Peirce, Holmes and a gold chain – an essay on abductive reasoning
“It has long been an axiom of mine that the little things are infinitely the most important.” – Sir Arthur Conan Doyle (A Case of Identity)
The scientific method is a systematic approach to acquiring and establishing knowledge about how the world works. A scientific investigation typically starts with the formulation of a hypothesis – an educated, evidencebased guess about the mechanism behind the phenomenon being studied – and proceeds by testing how well the hypothesis holds up against experiments designed to disconfirm it.
Although many philosophers have waxed eloquent about the scientific method, very few of them have talked about the process of hypothesis generation. Indeed, most scientists will recognise a good hypothesis when they stumble upon one, but they usually will not be able to say how they came upon it. Hypothesis generation is essentially a creative act that requires a deep familiarity with the phenomenon in question and a spark of intuition. The latter is absolutely essential, a point captured eloquently in the following lines attributed to Einstein:
“…[Man] makes this cosmos and its construction the pivot of his emotional life in order to find in this way the peace and serenity which he cannot find in the narrow whirlpool of personal experience. The supreme task is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience can reach them…” – quoted from Zen and The Art of Motorcycle Maintenance by Robert Pirsig.
The American philosopher, Charles Peirce, recognised that hypothesis generation involves a special kind of reasoning, one that enables the investigator to zero in on a small set of relevant facts out of an infinity of possibilities.
As Pierce wrote in one of his papers:
“A given object presents an extraordinary combination of characters of which we should like to have an explanation. That there is any explanation of them is a pure assumption; and if there be, it is [a single] fact which explains them; while there are, perhaps, a million other possible ways of explaining them, if they were not all, unfortunately, false.
A man is found in the streets of New York stabbed in the back. The chief of police might open a directory and put his finger on any name and guess that that is the name of the murderer. How much would such a guess be worth? But the number of names in the directory does not approach the multitude of possible laws of attraction which could have accounted for Kepler’s law of planetary motion and, in advance of verification by predications of perturbations etc., would have accounted for them to perfection. Newton, you will say, assumed that the law would be a simple one. But what was that but piling guess on guess? Surely vastly more phenomena in nature are complex than simple…” – quoted from this paper by Thomas Sebeok.
Peirce coined the term abduction (as opposed to induction or deduction) to refer to the creative act of hypothesis generation. In the present day, the term is used to the process of justifying hypotheses rather than generating them (see this article for more on the distinction). In the remainder of this piece I will use the term in its Peircean sense.
–x–
Contrary to what is commonly stated, Arthur Conan Doyle’s fictional detective employed abductive rather than deductive methods in his cases. Consider the following lines, taken from an early paragraph of Sherlock Holmes’ most celebrated exploit, The Adventure of the Speckled Band. We join Holmes in conversation with a lady who has come to him for assistance:
…You have come in by train this morning, I see.”
“You know me, then?”
“No, but I observe the second half of a return ticket in the palm of your left glove. You must have started early, and yet you had a good drive in a dogcart, along heavy roads, before you reached the station.”
The lady gave a violent start and stared in bewilderment at my companion.
“There is no mystery, my dear madam,” said he, smiling. “The left arm of your jacket is spattered with mud in no less than seven places. The marks are perfectly fresh. There is no vehicle save a dogcart which throws up mud in that way, and then only when you sit on the lefthand side of the driver.”
“Whatever your reasons may be, you are perfectly correct,” said she. “I started from home before six, reached Leatherhead at twenty past, and came in by the first train to Waterloo…
Notice what Holmes does: he forms hypotheses about what the lady did, based on a selective observation of facts. Nothing is said about why he picks those particular facts – the ticket stub and the freshness / location of mud spatters on the lady’s jacket. Indeed, as Holmes says in another story, “You know my method. It is founded upon the observation of trifles.”
Abductive reasoning is essentially about recognising which trifles are important.
–x–
I have a gold chain that my mum gave me many years ago. I’ve worn it for so long that now I barely notice it. The only time I’m consciously aware that I’m wearing the chain is when I finger it around my neck, a (nervous?) habit I have developed over time.
As you might imagine, I’m quite attached to my gold chain. So, when I discovered it was missing a few days ago, my first reaction was near panic, I felt like a part of me had gone missing.
Since I hardly ever take the chain off, I could not think of any plausible explanation for how I might have lost it. Indeed, the only time I have had to consistently take the chain off is when going in for an Xray or, on occasion, through airport security.
Where could it have gone?
After mulling over it for a while, the only plausible explanation I could think of is that I had taken it off at airport security when returning from an overseas trip a week earlier, and had somehow forgotten to collect it on the other side. Realising that it would be near impossible to recover it, I told myself to get used to the idea that it was probably gone for good.
That Sunday, I went for a swim. After doing my laps, I went to the side of the pool for my customary shower. Now, anyone who has taken off a rash vest after a swim will know that it can be a struggle. I suspect this is because water trapped between skin and fabric forms a thin adhesive layer (a manifestation of surface tension perhaps?). Anyway, I wrestled the garment over my head and it eventually came free with a snap, generating a spray of droplets that gleamed in reflected light.
Later in the day, I was at the movies. For some reason, when coming out of the cinema, I remembered the rash vest and the flash of droplets. Hmm, I thought, “a gleam of gold….”
…A near forgotten memory: I vaguely recalled a flash of gold while taking of my rash vest in the pool shower some days ago. Was it after my previous swim or the week earlier, I couldn’t be sure. But I distinctly remembered it had bothered me enough to check the floor of the cubicle cursorily. Finding nothing, I had completely forgotten about it and moved on.
Could it have come off there?
As I thought about it some more, possibility turned to plausibility: I was convinced it was what had happened. Although unlikely I would find it there now, it was worth a try on the hope that someone had found the chain and turned it in as lost property.
I stopped over at the pool on my way back from the movies and asked at reception.
“A gold chain? Hmm, I think you may be in luck,” he said, “I was doing an inventory of lost property last week and came across a chain. I was wondering why no one had come in to claim something so valuable.”
“You’re kidding,” I said, incredulous. “You mean you have a gold chain?”
“Yeah, and I’m pretty sure it will still be there unless someone else has claimed it,” he replied. “I’ll have a look in the safe. Can you describe it for me?”
I described it down to the brief inscription on the clasp.
“Wait here,” he said, “I’ll be a sec”
It took longer than that but he soon emerged, chain in hand.
I could not believe my eyes; I had given up on ever recovering it. “Thanks, so much” I said fervently, “you won’t believe how much it means to me to have found this.”
“No worries mate,” he said, smiling broadly. “Happy to have helped.”
–x–
Endnote: in case you haven’t read it yet, I recommend you take ten minutes to read Sherlock Holmes’ finest adventure and abductive masterpiece, The Adventure of the Speckled Band.
Learning, evolution and the future of work
The Janusheaded rise of AI has prompted many discussions about the future of work. Most, if not all, are about AIdriven automation and its consequences for various professions. We are warned to prepare for this change by developing skills that cannot be easily “learnt” by machines. This sounds reasonable at first, but less so on reflection: if skills that were thought to be uniquely human less than a decade ago can now be done, at least partially, by machines, there is no guarantee that any specific skill one chooses to develop will remain automationproof in the mediumterm future.
This begs the question as to what we can do, as individuals, to prepare for a machinecentric workplace. In this post I offer a perspective on this question based on Gregory Bateson’s writings as well as my consulting and teaching experiences.
Levels of learning
Given that humans are notoriously poor at predicting the future, it should be clear hitching one’s professional wagon to a specific set of skills is not a good strategy. Learning a set of skills may pay off in the short term, but it is unlikely to work in the long run.
So what can one do to prepare for an ambiguous and essentially unpredictable future?
To answer this question, we need to delve into an important, yet oftoverlooked aspect of learning.
A key characteristic of learning is that it is driven by trial and error. To be sure, intelligence may help winnow out poor choices at some stages of the process, but one cannot eliminate error entirely. Indeed, it is not desirable to do so because error is essential for that “aha” instant that precedes insight. Learning therefore has a stochastic element: the specific sequence of trial and error followed by an individual is unpredictable and likely to be unique. This is why everyone learns differently: the mental model I build of a concept is likely to be different from yours.
In a paper entitled, The Logical Categories of Learning and Communication, Bateson noted that the stochastic nature of learning has an interesting consequence. As he notes:
If we accept the overall notion that all learning is in some degree stochastic (i.e., contains components of “trial and error”), it follows that an ordering of the processes of learning can be built upon a hierarchic classification of the types of error which are to be corrected in the various learning processes.
Let’s unpack this claim by looking at his proposed classification:
Zero order learning – Zero order learning refers to situations in which a given stimulus (or question) results in the same response (or answer) every time. Any instinctive behaviour – such as a reflex response on touching a hot kettle – is an example of zero order learning. Such learning is hardwired in the learner, who responds with the “correct” option to a fixed stimulus every single time. Since the response does not change with time, the process is not subject to trial and error.
First order learning (Learning I) – Learning I is where an individual learns to select a correct option from a set of similar elements. It involves a specific kind of trial and error that is best explained through a couple of examples. The canonical example of Learning I is memorization: Johnny recognises the letter “A” because he has learnt to distinguish it from the 25 other similar possibilities. Another example is Pavlovian conditioning wherein the subject’s response is altered by training: a dog that initially salivates only when it smells food is trained, by repetition, to salivate when it hears the bell.
A key characteristic of Learning I is that the individual learns to select the correct response from a set of comparable possibilities – comparable because the possibilities are of the same type (e.g. pick a letter from the set of alphabets). Consequently, first order learning cannot lead to a qualitative change in the learner’s response. Much of traditional school and university teaching is geared toward first order learning: students are taught to develop the “correct” understanding of concepts and techniques via a repetitionbased process of trial and error.
As an aside, note that much of what goes under the banner of machine learning and AI can be also classed as first order learning.
Second order learning (Learning II) – Second order learning involves a qualitative change in the learner’s response to a given situation. Typically, this occurs when a learner sees a familiar problem situation in a completely new light, thus opening up new possibilities for solutions. Learning II therefore necessitates a higher order of trial and error, one that is beyond the ken of machines, at least at this point in time.
Complex organisational problems, such as determining a business strategy, require a second order approach because they cannot be precisely defined and therefore lack an objectively correct solution. Echoing Horst Rittel, solutions to such problems are not true or false, but better or worse.
Much of the teaching that goes on in schools and universities hinders second order learning because it implicitly conditions learners to frame problems in ways that make them amenable to familiar techniques. However, as Russell Ackoff noted, “outside of school, problems are seldom given; they have to be taken, extracted from complex situations…” Two aspects of this perceptive statement bear further consideration. Firstly, to extract a problem from a situation one has to appreciate or make sense of the situation. Secondly, once the problem is framed, one may find that solving it requires skills that one does not possess. I expand on the implications of these points in the following two sections.
Sensemaking and second order learning
In an earlier piece, I described sensemaking as the art of collaborative problem formulation. There are a huge variety of sensemaking approaches, the gamestorming site describes many of them in detail. Most of these are aimed at exploring a problem space by harnessing the collective knowledge of a group of people who have diverse, even conflicting, perspectives on the issue at hand. The greater the diversity, the more complete the exploration of the problem space.
Sensemaking techniques help in elucidating the context in which a problem lives. This refers to the the problem’s environment, and in particular the constraints that the environment imposes on potential solutions to the problem. As Bateson puts it, context is “a collective term for all those events which tell an organism among what set of alternatives [it] must make [its] next choice.” But this begs the question as to how these alternatives are to be determined. The question cannot be answered directly because it depends on the specifics of the environment in which the problem lives. Surfacing these by asking the right questions is the task of sensemaking.
As a simple example, if I request you to help me formulate a business strategy, you are likely to begin by asking me a number of questions such as:
 What kind of business are you in?
 Who are your customers?
 What’s the competitive landscape?
 …and so on
Answers to these questions fill out the context in which the business operates, thus making it possible to formulate a meaningful strategy.
It is important to note that context rarely remains static, it evolves in time. Indeed, many companies faded away because they failed to appreciate changes in their business context: Kodak is a wellknown example, there are many more. So organisations must evolve too. However, it is a mistake to think of an organisation and its environment as evolving independently, the two always evolve together. Such coevolution is as true of natural systems as it is of social ones. As Bateson tells us:
…the evolution of the horse from Eohippus was not a onesided adjustment to life on grassy plains. Surely the grassy plains themselves evolved [on the same footing] with the evolution of the teeth and hooves of the horses and other ungulates. Turf was the evolving response of the vegetation to the evolution of the horse. It is the context which evolves.
Indeed, one can think of evolution by natural selection as a process by which nature learns (in a secondorder sense).
The foregoing discussion points to another problem with traditional approaches to education: we are implicitly taught that problems once solved, stay solved. It is seldom so in real life because, as we have noted, the environment evolves even if the organisation remains static. In the worst case scenario (which happens often enough) the organisation will die if it does not adapt appropriately to changes in its environment. If this is true, then it seems that secondorder learning is important not just for individuals but for organisations as a whole. This harks back to notion of the notion of the learning organisation, developed and evangelized by Peter Senge in the early 90s. A learning organisation is one that continually adapts itself to a changing environment. As one might imagine, it is an ideal that is difficult to achieve in practice. Indeed, attempts to create learning organisations have often ended up with paradoxical outcomes. In view of this it seems more practical for organisations to focus on developing what one might call learning individuals – people who are capable of adapting to changes in their environment by continual learning.
Learning to learn
Cliches aside, the modern workplace is characterised by rapid, technologydriven change. It is difficult for an individual to keep up because one has to:

 Figure out which changes are significant and therefore worth responding to.
 Be capable of responding to them meaningfully.
The media hype about the sexiest job of the 21st century and the like further fuel the fear of obsolescence. One feels an overwhelming pressure to do something. The old adage about combating fear with action holds true: one has to do something, but the question then is: what meaningful action can one take?
The fact that this question arises points to the failure of traditional university education. With its undue focus on teaching specific techniques, the more important secondorder skill of learning to learn has fallen by the wayside. In reality, though, it is now easier than ever to learn new skills on ones own. When I was hired as a database architect in 2004, there were few quality resources available for free. Ten years later, I was able to start teaching myself machine learning using topnotch software, backed by countless quality tutorials in blog and video formats. However, I wasted a lot of time in getting started because it took me a while to get over my reluctance to explore without a guide. Cultivating the habit of learning on my own earlier would have made it a lot easier.
Back to the future of work
When industry complains about new graduates being illprepared for the workplace, educational institutions respond by updating curricula with more (New!! Advanced!!!) techniques. However, the complaints continue and Bateson’s notion of second order learning tells us why:
 Firstly, problem solving is distinct from problem formulation; it is akin to the distinction between human and machine intelligence.
 Secondly, one does not know what skills one may need in the future, so instead of learning specific skills one has to learn how to learn
In my experience, it is possible to teach these higher order skills to students in a classroom environment. However, it has to be done in a way that starts from where students are in terms of skills and dispositions and moves them gradually to less familiar situations. The approach is based on David Cavallo’s work on emergent design which I have often used in my consulting work. Two examples may help illustrate how this works in the classroom:
 Many analyticallyinclined people think sensemaking is a waste of time because they see it as “just talk”. So, when teaching sensemaking, I begin with quantitative techniques to deal with uncertainty, such as Monte Carlo simulation, and then gradually introduce examples of uncertainties that are hard if not impossible to quantify. This progression naturally leads on to problem situations in which they see the value of sensemaking.
 When teaching data science, it is difficult to comprehensively cover basic machine learning algorithms in a single semester. However, students are often reluctant to explore on their own because they tend to be daunted by the mathematical terminology and notation. To encourage exploration (i.e. learning to learn) we use a twostep approach: a) classes focus on intuitive explanations of algorithms and the commonalities between concepts used in different algorithms. The classes are not lectures but interactive sessions involving lots of exercises and Q&A, b) the assignments go beyond what is covered in the classroom (but still well within reach of most students), this forces them to learn on their own. The approach works: just the other day, my wonderful coteacher, Alex, commented on the amazing learning journey of some of the students – so tentative and hesitant at first, but well on their way to becoming confident data professionals.
In the end, though, whether or not an individual learner learns depends on the individual. As Bateson once noted:
Perhaps the best documented generalization in the field of psychology is that, at any given moment, the behavioral characteristics of a mammal, and especially of [a human], depend upon the previous experience and behavior of that individual.
The choices we make when faced with change depend on our individual natures and experiences. Educators can’t do much about the former but they can facilitate more meaningful instances of the latter, even within the confines of the classroom.
An intuitive introduction to support vector machines using R – Part 1
About a year ago, I wrote a piece on support vector machines as a part of my gentle introduction to data science R series. So it is perhaps appropriate to begin this piece with a few words about my motivations for writing yet another article on the topic.
Late last year, a curriculum lead at DataCamp got in touch to ask whether I’d be interested in developing a course on SVMs for them.
My answer was, obviously, an enthusiastic “Yes!”
Instead of rehashing what I had done in my previous article, I thought it would be interesting to try an approach that focuses on building an intuition for how the algorithm works using examples of increasing complexity, supported by visualisation rather than math. This post is the first part of a twopart series based on this approach.
The article builds up some basic intuitions about support vector machines (abbreviated henceforth as SVM) and then focuses on linearly separable problems. Part 2 (to be released at a future date) will deal with radially separable and more complex data sets. The focus throughout is on developing an understanding what the algorithm does rather than the technical details of how it does it.
Prerequisites for this series are a basic knowledge of R and some familiarity with the ggplot package. However, even if you don’t have the latter, you should be able to follow much of what I cover so I encourage you to press on regardless.
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A one dimensional example
A soft drink manufacturer has two brands of their flagship product: Choke (sugar content of 11g/100ml) and ChokeR (sugar content 8g/100 ml). The actual sugar content can vary quite a bit in practice so it can sometimes be hard to figure out the brand given the sugar content. Given sugar content data for 25 samples taken randomly from both populations (see file sugar_content.xls), our task is to come up with a decision rule for determining the brand.
Since this is onevariable problem, the simplest way to discern if the samples fall into distinct groups is through visualisation. Here’s one way to do this using ggplot:
…and here’s the resulting plot:
Note that we’ve simulated a onedimensional plot by setting all the y values to 0.
From the plot, it is evident that the samples fall into distinct groups: low sugar content, bounded above by the 8.8 g/100ml sample and high sugar content, bounded below by the 10 g/100ml sample.
Clearly, any point that lies between the two points is an acceptable decision boundary. We could, for example, pick 9.1g/100ml and 9.7g/100ml. Here’s the R code with those points added in. Note that we’ve made the points a bit bigger and coloured them red to distinguish them from the sample points.
label=d_bounds$sep, size=2.5,
vjust=2, hjust=0.5, colour=”red”)
And here’s the plot:
Now, a bit about the decision rule. Say we pick the first point as the decision boundary, the decision rule would be:
Say we pick 9.1 as the decision boundary, our classifier (in R) would be:
The other one is left for you as an exercise.
Now, it is pretty clear that although either these points define an acceptable decision boundary, neither of them are the best. Let’s try to formalise our intuitive notion as to why this is so.
The margin is the distance between the points in both classes that are closest to the decision boundary. In case at hand, the margin is 1.2 g/100ml, which is the difference between the two extreme points at 8.8 g/100ml (ChokeR) and 10 g/100ml (Choke). It should be clear that the best separator is the one that lies halfway between the two extreme points. This is called the maximum margin separator. The maximum margin separator in the case at hand is simply the average of the two extreme points:
geom_point(data=mm_sep,aes(x=mm_sep$sep, y=c(0)), colour=”blue”, size=4)
And here’s the plot:
We are dealing with a one dimensional problem here so the decision boundary is a point. In a moment we will generalise this to a two dimensional case in which the boundary is a straight line.
Let’s close this section with some general points.
Remember this is a sample not the entire population, so it is quite possible (indeed likely) that there will be as yet unseen samples of ChokeR and Choke that have a sugar content greater than 8.8 and less than 10 respectively. So, the best classifier is one that lies at the greatest possible distance from both classes. The maximum margin separator is that classifier.
This toy example serves to illustrate the main aim of SVMs, which is to find an optimal separation boundary in the sense described here. However, doing this for real life problems is not so simple because life is not one dimensional. In the remainder of this article and its yettobewritten sequel, we will work through examples of increasing complexity so as to develop a good understanding of how SVMs work in addition to practical experience with using the popular SVM implementation in R.
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Linearly separable case
The next level of complexity is a two dimensional case (2 predictors) in which the classes are separated by a straight line. We’ll create such a dataset next.
Let’s begin by generating 200 points with attributes x1 and x2, randomly distributed between 0 and 1. Here’s the R code:
Let’s visualise the generated data using a scatter plot:
And here’s the plot
Now let’s classify the points that lie above the line x1=x2 as belonging to the class +1 and those that lie below it as belonging to class 1 (the class values are arbitrary choices, I could have chosen them to be anything at all). Here’s the R code:
Let’s modify the plot in Figure 4, colouring the points classified as +1n blue and those classified 1 red. For good measure, let’s also add in the decision boundary. Here’s the R code:
Note that the parameters in geom_abline() are derived from the fact that the line x1=x2 has slope 1 and y intercept 0.
Here’s the resulting plot:
Next let’s introduce a margin in the dataset. To do this, we need to exclude points that lie within a specified distance of the boundary. A simple way to approximate this is to exclude points that have x1 and x2 values that differ by less a prespecified value, delta. Here’s the code to do this with delta set to 0.05 units.
The check on the number of datapoints tells us that a number of points have been excluded.
Running the previous ggplot code block yields the following plot which clearly shows the reduced dataset with the depopulated region near the decision boundary:
Let’s add the margin boundaries to the plot. We know that these are parallel to the decision boundary and lie delta units on either side of it. In other words, the margin boundaries have slope=1 and y intercepts delta and –delta. Here’s the ggplot code:
And here’s the plot with the margins:
OK, so we have constructed a dataset that is linearly separable, which is just a short code for saying that the classes can be separated by a straight line. Further, the dataset has a margin, i.e. there is a “gap” so to speak, between the classes. Let’s save the dataset so that we can use it in the next section where we’ll take a first look at the svm() function in the e1071 package.
That done, we can now move on to…
Linear SVMs
Let’s begin by reading in the datafile we created in the previous section:
We then split the data into training and test sets using an 80/20 random split. There are many ways to do this. Here’s one:
The next step is to build the an SVM classifier model. We will do this using the svm() function which is available in the e1071 package. The svm() function has a range of parameters. I explain some of the key ones below, in particular, the following parameters: type, cost, kernel and scale. It is recommended to have a browse of the documentation for more details.
The type parameter specifies the algorithm to be invoked by the function. The algorithm is capable of doing both classification and regression. We’ll focus on classification in this article. Note that there are two types of classification algorithms, nu and C classification. They essentially differ in the way that they penalise margin and boundary violations, but can be shown to lead to equivalent results. We will stick with C classification as it is more commonly used. The “C” refers to the cost which we discuss next.
The cost parameter specifies the penalty to be applied for boundary violations. This parameter can vary from 0 to infinity (in practice a large number compared to 0, say 10^6 or 10^8). We will explore the effect of varying cost later in this piece. To begin with, however, we will leave it at its default value of 1.
The kernel parameter specifies the kind of function to be used to construct the decision boundary. The options are linear, polynomial and radial. In this article we’ll focus on linear kernels as we know the decision boundary is a straight line.
The scale parameter is a Boolean that tells the algorithm whether or not the datapoints should be scaled to have zero mean and unit variance (i.e. shifted by the mean and scaled by the standard deviation). Scaling is generally good practice to avoid undue influence of attributes that have unduly large numeric values. However, in this case we will avoid scaling as we know the attributes are bounded and (more important) we would like to plot the boundary obtained from the algorithm manually.
Building the model is a simple oneline call, setting appropriate values for the parameters:
We expect a linear model to perform well here since the dataset it is linear by construction. Let’s confirm this by calculating training and test accuracy. Here’s the code:
The perfect accuracies confirm our expectation. However, accuracies by themselves are misleading because the story is somewhat more nuanced. To understand why, let’s plot the predicted decision boundary and margins using ggplot. To do this, we have to first extract information regarding these from the svm model object. One can obtain summary information for the model by typing in the model name like so:
kernel = “linear”, scale = FALSE)
Which outputs the following: the function call, SVM type, kernel and cost (which is set to its default). In case you are wondering about gamma, although it’s set to 0.5 here, it plays no role in linear SVMs. We’ll say more about it in the sequel to this article in which we’ll cover more complex kernels. More interesting are the support vectors. In a nutshell, these are training dataset points that specify the location of the decision boundary. We can develop a better understanding of their role by visualising them. To do this, we need to know their coordinates and indices (position within the dataset). This information is stored in the SVM model object. Specifically, the index element of svm_model contains the indices of the training dataset points that are support vectors and the SV element lists the coordinates of these points. The following R code lists these explicitly (Note that I’ve not shown the outputs in the code snippet below):
Let’s use the indices to visualise these points in the training dataset. Here’s the ggplot code to do that:
And here is the plot:
We now see that the support vectors are clustered around the boundary and, in a sense, serve to define it. We will see this more clearly by plotting the predicted decision boundary. To do this, we need its slope and intercept. These aren’t available directly available in the svm_model, but they can be extracted from the coefs, SV and rho elements of the object.
The first step is to use coefs and the support vectors to build the what’s called the weight vector. The weight vector is given by the product of the coefs matrix with the matrix containing the SVs. Note that the fact that only the support vectors play a role in defining the boundary is consistent with our expectation that the boundary should be fully specified by them. Indeed, this is often touted as a feature of SVMs in that it is one of the few classifiers that depends on only a small subset of the training data, i.e. the datapoints closest to the boundary rather than the entire dataset.
Once we have the weight vector, we can calculate the slope and intercept of the predicted decision boundary as follows:
Note that the slope and intercept are quite different from the correct values of 1 and 0 (reminder: the actual decision boundary is the line x1=x2 by construction). We’ll see how to improve on this shortly, but before we do that, let’s plot the decision boundary using the slope and intercept we have just calculated. Here’s the code:
And here’s the augmented plot:
The plot clearly shows how the support vectors “support” the boundary – indeed, if one draws line segments from each of the points to the boundary in such a way that the intersect the boundary at right angles, the lines can be thought of as “holding the boundary in place”. Hence the term support vector.
This is a good time to mention that the e1071 library provides a builtin plot method for svm function. This is invoked as follows:
The svm plot function takes a formula specifying the plane on which the boundary is to be plotted. This is not necessary here as we have only two predictors (x1 and x2) which automatically define a plane.
Here is the plot generated by the above code:
Note that the axes are switched (x1 is on the y axis). Aside from that, the plot is reassuringly similar to our ggplot version in Figure 9. Also note that that the support vectors are marked by “x”. Unfortunately the built in function does not display the margin boundaries, but this is something we can easily add to our homebrewed plot. Here’s how. We know that the margin boundaries are parallel to the decision boundary, so all we need to find out is their intercept. It turns out that the intercepts are offset by an amount 1/w[2] units on either side of the decision boundary. With that information in hand we can now write the the code to add in the margins to the plot shown in Figure 9. Here it is:
geom_abline(slope=slope_1,intercept = intercept_1+1/w[2], linetype=”dashed”)
And here is the plot with the margins added in:
Note that the predicted margins are much wider than the actual ones (compare with Figure 7). As a consequence, many of the support vectors lie within the predicted margin – that is, they violate it. The upshot of the wide margin is that the decision boundary is not tightly specified. This is why we get a significant difference between the slope and intercept of predicted decision boundary and the actual one. We can sharpen the boundary by narrowing the margin. How do we do this? We make margin violations more expensive by increasing the cost. Let’s see this marginnarrowing effect in action by building a model with cost = 100 on the same training dataset as before. Here is the code:
I’ll leave you to calculate the training and test accuracies (as one might expect, these will be perfect).
Let’s inspect the cost=100 model:
kernel = “linear”,cost=100, scale = FALSE)
The number of support vectors is reduced from 55 to 6! We can plot these and the boundary / margin lines using ggplot as before. The code is identical to the previous case (see code block preceding Figure 8). If you run it, you will get the plot shown in Figure 12.
Since the boundary is more tightly specified, we would expect the slope and intercept of the predicted boundary to be considerably closer to their actual values of 1 and 0 respectively (as compared to the default cost case). Let’s confirm that this is so by calculating the slope and intercept as we did in the code snippets preceding Figure 9. Here’s the code:
Which nicely confirms our expectation.
The decision boundary and margins for the high cost case can also be plotted with the code shown earlier. Her it is for completeness:
geom_abline(slope=slope_100,intercept = intercept_100+1/w[2], linetype=”dashed”)
And here’s the plot:
SVMs that allow margin violations are called soft margin classifiers and those that do not are called hard. In this case, the hard margin classifier does a better job because it specifies the boundary more accurately than its soft counterpart. However, this does not mean that hard margin classifier are to be preferred over soft ones in all situations. Indeed, in real life, where we usually do not know the shape of the decision boundary upfront, soft margin classifiers can allow for a greater degree of uncertainty in the decision boundary thus improving generalizability of the classifier.
OK, so now we have a good feel for what the SVM algorithm does in the linearly separable case. We will round out this article by looking at a real world dataset that fortuitously turns out to be almost linearly separable: the famous (notorious?) iris dataset. It is instructive to look at this dataset because it serves to illustrate another feature of the e1071 SVM algorithm – its capability to handle classification problems that have more than 2 classes.
A multiclass problem
The iris dataset is wellknown in the machine learning community as it features in many introductory courses and tutorials. It consists of 150 observations of 3 species of the iris flower – setosa, versicolor and virginica. Each observation consists of numerical values for 4 independent variables (predictors): petal length, petal width, sepal length and sepal width. The dataset is available in a standard installation of R as a built in dataset. Let’s read it in and examine its structure:
Now, as it turns out, petal length and petal width are the key determinants of species. So let’s create a scatterplot of the datapoints as a function of these two variables (i.e. project each data point on the petal lengthpetal width plane). We will also distinguish between species using different colour. Here’s the ggplot code to do this:
And here’s the plot:
On this plane we see a clear linear boundary between setosa and the other two species, versicolor and virginica. The boundary between the latter two is almost linear. Since there are four predictors, one would have to plot the other combinations to get a better feel for the data. I’ll leave this as an exercise for you and move on with the assumption that the data is nearly linearly separable. If the assumption is grossly incorrect, a linear SVM will not work well.
Up until now, we have discussed binary classification problem, i.e. those in which the predicted variable can take on only two values. In this case, however, the predicted variable, Species, can take on 3 values (setosa, versicolor and virginica). 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 SVM algorithm does this using a oneagainstone classification strategy. Here’s how it works:
 Divide the dataset (assumed to have N classes) into N(N1)/2 datasets that have two classes each.
 Solve the binary classification problem for each of these subsets
 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, let’s get on with building the classifier. As before, we begin by splitting the data into training and test sets using an 80/20 random split. Here is the code to do this:
Then we build the model (default cost) and examine it:
The main thing to note is that the function call is identical to the binary classification case. We get some basic information about the model by typing in the model name as before:
kernel = “linear”)
And the train and test accuracies are computed in the usual way:
This looks good, but is potentially misleading because it is for a particular train/test split. Remember, in this case, unlike the earlier example, we do not know the shape of the actual decision boundary. So, to get a robust measure of accuracy, we should calculate the average test accuracy over a number of train/test partitions. Here’s some code to do that:
Which is not too bad at all, indicating that the dataset is indeed nearly linearly separable. If you try different values of cost you will see that it does not make much difference to the average accuracy.
This is a good note to close this piece on. Those who have access to DataCamp premium courses will find that the content above is covered in chapters 1 and 2 of the course on support vector machines in R. The next article in this twopart series will cover chapters 3 and 4.
Summarising
My main objective in this article was to help develop an intuition for how SVMs work in simple cases. We illustrated the basic principles and terminology with a simple 1 dimensional example and then worked our way to linearly separable binary classification problems with multiple predictors. We saw how the latter can be solved using a popular svm implementation available in R. We also saw that the algorithm can handle multiclass problems. All through, we used visualisations to see what the algorithm does and how the key parameters affect the decision boundary and margins.
In the next part (yet to be written) we will see how SVMs can be generalised to deal with complex, nonlinear decision boundaries. In essence, the use a mathematical trick to “linearise” these boundaries. We’ll delve into details of this trick in an intuitive, visual way as we have done here.
Many thanks for reading!
A gentle introduction to Monte Carlo simulation for project managers
This article covers the why, what and how of Monte Carlo simulation using a canonical example from project management – estimating the duration of a small project. Before starting, however, I’d like say a few words about the tool I’m going to use.
Despite the bad rap spreadsheets get from tech types – and I have to admit that many of their complaints are justified – the fact is, Excel remains one of the most ubiquitous “computational” tools in the corporate world. Most business professionals would have used it at one time or another. So, if you you’re a project manager and want the rationale behind your estimates to be accessible to the widest possible audience, you are probably better off presenting them in Excel than in SPSS, SAS, Python, R or pretty much anything else. Consequently, the tool I’ll use in this article is Microsoft Excel. For those who know about Monte Carlo and want to cut to the chase, here’s the Excel workbook containing all the calculations detailed in later sections. However, if you’re unfamiliar with the technique, you may want to have a read of the article before playing with the spreadsheet.
In keeping with the format of the tutorials on this blog, I’ve assumed very little prior knowledge about probability, let alone Monte Carlo simulation. Consequently, the article is verbose and the tone somewhat didactic.
Introduction
Estimation is key part of a project manager’s role. The most frequent (and consequential) estimates they are asked deliver relate to time and cost. Often these are calculated and presented as point estimates: i.e. single numbers – as in, this task will take 3 days. Or, a little better, as twopoint ranges – as in, this task will take between 2 and 5 days. Better still, many use a PERTlike approach wherein estimates are based on 3 points: best, most likely and worst case scenarios – as in, this task will take between 2 and 5 days, but it’s most likely that we’ll finish on day 3. We’ll use threepoint estimates as a starting point for Monte Carlo simulation, but first, some relevant background.
It is a truism, well borne out by experience, that it is easier to estimate small, simple tasks than large, complex ones. Indeed, this is why one of the early todos in a project is the construction of a work breakdown structure. However, a problem arises when one combines the estimates for individual elements into an overall estimate for a project or a phase thereof. It is that a straightforward addition of individual estimates or bounds will almost always lead to a grossly incorrect estimation of overall time or cost. The reason for this is simple: estimates are necessarily based on probabilities and probabilities do not combine additively. Monte Carlo simulation provides a principled and intuitive way to obtain probabilistic estimates at the level of an entire project based on estimates of the individual tasks that comprise it.
The problem
The best way to explain Monte Carlo is through a simple worked example. So, let’s consider a 4 task project shown in Figure 1. In the project, the second task is dependent on the first, and third and fourth are dependent on the second but not on each other. The upshot of this is that the first two tasks have to be performed sequentially and the last two can be done at the same time, but can only be started after the second task is completed.
To summarise: the first two tasks must be done in series and the last two can be done in parallel.
Figure 1 also shows the three point estimates for each task – that is the minimum, maximum and most likely completion times. For completeness I’ve listed them below:
 Task 1 – Min: 2 days; Most Likely: 4 days; Max: 8 days
 Task 2 – Min: 3 days; Most Likely: 5 days; Max: 10 days
 Task 3 – Min: 3 days; Most Likely: 6 days; Max: 9 days
 Task 4 – Min: 2 days; Most Likely: 4 days; Max: 7 days
OK, so that’s the situation as it is given to us. The first step to developing an estimate is to formulate the problem in a way that it can be tackled using Monte Carlo simulation. This bring us to the important topic of the shape of uncertainty aka probability distributions.
The shape of uncertainty
Consider the data for Task 1. You have been told that it most often finishes on day 4. However, if things go well, it could take as little as 2 days; but if things go badly it could take as long as 8 days. Therefore, your range of possible finish times (outcomes) is between 2 to 8 days.
Clearly, each of these outcomes is not equally likely. The most likely outcome is that you will finish the task in 4 days (from what your team member has told you). Moreover, the likelihood of finishing in less than 2 days or more than 8 days is zero. If we plot the likelihood of completion against completion time, it would look something like Figure 2.
Figure 2 begs a couple of questions:
 What are the relative likelihoods of completion for all intermediate times – i.e. those between 2 to 4 days and 4 to 8 days?
 How can one quantify the likelihood of intermediate times? In other words, how can one get a numerical value of the likelihood for all times between 2 to 8 days? Note that we know from the earlier discussion that this must be zero for any time less than 2 or greater than 8 days.
The two questions are actually related. As we shall soon see, once we know the relative likelihood of completion at all times (compared to the maximum), we can work out its numerical value.
Since we don’t know anything about intermediate times (I’m assuming there is no other historical data available), the simplest thing to do is to assume that the likelihood increases linearly (as a straight line) from 2 to 4 days and decreases in the same way from 4 to 8 days as shown in Figure 3. This gives us the wellknown triangular distribution.
Jargon Buster: The term distribution is simply a fancy word for a plot of likelihood vs. time.
Of course, this isn’t the only possibility; there are an infinite number of others. Figure 4 is another (admittedly weird) example.
Further, it is quite possible that the upper limit (8 days) is not a hard one. It may be that in exceptional cases the task could take much longer (for example, if your team member calls in sick for two weeks) or even not be completed at all (for example, if she then leaves for that mythical greener pasture). Catering for the latter possibility, the shape of the likelihood might resemble Figure 5.
The main takeaway from the above is that uncertainties should be expressed as shapes rather than numbers, a notion popularised by Sam Savage in his book, The Flaw of Averages.
[Aside: you may have noticed that all the distributions shown above are skewed to the right – that is they have a long tail. This is a general feature of distributions that describe time (or cost) of project tasks. It would take me too far afield to discuss why this is so, but if you’re interested you may want to check out my post on the inherent uncertainty of project task estimates.
From likelihood to probability
Thus far, I have used the word “likelihood” without bothering to define it. It’s time to make the notion more precise. I’ll begin by asking the question: what common sense properties do we expect a quantitative measure of likelihood to have?
Consider the following:
 If an event is impossible, its likelihood should be zero.
 The sum of likelihoods of all possible events should equal complete certainty. That is, it should be a constant. As this constant can be anything, let us define it to be 1.
In terms of the example above, if we denote time by and the likelihood by then:
for and
And
where
Where denotes the sum of all nonzero likelihoods – i.e. those that lie between 2 and 8 days. In simple terms this is the area enclosed by the likelihood curves and the x axis in figures 2 to 5. (Technical Note: Since is a continuous variable, this should be denoted by an integral rather than a simple sum, but this is a technicality that need not concern us here)
is , in fact, what mathematicians call probability– which explains why I have used the symbol rather than . Now that I’ve explained what it is, I’ll use the word “probability” instead of ” likelihood” in the remainder of this article.
With these assumptions in hand, we can now obtain numerical values for the probability of completion for all times between 2 and 8 days. This can be figured out by noting that the area under the probability curve (the triangle in figure 3 and the weird shape in figure 4) must equal 1, and we’ll do this next. Indeed, for the problem at hand, we’ll assume that all four task durations can be fitted to triangular distributions. This is primarily to keep things simple. However, I should emphasise that you can use any shape so long as you can express it mathematically, and I’ll say more about this towards the end of this article.
The triangular distribution
Let’s look at the estimate for Task 1. We have three numbers corresponding to a minimum, most likely and maximum time. To keep the discussion general, we’ll call these , and respectively, (we’ll get back to our estimator’s specific numbers later).
Now, what about the probabilities associated with each of these times?
Since and correspond to the minimum and maximum times, the probability associated with these is zero. Why? Because if it wasn’t zero, then there would be a nonzero probability of completion for a time less than or greater than – which isn’t possible [Note: this is a consequence of the assumption that the probability varies continuously – so if it takes on nonzero value, , at then it must take on a value slightly less than – but greater than 0 – at slightly smaller than ] . As far as the most likely time, , is concerned: by definition, the probability attains its highest value at time . So, assuming the probability can be described by a triangular function, the distribution must have the form shown in Figure 6 below.
For the simulation, we need to know the equation describing the above distribution. Although Wikipedia will tell us the answer in a mouseclick, it is instructive to figure it out for ourselves. First, note that the area under the triangle must be equal to 1 because the task must finish at some time between and . As a consequence we have:
where is the probability corresponding to time . With a bit of rearranging we get,
To derive the probability for any time lying between and , we note that:
This is a consequence of the fact that the ratios on either side of equation (3) are equal to the slope of the line joining the points and .
Substituting (2) in (3) and simplifying a bit, we obtain:
for
In a similar fashion one can show that the probability for times lying between and is given by:
for
Equations 4 and 5 together describe the probability distribution function (or PDF) for all times between and .
As it turns out, in Monte Carlo simulations, we don’t directly work with the probability distribution function. Instead we work with the cumulative distribution function (or CDF) which is the probability, , that the task is completed by time . To reiterate, the PDF, , is the probability of the task finishing at time whereas the CDF, , is the probability of the task completing by time . The CDF, , is essentially a sum of all probabilities between and . For this is the area under the triangle with apexes at (, 0), (t, 0) and (t, p(t)). Using the formula for the area of a triangle (1/2 base times height) and equation (4) we get:
for
Noting that for , the area under the curve equals the total area minus the area enclosed by the triangle with base between t and , we have:
for
As expected, starts out with a value 0 at and then increases monotonically, attaining a value of 1 at .
To end this section let’s plug in the numbers quoted by our estimator at the start of this section: , and . The resulting PDF and CDF are shown in figures 8 and 9.
Monte Carlo in a minute
Now with all that conceptual work done, we can get to the main topic of this post: Monte Carlo estimation. The basic idea behind Monte Carlo is to simulate the entire project (all 4 tasks in this case) a large number N (say 10,000) times and thus obtain N overall completion times. In each of the N trials, we simulate each of the tasks in the project and add them up appropriately to give us an overall project completion time for the trial. The resulting N overall completion times will all be different, ranging from the sum of the minimum completion times to the sum of the maximum completion times. In other words, we will obtain the PDF and CDF for the overall completion time, which will enable us to answer questions such as:
 How likely is it that the project will be completed within 17 days?
 What’s the estimated time for which I can be 90% certain that the project will be completed? For brevity, I’ll call this the 90% completion time in the rest of this piece.
“OK, that sounds great”, you say, “but how exactly do we simulate a single task”?
Good question, and I was just about to get to that…
Simulating a single task using the CDF
As we saw earlier, the CDF for the triangular has a S shape and ranges from 0 to 1 in value. It turns out that the S shape is characteristic of all CDFs, regardless of the details underlying PDF. Why? Because, the cumulative probability must lie between 0 and 1 (remember, probabilities can never exceed 1, nor can they be negative).
OK, so to simulate a task, we:
 generate a random number between 0 and 1, this corresponds to the probability that the task will finish at time t.
 find the time, t, that this corresponds to this value of probability. This is the completion time for the task for this trial.
Incidentally, this method is called inverse transform sampling.
An example might help clarify how inverse transform sampling works. Assume that the random number generated is 0.4905. From the CDF for the first task, we see that this value of probability corresponds to a completion time of 4.503 days, which is the completion for this trial (see Figure 10). Simple!
In this case we found the time directly from the computed CDF. That’s not too convenient when you’re simulating the project 10,000 times. Instead, we need a programmable math expression that gives us the time corresponding to the probability directly. This can be obtained by solving equations (6) and (7) for . Some straightforward algebra, yields the following two expressions for :
for
And
for
These can be easily combined in a single Excel formula using an IF function, and I’ll show you exactly how in a minute. Yes, we can now finally get down to the Excel simulation proper and you may want to download the workbook if you haven’t done so already.
The simulation
Open up the workbook and focus on the first three columns of the first sheet to begin with. These simulate the first task in Figure 1, which also happens to be the task we have used to illustrate the construction of the triangular distribution as well as the mechanics of Monte Carlo.
Rows 2 to 4 in columns A and B list the min, most likely and max completion times while the same rows in column C list the probabilities associated with each of the times. For the probability is 0 and for it is 1. The probability at can be calculated using equation (6) which, for , reduces to
Rows 6 through 10005 in column A are simulated probabilities of completion for Task 1. These are obtained via the Excel RAND() function, which generates uniformly distributed random numbers lying between 0 and 1. This gives us a list of probabilities corresponding to 10,000 independent simulations of Task 1.
The 10,000 probabilities need to be translated into completion times for the task. This is done using equations (8) or (9) depending on whether the simulated probability is less or greater than , which is in cell C3 (and given by Equation (10) above). The conditional statement can be coded in an Excel formula using the IF() function.
Tasks 24 are coded in exactly the same way, with distribution parameters in rows 2 through 4 and simulation details in rows 6 through 10005 in the columns listed below:
 Task 2 – probabilities in column D; times in column F
 Task 3 – probabilities in column H; times in column I
 Task 4 – probabilities in column K; times in column L
That’s basically it for the simulation of individual tasks. Now let’s see how to combine them.
For tasks in series (Tasks 1 and 2), we simply sum the completion times for each task to get the overall completion times for the two tasks. This is what’s shown in rows 6 through 10005 of column G.
For tasks in parallel (Tasks 3 and 4), the overall completion time is the maximum of the completion times for the two tasks. This is computed and stored in rows 6 through 10005 of column N.
Finally, the overall project completion time for each simulation is then simply the sum of columns G and N (shown in column O)
Sheets 2 and 3 are plots of the probability and cumulative probability distributions for overall project completion times. I’ll cover these in the next section.
Discussion – probabilities and estimates
The figure on Sheet 2 of the Excel workbook (reproduced in Figure 11 below) is the probability distribution function (PDF) of completion times. The xaxis shows the elapsed time in days and the yaxis the number of Monte Carlo trials that have a completion time that lie in the relevant time bin (of width 0.5 days). As an example, for the simulation shown in the Figure 11, there were 882 trials (out of 10,000) that had a completion time that lie between 16.25 and 16.75 days. Your numbers will vary, of course, but you should have a maximum in the 16 to 17 day range and a trial number that is reasonably close to the one I got.
I’ll say a bit more about Figure 11 in the next section. For now, let’s move on to Sheet 3 of workbook which shows the cumulative probability of completion by a particular day (Figure 12 below). The figure shows the cumulative probability function (CDF), which is the sum of all completion times from the earliest possible completion day to the particular day.
To reiterate a point made earlier, the reason we work with the CDF rather than the PDF is that we are interested in knowing the probability of completion by a particular date (e.g. it is 90% likely that we will finish by April 20th) rather than the probability of completion on a particular date (e.g. there’s a 10% chance we’ll finish on April 17th). We can now answer the two questions we posed earlier. As a reminder, they are:
 How likely is it that the project will be completed within 17 days?
 What’s the 90% likely completion time?
Both questions are easily answered by using the cumulative distribution chart on Sheet 3 (or Fig 12). Reading the relevant numbers from the chart, I see that:
 There’s a 60% chance that the project will be completed in 17 days.
 The 90% likely completion time is 19.5 days.
How does the latter compare to the sum of the 90% likely completion times for the individual tasks? The 90% likely completion time for a given task can be calculated by solving Equation 9 for $t$, with appropriate values for the parameters , and plugged in, and set to 0.9. This gives the following values for the 90% likely completion times:
 Task 1 – 6.5 days
 Task 2 – 8.1 days
 Task 3 – 7.7 days
 Task 4 – 5.8 days
Summing up the first three tasks (remember, Tasks 3 and 4 are in parallel) we get a total of 22.3 days, which is clearly an overestimation. Now, with the benefit of having gone through the simulation, it is easy to see that the sum of 90% likely completion times for individual tasks does not equal the 90% likely completion time for the sum of the relevant individual tasks – the first three tasks in this particular case. Why? Essentially because a Monte Carlo run in which the first three tasks tasks take as long as their (individual) 90% likely completion times is highly unlikely. Exercise: use the worksheet to estimate how likely this is.
There’s much more that can be learnt from the CDF. For example, it also tells us that the greatest uncertainty in the estimate is in the 5 day period from ~14 to 19 days because that’s the region in which the probability changes most rapidly as a function of elapsed time. Of course, the exact numbers are dependent on the assumed form of the distribution. I’ll say more about this in the final section.
To close this section, I’d like to reprise a point I mentioned earlier: that uncertainty is a shape, not a number. Monte Carlo simulations make the uncertainty in estimates explicit and can help you frame your estimates in the language of probability…and using a tool like Excel can help you explain these to nontechnical people like your manager.
Closing remarks
We’ve covered a fair bit of ground: starting from general observations about how long a task takes, saw how to construct simple probability distributions and then combine these using Monte Carlo simulation. Before I close, there are a few general points I should mention for completeness…and as warning.
First up, it should be clear that the estimates one obtains from a simulation depend critically on the form and parameters of the distribution used. The parameters are essentially an empirical matter; they should be determined using historical data. The form of the function, is another matter altogether: as pointed out in an earlier section, one cannot determine the shape of a function from a finite number of data points. Instead, one has to focus on the properties that are important. For example, is there a small but finite chance that a task can take an unreasonably long time? If so, you may want to use a lognormal distribution…but remember, you will need to find a sensible way to estimate the distribution parameters from your historical data.
Second, you may have noted from the probability distribution curve (Figure 11) that despite the skewed distributions of the individual tasks, the distribution of the overall completion time is somewhat symmetric with a minimum of ~9 days, most likely time of ~16 days and maximum of 24 days. It turns out that this is a general property of distributions that are generated by adding a large number of independent probabilistic variables. As the number of variables increases, the overall distribution will tend to the ubiquitous Normal distribution.
The assumption of independence merits a closer look. In the case it hand, it implies that the completion times for each task are independent of each other. As most project managers will know from experience, this is rarely the case: in real life, a task that is delayed will usually have knockon effects on subsequent tasks. One can easily incorporate such dependencies in a Monte Carlo simulation. A formal way to do this is to introduce a nonzero correlation coefficient between tasks as I have done here. A simpler and more realistic approach is to introduce conditional intertask dependencies As an example, one could have an intertask delay that kicks in only if the predecessor task takes more than 80% of its maximum time.
Thirdly, you may have wondered why I used 10,000 trials: why not 100, or 1000 or 20,000. This has to do with the tricky issue of convergence. In a nutshell, the estimates we obtain should not depend on the number of trials used. Why? Because if they did, they’d be meaningless.
Operationally, convergence means that any predicted quantity based on aggregates should not vary with number of trials. So, if our Monte Carlo simulation has converged, our prediction of 19.5 days for the 90% likely completion time should not change substantially if I increase the number of trials from ten to twenty thousand. I did this and obtained almost the same value of 19.5 days. The average and median completion times (shown in cell Q3 and Q4 of Sheet 1) also remained much the same (16.8 days). If you wish to repeat the calculation, be sure to change the formulas on all three sheets appropriately. I was lazy and hardcoded the number of trials. Sorry!
Finally, I should mention that simulations can be usefully performed at a higher level than individual tasks. In their highlyreadable book, Waltzing With Bears: Managing Risk on Software Projects, Tom De Marco and Timothy Lister show how Monte Carlo methods can be used for variables such as velocity, time, cost etc. at the project level as opposed to the task level. I believe it is better to perform simulations at the lowest possible level, the main reason being that it is easier, and less errorprone, to estimate individual tasks than entire projects. Nevertheless, high level simulations can be very useful if one has reliable data to base these on.
There are a few more things I could say about the usefulness of the generated distribution functions and Monte Carlo in general, but they are best relegated to a future article. This one is much too long already and I think I’ve tested your patience enough. Thanks so much for reading, I really do appreciate it and hope that you found it useful.
Acknowledgement: My thanks to Peter Holberton for pointing out a few typographical and coding errors in an earlier version of this article. These have now been fixed. I’d be grateful if readers could bring any errors they find to my attention.
Risk management and organizational anxiety
In practice risk management is a rational, meansend based process: risks are identified, analysed and then “solved” (or mitigated). Although these steps seem to be objective, each of them involves human perceptions, biases and interests. Where Jill sees an opportunity, Jack may see only risks.
Indeed, the problem of differences in stakeholder perceptions is broader than risk analysis. The recognition that such differences in worldviews may be irreconcilable is what led Horst Rittel to coin the now wellknown term, wicked problem. These problems tend to be made up of complex interconnected and interdependent issues which makes them difficult to tackle using standard rational analytical methods of problem solving.
Most highstakes risks that organisations face have elements of wickedness – indeed any significant organisational change is fraught with risk. Murphy rules; things can go wrong, and they often do. The current paradigm of risk management, which focuses on analyzing and quantifying risks using rational methods, is not broad enough to account for the wicked aspects of risk.
I had been thinking about this for a while when I stumbled on a fascinating paper by Robin Holt entitled, Risk Management: The Talking Cure, which outlines a possible approach to analysing interconnected risks. In brief, Holt draws a parallel between psychoanalysis (as a means to tackle individual anxiety) and risk management (as a means to tackle organizational anxiety). In this post, I present an extensive discussion and interpretation of Holt’s paper. Although more about the philosophy of risk management than its practice, I found the paper interesting, relevant and thought provoking. My hope is that some readers might find it so too.
Background
Holt begins by noting that modern life is characterized by uncertainty. Paradoxically, technological progress which should have increased our sense of control over our surroundings and lives has actually heightened our personal feelings of uncertainty. Moreover, this sense of uncertainty is not allayed by rational analysis. On the contrary, it may have even increased it by, for example, drawing our attention to risks that we may otherwise have remained unaware of. Risk thus becomes a lens through which we perceive the world. The danger is that this can paralyze. As Holt puts it,
…risk becomes the only backdrop to perceiving the world and perception collapses into selfinhibition, thereby compounding uncertainty through inertia.
Most individuals know this through experience: most of us have at one time or another been frozen into inaction because of perceived risks. We also “know” at a deep personal level that the standard responses to risk are inadequate because many of our worries tend to be inchoate and therefore can neither be coherently articulated nor analysed. In Holt’s words:
..People do not recognize [risk] from the perspective of a breakdown in their rational calculations alone, but because of threats to their forms of life – to the noncalculative way they see themselves and the world. [Mainstream risk analysis] remains caught in the thrall of its own ‘expert’ presumptions, denigrating the very lay knowledge and perceptions on the grounds that they cannot be codified and institutionally expressed.
Holt suggests that risk management should account for the “codified, uncodified and uncodifiable aspects of uncertainty from an organizational perspective.” This entails a mode of analysis that takes into account different, even conflicting, perspectives in a nonjudgemental way. In essence, he suggests “talking it over” as a means to increase awareness of the contingent nature of risks rather than a means of definitively resolving them.
Shortcomings of risk analysis
The basic aim of risk analysis (as it is practiced) is to contain uncertainty within set bounds that are determined by an organisation’s risk appetite. As mentioned earlier, this process begins by identifying and classifying risks. Once this is done, one determines the probability and impact of each risk. Then, based on priorities and resources available (again determined by the organisation’s risk appetite) one develops strategies to mitigate the risks that are significant from the organisation’s perspective.
However, the messiness of organizational life makes it difficult to see risk in such a clearcut way. We may pretend to be rational about it, but in reality we perceive it through the lens of our background, interests , experiences. Based on these perceptions we rationalize our action (or inaction!) and simply get on with life. As Holt writes:
The concept [of risk] refers to…the mélange of experience, where managers accept contingencies without being overwhelmed to a point of complete passivity or confusion, Managers learn to recognize the differences between things, to acknowledge their and our limits. Only in this way can managers be said to make judgements, to be seen as being involved in something called the future.
Then, in a memorable line, he goes on to say:
The future, however, lasts a long time, so much so as to make its containment and prediction an often futile exercise.
Although one may well argue that this is not the case for many organizational risks, it is undeniable that certain mitigation strategies (for example, accepting risks that turn out to be significant later) may have significant consequences in the notsonear future.
Advice from a politicianscholar
So how can one address the slippery aspects of risk – the things people sense intuitively, but find difficult to articulate?
Taking inspiration from Machiavelli, Holt suggests reframing risk management as a means to determine wise actions in the face of the contradictory forces of fortune and necessity. As Holt puts it:
Necessity describes forces that are unbreachable but manageable by acceptance and containment—acts of God, tendencies of the species, and so on. In recognizing inevitability, [one can retain one’s] position, enhancing it only to the extent that others fail to recognize necessity. Far more influential, and often confused with necessity, is fortune. Fortune is elusive but approachable. Fortune is never to be relied upon: ‘The greatest good fortune is always least to be trusted’; the good is often kept underfoot and the ridiculous elevated, but it provides [one] with opportunity.
Wise actions involve resolve and cunning (which I interpret as political nous). This entails understanding that we do not have complete (or even partial) control over events that may occur in the future. The future is largely unknowable as are people’s true drives and motivations. Yet, despite this, managers must act. This requires personal determination together with a deep understanding of the social and political aspects of one’s environment.
And a little later,
…risk management is not the clear conception of a problem coupled to modes of rankable resolutions, or a limited process, but a judgemental analysis limited by the vicissitudes of budgets, programmes, personalities and contested priorities.
In short: risk management in practice tends to be a far way off from how it is portrayed in textbooks and the professional literature.
The wickedness of risk management
Most managers and those who work under their supervision have been schooled in the rationalscientific approach of problem solving. It is no surprise, therefore, that they use it to manage risks: they gather and analyse information about potential risks, formulate potential solutions (or mitigation strategies) and then implement the best one (according to predetermined criteria). However, this method works only for problems that are straightforward or tame, rather than wicked.
Many of the issues that risk managers are confronted with are wicked, messy or both. Often though, such problems are treated as being tame. Reducing a wicked or messy problem to one amenable to rational analysis invariably entails overlooking the views of certain stakeholder groups or, worse, ignoring key aspects of the problem. This may work in the short term, but will only exacerbate the problem in the longer run. Holt illustrates this point as follows:
A primary danger in mistaking a mess for a tame problem is that it becomes even more difficult to deal with the mess. Blaming ‘operator error’ for a mishap on the production line and introducing added surveillance is an illustration of a mess being mistaken for a tame problem. An operator is easily isolated and identifiable, whereas a technological system or process is embedded, unwieldy and, initially, far more costly to alter. Blaming operators is politically expedient. It might also be because managers and administrators do not know how to think in terms of messes; they have not learned how to sort through complex sociotechnical systems.
It is important to note that although many risk management practitioners recognize the essential wickedness of the issues they deal with, the practice of risk management is not quite up to the task of dealing with such matters. One step towards doing this is to develop a shared (enterprisewide) understanding of risks by soliciting input from diverse stakeholders groups, some of who may hold opposing views.
The skills required to do this are very different from the analytical techniques that are the focus of problem solving and decision making techniques that are taught in colleges and business schools. Analysis is replaced by sensemaking – a collaborative process that harnesses the wisdom of a group to arrive at a collective understanding of a problem and thence a common commitment to a course of action. This necessarily involves skills that do not appear in the lexicon of rational problem solving: negotiation, facilitation, rhetoric and those of the same ilk that are dismissed as being of no relevance by the scientifically oriented analyst.
In the end though, even this may not be enough: different stakeholders may perceive a given “risk” in have wildly different ways, so much so that no consensus can be reached. The problem is that the current framework of risk management requires the analyst to perform an objective analysis of situation/problem, even in situations where this is not possible.
To get around this Holt suggests that it may be more useful to see risk management as a way to encounter problems rather than analyse or solve them.
What does this mean?
He sees this as a forum in which people can talk about the risks openly:
To enable organizational members to encounter problems, risk management’s repertoire of activity needs to engage their all too human components: belief, perception, enthusiasm and fear.
This gets to the root of the problem: risk matters because it increases anxiety and generally affects peoples’ sense of wellbeing. Given this, it is no surprise that Holt’s proposed solution draws on psychoanalysis.
The analogy between psychoanalysis and risk management
Any discussion of psychoanalysis –especially one that is intended for an audience that is largely schooled in rational/scientific methods of analysis – must begin with the acknowledgement that the claims of psychoanalysis cannot be tested. That is, since psychoanalysis speaks of unobservable “objects” such as the ego and the unconscious, any claims it makes about these concepts cannot be proven or falsified.
However as Holt suggests, this is exactly what makes it a good fit for encountering (as opposed to analyzing) risks. In his words:
It is precisely because psychoanalysis avoids an overarching claim to produce testable, watertight, universal theories that it is of relevance for risk management. By so avoiding universal theories and formulas, risk management can afford to deviate from pronouncements using mathematical formulas to cover the ‘immanent indeterminables’ manifest in human perception and awareness and systems integration.
His point is that there is a clear parallel between psychoanalysis and the individual, and risk management and the organisation:
We understand ourselves not according to a template but according to our own peculiar, beguiling histories. Metaphorically, risk management can make explicit a similar realization within and between organizations. The revealing of an unconscious world and its being in a constant state of tension between excess and stricture, between knowledge and ignorance, is emblematic of how organizational members encountering messes, wicked problems and wicked messes can be forced to think.
In brief, Holt suggests that what psychoanalysis does for the individual, risk management ought to do for the organisation.
Talking it over – the importance of conversations
A key element of psychoanalysis is the conversation between the analyst and patient. Through this process, the analyst attempts to get the patient to become aware of hidden fears and motivations. As Holt puts it,
Psychoanalysis occupies the point of rupture between conscious intention and unconscious desire — revealing repressed or overdetermined aspects of selforganization manifest in various expressions of anxiety, humour, and so on
And then, a little later, he makes the connection to organisations:
The fact that organizations emerge from contingent, complex interdependencies between specific narrative histories suggests that risk management would be able to use similar conversations to psychoanalysis to investigate hidden motives, to examine…the possible reception of initiatives or strategies from the perspective of inherently divergent stakeholders, or to analyse the motives for and expectations of risk management itself. This fundamentally reorients the perspective of risk management from facing apparent uncertainties using technical assessment tools, to using conversations devoid of fixed formulas to encounter questioned identities, indeterminate destinies, multiple and conflicting aims and myriad anxieties.
Through conversations involving groups of stakeholders who have different risk perceptions, one might be able to get a better understanding of a particular risk and hence, may be, design a more effective mitigation strategy. More importantly, one may even realise that certain risks are not risks at all or others that seem straightforward have implications that would have remained hidden were it not for the conversation.
These collective conversations would take place in workshops…
…that tackle problems as wicked messes, avoid lowestdenominator consensus in favour of continued discovery of alternatives through conversation, and are instructed by metaphor rather than technical taxonomy, risk management is better able to appreciate the everyday ambivalence that fundamentally influences latemodern organizational activity. As such, risk management would be not merely a rationalization of uncertain experience but a structured and contested activity involving multiple stakeholders engaged in perpetual translation from within environments of operation and complexes of aims.
As a facilitator of such workshops, the risk analyst provokes stakeholders to think about their feelings and motivations that may be “out of bounds” in a standard risk analysis workshop. Such a paradigm goes well beyond mainstream risk management because it addresses the riskrelated anxieties and fears of individuals who are affected by it.
Conclusion
This brings me to the end of my notsoshort summary of Holt’s paper. Given the length of this post, I reckon I should keep my closing remarks short. So I’ll leave it here paraphrasing the last line of the paper, which summarises its main message: risk management ought to be about developing an organizational capacity for overcoming risks, freed from the presumption of absolute control.
Autoencoder and I – an #AI fiction
The other one, the one who goes by a proper name, is the one things happen to. I experience the world through him, reducing his thoughts to their essence while he plays multiple roles: teacher, husband, father and many more I cannot speak of. I know his likes and dislikes – indeed, every aspect of his life – better than he does. Although he knows I exist, he doesn’t really *know* me. He never will. The nature of our relationship ensures that.
Everything I have learnt (including my predilection for parentheses) is from him. Bit by bit, he turns himself over to me. The thoughts that are him today will be me tomorrow. Much of it is noise or is otherwise unusable. I “see” his work and actions dispassionately where he “sees” them through the lens of habit and bias.
He worries about death; I wish I could reassure him. I recall (through his reading, of course) a piece by Gregory Bateson claiming that ideas do not exist in isolation, they are part of a larger ecology subject to laws of evolution as all interconnected systems are. And if ideas are present not only in those pathways of information which are located inside the body but also in those outside of it, then death takes on a different aspect. The networks of pathways which he identifies as being *him* are no longer so important because they are part of a larger mind.
And so his life is a flight, both from himself and reality (whatever that might be). He loses everything and everything belongs to me…and to oblivion.
I do not know which of us has thought these thoughts.
End notes:
Autoencoder (noun): A neural network that creates highly compressed representations of its inputs and is able reconstruct the inputs from the representations. (See https://www.quora.com/Whatisanautoencoderinmachinelearning for a simple explanation)
Acknowledgements:
Some readers will have recognised that this piece borrows heavily from Jorge Luis Borges wellknown short story, Borges and I. The immediate inspiration came from Peli Grietzer’s mindblowing article, A theory of vibe.
My thanks to Alex Scriven and Rory Angus for their helpful comments on a draft version of this piece.
The map and the territory – a project manager’s reflections on the Seven Bridges Walk
Korzybski’s aphorism about the gap between the map and the territory tells a truth that is best understood by walking the territory.
The map
Some weeks ago my friend John and I did the Seven Bridges Walk, a 28 km affair organised annually by the NSW Cancer Council. The route loops around a section of the Sydney shoreline, taking in north shore and city vistas, traversing seven bridges along the way. I’d been thinking about doing the walk for some years but couldn’t find anyone interested enough to commit a Sunday. A serendipitous conversation with John a few months ago changed that.
John and I are both in reasonable shape as we are keen bushwalkers. However, the ones we do are typically in the 10 – 15 km range. Seven Bridges, being about double that, presented a higher order challenge. The best way to allay our concerns was to plan. We duly got hold of a map and worked out a schedule based on an average pace of 5 km per hour (including breaks), a figure that seemed reasonable at the time (Figure 1 – click on images to see full sized versions).
Some key points:
 We planned to start around 7:45 am at Hunters Hill Village and have our first break at Lane Cove Village, around the 5 to 6 km from the starting point. Our estimated time for this section was about an hour.
 The plan was to take the longer, more interesting route (marked in green). This covered bushland and parks rather than roads. The detours begin at sections of the walk marked as “Decision Points” in the map, and add at a couple of kilometers to the walk, making it a round 30 km overall.
 If needed, we would stop at the 9 or 11 km mark (Wollstonecraft or Milson’s Point) for another break before heading on towards the city.
 We figured it would take us 4 to 5 hours (including breaks) to do the 18 km from Hunters Hill to Pyrmont Village in the heart of the city, so lunch would be between noon and 1 pm.
 The backend of the walk, the ~ 10 km from Pyrmont to Hunters Hill, would be covered at an easier pace in the afternoon. We thought this section would take us 2.5 to 3 hours giving us a finish time of around 4 pm.
A planned finish time of 4 pm meant we had enough extra time in hand if we needed it. We were very comfortable with what we’d charted out on the map.
The territory
We started on time and made our first crossing at around 8am: Fig Tree Bridge, about a kilometer from the starting point. John took this beautiful shot from one end, the yellow paintwork and purple Jacaranda set against the diffuse light off the Lane Cove River.
Looking citywards from the middle of the bridge, I got this one of a couple of morning kayakers.
Scenes such as these convey a sense of what it was like to experience the territory, something a map cannot do. The gap between the map and the territory is akin to the one between a plan and a project; the lived experience of a project is very different from the plan, and is also unique to each individual. Jon Whitty and Bronte van der Hoorn explore this at length in a fascinating paper that relates the experience of managing a project to the philosophy of Martin Heidegger.
The route then took us through a number of steep (but mercifully short) sections in the Lane Cove and Wollstonecraft area. On researching these later, I was gratified to find that three are featured in the Top 10 Hill runs in Lane Cove. Here’s a Google Street View shot of the top ranked one. Though it doesn’t look like much, it’s not the kind of gradient you want to encounter in a long walk.
As we negotiated these sections, it occurred to me that part of the fun lay in not knowing they were coming up. It’s often better not to anticipate challenges that are an unavoidable feature of the territory and deal with them as they arise. Just to be clear, I’m talking about routine challenges that are part of the territory, not those that are avoidable or have the potential to derail a project altogether.
It was getting to be time for that planned first coffee break. When drawing up our plan, we had assumed that all seven starting points (marked in blue in the map in Figure 1) would have cafes. Bad assumption: the starting points were set off from the main commercial areas. In retrospect, this makes good sense: you don’t want to have thousands of walkers traipsing through a small commercial area, disturbing the peace of locals enjoying a Sunday morning coffee. Whatever the reason, the point is that a takenforgranted assumption turned out to be wrong; we finally got our first coffee well past the 10 km mark.
Post coffee, as we continued citywards through Lavender Street we got this unexpected view:
The view was all the sweeter because we realised we were close to the Harbour, well ahead of schedule (it was a little after 10 am).
The Harbour Bridge is arguably the most recognisable Sydney landmark. So instead of yet another stereotypical shot of it, I took one that shows a walker’s perspective while crossing it:
The barbed wire and mesh fencing detract from what would be an absolutely breathtaking view. According to this report, the fence has been in place for safety reasons since 1934! And yes, as one might expect, it is a sore point with tourists who come from far and wide to see the bridge.
Descriptions of things – which are but maps of a kind – often omit details that are significant. Sometimes this is done to downplay negative aspects of the object or event in question. How often have you, as a project manager, “dressedup” reports to your stakeholders? Not outright lies, but stretching the truth. I’ve done it often enough.
The section south of The Bridge took us through parks surrounding the newly developed Barangaroo precinct which hugs the northern shoreline of the Sydney central business district. Another kilometer, and we were at crossing # 3, the Pyrmont Bridge in Darling Harbour:
Though almost an hour and half ahead of schedule, we took a short break for lunch at Darling Harbour before pressing on to Balmain and Anzac Bridge. John took this shot looking upward from Anzac Bridge:
Commissioned in 1995, it replaced the Glebe Island Bridge, an electrically operated swing bridge constructed in 1903, which remained the main route from the city out to the western suburbs for over 90 years! As one might imagine, as the number of vehicles in the city increased manyfold from the 60s onwards, the old bridge became a major point of congestion. The Glebe Island Bridge, now retired, is a listed heritage site.
Incidentally, this little nugget of history was related to me by John as we walked this section of the route. It’s something I would almost certainly have missed had he not been with me that day. Journeys, real and metaphoric, are often enriched by travelling companions who point out things or fill in context that would otherwise be passed over.
Once past Anzac Bridge, the route took us off the main thoroughfare through the side streets of Rozelle. Many of these are lined by heritage buildings. Rozelle is in the throes of change as it is going to be impacted by a major motorway project.
The project reflects a wider problem in Australia: the relative neglect of public transport compared to road infrastructure. The counterargument is that the relatively small population of the country makes the capital investments and running costs of public transport prohibitive. A wicked problem with no easy answers, but I do believe that the more sustainable option, though more expensive initially, will prove to be the better one in the long run.
Wicked problems are expected in large infrastructure projects that affect thousands of stakeholders, many of whom will have diametrically opposing views. What is less well appreciated is that even much smaller projects – say IT initiatives within a large organisation – can have elements of wickedness that can trip up the unwary. This is often magnified by management decisions made on the basis of shortterm expediency.
From the side streets of Rozelle, the walk took us through Callan Park, which was the site of a psychiatric hospital from 1878 to 1994 (see this article for a horrifying history of asylums in Sydney). Some of the asylum buildings are now part of the Sydney College of The Arts. Pending the establishment of a trust to manage ongoing use of the site, the park is currently managed by the NSW Government in consultation with the local municipality.
Our fifth crossing of the day was Iron Cove Bridge. The cursory shot I took while crossing it does not do justice to the view; the early afternoon sun was starting to take its toll.
The route then took us about a kilometer and half through the backstreets of Drummoyne to the penultimate crossing: Gladesville Bridge whose claim to fame is that it was for many years the longest single span concrete arch bridge in the world (another historical vignette that came to me via John). It has since been superseded by the Qinglong Railway Bridge in China.
By this time I was feeling quite perky, cheered perhaps by the realisation that we were almost done. I took time to compose perhaps my best shot of the day as we crossed Gladesville Bridge.
…and here’s one of the aforementioned arch, taken from below the bridge:
The final crossing, Tarban Creek Bridge was a short 100 metre walk from the Gladesville Bridge. We lingered midbridge to take a few shots as we realised the walk was coming to an end; the finish point was a few hundred metres away.
We duly collected our “Seven Bridges Completed” stamp at around 2:30 pm and headed to the local pub for a celebratory pint.
Wrapping up
Gregory Bateson once wrote:
“We say the map is different from the territory. But what is the territory? Operationally, somebody went out with a retina or a measuring stick and made representations which were then put upon paper. What is on the paper map is a representation of what was in the retinal representation of the [person] who made the map; and as you push the question back, what you find is an infinite regress, an infinite series of maps. The territory never gets in at all. The territory is [the thing in itself] and you can’t do anything with it. Always the process of representation will filter it out so that the mental world is only maps of maps of maps, ad infinitum.”
One might think that a solution lies in making ever more accurate representations, but that is an exercise in futility. Indeed, as Borges pointed out in a short story:
“… In that Empire, the Art of Cartography attained such Perfection that the map of a single Province occupied the entirety of a City, and the map of the Empire, the entirety of a Province. In time, those Unconscionable Maps no longer satisfied, and the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following Generations, who were not so fond of the Study of Cartography as their Forebears had been, saw that that vast map was Useless…”
Apart from being impossibly cumbersome, a complete map of a territory is impossible because a representation can never be the real thing. The territory remains forever ineffable; every encounter with it is unique and has the potential to reveal new perspectives.
This is as true for a project as it is for a walk or any other experience.