He's one of the content for the very first module. And we'll start off by looking at some examples and I discussed 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 the straight line function, the power functions thinks 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 the 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 and 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 hand, 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 underlie the model and it's important to be able to articulate those assumptions and 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 of questions they're able to answer. So, one of the things you might be interested in thinking about if we were into the jewelry business is how a 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 based on 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 overtime, 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 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 to, so there's a relationship we'll be interested in modelling. The relationship between the price and demand., if I'm more into 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 defuses through a market is an idea that lends itself to qualitative modeling. So those are some examples in disparate areas but all can be addressed to 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 modelling itself. And I've actually chosen these examples so many of the functions that we're going to see are going to be a precursor to the mathematical functions that we're going to 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 to go for. And so, often times we're going to 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'll have the expected price of the diamond and what I'm looking at here is a potential model. It's a very straight forward model, it's what we termed 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 it much more detail such the liner 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 weights 0.3 of a carat. All that I need to do is go into this graph Identify the point 3 on the horizontal axis. Go up to the graph itself of 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 have 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, is the realm that I'm going to apply this model. I'm not saying that it, necessarily, applies to a diamond that weighs one carat or two carats way outside of 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, and exponential model. And here, I have a graph of an 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 term a nonlinear relationship. It is growing very quickly with term exponential growth and it might be more appropriate for the spread of an epidemic in its early phases. Now we would really hope that the exponential graph does not continue on for long because the thing about these exponential graphs. They're sometimes called hockey sticks when one refers to them in the business context, is that they shoot up very, very quickly. And I would not hear a claim that this would be a reasonable model over a long period of time. But in the initial phase is of an epidemic, it might well serve as reasonable approximation. And again, with such a structure, by which I mean the graph itself, let's say, with 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 there are 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 turned the negative association. The previous two examples, the graph one was a straight line the other was an exponential function, would both going from bottom left to top right. We term 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, so the quantity solved is actually going to decrease then so that's why we 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 a power function. In terms of the model that we're using, we have the quantity demanded is equal to some multiplicative constant that's a 60,000 times the price to the power minus 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 described. 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, viewing increase the price the, one unit of this product is going to bring in 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 is, that it is 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 being exposed to the product. But it's the product the 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 adapters picking up the product. Then, there's a rapid take up of the product as more and more people get to know about it. 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, there's a special curve that is able to capture these intrinsic features of the outcome variable that I am 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 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. Linear modelling exponential model, we saw the power function and here we've finished off by having a look at the logistic model. So these would all be quantitative models that would suddenly have a role in the business setting.
