Monday, October 14, 2024
Quantitative Methods: It takes a number!
Numbers are a PM's best friend
Is this news?
I hope not; I wrote the book Quantitative Methods in Project Management some years ago. (Still a good seller)
So, here's a bit of information you can use:
Real numbers: (**)
- Useful for day-to-day project management
- 'Real numbers' are what you count with, measure with, budget with, and schedule with.You can do all manner of arithmetic with them, just as you learned in elementary school.
- Real numbers are continuous, meaning every number in between is also a real number
- Real numbers can be plotted on a line, and there is no limit to how long the line can extend, so a real number can be a decimal of infinite length
- Real numbers are both rational (a ratio of two numbers) or irrational (like 'pi', not a ratio of two numbers)
Random numbers
- Essential for risk management subject to random effects
- Not a number exactly, but a number probably
- Random numbers underlie all of probability and statistics, and thus are key to risk management
- Random numbers are not a point on a line -- like 2.0 -- but rather a range on a line like 'from 1.7 to 2.3'
- The 'distribution' of the random number describes the probability that the actual value is more likely 1.7 than 1.9, etc
- Mathematically,
distributions are expressed in functional form, as for example the
value of Y is a consequence of the value of X.
- Arithmetic can not be done with random numbers per se, but arithmetic can be done on the functions that represent random numbers. This is very complex business, and is usually best done by simulation rather than an a direct calculus on the distributions.
- Monte Carlo tools have made random numbers practical in project management risk evaluations.
Rational numbers:
- A number that is a ratio of two numbers
- In project management, ratios are tricky: both the numerator and the denominator can move about, but if you are looking only at the ratio, like a percentage, you may not have visibility into what is actually moving.
- A number that is not a ratio, and thus is likely to have an infinite number of digits, like 'pi'
- Mostly these show up in science and engineering, and so less likely in the project office
- Many 'constants' in mathematics are irrational .... they just are what they are
- A number that expresses position, like 1st or 2nd
- You can not do arithmetic with ordinal numbers: No one would try to add 1st and 2nd place to get 3rd place
- Ordinal numbers show up in risk management a lot. Instead of 'red' 'yellow' 'green' designations or ranks for risk ranking, often a ordinal rank like 1, 5, 10 are used to rank risks. BUT, such are really labels, where 1 = green etc. You can not do arithmetic on 1,5,10 labels no more than you can add red + green. At best 1, 5, 10 are ordinal; they are not continuous like real numbers, so arithmetic is disallowed.
- Cardinality refers to the number of units in a container. If a set, or box, or a team contains 10 units, it is said it's cardinality is 10.
- Cardinal numbers are the integers (whole numbers) used to express cardinality
- In project management, you could think of a team with a cardinality of 5, meaning 5 full-time equivalents (whole number equivalent of members)
Exponents and exponential performance
- All real numbers have an exponent. If the exponent is '0', then the value is '1'. Example: 3exp0 = 1
- An exponent tells us how many times a number is multiplied by itself: 2exp3 means: 2x2x2 (*)
- In the project office, exponential growth is often encountered. Famously, the number of communication paths between N communicators (team members) is approximately Nexp2. Thus, as you add team members, you add communications exponentially such that some say: "adding team members actually detracts from productivity and throughput!"
- Got a graphics project? You may have vector graphics in your project solution
- Vectors are numbers with more than one constituent; in effect a vector is a set of numbers or parameters
- Example: [20mph, North] is a two-dimensional vector describing magnitude (speed) and direction
- In vector graphics, the 'vector' has the starting point and the ending point of an image component, like a line, curve, box, color, or even text. There are no pixels ... so the image can scale (enlarge) without the blurriness of pixels.
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(*) It gets tricky, but exponents can be decimal, like 2.2. How do you
multiply a number by itself 2.2 times? It can be done, but you have to
use logarithms which work by adding exponents.
(**) This begs the question: are there 'un-real' numbers? Yes, there are, but mathematicians call them 'imaginary numbers'. When a number is imaginary, it is denoted with an 'i', as 5i. These are useful for handling vexing problems like the square root of a negative number, because iexp2 = -1; thus i = square root of -1.
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Thursday, October 10, 2024
Bayes Thinking Part II
In Part I of this series, we developed the idea that Thomas Bayes was a rebel in his time, looking at probability problems in a different light, specifically from the proposition of dependencies between probabilistic events.
In Part I we posed the project situation of 'A' and 'B', where 'A' is a probabilistic event--in our example 'A' is the weather--and 'B' is another probabilistic event, the results of tests. We hypothesized that 'B' had a dependency on 'A', but not the other way 'round.
Bayes' Grid
The Figure below is a Bayes' Grid for this situation. 'A+' is good weather, and 'B+' is a good test result. 'A' is independent of 'B', but 'B' has dependencies on 'A'. The notation, 'B+ | A' means a good test result given any conditions of the weather, whereas 'B+ | A+' [shown in another figure] means a good test result given the condition of good weather. 'B+ and A+' means a good test result when at the same time the weather is good. Note the former is a dependency and the latter is a intersection of two conditions; they are not the same.
There are a few basic math rules that govern Bayes' Grid.
- The dark blue space [4 cells] is every condition of 'A' and 'B', so the numbers in this 'space' must sum 1.0, representing the total 'A' and 'B' union
- The light blue row just under the 'A' is every condition of 'A', so this row must sum to 1.0
- The light blue column just adjacent to 'B' is every condition of 'B' so this column must sum to 1.0
- The dark blue columns or rows must sum to their light blue counter parts
Empirical Data
First, let's say the empirical observations of the weather are that 60% of the time it is good and 40% of the time it is bad. Going forward, using the empirical observations, we can say that our 'confidence' of good weather is 60%-or-less. We can begin to fill in the grid, as shown below.
In spite of the intersections of A and B shown on the grid, it's very rare for the project to observe them. More commonly, observations are made of conditional results. Suppose we observe that given good weather, 90% of the test results are good. This is a conditional statement of the form P(B+ | A+) which is read: "probability of B+ given the condition of A+". Now, the situation of 'B+ | A+' per se is not shown on the grid. What is shown is 'B+ and A+'. However, our friend Bayes gave us this equation:
P(B+ | A+) * P(A+) = P (B+ and A+) = 0.9 * 0.6 = 0.54
Take note: B+ is not 90%; in fact, we don't know yet what B+ is. However, we know the value of 'B+ and A+' is 0.54 because of Bayes' equation given above.
Now, since the grid has to add in every direction, we also know that the second number in the A+ column is 0.06, P(B- and A+).
However, we can go no farther until we obtain another independent emprical observation.
To be continued
In the next posting in this series, we will examine how the project risk manager uses the rest of the grid to estimate other conditional situations.
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Monday, October 7, 2024
Bayes thinking, Part I
There was already a body of knowledge about probabilities by his time, so curious Bayes went at probability in a different way. Until Bayes came along, probability was a matter of frequency:
"How many times did an event happen/how many times could an event happen". In other words, "actual/opportunity".
To apply this definition in practice, certain, or "calibrated", information is needed about the opportunity, and of course actual outcomes are needed, often several trials of actual outcomes.
Bayes' Insight
Recognizing the practicalities of obtaining the requisite information, brother Bayes decided, more or less, to look backward from actual observations to ascertain and understand conditions that influenced the actual outcomes, and might influence future outcomes.
So Bayes developed his own definition of probability that is not frequency and trials oriented, but it does require an actual observation. Bayes’ definition of probability, somewhat paraphrased, is that probability is...
The ratio of expected value before an event happens to the actual observed value at the time the event happens.
This way of looking at probability is really a bet on an outcome based on [mostly subjective] evaluations of circumstances that might lead to that outcome. It's a ratio of values, rather than a frequency ratio.
Bayes' Theorem
He developed a widely known explanation of his ideas [first published after his death] that have become known as Bayes' Theorem. Used quantitatively [rather qualitatively as Bayes himself reasoned], Bayesian reasoning begins with an observation, hypothesis, or "guess" and works backward through a set of mathematical functions to arrive at the underlying probabilities.
To use his theorem, information about two probabilistic events is needed:
One event, call it 'A', must be independent of outcomes, but otherwise has some influence over outcomes. For example, 'A' could be the weather. The weather seems to go its own way most of the time. Specifically 'good weather' is the event 'A+', and 'bad weather' is the event 'A-'.
The second event, call it 'B', is hypothesized to have some dependency on 'A'. [This is Bayes' 'bet' on the future value] For example, project test results in some cases could be weather dependent. Specifically, 'B+' is the event 'good test result' and 'B-' is a bad test result; test results could depend on the weather, but not the other way 'round.
Project Questions
Now situation we have described raises some interesting questions:
- What is the likelihood of B+, given A+?
- What are the prospects for B+ if A+ doesn't happen?
- Is there a way to estimate the likelihood of B+ or B- given any condition of A?
- Can we validate that B indeed depends on A?
Bayes' Grid
Curious Bayes [or those who came after him] realized that a "Bayes' Grid", a 2x2 matrix, could help sort out functional relationships between the 'A' space and the 'B' space. Bayes' Grid is a device that simplifies the reasoning, provides a visualization of the relationships, and avoids dealing directly with equations of probabilities.
Since there's a lot detail behind Bayes' Grid, we'll take up those details in Part II of this series.
Photo credit: Wikipedia
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