So now that we've had a look and we understand what a quantitative model is about let me tell you some of the specific activities where these models are used. So one thing that we can do once we've got a quantitative model is prediction. And prediction is basically taking the model, putting in an input, and calculating the output. And so going back to the diamonds ring example, what I mean by prediction. What's the expected price of a diamond ring that weighs 0.2 of a carats? If I got a model, I can create that prediction. And certainly one of the most often used places for these quantitative models is what we call predictive analytics. And so if you want to do predictive analytics, you need some underlying model for the process, typically. Another place that we use these models is in forecasting. And when I'm talking about forecasting I'm really thinking about a time series. And I'm thinking about trying to make a comment about what's happening in the future. And so forecasting is an activity that most businesses and entities go through at some stage, often to do with resource planning. And going back to the example of the epidemic if one was involved in a public health situation, a key question that you would be asking, for example, would be how many people are expected to be infected in six weeks time. Because that, the answer to that sort of question is going to help me in terms of resource planning. Here's another forecasting type problem where quantitative models can be very, very useful. So imagine that you're running a hospital. And you are trying to schedule patients for appointments. One of the truths of the matter here is that not everybody shows up for their appointment. And that leads to inefficiencies in the system. The system could be improved if we had a sense of who was likely to not show up. And perhaps we could tweak the schedule as a result of that. And so that would be another example where we would like to do some forecasting, forecasting whether or not someone's likely to show up for an appointment. Another activity that we use these quantitative models for is optimization. And so optimization takes up a lot of business thinking. And the example that I've got here is the demand and price. And so It's a very legitimate question to ask. I wonder what price is going to maximize the profit? And the key word there is maximize, that's to optimize an output, the profit. And optimization activities are going to typically require an underlying quantitative model. And so that's one of the places where quantitative models go to work as inputs for optimization, trying to make your business as good as it can be, optimizing price, optimizing the supply chain etc. So we use models for optimization. Another activity that we go through and models help us are ranking and targeting. And so what I mean there is that we're often looking at a list. It might be customers or it might be diamonds, for example. And we'd like to have a look at these diamonds, perhaps, and figure out which ones we'd be interested in purchasing if we're a diamond merchant, for example. And so I can't look at all the diamonds that are out there in the world because I simply don't have the resources. And so that's the idea of given limited resources. It would be really nice if I could identify potential targets of opportunity. And that essentially is a ranking and targeting exercise. And if we have a model we can create a set of predictions. We can sort those predictions and that creates a ranking. And then we can work our way down that list of predictions in order there, that rank list, in order to optimize our own time. And so another example would be that I'm interested in real estate. And I'm considering potential properties to buy. There are millions of properties potentially for sale in the country at any one point in time. I don't have an opportunity to look at all of them. Be nice if I could create a model that would help me identify those properties that seem the, are of the most interest to me. And that's something that a model can help you do. Here are some more things that we can use our quantitative models for. What-if scenarios, scenario planning, we will often like to understand what might happen to the world if certain things change. And so going back to the epidemic we might want to ask ourselves the question well, what if the growth rate changed and increased to 20% per week? Then how may infections are we going to expect in the next ten weeks? So if we have a model, we are able to examine the consequences of the change of some of our assumptions. And that's the idea of scenario planning and what-if analysis. So that's something else that a model can help you do. Certain models lend themselves to interpretation. And in terms of the price and quantity demanded model, there was a coefficient In the equation, if you look back. And that coefficient was the power, which was -2.5. So it's a number, but sometimes these coefficients have interpretations. And that interpretation can be helpful. And turns out that the interpretation of that -2.5 is what's known as an elasticity. And it tells us that as the price goes up by 1%, we can anticipate the demand to fall by 2.5%. And so that's what I mean by interpreting a coefficient in a model. And that can give us additional insight and help us explain the model to other people. So our models involve mathematical equations. The mathematical equations often have coefficients in them. The coefficients can have real meanings and interpretations. Another task that models are used for is to conduct a sensitivity analysis. And so I pointed out earlier on that pretty much every model you create is going to rely on some assumptions. And a sensitivity analysis is a process where we look to see how sensitive the outputs of the model are to some of those assumptions. And if we find that the model that's particularly sensitive to an assumption, then that tells us that we need to think a little more carefully about that assumption. Maybe we'll try and confirm that the assumption is realistic or collect more information to try and tie that assumption down more precisely. So there's a sensitivity analysis typically with a model that helps us figure out which of the assumptions are really important and which aren't so important, and therefore how we might want to use our time in confirming that the underlying assumptions are reasonable. After having gone through this whole modeling business, hopefully there are some benefits. And here I've listed out a set of benefits. They're undoubtedly more than up here on this slide. But these are benefits that I think and have experienced myself, after having gone through this modeling activity. So one of the things that can be the outcome of a model is the identification that there's some gaps in the current understanding. You've tried to lay out your understanding of the business process. And theres just some big gap sitting there in the middle. And that's only become apparent because you've taken the time to lay out your current understanding. So identifying gaps is certainly a benefit. It's often the case that people are using models without having made underlying assumptions explicit. And so one of the benefits of creating a useful model is that you will have explicated what those assumptions are. So they're on the table, everybody can see what you're assuming as you come up with your recommendations from that model. And sometimes those assumptions have been implicitly foreign. Not everyone is aware of them. And so explicating the assumptions can be very useful. You will also have at the end of the modeling process a well-defined description of the business process, how the pieces fit together. And that can be of benefit its own right. Fourth one I've got here isn't entirely obvious. But one of the things a model can do is create what I call an institutional memory. So, many businesses will have some smart person who is relied on for doing certain things, certain forecasting activities. What do you think we're going to sell next month? They're someone who's worked for the company 25 years. You go to them, they have a good sense of what's going on. But what happens if that person leaves? Knowledge goes with them. And so you can think of a model as creating an institutional memory. Because that model is going to be a set of equations, a set of inputs and outputs. And that's going to stick around beyond any individual. And so I think models can be useful from that point of view. Ultimately, the model is going to be used as a decision support tool. And I've bolded the word support here because it's a little naive to think that the model is going to reveal truth. They're always approximations. And the model is typically going to be used as one of a suite of tools to help support the decision making within a company. And so it's very much not an end in its own right but a support to other activities. My final comment here is that sometimes I think that models are serendipitous insight generators. By going through the modeling process you learn something that you hadn't thought about at all, something very unexpected. And that happens quite often. And so that's another benefit of modeling.