17 in which the different remoteness categories are presented in separate plots (facets) against a common y axis. It shows, the sharp differences between student numbers between remote and other regions.

Figure 17: Horizontally laid out facet plots of ICSEA scores for different remoteness categories

To get a vertically laid out plot, switch the faceted variable to other side of the formula (left as an exercise for you).

If one has too many categories to fit into a single row, one can wrap the facets using facet_wrap like so:

#faceting – wrapping facets in 2 columns
p <- ggplot(data=nsw_schools) +
geom_point(mapping = aes(x=student_number,y=ICSEA_Value))+
facet_wrap(~ASGS_remoteness, ncol= 2)
#display plot
p

The resulting plot is shown in Figure 18.

Figure 18: Same data as in Figure 17, with facets wrapped in a 2 column format

One can specify the number of rows instead of columns. I won’t illustrate that as the change in syntax is quite obvious.

…and I think that’s a good place to stop.

Wrapping up

Data visualisation has a reputation of being a dark art, masterable only by the visually gifted. This may have been partially true some years ago, but in this day and age it definitely isn’t. Versatile  packages such as ggplot, that use a consistent syntax have made  the art much more accessible to visually ungifted folks like myself. In this post I have attempted to provide a brief and (hopefully) logical introduction to ggplot.  In closing I note that  although some of the illustrative examples  violate the  principles of good data visualisation, I hope this article will serve its primary purpose which is pedagogic rather than artistic.

Further reading:

Where to go for more? Two of the best known references are Hadley Wickham’s books:

I highly recommend his R for Data Science , available online here. Apart from providing a good overview of ggplot, it is an excellent introduction to R for data scientists.  If you haven’t read it, do yourself a favour and buy it now.

People tell me his ggplot book is an excellent book for those wanting to learn the ins and outs of ggplot . I have not read it myself, but if his other book is anything to go by, it should be pretty damn good.

Written by K

October 10, 2017 at 8:17 pm

The two tributaries of time

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How time flies. Ten years ago this month, I wrote my first post on Eight to Late.  The anniversary gives me an excuse to post something a little different. When rummaging around in my drafts folder for something suitable, I came across this piece that I wrote some years ago (2013) but didn’t publish.   It’s about our strange relationship with time, which I thought makes it a perfect piece to mark the occasion.

Introduction

The metaphor of time as a river resonates well with our subjective experiences of time.  Everyday phrases that evoke this metaphor include the flow of time and time going by, or the somewhat more poetic currents of time.  As Heraclitus said, no [person] can step into the same river twice – and so it is that a particular instant in time …like right now…is ephemeral, receding into the past as we become aware of it.

On the other hand, organisations have to capture and quantify time because things have to get done within fixed periods, the financial year being a common example. Hence, key organisational activities such as projects, strategies and budgets are invariably time-bound affairs. This can be problematic because there is a mismatch between the ways in which organisations view time and individuals experience it.

Organisational time

The idea that time is an objective entity is most clearly embodied in the notion of a timeline: a graphical representation of a time period, punctuated by events. The best known of these is perhaps the ubiquitous Gantt Chart, loved (and perhaps equally, reviled) by managers the world over.

Timelines are interesting because, as Elaine Yakura states in this paper, “they seem to render time, the ultimate abstraction, visible and concrete.”   As a result, they can serve as boundary objects that make it possible to negotiate and communicate what is to be accomplished in the specified time period. They make this possible because they tell a story with a clear beginning, middle and end, a narrative of what is to come and when.

For the reasons mentioned in the previous paragraph, timelines are often used to manage time-bound organisational initiatives. Through their use in scheduling and allocation, timelines serve to objectify time in such a way that it becomes a resource that can be measured and rationed, much like other resources such as money, labour etc.

At our workplaces we are governed by many overlapping timelines – workdays, budgeting cycles and project schedules being examples. From an individual perspective, each of these timelines are different representations of how one’s time is to be utilised, when an activity should be started and when it must be finished. Moreover, since we are generally committed to multiple timelines, we often find ourselves switching between them. They serve to remind us what we should be doing and when.

But there’s more: one of the key aims of developing a timeline is to enable all stakeholders to have a shared understanding of time as it pertains to the initiative. In this view, a timeline is a consensus representation of how a particular aspect of the future will unfold.  Timelines thus serve as coordinating mechanisms.

In terms of the metaphor, a timeline is akin to a map of the river of time. Along the map we can measure out and apportion it; we can even agree about way-stops at various points in time. However, we should always be aware that it remains a representation of time, for although we might treat a timeline as real, the fact is no one actually experiences time as it is depicted in a timeline. Mistaking one for the other is akin to confusing the map with the territory.

This may sound a little strange so I’ll try to clarify.  I’ll start with the observation that we experience time through events and processes – for example the successive chimes of a clock, the movement of the second hand of a watch (or the oscillations of a crystal), the passing of seasons or even the greying of one’s hair. Moreover, since these events and processes can be objectively agreed on by different observers, they can also be marked out on a timeline.  Yet the actual experience of living these events is unique to each individual.

Individual perception of time

As we have seen, organisations treat time as an objective commodity that can be represented, allocated and used much like any tangible resource.  On the other hand our experience of time is intensely personal.  For example, I’m sitting in a cafe as I write these lines. My perception of the flow of time depends rather crucially on my level of engagement in writing: slow when I’m struggling for words but zipping by when I’m deeply involved. This is familiar to us all: when we are deeply engaged in an activity, we lose all sense of time but when our involvement is superficial we are acutely aware of the clock.

This is true at work as well. When I’m engaged in any kind of activity at work, be it a group activity such as a meeting, or even an individual one such as developing a business case, my perception of time has little to do with the actual passage of seconds, minutes and hours on a clock. Sure, there are things that I will do habitually at a particular time – going to lunch, for example – but my perception of how fast the day goes is governed not by the clock but by the degree of engagement with my work.

I can only speak for myself, but I suspect that this is the case with most people. Though our work lives are supposedly governed by “objective” timelines, the way we actually live out our workdays depends on a host of things that have more to do with our inner lives than visible outer ones.  Specifically, they depend on things such as feelings, emotions, moods and motivations.

Flow and engagement

OK, so you may be wondering where I’m going with this. Surely, my subjective perception of my workday should not matter as long as I do what I’m required to do and meet my deadlines, right?

As a matter of fact, I think the answer to the above question is a qualified, “No”. The quality of the work we do depends on our level of commitment and engagement. Moreover, since a person’s perception of the passage of time depends rather sensitively on the degree of their involvement in a task, their subjective sense of time is a good indicator of their engagement in work.

In his book, Finding Flow, Mihalyi Csikszentmihalyi describes such engagement as an optimal experience in which a person is completely focused on the task at hand.  Most people would have experienced flow when engaged in activities that they really enjoy. As Anthony Reading states in his book, Hope and Despair: How Perceptions of the Future Shape Human Behaviour, “…most of what troubles us resides in our concerns about the past and our apprehensions about the future.”  People in flow are entirely focused on the present and are thus (temporarily) free from troubling thoughts. As Csikszentmihalyi puts it, for such people, “the sense of time is distorted; hours seem to pass by in minutes.”

All this may seem far removed from organisational concerns, but it is easy to see that it isn’t: a Google search on the phrase “increase employee engagement” will throw up many articles along the lines of “N ways to increase employee engagement.”  The sense in which the term is used in these articles is essentially the same as the one Csikszentmihalyi talks about: deep involvement in work.

So, the advice of management gurus and business school professors notwithstanding, the issue is less about employee engagement or motivation than about creating conditions that are conducive to flow.   All that is needed for the latter is a deep understanding how the particular organisation functions, the task at hand and (most importantly) the people who will be doing it.  The best managers I’ve worked with have grokked this, and were able to create the right conditions in a seemingly effortless and unobtrusive way. It is a skill that cannot be taught, but can be learnt by observing how such managers do what they do.

Time regained

Organisations tend to treat their employees’ time as though it were a commodity or resource that can be apportioned and allocated for various tasks. This view of time is epitomised by the timeline as depicted in a Gantt Chart or a resource-loaded project schedule.

In contrast, at an individual level, the perception of time depends rather critically on the level of engagement that a person feels with the task he or she is performing. Ideally organisations would (or ought to!) want their employees to be in that optimal zone of engagement that Csikszentmihalyi calls flow, at least when they are involved in creative work. However, like spontaneity, flow is a state that cannot be achieved by corporate decree; the best an organisation can do is to create the conditions that encourage it.

The organisational focus on timelines ought to be balanced by actions that are aimed at creating the conditions that are conducive to employee engagement and flow.  It may then be possible for those who work in organisation-land to experience, if only fleetingly, that Blakean state in which eternity is held in an hour.

Written by K

September 20, 2017 at 9:17 pm

A gentle introduction to logistic regression and lasso regularisation using R

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In this day and age of artificial intelligence and deep learning, it is easy to forget that simple algorithms can work well for a surprisingly large range of practical business problems.  And the simplest place to start is with the granddaddy of data science algorithms: linear regression and its close cousin, logistic regression. Indeed, in his acclaimed MOOC and accompanying textbook, Yaser Abu-Mostafa spends a good portion of his time talking about linear methods, and with good reason too: linear methods are not only a good way to learn the key principles of machine learning, they can also be remarkably helpful in zeroing in on the most important predictors.

My main aim in this post is to provide a beginner level introduction to logistic regression using R and also introduce LASSO (Least Absolute Shrinkage and Selection Operator), a powerful feature selection technique that is very useful for regression problems. Lasso is essentially a regularization method. If you’re unfamiliar with the term, think of it as a way to reduce overfitting using less complicated functions (and if that means nothing to you, check out my prelude to machine learning).  One way to do this is to toss out less important variables, after checking that they aren’t important.  As we’ll discuss later, this can be done manually by examining p-values of coefficients and discarding those variables whose coefficients are not significant. However, this can become tedious for classification problems with many independent variables.  In such situations, lasso offers a neat way to model the dependent variable while automagically selecting significant variables by shrinking the coefficients of unimportant predictors to zero.  All this without having to mess around with p-values or obscure information criteria. How good is that?

Why not linear regression?

In linear regression one attempts to model a dependent variable (i.e. the one being predicted) using the best straight line fit to a set of predictor variables.  The best fit is usually taken to be one that minimises the root mean square error,  which is the sum of square of the differences between the actual and predicted values of the dependent variable. One can think of logistic regression as the equivalent of linear regression for a classification problem.  In what follows we’ll look at binary classification – i.e. a situation where the dependent variable takes on one of two possible values (Yes/No, True/False, 0/1 etc.).

First up, you might be wondering why one can’t use linear regression for such problems. The main reason is that classification problems are about determining class membership rather than predicting variable values, and linear regression is more naturally suited to the latter than the former. One could, in principle, use linear regression for situations where there is a natural ordering of categories like High, Medium and Low for example. However, one then has to map sub-ranges of the predicted values to categories. Moreover, since predicted values are potentially unbounded (in data as yet unseen) there remains a degree of arbitrariness associated with such a mapping.

Logistic regression sidesteps the aforementioned issues by modelling class probabilities instead.  Any input to the model yields a number lying between 0 and 1, representing the probability of class membership. One is still left with the problem of determining the threshold probability, i.e. the probability at which the category flips from one to the other.  By default this is set to p=0.5, but in reality it should be settled based on how the model will be used.  For example, for a marketing model that identifies potentially responsive customers, the threshold for a positive event might be set low (much less than 0.5) because the client does not really care about mailouts going to a non-responsive customer (the negative event). Indeed they may be more than OK with it as there’s always a chance – however small – that a non-responsive customer will actually respond.  As an opposing example, the cost of a false positive would be high in a machine learning application that grants access to sensitive information. In this case, one might want to set the threshold probability to a value closer to 1, say 0.9 or even higher. The point is, the setting an appropriate threshold probability is a business issue, not a technical one.

Logistic regression in brief

So how does logistic regression work?

For the discussion let’s assume that the outcome (predicted variable) and predictors are denoted by Y and X respectively and the two classes of interest are denoted by + and – respectively.  We wish to model the conditional probability that the outcome Y is +, given that the input variables (predictors) are X. The conditional probability is denoted by p(Y=+|X)   which we’ll abbreviate as p(X) since we know we are referring to the positive outcome Y=+.

As mentioned earlier, we are after the probability of class membership so we must ensure that the hypothesis function (a fancy word for the model) always lies between 0 and 1. The function assumed in logistic regression is:

p(X) = \dfrac{\exp^{\beta_0+\beta_1 X}}{1+\exp^{\beta_0 + \beta_1 X}} .....(1)

You can verify that p(X) does indeed lie between 0 and  1 as X varies from -\infty to \infty.  Typically, however, the values of X that make sense are bounded as shown in the example (stolen from Wikipedia) shown in Figure 1. The figure also illustrates the typical S-shaped  curve characteristic of logistic regression.

Figure 1: Logistic function

As an aside, you might be wondering where the name logistic comes from. An equivalent way of expressing the above equation is:

\log(\dfrac{p(X)}{1-p(X)}) = \beta_0+\beta_1 X .....(2)

The quantity on the left is the logarithm of the odds. So, the model is a linear regression of the log-odds, sometimes called logit, and hence the name logistic.

The problem is to find the values of \beta_0  and \beta_1 that results in a p(X) that most accurately classifies all the observed data points – that is, those that belong to the positive class have a probability as close as possible to 1 and those that belong to the negative class have a probability as close as possible to 0. One way to frame this problem is to say that we wish to maximise the product of these probabilities, often referred to as the likelihood:

\displaystyle\log ( {\prod_{i:Y_i=+} p(X_{i}) \prod_{j:Y_j=-}(1-p(X_{j}))})

Where \prod represents the products over i and j, which run over the +ve and –ve classed points respectively. This approach, called maximum likelihood estimation, is quite common in many machine learning settings, especially those involving probabilities.

It should be noted that in practice one works with the log likelihood because it is easier to work with mathematically. Moreover, one minimises the negative  log likelihood which, of course, is the same as maximising the log likelihood.  The quantity one minimises is thus:

L = - \displaystyle\log ( {\prod_{i:Y_i=+} p(X_{i}) \prod_{j:Y_j=-}(1-p(X_{j}))}).....(3)

However, these are technical details that I mention only for completeness. As you will see next, they have little bearing on the practical use of logistic regression.

Logistic regression in R – an example

In this example, we’ll use the logistic regression option implemented within the glm function that comes with the base R installation. This function fits a class of models collectively known as generalized linear models. We’ll apply the function to the Pima Indian Diabetes dataset that comes with the mlbench package. The code is quite straightforward – particularly if you’ve read earlier articles in my “gentle introduction” series – so I’ll just list the code below  noting that the logistic regression option is invoked by setting family=”binomial”  in the glm function call.

Here we go:

#set working directory if needed (modify path as needed)
#setwd(“C:/Users/Kailash/Documents/logistic”)
#load required library
library(mlbench)
#load Pima Indian Diabetes dataset
data(“PimaIndiansDiabetes”)
#set seed to ensure reproducible results
set.seed(42)
#split into training and test sets
PimaIndiansDiabetes[,”train”] <- ifelse(runif(nrow(PimaIndiansDiabetes))<0.8,1,0)
#separate training and test sets
trainset <- PimaIndiansDiabetes[PimaIndiansDiabetes$train==1,]
testset <- PimaIndiansDiabetes[PimaIndiansDiabetes$train==0,]
#get column index of train flag
trainColNum <- grep(“train”,names(trainset))
#remove train flag column from train and test sets
trainset <- trainset[,-trainColNum]
testset <- testset[,-trainColNum]
#get column index of predicted variable in dataset
typeColNum <- grep(“diabetes”,names(PimaIndiansDiabetes))
#build model
glm_model <- glm(diabetes~.,data = trainset, family = binomial)
summary(glm_model)
Call:
glm(formula = diabetes ~ ., family = binomial, data = trainset)
<<output edited>>
Coefficients:
            Estimate  Std. Error z value Pr(>|z|)
(Intercept)-8.1485021 0.7835869 -10.399  < 2e-16 ***
pregnant    0.1200493 0.0355617   3.376  0.000736 ***
glucose     0.0348440 0.0040744   8.552  < 2e-16 ***
pressure   -0.0118977 0.0057685  -2.063  0.039158 *
triceps     0.0053380 0.0076523   0.698  0.485449
insulin    -0.0010892 0.0009789  -1.113  0.265872
mass        0.0775352 0.0161255   4.808  1.52e-06 ***
pedigree    1.2143139 0.3368454   3.605  0.000312 ***
age         0.0117270 0.0103418   1.134  0.256816
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#predict probabilities on testset
#type=”response” gives probabilities, type=”class” gives class
glm_prob <- predict.glm(glm_model,testset[,-typeColNum],type=”response”)
#which classes do these probabilities refer to? What are 1 and 0?
contrasts(PimaIndiansDiabetes$diabetes)
    pos
neg 0
pos 1
#make predictions
##…first create vector to hold predictions (we know 0 refers to neg now)
glm_predict <- rep(“neg”,nrow(testset))
glm_predict[glm_prob>.5] <- “pos”
#confusion matrix
table(pred=glm_predict,true=testset$diabetes)
glm_predict  neg pos
        neg    90 22
        pos     8 33
#accuracy
mean(glm_predict==testset$diabetes)
[1] 0.8039216

 

Although this seems pretty good, we aren’t quite done because there is an issue that is lurking under the hood. To see this, let’s examine the information output from the model summary, in particular the coefficient estimates (i.e. estimates for \beta) and their significance. Here’s a summary of the information contained in the table:

  • Column 2 in the table lists coefficient estimates.
  • Column 3 list s the standard error of the estimates (the larger the standard error, the less confident we are about the estimate)
  • Column 4 the z statistic (which is the coefficient estimate (column 2) divided by the standard error of the estimate (column 3)) and
  • The last column (Pr(>|z|) lists the p-value, which is the probability of getting the listed estimate assuming the predictor has no effect. In essence, the smaller the p-value, the more significant the estimate is likely to be.

From the table we can conclude that only 4 predictors are significant – pregnant, glucose, mass and pedigree (and possibly a fifth – pressure). The other variables have little predictive power and worse, may contribute to overfitting.  They should, therefore, be eliminated and we’ll do that in a minute. However, there’s an important point to note before we do so…

In this case we have only 9 variables, so are able to identify the significant ones by a manual inspection of p-values.  As you can well imagine, such a process will quickly become tedious as the number of predictors increases. Wouldn’t it be be nice if there were an algorithm that could somehow automatically shrink the coefficients of these variables or (better!) set them to zero altogether?  It turns out that this is precisely what  lasso and its close cousin, ridge regression, do.

Ridge and Lasso

Recall that the values of the logistic regression coefficients \beta_0  and \beta_1 are found by minimising the negative log likelihood described in equation (3).  Ridge and lasso regularization work by adding a penalty term to the log likelihood function.  In the case of ridge regression, the penalty term is \beta_1^2 and in the case of lasso, it is |\beta_1| (Remember, \beta_1  is a vector, with as many components as there are predictors).  The quantity to be minimised in the two cases is thus:

L +\lambda \sum \beta_1^2.....(4) – for ridge regression,

and

L +\lambda \sum |\beta_1|.....(5) – for lasso regression.

Where \lambda is a free parameter which is usually selected in such a way that the resulting model minimises the out of sample error. Typically, the optimal value of \lambda is found using grid search with cross-validation, a process akin to the one described in my discussion on cost-complexity parameter  estimation in decision trees. Most canned algorithms provide methods to do this; the one we’ll use in the next section is no exception.

In the case of ridge regression, the effect of the penalty term is to shrink the coefficients that contribute most to the error. Put another way, it reduces the magnitude of the coefficients that contribute to increasing L.  In contrast, in  the case of lasso regression, the effect of the penalty term is to set the these coefficients exactly to zero! This is cool because what it mean that lasso regression works like a feature selector that picks out the most important coefficients, i.e. those that are most predictive (and have the lowest p-values).

Let’s illustrate this through an example. We’ll use the glmnet package which implements a combined version of ridge and lasso (called elastic net). Instead of minimising (4) or (5) above, glmnet minimises:

L +\lambda[ (1-\alpha)\sum [\beta_1^2 + \alpha\sum|\beta_1|]....(6)

where \alpha controls the “mix” of ridge and lasso regularisation, with \alpha=0 being “pure” ridge and  \alpha=1 being “pure” lasso.

Lasso regularisation using glmnet

Let’s reanalyse the Pima Indian Diabetes dataset using glmnet with \alpha=1 (pure lasso). Before diving into code, it is worth noting that glmnet:

  • does not have a formula interface, so one has to input the predictors as a matrix and the class labels as a vector.
  • does not accept categorical predictors, so one has to convert these to numeric values before passing them to glmnet.

The glmnet function model.matrix creates the matrix and also converts categorical predictors to appropriate dummy variables.

Another important point to note is that we’ll use the function cv.glmnet, which automatically performs a grid search to find the optimal value of \lambda.

OK, enough said, here we go:

#load required library
library(glmnet)
#convert training data to matrix format
x <- model.matrix(diabetes~.,trainset)
#convert class to numerical variable
y <- ifelse(trainset$diabetes==”pos”,1,0)
#perform grid search to find optimal value of lambda
#family= binomial => logistic regression, alpha=1 => lasso
# check docs to explore other type.measure options
cv.out <- cv.glmnet(x,y,alpha=1,family=”binomial”,type.measure = “mse” )
#plot result
plot(cv.out)

 

The plot is shown in Figure 2 below:

Figure 2: Error as a function of lambda (select lambda that minimises error)

The plot shows that the log of the optimal value of lambda (i.e. the one that minimises the root mean square error) is approximately -5. The exact value can be viewed by examining the variable lambda_min in the code below. In general though, the objective of regularisation is to balance accuracy and simplicity. In the present context, this means a model with the smallest number of coefficients that also gives a good accuracy.  To this end, the cv.glmnet function  finds the value of lambda that gives the simplest model but also lies within one standard error of the optimal value of lambda. This value of lambda (lambda.1se) is what we’ll use in the rest of the computation. Interested readers should have a look at this article for more on lambda.1se vs lambda.min.

#min value of lambda
lambda_min <- cv.out$lambda.min
#best value of lambda
lambda_1se <- cv.out$lambda.1se
#regression coefficients
coef(cv.out,s=lambda_1se)
10 x 1 sparse Matrix of class “dgCMatrix”
                      1
(Intercept) -4.61706681
(Intercept)  .
pregnant     0.03077434
glucose      0.02314107
pressure     .
triceps      .
insulin      .
mass         0.02779252
pedigree     0.20999511
age          .

 

The output shows that only those variables that we had determined to be significant on the basis of p-values have non-zero coefficients. The coefficients of all other variables have been set to zero by the algorithm! Lasso has reduced the complexity of the fitting function massively…and you are no doubt wondering what effect this  has on accuracy. Let’s see by running the model against our test data:

 

#get test data
x_test <- model.matrix(diabetes~.,testset)
#predict class, type=”class”
lasso_prob <- predict(cv.out,newx = x_test,s=lambda_1se,type=”response”)
#translate probabilities to predictions
lasso_predict <- rep(“neg”,nrow(testset))
lasso_predict[lasso_prob>.5] <- “pos”
#confusion matrix
table(pred=lasso_predict,true=testset$diabetes)
pred  neg pos
   neg 94 28
  pos  4 27
#accuracy
mean(lasso_predict==testset$diabetes)
[1] 0.7908497

 

Which is a bit less than what we got with the more complex model. So, we get  a similar out-of-sample accuracy as we did before, and we do so using a way simpler function (4 non-zero coefficients) than the original one (9  nonzero coefficients). What this means is that the simpler function does at least as good a job fitting the signal in the data as the more complicated one.  The bias-variance tradeoff tells us that the simpler function should be preferred because it is less likely to overfit the training data.

Paraphrasing William of Ockhamall other things being equal, a simple hypothesis should be preferred over a complex one.

Wrapping up

In this post I have tried to provide a detailed introduction to logistic regression, one of the simplest (and oldest) classification techniques in the machine learning practitioners arsenal. Despite it’s simplicity (or I should say, because of it!) logistic regression works well for many business applications which often have a simple decision boundary. Moreover, because of its simplicity it is less prone to overfitting than flexible methods such as decision trees. Further, as we have shown, variables that contribute to overfitting can be eliminated using lasso (or ridge) regularisation, without compromising out-of-sample accuracy. Given these advantages and its inherent simplicity, it isn’t surprising that logistic regression remains a workhorse for data scientists.

Written by K

July 11, 2017 at 10:00 pm

The improbability of success

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Anyone who has tidied up after a toddler intuitively understands that making a mess is far easier than creating order. The fundamental reason for this is that the number of messy states in the universe (or a toddler’s room) far outnumbers the ordered ones.  As this point might not be obvious, I’ll demonstrate it via a simple thought experiment involving marbles:

Throw three marbles onto a flat surface.  When the marbles come to rest, you are most likely to end up with a random configuration  as in Figure 1.

Figure 1: A random configuration of 3 marbles

Indeed, you’d be extremely surprised if the three ended up being collinear as in Figure 2.   Note that Figure 2 is just one example of many collinear possibilities, but the point I’m making is that if the marbles are thrown randomly, they are more likely to end up in a random state than a lined-up one.

Figure 2: an unlikely (ordered) configuration

This raises a couple of questions:

Question: On what basis can one claim that the collinear configuration is tidier or more ordered than the non-collinear one?

Naive answer:  It looks more ordered. Yes, tidiness is in the eye of the beholder so it is necessarily subjective. However, I’ll wager that if one took a poll, an overwhelming number of people would say that the configuration in Figure 2 is more ordered than the one in Figure 1.

More sophisticated answer : The “state” of collinear marbles can be described using 2 parameters, the slope and intercept of the straight line that three marbles lie on (in any coordinate system) whereas the description of the nonlinear state requires 3 parameters. The first state is tidier because it requires fewer parameters.  Another way to think about is that the line can be described by two marbles; the third one is redundant as far as the description of the state is concerned.

Question: Why is a tidier configuration less likely than a messy one?

Answer:  May be you see this intuitively and need no proof, but here’s one just in case. Imagine rolling the three marbles one after the other. The first two, regardless of where they end up, will necessarily lie along a line (two points lie on the straight line joining them). Now, I think it is easy to see that if we throw the third marble randomly, it is highly unlikely end up on that line. Indeed, for the third marble to end up exactly on the same straight line requires a coincidence of near cosmic proportions.

I know, I know, this is not a proof, but I trust it makes the point.

Now, although it is near impossible to get to a collinear end state via random throws, it is possible to approximate it by changing the way we throw the marbles. Here’s how:

  1. Throw the marbles consecutively rather than in one go.
  2. When throwing the third marble, adjust its initial speed and direction in a way that takes into account the positions of the two marbles that are already on the surface. Remember these two already define a straight line.

The third throw is no longer random because it is designed to maximise the chance that the last marble will get as close as possible to the straight line defined by the first two. Done right, you’ll end up with something closer to the configuration in Figure 3 rather than the one in Figure 2.

Figure 3: an “approximately ordered” state

Now you’re probably wondering what this has to do with success. I’ll make the connection via an example that will be familiar to many readers of this blog: an organisation’s strategy. However, as I will reiterate later, the arguments I present are very general and can be applied to just about any initiative or situation.

Typically, a strategy sets out goals for an organisation and a plan to achieve them in a specified timeframe. The goals define a number of desirable outcomes, or states which, by design, are constrained to belong to a (very) small subset of all possible states the organisation can end up in.  In direct analogy with the simple model discussed above it is clear that, left to its own devices, the organisation is more likely to end up in one of the much overwhelmingly larger number of “failed states” than one of the successful ones.  Notwithstanding the popular quote about there being many roads to success, in reality there are a great many more roads to failure.

Of course, that’s precisely why organisations are never “left to their own devices.” Indeed, a strategic plan specifies actions that are intended to make a successful state more likely than an unsuccessful one. However, no plan can guarantee success; it can, at best, make it more likely. As in the marble game, success is ultimately a matter of chance, even when we take actions to make it more likely.

If we accept this, the key question becomes: how can one design a strategy that improves the odds of success?  The marble analogy suggests a way to do this is to:

  1. Define success in terms of an end state that is a natural extension of your current state.
  2. Devise a plan to (approximately) achieve that end state. Such a plan will necessarily build on the current state rather than change it wholesale. Successful change is an evolutionary process rather than a revolutionary one.

My contention is that these points are often ignored by management strategists. More often than not, they will define an end state based on a textbook idealisation, consulting model or (horror!) best practice. The marble analogy shows why copying others is unlikely to succeed.

Figure 4 shows a variant of the marble game in which we have two sets of marbles (or organisations!), one blue, as before, and the other red.


Figure 4: Two distinct configurations of marbles (or organisations)

Now, it is considerably harder to align an additional marble with both sets of marbles than the blue one alone. Here’s why…

To align with both sets, the new marble has to end up close to the point that lies at the intersection of the blue and red lines in Figure 5. In contrast, to align with the blue set alone, all that’s needed is for it to get close to any point on the blue line.

QED!

Figure 5: Why copying others is not a good idea (see text for explanation)

Finally, on a broader note, it should be clear that the arguments made above go beyond organisational strategies. They apply to pretty much any planned action, whether at work or in one’s personal life.

So, to sum up: when developing an organisational (or personal) strategy, the first step is to understand where you are and then identify the minimal actions you need to take in order to get to an “improved” state that is consistent with  your current one. Yes, this is akin to the incremental and evolutionary approach that Agilistas and Leaners have been banging on about for years. However, their prescriptions focus on specific areas: software development and process improvement.  My point is that the basic principles are way broader because they are a direct consequence of a fundamental fact regarding the relative likelihood of order and disorder in a toddler’s room, an organisation, or even the universe at large.

Written by K

April 4, 2017 at 9:16 pm

Uncertainty, ambiguity and the art of decision making

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A common myth about decision making in organisations is that it is, by and large, a rational process.   The term rational refers to decision-making methods that are based on the following broad steps:

  1. Identify available options.
  2. Develop criteria for rating options.
  3. Rate options according to criteria developed.
  4. Select the top-ranked option.

Although this appears to be a logical way to proceed it is often difficult to put into practice, primarily because of uncertainty about matters relating to the decision.

Uncertainty can manifest itself in a variety of ways: one could be uncertain about facts, the available options, decision criteria or even one’s own preferences for options.

In this post, I discuss the role of uncertainty in decision making and, more importantly, how one can make well-informed decisions in such situations.

A bit about uncertainty

It is ironic that the term uncertainty is itself vague when used in the context of decision making. There are at least five distinct senses in which it is used:

  1. Uncertainty about decision options.
  2. Uncertainty about one’s preferences for options.
  3. Uncertainty about what criteria are relevant to evaluating the options.
  4. Uncertainty about what data is needed (data relevance).
  5. Uncertainty about the data itself (data accuracy).

Each of these is qualitatively different: uncertainty about data accuracy (item 5 above) is very different from uncertainty regarding decision options (item 1). The former can potentially be dealt with using statistics whereas the latter entails learning more about the decision problem and its context, ideally from different perspectives. Put another way, the item 5 is essentially a technical matter whereas item 1 is a deeper issue that may have social, political and – as we shall see – even behavioural dimensions. It is therefore reasonable to expect that the two situations call for vastly different approaches.

Quantifiable uncertainty

A common problem in project management is the estimation of task durations. In this case, what’s requested is a “best guess” time (in hours or days) it will take to complete a task. Many project schedules represent task durations by point estimates, i.e.  by single numbers. The Gantt Chart shown in Figure 1 is a common example. In it, each task duration is represented by its expected duration. This is misleading because the single number conveys a sense of certainty that is unwarranted.  It is far more accurate, not to mention safer, to quote a range of possible durations.

Figure 1: Gantt Chart (courtesy Wikimedia)

Figure 1: Gantt Chart (courtesy Wikimedia)

In general, quantifiable uncertainties, such as those conveyed in estimates, should always be quoted as ranges – something along the following lines: task A may take anywhere between 2 and 8 days, with a most likely completion time of 4 days (Figure 2).

Figure 2: Task completion likelihood (3 point estimates)

Figure 2: Task completion likelihood (3 point estimates)

In this example, aside from stating that the task will finish sometime between 2 and 4 days, the estimator implicitly asserts that the likelihood of finishing before 2 days or after 8 days is zero.  Moreover, she also implies that some completion times are more likely than others. Although it may be difficult to quantify the likelihood exactly, one can begin by making simple (linear!) approximations as shown in Figure 3.

Figure 3: Simple probability distribution based on the estimates in Figure 2

Figure 3: Simple probability distribution based on the estimates in Fig 2

The key takeaway from the above is that quantifiable uncertainties are shapes rather than single numbers.  See this post and this one for details for how far this kind of reasoning can take you. That said, one should always be aware of the assumptions underlying the approximations. Failure to do so can be hazardous to the credibility of estimators!

Although I haven’t explicitly said so, estimation as described above has a subjective element. Among other things, the quality of an estimate depends on the judgement and experience of the estimator. As such, it is prone to being affected by errors of judgement and cognitive biases.  However, provided one keeps those caveats in mind, the probability-based approach described above is suited to situations in which uncertainties are quantifiable, at least in principle. That said, let’s move on to more complex situations in which uncertainties defy quantification.

Introducing ambiguity

The economist Frank Knight was possibly the first person to draw the distinction between quantifiable and unquantifiable uncertainties.  To make things really confusing, he called the former risk and the latter uncertainty. In his doctoral thesis, published in 1921, wrote:

…it will appear that a measurable uncertainty, or “risk” proper, as we shall call the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all. We shall accordingly restrict the term “uncertainty” to cases of the non-quantitative type (p.20)

Terminology has moved on since Knight’s time, the term uncertainty means lots of different things, depending on context. In this piece, we’ll use the term uncertainty to refer to quantifiable uncertainty (as in the task estimate of the previous section) and use ambiguity to refer to nonquantifiable uncertainty. In essence, then, we’ll use the term uncertainty for situations where we know what we’re measuring (i.e. the facts) but are uncertain about its numerical or categorical values whereas we’ll use the word ambiguity to refer to situations in which we are uncertain about what the facts  are or which facts are relevant.

As a test of understanding, you may want to classify each of the five points made in the second section of this post as either uncertain or ambiguous (Answers below)

Answer: 1 through 4 are ambiguous and 5 is uncertain.

How ambiguity manifests itself in decision problems

The distinction between uncertainty and ambiguity points to a problem with quantitative decision-making techniques such as cost-benefit analysis, multicriteria decision making methods or analytic hierarchy process. All these methods assume that decision makers are aware of all the available options, their preferences for them, the relevant evaluation criteria and the data needed. This is almost never the case for consequential decisions. To see why, let’s take a closer look at the different ways in which ambiguity can play out in the rational decision making process mentioned at the start of this article.

  1. The first step in the process is to identify available options. In the real world, however, options often cannot be enumerated or articulated fully. Furthermore, as options are articulated and explored, new options and sub-options tend to emerge. This is particularly true if the options depend on how future events unfold.
  2. The second step is to develop criteria for rating options. As anyone who has been involved in deciding on a contentious issue will confirm, it is extremely difficult to agree on a set of decision criteria for issues that affect different stakeholders in different ways.  Building a new road might improve commute times for one set of stakeholders but result in increased traffic in a residential area for others. The two criteria will be seen very differently by the two groups. In this case, it is very difficult for the two groups to agree on the relative importance of the criteria or even their legitimacy. Indeed, what constitutes a legitimate criterion is a matter of opinion.
  3. The third step is to rate options. The problem here is that real-world options often cannot be quantified or rated in a meaningful way. Many of life’s dilemmas fall into this category. For example, a decision to accept or decline a job offer is rarely made on the basis of material gain alone. Moreover, even where ratings are possible, they can be highly subjective. For example, when considering a job offer, one candidate may give more importance to financial matters whereas another might consider lifestyle-related matters (flexi-hours, commuting distance etc.) to be paramount. Another complication here is that there may not be enough information to settle the matter conclusively. As an example, investment decisions are often made on the basis of quantitative information that is based on questionable assumptions.

A key consequence of the above is that such ambiguous decision problems are socially complex – i.e. different stakeholders could have wildly different perspectives on the problem itself.   One could say the ambiguity experienced by an individual is compounded by the group.

Before going on I should point out that acute versions of such ambiguous decision problems go by many different names in the management literature. For example:

All these terms are more or less synonymous:  the root cause of the difficulty in every case is ambiguity (or unquantifiable uncertainty), which prevents a clear formulation of the problem.

Social complexity is hard enough to tackle as it is, but there’s another issue that makes things even harder: ambiguity invariably triggers negative emotions such as fear and anxiety in individuals who make up the group.  Studies in neuroscience have shown that in contrast to uncertainty, which evokes logical responses in people, ambiguity tends to stir up negative emotions while simultaneously suppressing the ability to think logically.  One can see this playing out in a group that is debating a contentious decision: stakeholders tend to get worked up over issues that touch on their values and identities, and this seems to limit their ability to look at the situation objectively.

Tackling ambiguity

Summarising the discussion thus far: rational decision making approaches are based on the assumption that stakeholders have a shared understanding of the decision problem as well as the facts and assumptions around it. These conditions are clearly violated in the case of ambiguous decision problems. Therefore, when confronted with a decision problem that has even a hint of ambiguity, the first order of the day is to help the group reach a shared understanding of the problem.  This is essentially an exercise in sensemaking, the art of collaborative problem formulation. However, this is far from straightforward because ambiguity tends to evoke negative emotions and attendant defensive behaviours.

The upshot of all this is that any approach to tackle ambiguity must begin by taking the concerns of individual stakeholders seriously.  Unless this is done, it will be impossible for the group to coalesce around a consensus decision. Indeed, ambiguity-laden decisions in organisations invariably fail when they overlook concerns of specific stakeholder groups.  The high failure rate of organisational change initiatives (60-70% according to this Deloitte report) is largely attributable to this point

There are a number of techniques that one can use to gather and synthesise diverse stakeholder viewpoints and thus reach a shared understanding of a complex or ambiguous problem. These techniques are often referred to as problem structuring methods (PSMs). I won’t go into these in detail here; for an example check out Paul Culmsee’s articles on dialogue mapping and Barry Johnson’s introduction to polarity management. There are many more techniques in the PSM stable. All of them are intended to help a group reconcile different viewpoints and thus reach a common basis from which one can proceed to the next step (i.e., make a decision on what should be done). In other words, these techniques help reduce ambiguity.

But there’s more to it than a bunch of techniques.  The main challenge is to create a holding environment that enables such techniques to work. I am sure readers have been involved in a meeting or situation where the outcome seems predetermined by management or has been undermined by self- interest. When stakeholders sense this, no amount of problem structuring is going to help.  In such situations one needs to first create the conditions for open dialogue to occur. This is precisely what a holding environment provides.

Creating such a holding environment is difficult in today’s corporate world, but not impossible. Note that this is not an idealist’s call for an organisational utopia. Rather, it involves the application of a practical set of tools that address the diverse, emotion-laden reactions that people often have when confronted with ambiguity.   It would take me too far afield to discuss PSMs and holding environments any further here. To find out more, check out my papers on holding environments and dialogue mapping in enterprise IT projects, and (for a lot more) the Heretic’s Guide series of books that I co-wrote with Paul Culmsee.

The point is simply this: in an ambiguous situation, a good decision – whatever it might be – is most likely to be reached by a consultative process that synthesises diverse viewpoints rather than by an individual or a clique.  However, genuine participation (the hallmark of a holding environment) in such a process will occur only after participants’ fears have been addressed.

Wrapping up

Standard approaches to decision making exhort managers and executives to begin with facts, and if none are available, to gather them diligently prior to making a decision. However, most real-life decisions are fraught with uncertainty so it may be best to begin with what one doesn’t know, and figure out how to make the possible decision under those “constraints of ignorance.” In this post I’ve attempted to outline what such an approach would entail. The key point is to figure out the kind uncertainty one is dealing with and choosing an approach that works for it. I’d argue that most decision making debacles stem from a failure to appreciate this point.

Of course, there’s a lot more to this approach than I can cover in the span of a post, but that’s a story for another time.

Note: This post is written as an introduction to the Data and Decision Making subject that is part of the core curriculum of the Master of Data Science and Innovation program, run by the Connected Intelligence Centre at UTS. I’m coordinating the subject this semester, and am honoured to be co-teaching it with my erstwhile colleague Sean Heffernan and my longtime collaborator Paul Culmsee.

Written by K

March 9, 2017 at 10:04 am

A prelude to machine learning

with 6 comments

What is machine learning?

The term machine learning gets a lot of airtime in the popular and trade press these days. As I started writing this article, I did a quick search for recent news headlines that contained this term. Here are the top three results with datelines within three days of the search:

http://venturebeat.com/2017/02/01/beyond-the-gimmick-implementing-effective-machine-learning-vb-live/

http://www.infoworld.com/article/3164249/artificial-intelligence/new-big-data-tools-for-machine-learning-spring-from-home-of-spark-and-mesos.html

http://www.infoworld.com/article/3163525/analytics/review-the-best-frameworks-for-machine-learning-and-deep-learning.html

The truth about hype usually tends to be quite prosaic and so it is in this case. Machine learning, as Professor Yaser Abu-Mostafa  puts it, is simply about “learning from data.”  And although the professor is referring to computers, this is so for humans too – we learn through patterns discerned from sensory data. As he states in the first few lines of his wonderful (but mathematically demanding!) book entitled, Learning From Data:

If you show a picture to a three-year-old and ask if there’s a tree in it, you will likely get a correct answer. If you ask a thirty year old what the definition of a tree is, you will likely get an inconclusive answer. We didn’t learn what a tree is by studying a [model] of what trees [are]. We learned by looking at trees. In other words, we learned from data.

In other words, the three year old forms a model of what constitutes a tree through a process of discerning a common pattern between all objects that grown-ups around her label “trees.” (the data). She can then “predict” that something is (or is not) a tree by applying this model to new instances presented to her.

This is exactly what happens in machine learning: the computer (or more correctly, the algorithm) builds a predictive model of a variable (like “treeness”) based on patterns it discerns in data.  The model can then be applied to predict the value of the variable (e.g. is it a tree  or not) in new instances.

With that said for an introduction, it is worth contrasting this machine-driven process of model building with the traditional approach of building mathematical models to predict phenomena as in, say,  physics and engineering.

What are models good for?

Physicists and engineers model phenomena using physical laws and mathematics. The aim of such modelling is both to understand and predict natural phenomena.  For example, a physical law such as Newton’s Law of Gravitation is itself a model – it helps us understand how gravity works and make predictions about (say) where Mars is going to be six months from now.  Indeed, all theories and laws of physics are but models that have wide applicability.

(Aside: Models are typically expressed via differential equations.  Most differential equations are hard to solve analytically (or exactly), so scientists use computers to solve them numerically.  It is important to note that in this case computers are used as calculation tools, they play no role in model-building.)

As mentioned earlier, the role of models in the sciences is twofold – understanding and prediction. In contrast, in machine learning the focus is usually on prediction rather than understanding.  The predictive successes of machine learning have led certain commentators to claim that scientific theory building is obsolete and science can advance by crunching data alone.  Such claims are overblown, not to mention, hubristic, for although a data scientist may be able to predict with accuracy, he or she may not be able to tell you why a particular prediction is obtained. This lack of understanding can mislead and can even have harmful consequences, a point that’s worth unpacking in some detail…

Assumptions, assumptions

A model of a real world process or phenomenon is necessarily a simplification. This is essentially because it is impossible to isolate a process or phenomenon from the rest of the world. As a consequence it is impossible to know for certain that the model one has built has incorporated all the interactions that influence the process / phenomenon of interest. It is quite possible that potentially important variables have been overlooked.

The selection of variables that go into a model is based on assumptions. In the case of model building in physics, these assumptions are made upfront and are thus clear to anybody who takes the trouble to read the underlying theory. In machine learning, however, the assumptions are harder to see because they are implicit in the data and the algorithm. This can be a problem when data is biased or an algorithm opaque.

Problem of bias and opacity become more acute as datasets increase in size and algorithms become more complex, especially when applied to social issues that have serious human consequences. I won’t go into this here, but for examples the interested reader may want to have a look at Cathy O’Neil’s book, Weapons of Math Destruction, or my article on the dark side of data science.

As an aside, I should point out that although assumptions are usually obvious in traditional modelling, they are often overlooked out of sheer laziness or, more charitably, lack of awareness. This can have disastrous consequences. The global financial crisis of 2008 can – to some extent – be blamed on the failure of trading professionals to understand assumptions behind the model that was used to calculate the value of collateralised debt obligations.

It all starts with a straight line….

Now that we’ve taken a tour of some of the key differences between model building in the old and new worlds, we are all set to start talking about machine learning proper.

I should begin by admitting that I overstated the point about opacity: there are some machine learning algorithms that are transparent as can possibly be. Indeed, chances are you know the  algorithm I’m going to discuss next, either from an introductory statistics course in university or from plotting relationships between two variables in your favourite spreadsheet.  Yea, you may have guessed that I’m referring to linear regression.

In its simplest avatar, linear regression attempts to fit a straight line to a set of data points in two dimensions. The two dimensions correspond to a dependent variable (traditionally denoted by y) and an independent variable (traditionally denoted by x).    An example of such a fitted line is shown in Figure 1.  Once such a line is obtained, one can “predict” the value of the dependent variable for any value of the independent variable.  In terms of our earlier discussion, the line is the model.

Figure 1: Linear Regression

Figure 1: Linear Regression

Figure 1 also serves to illustrate that linear models are going to be inappropriate in most real world situations (the straight line does not fit the data well). But it is not so hard to devise methods to fit more complicated functions.

The important point here is that since machine learning is about finding functions that accurately predict dependent variables for as yet unknown values of the independent variables, most algorithms make explicit or implicit choices about the form of these functions.

Complexity versus simplicity

At first sight it seems a no-brainer that complicated functions will work better than simple ones. After all, if we choose a nonlinear function with lots of parameters, we should be able to fit a complex data set better than a linear function can (See Figure 2 – the complicated function fits the datapoints better than the straight line).   But there’s catch: although the ability to fit a dataset increases with the flexibility of the fitting function,  increasing complexity beyond a point will invariably reduce predictive power.  Put another way, a complex enough function may fit the known data points perfectly but, as a consequence, will inevitably perform poorly on unknown data. This is an important point so let’s look at it in greater detail.

Figure 2: Simple and complex fitting functions

Figure 2: Simple and complex fitting function (courtesy: Wikimedia)

Recall that the aim of machine learning is to predict values of the dependent variable for as yet unknown values of the independent variable(s).  Given a finite (and usually, very limited) dataset, how do we build a model that we can have some confidence in? The usual strategy is to partition the dataset into two subsets, one containing 60 to 80% of the data (called the training set) and the other containing the remainder (called the test set). The model is then built – i.e. an appropriate function fitted – using the training data and verified against the test data. The verification process consists of comparing the predicted values of the dependent variable with the known values for the test set.

Now, it should be intuitively clear that the more complicated the function, the better it will fit the training data.

Question: Why?

Answer: Because complicated functions have more free parameters – for example, linear functions of a single (dependent) variable have two parameters (slope and intercept), quadratics have three, cubics four and so on.  The mathematician, John von Neumann is believed to have said, “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” See this post for a nice demonstration of the literal truth of his words.

Put another way, complex functions are wrigglier than simple ones, and – by suitable adjustment of parameters – their “wriggliness” can be adjusted to fit the training data better than functions that are less wriggly. Figure 2 illustrates this point well.

This may sound like you can have your cake and eat it too: choose a complicated enough function and you can fit both the training and test data well. Not so! Keep in mind that the resulting model (fitted function) is built using the training set alone, so a good fit to the test data is not guaranteed.  In fact, it is intuitively clear that a function that fits the training data perfectly (as in Figure 2) is likely to do a terrible job on the test data.

Question: Why?

Answer:  Remember, as far as the model is concerned, the test data is unknown. Hence, the greater the wriggliness in the trained model, the less likely it is to fit the test data well. Remember, once the model is fitted to the training data, you have no freedom to tweak parameters any further.

This tension between simplicity and complexity of models is one of the key principles of machine learning and is called the bias-variance tradeoff. Bias here refers to lack of flexibility and variance, the reducible error. In general simpler functions have greater bias and lower variance and complex functions, the opposite.  Much of the subtlety of machine learning lies in developing an understanding of how to arrive at the right level of complexity for the problem at hand –  that is, how to tweak parameters so that the resulting function fits the training data just well enough so as to generalise well to unknown data.

Note: those who are curious to learn more about the bias-variance tradeoff may want to have a look at this piece.  For details on how to achieve an optimal tradeoff, search for articles on regularization in machine learning.

Unlocking unstructured data

The discussion thus far has focused primarily on quantitative or enumerable data (numbers and categories) that’s stored in  a structured format – i.e. as columns and rows in a spreadsheet or database table). This is fine as it goes, but the fact is that much of the data in organisations is unstructured, the most common examples being text documents and audio-visual media. This data is virtually impossible to analyse computationally using relational database technologies  (such as SQL) that are commonly used by organisations.

The situation has changed dramatically in the last decade or so. Text analysis techniques that once required expensive software and high-end computers have now been implemented in open source languages such as Python and R, and can be run on personal computers.  For problems that require computing power and memory beyond that, cloud technologies make it possible to do so cheaply. In my opinion, the ability to analyse textual data is the most important advance in data technologies in the last decade or so. It unlocks a world of possibilities for the curious data analyst. Just think, all those comment fields in your survey data can now be analysed in a way that was never possible in the relational world!

There is a general impression that text analysis is hard.  Although some of the advanced techniques can take a little time to wrap one’s head around, the basics are simple enough. Yea, I really mean that – for proof, check out my tutorial on the topic.

Wrapping up

I could go on for a while. Indeed, I was planning to delve into a few algorithms of increasing complexity (from regression to trees and forests to neural nets) and then close with a brief peek at some of the more recent headline-grabbing developments like deep learning. However, I realised that such an exploration would be too long and (perhaps more importantly) defeat the main intent of this piece which is to give starting students an idea of what machine learning is about, and how it differs from preexisting techniques of data analysis. I hope I have succeeded, at least partially, in achieving that aim.

For those who are interested in learning more about machine learning algorithms, I can suggest having a look at my “Gentle Introduction to Data Science using R” series of articles. Start with the one on text analysis (link in last line of previous section) and then move on to clustering, topic modelling, naive Bayes, decision trees, random forests and support vector machines. I’m slowly adding to the list as I find the time, so please do check back again from time to time.

Note: This post is written as an introduction to the Data, Algorithms and Meaning subject that is part of the core curriculum of the Master of Data Science and Innovation program, run by the Connected Intelligence Centre at UTS. I’m coordinating the subject this semester, and will be co-teaching it with Stephan Curiskis.

Written by K

February 23, 2017 at 3:12 pm

 A gentle introduction to support vector machines using R

with 5 comments

Introduction

Most machine learning algorithms involve minimising an error measure of some kind (this measure is often called an objective function or loss function).  For example, the error measure in linear regression problems is the famous mean squared error – i.e. the averaged sum of the squared differences between the predicted and actual values. Like the mean squared error, most objective functions depend on all points in the training dataset.  In this post, I describe the support vector machine (SVM) approach which focuses instead on finding the optimal separation boundary between datapoints that have different classifications.  I’ll elaborate on what this means in the next section.

Here’s the plan in brief. I’ll begin with the rationale behind SVMs using a simple case of a binary (two class) dataset with a simple separation boundary (I’ll clarify what “simple” means in a minute).  Following that, I’ll describe how this can be generalised to datasets with more complex boundaries. Finally, I’ll work through a couple of examples in R, illustrating the principles behind SVMs. In line with the general philosophy of my “Gentle Introduction to Data Science Using R” series, the focus is on developing an intuitive understanding of the algorithm along with a practical demonstration of its use through a toy example.

The rationale

The basic idea behind SVMs is best illustrated by considering a simple case:  a set of data points that belong to one of two classes, red and blue, as illustrated in figure 1 below. To make things simpler still, I have assumed that the boundary separating the two classes is a straight line, represented by the solid green line in the diagram.  In the technical literature, such datasets are called linearly separable.

Figure 1:

Figure 1: Linearly separable data

In the linearly separable case, there is usually a fair amount of freedom in the way a separating line can be drawn. Figure 2 illustrates this point: the two broken green lines are also valid separation boundaries. Indeed, because there is a non-zero distance between the two closest points between categories, there are an infinite number of possible separation lines. This, quite naturally, raises the question as to whether it is possible to choose a separation boundary that is optimal.

Figure 2: Illustrating multiple separation boundaries

Figure 2: Illustrating multiple separation boundaries

The short answer is, yes there is. One way to do this is to select a boundary line that maximises the margin, i.e. the distance between the separation boundary and the points that are closest to it.  Such an optimal boundary is illustrated by the black brace in Figure 3.  The really cool thing about this criterion is that the location of the separation boundary depends only on the points that are closest to it. This means, unlike other classification methods, the classifier does not depend on any other points in dataset. The directed lines between the boundary and the closest points on either side are called support vectors (these are the solid black lines in figure 3). A direct implication of this is that the fewer the support vectors, the better the generalizability of the boundary.

Figure 3: Optimal separation boundary in linearly separable case

Although the above sounds great, it is of limited practical value because real data sets are seldom (if ever) linearly separable.

So, what can we do when dealing with real (i.e. non linearly separable) data sets?

A simple approach to tackle small deviations from linear separability is to allow a small number of points (those that are close to the boundary) to be misclassified.  The number of possible misclassifications is governed by a free parameter C, which is called the cost.  The cost is essentially the penalty associated with making an error: the higher the value of C, the less likely it is that the algorithm will misclassify a point.

This approach – which is called soft margin classification – is illustrated in Figure 4. Note the points on the wrong side of the separation boundary.  We will demonstrate soft margin SVMs in the next section.  (Note:  At the risk of belabouring the obvious, the purely linearly separable case discussed in the previous para is simply is a special case of the soft margin classifier.)

Figure 3: Soft margin classifier (linearly separable data)

Figure 4: Soft margin classifier (linearly separable data)

Real life situations are much more complex and cannot be dealt with using soft margin classifiers. For example, as shown in Figure 5, one could have widely separated clusters of points that belong to the same classes. Such situations, which require the use of multiple (and nonlinear) boundaries, can sometimes be dealt with using a clever approach called the kernel trick.

Figure 5: Non-linearly separable data

Figure 5: Non-linearly separable data

The kernel trick

Recall that in the linearly separable (or soft margin) case, the SVM algorithm works by finding a separation boundary that maximises the margin, which is the distance between the boundary and the points closest to it. The distance here is the usual straight line distance between the boundary and the closest point(s). This is called the Euclidean distance in honour of the great geometer of antiquity. The point to note is that this process results in a separation boundary that is a straight line, which as Figure 5 illustrates, does not always work. In fact in most cases it won’t.

So what can we do? To answer this question, we have to take a bit of a detour…

What if we were able to generalize the notion of distance in a way that generates nonlinear separation boundaries? It turns out that this is possible. To see how, one has to first understand how the notion of distance can be generalized.

The key properties that any measure of distance must satisfy are:

  1. Non-negativity – a distance cannot be negative, a point that needs no further explanation I reckon 🙂
  2. Symmetry – that is, the distance between point A and point B is the same as the distance between point B and point A.
  3. Identity– the distance between a point and itself is zero.
  4. Triangle inequality – that is the sum of distances between point A and B and points B and C must be less than or equal to the distance between A and C (equality holds only if all three points lie along the same line).

Any mathematical object that displays the above properties is akin to a distance. Such generalized distances are called metrics and the mathematical space in which they live is called a metric space. Metrics are defined using special mathematical functions designed to satisfy the above conditions. These functions are known as kernels.

The essence of the kernel trick lies in mapping the classification problem to a  metric space in which the problem is rendered separable via a separation boundary that is simple in the new space, but complex – as it has to be – in the original one. Generally, the transformed space has a higher dimensionality, with each of the dimensions being (possibly complex) combinations of the original problem variables. However, this is not necessarily a problem because in practice one doesn’t actually mess around with transformations, one just tries different kernels (the transformation being implicit in the kernel) and sees which one does the job. The check is simple: we simply test the predictions resulting from using different kernels against a held out subset of the data (as one would for any machine learning algorithm).

It turns out that a particular function – called the radial basis function kernel  (RBF kernel) – is very effective in many cases.  The RBF kernel is essentially a Gaussian (or Normal) function with the Euclidean distance between pairs of points as the variable (see equation 1 below).   The basic rationale behind the RBF kernel is that it creates separation boundaries that it tends to classify points close together (in the Euclidean sense) in the original space in the same way. This is reflected in the fact that the kernel decays (i.e. drops off to zero) as the Euclidean distance between points increases.

\exp (-\gamma |\mathbf{x-y}|)....(1)

The rate at which a kernel decays is governed by the parameter \gamma – the higher the value of \gamma, the more rapid the decay.  This serves to illustrate that the RBF kernel is extremely flexible….but the flexibility comes at a price – the danger of overfitting for large values of \gamma .  One should choose appropriate values of C and \gamma so as to ensure that the resulting kernel represents the best possible balance between flexibility and accuracy. We’ll discuss how this is done in practice later in this article.

Finally, though it is probably obvious, it is worth mentioning that the separation boundaries for arbitrary kernels are also defined through support vectors as in Figure 3.  To reiterate a point made earlier, this means that a solution that has fewer support vectors is likely to be more robust than one with many. Why? Because the data points defining support vectors are ones that are most sensitive to noise- therefore the fewer, the better.

There are many other types of kernels, each with their own pros and cons. However, I’ll leave these for adventurous readers to explore by themselves.  Finally, for a much more detailed….and dare I say, better… explanation of the kernel trick, I highly recommend this article by Eric Kim.

Support vector machines in R

In this demo we’ll use the svm interface that is implemented in the e1071 R package. This interface provides R programmers access to the comprehensive libsvm library written by Chang and Lin. I’ll use two toy datasets: the famous iris dataset available with the base R package and the sonar dataset from the mlbench package. I won’t describe details of the datasets as they are discussed at length in the documentation that I have linked to. However, it is worth mentioning the reasons why I chose these datasets:

  1. As mentioned earlier, no real life dataset is linearly separable, but the iris dataset is almost so. Consequently, it is a good illustration of using linear SVMs. Although one almost never uses these in practice, I have illustrated their use primarily for pedagogical reasons.
  2. The sonar dataset is a good illustration of the benefits of using RBF kernels in cases where the dataset is hard to visualise (60 variables in this case!). In general, one would almost always use RBF (or other nonlinear) kernels in practice.

With that said, let’s get right to it. I assume you have R and RStudio installed. For instructions on how to do this, have a look at the first article in this series. The processing preliminaries – loading libraries, data and creating training and test datasets are much the same as in my previous articles so I won’t dwell on these here. For completeness, however, I’ll list all the code so you can run it directly in R or R studio (a complete listing of the code can be found here):

#set working directory if needed (modify path as needed)
setwd(“C:/Users/Kailash/Documents/svm”)
#load required library
library(e1071)
#load built-in iris dataset
data(iris)
#set seed to ensure reproducible results
set.seed(42)
#split into training and test sets
iris[,”train”] <- ifelse(runif(nrow(iris))<0.8,1,0)
#separate training and test sets
trainset <- iris[iris$train==1,]
testset <- iris[iris$train==0,]
#get column index of train flag
trainColNum <- grep("train",names(trainset))
#remove train flag column from train and test sets
trainset <- trainset[,-trainColNum]
testset <- testset[,-trainColNum]
#get column index of predicted variable in dataset
typeColNum <- grep("Species",names(iris))
#build model – linear kernel and C-classification (soft margin) with default cost (C=1)
svm_model <- svm(Species~ ., data=trainset, method="C-classification", kernel="linear")
svm_model
Call:
svm(formula = Species ~ ., data = trainset, method = “C-classification”, kernel = “linear”)
Parameters:
SVM-Type: C-classification
SVM-Kernel: linear
cost: 1
gamma: 0.25
Number of Support Vectors: 24
#training set predictions
pred_train <-predict(svm_model,trainset)
mean(pred_train==trainset$Species)
[1] 0.9826087
#test set predictions
pred_test <-predict(svm_model,testset)
mean(pred_test==testset$Species)
[1] 0.9142857

 

The output from the SVM model show that there are 24 support vectors. If desired, these can be examined using the SV variable in the model – i.e via svm_model$SV.

The test prediction accuracy indicates that the linear performs quite well on this dataset, confirming that it is indeed near linearly separable. To check performance by class, one can create a confusion matrix as described in my post on random forests. I’ll leave this as an exercise for you.  Another point is that  we have used a soft-margin classification scheme with a cost C=1. You can experiment with this by explicitly changing the value of C. Again, I’ll leave this for you an exercise.

Before proceeding to the RBF kernel, I should mention a point that an alert reader may have noticed. The predicted variable, Species, can take on 3 values (setosa, versicolor and virginica). However, our discussion above dealt with a binary (2 valued) classification problem. This brings up the question as to how the algorithm deals multiclass classification problems – i.e those involving datasets with more than two classes. The libsvm algorithm (which svm uses) does this using a one-against-one classification strategy. Here’s how it works:

  1. Divide the dataset (assumed to have N classes) into N(N-1)/2 datasets that have two classes each.
  2. Solve the binary classification problem for each of these subsets
  3. Use a simple voting mechanism to assign a class to each data point.

Basically, each data point is assigned the most frequent classification it receives from all the binary classification problems it figures in.

With that said for the unrealistic linear classifier, let’s move to the real world.  In the code below, I build SVM models using three different kernels

  1.  Linear kernel (this is for comparison with the following 2 kernels).
  2. RBF kernel with default values for the parameters C and \gamma.
  3. RBF kernel with optimal values for C and \gamma. The optimal values are obtained using the tune.svm function (also available in e1071), which essentially builds models for multiple combinations of parameter values and selects the best.

OK, lets go:

#load required library (assuming e1071 is already loaded)
library(mlbench)
#load Sonar dataset
data(Sonar)
#set seed to ensure reproducible results
set.seed(42)
#split into training and test sets
Sonar[,”train”] <- ifelse(runif(nrow(Sonar))<0.8,1,0)
#separate training and test sets
trainset <- Sonar[Sonar$train==1,]
testset <- Sonar[Sonar$train==0,]
#get column index of train flag
trainColNum <- grep("train",names(trainset))
#remove train flag column from train and test sets
trainset <- trainset[,-trainColNum]
testset <- testset[,-trainColNum]
#get column index of predicted variable in dataset
typeColNum <- grep("Class",names(Sonar))
#build model – linear kernel and C-classification with default cost (C=1)
svm_model <- svm(Class~ ., data=trainset, method="C-classification", kernel="linear")
#training set predictions
pred_train <-predict(svm_model,trainset)
mean(pred_train==trainset$Class)
[1] 0.969697
#test set predictions
pred_test <-predict(svm_model,testset)
mean(pred_test==testset$Class)
[1] 0.6046512

I’ll leave you to examine the contents of the model. The important point to note here is that the performance of the model with the test set is quite dismal compared to the previous case. This simply indicates that the linear kernel is not appropriate here.  Let’s take a look at what happens if we use the RBF kernel with default values for the parameters:

#build model: radial kernel, default params
svm_model <- svm(Class~ ., data=trainset, method="C-classification", kernel="radial")
#print params
svm_model$cost
[1] 1
svm_model$gamma
[1] 0.01666667
#training set predictions
pred_train <-predict(svm_model,trainset)
mean(pred_train==trainset$Class)
[1] 0.9878788
#test set predictions
pred_test <-predict(svm_model,testset)
mean(pred_test==testset$Class)
[1] 0.7674419

That’s a pretty decent improvement from the linear kernel. Let’s see if we can do better by doing some parameter tuning. To do this we first invoke tune.svm and use the parameters it gives us in the call to svm:

#find optimal parameters in a specified range
tune_out <- tune.svm(x=trainset[,-typeColNum],y=trainset[,typeColNum],gamma=10^(-3:3),cost=c(0.01,0.1,1,10,100,1000),kernel="radial")
#print best values of cost and gamma
tune_out$best.parameters$cost
[1] 10
tune_out$best.parameters$gamma
[1] 0.01
#build model
svm_model <- svm(Class~ ., data=trainset, method="C-classification", kernel="radial",cost=tune_out$best.parameters$cost,gamma=tune_out$best.parameters$gamma)
#training set predictions
pred_train <-predict(svm_model,trainset)
mean(pred_train==trainset$Class)
[1] 1
#test set predictions
pred_test <-predict(svm_model,testset)
mean(pred_test==testset$Class)
[1] 0.8139535

Which is fairly decent improvement on the un-optimised case.

Wrapping up

This bring us to the end of this introductory exploration of SVMs in R. To recap, the distinguishing feature of SVMs in contrast to most other techniques is that they attempt to construct optimal separation boundaries between different categories.

SVMs  are quite versatile and have been applied to a wide variety of domains ranging from chemistry to pattern recognition. They are best used in binary classification scenarios. This brings up a question as to where SVMs are to be preferred to other binary classification techniques such as logistic regression. The honest response is, “it depends” – but here are some points to keep in mind when choosing between the two. A general point to keep in mind is that SVM  algorithms tend to be expensive both in terms of memory and computation, issues that can start to hurt as the size of the dataset increases.

Given all the above caveats and considerations, the best way  to figure out whether an SVM approach will work for your problem may be to do what most machine learning practitioners do: try it out!

Written by K

February 7, 2017 at 8:27 pm

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