I'm going to finish off this module now that we have been exposed to different sorts of models, the uses of models, the modeling process. We have been exposed to the terminology of models. And now I want to take a little bit of time to talk about these key mathematical functions. That you really do need to be familiar with if you're going to be successful at making quantitative models. So it's not as if you have to have a PhD in mathematics at this point, to be a useful modeller. I would never claim that. But you do have to have some facility with what I think of as the building blocks of quantitative models. So here are the four functions that I think you have to be comfortable with. And I'll explain, as I go through each of the four functions, what is so important about each one. And I'm going to try and characterize them in a way that lends itself to thinking about quantitative models. So here are the four functions. We're going to talk about linear functions, those are straight lines. We're going to talk about the power functions, things like quadratics and cubics. We'll talk about the exponential function. And we'll talk about the log function, which is formally the inverse of the exponential function. So let's have a look at these functions in turn now. So the linear function is probably the core building block of all models. And a linear function is simply a straight line. So you're looking now at a picture of a straight line function. It is characterized by two numbers, b and m, which are known as the intersect and the slope. So b is the height of the line above the origin that's at x equal to 0 on the graph. And m, the slope of the line. It tells you, as X moves by one unit, how much y the outcome has gone up by. So here's the equation for the straight line now. We're writing it as Y equals MX plus B. X would be the input to the model, and Y would be the output, and the two coefficients, or parameters are b, so I said the intercept, and m, the slope. Now, here's the essential characteristic of a straight line. It is that the slope is constant. Wherever you look on the graph, for any value of x, the slope of the graph, the value is always the same, it's always m. So as x changes by one unit, y goes up by m units regardless of the value of x. Now you have to ask yourself when you're modeling whether or not that assumption makes sense. Linear functions are the simplest functions that are out there. So they're often chosen for models. It doesn't necessarily mean that they're going to be right. And so, to use a linear function is to think carefully about whether or not this constant slope implication of a line is reasonable in practice. So let's think of an example here and we'll consider whether or not the linear function would be reasonable. Let's consider your salary as the y variable over time as you progress through your career. So x would be time, how long you've been working for, and y is your salary. Do you think a linear assumption there is going to be reasonable? And what would it imply? So a straight line implies that the slope is constant. That means, for every one-unit change in x, the change in y is always the same. So in the context of the example, you progressing through your career, and your salary increasing, if we used a straight line to model that, x is year, y is salary. It would be implying that your salary or pay rise was the same every year, all the way through your career. And you'd have to ask yourself, does that seem to be a realistic model for what is going on? I actually don't think it would be a realistic model. because I think at the beginning of your career salaries tend to go up faster and then much, much later on in your career, things tend to level off. And so that would be a sort of relationship that wouldn't necessarily lend itself to a linear function. So, I don't want to beat up on the linear functions. I don't want to say that they're not going to be useful. They're in fact incredibly useful, but you shouldn't be using one without asking yourself the critical question. Is it reasonable to expect this business process to exhibit linearity? And you say, you think of the word reasonable by the implication. Does it appear that the constant slope is viable in this situation. So that's the linear function. The next function we're going to talk about is the power function and I'm showing you here a graph that displays various power functions. Now we write the power function as y equals x to the power m. And what x to the power of m essentially means is we multiple x by itself m times. Examples that you might be familiar with are if we put in m equal to two we're going to get a quadratic. In fact, if we put in m equal to 1, any number raised to the power of 1 is itself. So you would get something linear coming out of this. And you can have fractional powers of m. You can have m equal to 1/2. Which is known as the square root. You can even have negative powers of m. M can equal minus one for example which would give you the graph of one over x, the reciprocal. So we can have various values for M and they will create different version of the power function. Now I have shown you power functions. Four various values of them, in fact. The ones I've just discussed on this graph and you can see the purple graph is m equal to 2. That's a quadratic. When m equals to 1, you see the blue graph, which is actually a straight line because if I say, any number raised to the power of one is itself. You can see m equal to 1/2 on here. That's the green curve on the picture. And that one is increasing, but it's increasing at a decreasing rate. And finally on here I've shown you the graph with m equal to minus 1. That's the sort of pinky colored one. And that shows a negative association, so what I'm showing you here on this slide is that in fact the power functions are very, very flexible. They can model all sorts of underlying relationships, increasing and decreasing, increasing at an increasing rate. That's the quadratic. Increasing at a decreasing rate. That's the square root function. So, a flexible family of functions. Language that we use for the power function, we will often term x the base and m the exponent. Now here comes the essential characteristic of the power function. Just as the essential characteristic of the straight line was that its slope was constant. There's something constant in a power function, but it's not the slope anymore, here's what it is. If x changes by 1%. Not 1 unit anymore, but 1%. Then, y is going to change by approximately n%. So the n in the exponent of the power function is relating percent change in x to percent change in y. And it's important that the word here is, I do have approximate in here. It is approximate. But it's a good approximation for small percent changes. And so, the key characteristic of a power function is that it relates percent change in x to percent change in y with the statement that that percent change is constant. So if I have x equal to 100 and I go up by 1%, then y is going to change by exactly the same percentage as if I had x equal to 200, and then took x up by 1% from 200. So it's an idea of this percent change, this proportionality, being constant percent change and x percent change and y, it's constant. Now, there are a couple of facts I just put in the math facts down at the bottom there, actually more, math facts than this, but these are really important ones about the power function. The x to the power m times x to the power n is, x to the m plus n, so the product here corresponds to the addition of the exponents. And x to the minus m is the same as 1 over x to the power m, and that's why we're able to use these negative powers to capture decreasing relationships. So there's the power function. Third up is the exponential function. Once again, I've drawn a graph with various exponential versions of the exponential function on here. They're all exponential functions, but they differ in their rate of growth and some of them are growing and some of them are decreasing. So we often talk about exponential growth for an increasing process and exponential decay for a decreasing process. The exponential function can capture both of these. The way it does it formulaically is we'll think of y = e to the power mx. Now, in this equation, e is standing for a very, very special number. That number is a mathematical constant that is approximately 2.71828. And so, rather than writing this number that technically has an infinite number of a decimal is associated with it we just call it e. And so, that's the base here and we're raising that number to the power mx and why this is different from the power function we think of it's different, it's where the x is. Here, it's in the exponent and not the base. The power function x was sitting in the base, now it's up there in the exponent. So we're letting the exponent vary this time around. And what's going to happen is that as you have different values for m, so we're going to get different relationships and on the slide of the exponential function I've put in some different values for m. The pink curve is m = -1, that's an exponential decay. If we take m to -3 then we decay more rapidly. You see the green curve is beneath the pink one. If we have m = 0.5, we've got an increasing exponential here. And if we have m = 1, because 1 is bigger than 0.5 where that's the purple graph we're increasing faster. So those are exponential functions and that was what I had been using to model the epidemic if you remember. Now, some facts about, or the essential characteristic about the exponential function is that the rate of change of y is proportional to y itself. And what that tells you is that there's an interpretation in the background here of m for small values, again, these are approximations for these interpretations. So let's say m is a small number, for example between -0.2 and positive 0.2. Then, what's going to come out of the exponential function is the idea that for every 1 unit change in x, there's going to be an approximate 100 times m% proportionate change in y. So what you're seeing in the exponential function, and it's differing from the power function, is now we're talking about absolute change in x being associated with percent or proportionate change in y. And we're claiming that that is a constant. You go back to the power function. We were looking at percent change in x, relating to percent change in y through the constant m. And if we go back to the linear function, we were seeing absolute change in x, being related to absolute change in y through the constant m. So these different functions that we're looking at are capturing how we're thinking about x and y changing. Are we thinking about them changing in an absolute sense or are we thinking about them changing in a relative sense? So just going back to this interpretation here of the constant m in the exponential function. We can, say for example, if m = 0.05, then a one-unit increase in x is associated with an approximate 5% increase in y and that 5% is cosmic, it doesn't matter. Or the value of x. So every time x goes up by 1 unit, y increases approximately by another 5%, a relative or proportionate change. So once again the exponential function lets us understand how absolute changes in x are related to relative changes in y. One more to go and that's the log function. This is the log transformation. It's probably the most commonly used transformation in quantitative modeling. We're not looking at the raw data, then often times we're looking at the log transform of the data. And this is what a log curve looks like. It's an increasing function, but the feature is that it's increasing at a decreasing rate. So the log function is extremely useful when it comes to modeling processes that exhibit diminishing returns to scale. So diminishing returns to scale says we're putting more into the process. But each time we put an extra thing into the process, you get more out. But not as much as we used to. And so, you might think of diminishing returns to scale as you've cooked a big meal at Thanksgiving. And it needs to be cleaned up. Now, if you're doing the cleanup by yourself, it takes quite a while. If you have some, one person help you, it's probably going to be a bit faster. Maybe if you had two people help you it's going to be even faster. But if you go up to ten people in the kitchen all trying to help you clear up that meal, at some point people start getting in the way of one another, and the benefits of those incremental people coming in to help you clear up really fall away quite quickly. And so, that's an idea of dimensions returns from scale. From a mathematical process point of view we think about the log function as increasing but at a decreasing rate. Now as I said, all of these functions that I'm introducing have an essential characteristics. And the essential characteristic of the log function is that a constant proportionate change in x is associated with the same absolute change in y. So notice how that's the flip side of the exponential function. The exponential function had absolute changes in x, being related to relative changes in y. The log function is doing it the other way around. We're talking about proportionate changes in x being associated with the same absolute change in y. Again, when you get to the stage of doing modeling, and you're thinking about the business process, you need to be thinking about these ideas as you choose your model, a functional representation of the process. How do you think things are changing? Do you think it's absolute change in x being related to absolute change in y as a constant? Or do you think it's relative change in x to relative change in y? Do you think it's relative change in x to absolute change in y, or absolute change in x to relative change in y? And here, in the log function, again, the essential characteristic, that constant proportion that changes in x are associated with the same absolute changes in y. If you think your business process looks like that, then the log function is a good candidate for a model. So, picking up the idea of the proportionate change in x being associated with the same constant change in y, you can see how that's working with the log function. And in this particular example, if we work our way up the steps that I've shown you here, the steps all have exactly the same height but the length in the step is different. So we start off on the bottom left hand side of this plot by going from one eighth to a quarter, that's a doubling. And when we do that we take a step up. Then we double again. We go from a quarter to a half. When we do that, the function steps up. But it steps up by exactly the same amount. Then, we double again. We go from 0.5 to 1. The log function increases, but by exactly the same amount as when we went from a quarter to a half. And an eighth to a quarter. And finally, the last step on these stairs here is another doubling from 1 to 2, and you can see that the height of the step is exactly the same again. So, the height of the step is constant. It's the length of the step that is varying. And the one that I've chosen here is a doubling from each period to the next. So, if you think that relative changes in x are being associated with absolute changes in y, the same constant absolute change in y, then you're already saying, I think that there's some kind of log relationship in the background here. So, that's our log function. And here are some facts about the log function. The way that we write it is log, l-o-g, but there is a subscript, b, which is called the base of the logarithm. There are lots of bases out there. The only ones, the only base that I'm going to be using in this course is the very special base, where we actually have the base as the number e, and that's called the natural log. And I choose to use that one because the interpretations of models with natural logs tend to be a little easier, these percent changes that I was talking about before. Now, it is the case that the log is formerly known as the inverse, it undoes the exponential function. And so the log of e to the x equals x itself, and e to the power log of x is x, too. So you can see that log and e are undoing, and the exponential function are undoing one another. The essential math fact about these logs is the log of a product is equal to the sum of the log. So you can see log of x y equal to the log of x plus the log of y. So that is used as one goes through the analysis of these models that have log terms, and as I say, in this course, I'm only going to be using the natural logarithm and we'll write that as log(x) and forget the subscript ultimately. So there's the log function. Here they are presented for you. On one slide, the four essential math functions that we'll be using, the linear, the exponential, the log, and the power. And you can see that these, taken as a whole, are going to provide us with a very flexible set of curves to get us started in the quantitative modeling of business processes.
