Here's the content for the very first module. And we'll start off by looking at some examples, and I discuss the uses of these models. I'm going to go through the key steps in the modeling process itself. I will start introducing the vocabulary for modeling, and we are going to have a look at the fundamental mathematical functions that you need to be comfortable with if you're going to be successful in implementing these quantitative models. And the four functions that will be appearing through the other modules in this course, the linear or straight line function, the power function with things like quadratics, the exponential function, and the logarithm. So we're going to do a review of those as well. Given that this is a course in modeling, we really need to get started by defining a model. Now in the business context, the models that we talk about are not physical models, so an architect might well create an architectural model of a building that they plan on creating, it's not that sort of model that we're talking about. What we're talking about, is a formal description of a business process, and so that's what we think of as a model. Now, that description is invariably going to involve a set of mathematical equations and incorporate what we term random variables. We'll discuss this in more detail later on in a later module, exactly what random variables are. But these are the elements typically, of a quantitative model. Now it's important to realize that it's almost always a simplification of the more complex business process, and so it's an art, as well as a science to achieve a suitable level of simplification. We don't want to over simplify, but on the other hands, if our models are overly complex, they will not be so useful. And, so, one needs to realize that they're not even striving, typically, to be an exact representation of what's going on. There's always a set of assumptions that underly the model, and it's important to be able to articulate those assumptions, and test the legitimacy of those assumptions. And in terms of implementation, within a business setting you'll find that most of the quantitative models are implemented using a spreadsheet tool like Excel, or Sheets, or potentially a custom computer program that is designed to specifically implement an individual model. So that's what we think about when we talk about a quantitative model in a business setting. Now, to provide some more concrete examples, I'm going to show you some models and illustrate the sorts questions that they're able to answer. So one of the things we might be interested in thinking about is, if we were into the jewelery business is how the price of a diamond varies as a function of its weight. We typically have a sense that heavier diamonds cost more money, but what exactly does that relationship look like? We could use a quantitative model to help us understand the form of that relationship. Now, if you're into public policy, and you're dealing perhaps with some outbreak of a disease, an epidemic, it's fundamental to be able to forecast, or anticipate the spread of that epidemic over time. Most importantly, you probably want to do some resource planning in the face of that epidemic. How many clinics do we need, how many physicians need to be available within the next six months, etc? And so that sort of question, understanding the spread of an epidemic over time, that's a place a quantitative model can be very useful. Going to the discipline of economics, one of the most fundamental ideas there, is to look at the association between the price of a product and the demand for that product. As I increase the price of my product, what happens to its demand? And ultimately, what's the best price to charge for my product if I want to maximize my profit? That's a question that we're going to come back too. So there's a relationship we would be interested in modeling. The relationship between price and demand. If I'm more in the marketing realm, I might be thinking about what's likely to happen in a market as I introduce a new product. What's the uptake of that product likely to be? Can I forecast the total number of units sold? And so understanding how a new product diffuses through a market is an idea that lends itself to quantitative modeling. So, those are some examples in disparate areas, but all can be addressed through the use of a quantitative model. I'm now going to go through each of the prior examples and illustrate how we might think about doing the modeling itself. And I've actually chosen these examples, so many of the functions that we're going to see are going to be a pre-cursor to the mathematical functions that we'll talk about later on in this module. And so, let's go back to thinking about the weight of a diamond, and the price that it's going to go for. And so, often times we think of representing, the model that we have through some graphical approach. And so, in this course I'm going to be using a lot of graphics because they are perhaps the most elegant way to produce, and represent, and share your models with other people. And so what you're looking at here is a graph where on the horizontal axis, we often call that the x-axis, you have the weight of the diamond that is measured in carats, and on the vertical axis you have the expected price of the diamond. And what I'm looking at here is a potential model. It's a very straightforward model, it's what we term a linear model because it's a straight line, and I have the equation associated with the model at the bottom of the slide here. And what I'll do later on is, discuss in much more detail such a linear equation, but right now I just want to show you that given such a model, you would be able to use it to help forecast the expected price of a diamond. And so if, for example, I'm looking at a diamond ring that weighs 0.3 of a carat, all that I need to do is go into this graph, identify the 0.3 on the horizontal axis, go up to the graph itself, the line, read off the value on the vertical axis that we often call the y-axis, and there I have an expected price for a diamond. And so in this particular case, we've got a linear model. It's not clear that that's going to work for all diamonds, but if you have a look at the range of the x-axis here, it's somewhat limited. These are diamonds between 0.15 and 0.35 of a carat, it's the realm that I'm going to apply this model. I'm not saying that it necessarily applies to the diamond that weighs one carat or two carats way outside the range, but it might be reasonable that within this limited range one would see a linear relationship. So that's an example of what we call a linear model. What about the spread of an epidemic? Now, depending on the nature of the disease, and the time at which we are following the epidemic, one of the basic models, at least to get started with, to think about a spread of an epidemic, is what we term an exponential model. And here I have a graph of a exponential function. On the bottom axis we have Week. And on the vertical axis we have the number of cases that have been reported. And notice now that this graph, it's no longer linear, it's what we would determine a nonlinear relationship. It is growing very quickly. We've termed this exponential growth, and it might be more appropriate for the spread of an epidemic in its early phases. Now we would really hope that that exponential graph does not continue on for long, because the thing about these exponential graphs, they're sometimes called hockey sticks, one that refers to them within, in the business context is that they shoot up very, very quickly. And I would not sit here and claim that this would be a reasonable model over a long period of time. But in the initial phases of an epidemic, it might well serve as a reasonable approximation. And again, with such a structure, by which I mean the graph itself, you can, let's say we're sitting at week 30, and we want to make a comment about what we think is going to happen at week 35. We can use the graph. We can use the equation to help us predict how many cases they're going to be. So that's an example of a non-linear relationship, and in particular it's called an exponential function, and I have presented the function at the bottom of the slide. We'll talk about it in more detail later on. Another example, the economics example, the idea of price and demand. Now in this situation we are looking at what is often termed a negative association. The previous two examples, the graph, one was a straight line, the other one was an exponential function, were both going from bottom left to top right. We termed that positive association. This time around, we're looking at something that has negative association, because typically for most goods, as the price increases, then sold, the quantity sold, is actually going to decrease. And so that's why we've got a graph that goes from top left to bottom right. Now, I'm using a different sort of mathematical function to capture this association. And the type of function that you're looking at here is called power function. In terms of the model that we're using, we have the quantity demanded is equal to some multiplicative constant, that's the 60,000 times the price to the power -2.5. And for the particular data that sat behind this example, this was a reasonable model to use. This is different from the exponential function, the power function that we're looking at here. And it has some very special features, this power function again to be describe, but it's an example of another place where these quantitative models can be very, very, useful and in particular one of the uses that one would be able to find for this model is to think about what an optimal price should be. Clearly if you'd increase the price, one unit of this product is going to bring a more money but you're going to be selling less units if you increase the price. So there's a trade off going on there. And the question is, how do we optimize that trade off? How do we find the best price? And so economics is a discipline that is full of quantitative models, and this is a basic quantitative model for demand. So, my final example here is a model for the uptake of a new product, and it's different from the previous examples that we've seen because this graph has a feature. The feature is that it's increasing, but then it starts to tail off, but the reason for that is because the variable, the outcome that I'm looking at is the proportion of a market that has been exposed to the product, that has bought the product, and a proportion can never be greater than one, so therefore the graph cannot keep going up and up. This particular function that we're looking at here is termed a logistic function, and it has the potential to map a process where, at the initial stages there's a slow start, that would be the early adopters picking up the product, then there's a rapid take up of the product, as more and more people get to know about it, and then, at some point, you can't have a proportion greater than one. So, the proportion of the market that has actually purchased the product, has to start to tail off, cannot go above one. And so this is a special sort of curve that is able to capture these intrinsic features of the outcome variable that I'm interested in here, the proportion. Proportions go between zero and one so I need a model that can reflect that. This logistic function has the ability to do that, and I've just presented at the bottom of the slide here what that logistic model looks like mathematically. So those are four examples of models, and you can see that from a qualitative perspective they're able to pick up different features in an underlying process. A linear model, an exponential model, we saw the power function, and here we have finished off by having a look at a logistics model. So these would all be quantitative models that would certainly have a role in a business setting.
