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 time to talk about this 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 would 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, cubics. We'll talk about the exponential function and we'll talk about the log function which is formerly 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 out of picture of a straight line function. It is characterized by two numbers, b and m which are known as the intercept and the slope. So b is the height of the line above the origin. That's 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, why the outcome has gone up by. So here's the equation for the straight line now. We're writing it as y = mx + b, x would be the input to the model and y would be the output, and the two coefficients or parameters, are b, so it's at the intercept, an 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 of that 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 take of an example here and we'll consider whether or not a linear function will be reasonable. Let's consider your salary as the y variable over time as you progress to your career. So x will be time or how long you'll be 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 mean's 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 well 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 then turn 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 they're not going to be useful matter of 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 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=x to the power m. And x to the power m essentially means is we modify x by itself m times. Examples that you might be familiar with if we put in m equal to 2 we're going to get a quadratic. In fact, if we put in m equal to 1 any number raised to the power 1 is itself. So you would get something linear coming out of this. And you can have fractional powers of m. You could have m equal to one-half which is known as the square root. You can even have negative powers of m, m can equal minus 1, for example, which would give you the graph of 1 over x, the reciprocal. So we can have various values for m and they will create different versions of the power function. Now I have shown you power functions for various values of m, in fact the ones that I 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 I said any number raised to the power 1 is itself. You can see m equal to a half 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, isn't that, in fact, the power functions are very,very flexible, that you can model all sorts of underlying relationships. Increasing and decreasing, increasing and increasing, right? 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 one unit anymore, but 1%, then y is going to change by approximately m%. So, the m 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 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 a idea of this percent change, this proportionality being constant. Percent change in x to percent change in y is constant. Now there are a couple of facts, I just put the math facts down at the bottom. There are actually more math facts than this but these are really important ones about the power function. The x to the power n times x to the power n = x to the m + n. So the product here corresponds to the addition of the exponents. And x to the- m = 1 / 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 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 decimals 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 as different is where the x is. Here it's in the exponent not the base. For 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 you're going to get different relationships. And on this slide of the exponential function, I 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 0.2. Then what's going to come out of the exponential function is the idea that for every one-unit change in x, there's going to be an approximate 100m% 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 = to 0.05, then a one-unit increase in x is associated with an approximate 5% increase in y. And that 5% is constant, is doesn't matter or the value of x. So every time x goes up by one 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, yeah, we 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 clean up by yourself, that takes quite a while. If you have one person help you, it's probably going to be a bit faster, and maybe 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 diminishing returns to 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 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 absolutely changing 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 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. 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 proportionate changes in x are associated with the same absolute change 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 of 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 point five to one, 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 one to two, 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 way, 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 really saying, I think that there's some kind of log relationship in the background here. So that's 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 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 formally known as the inverse it undoes the exponential function. And so the log of e to the x = x itself. And e to the power log of x = x too. So you can see that log and the 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( x,y) equal to the log(x) + log(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.
