0:12

I'm going to go through the key steps in the modeling process itself.

Â I will start introducing the vocabulary for modeling.

Â And we are going to have a look at the fundamental mathematical

Â functions that you need to be comfortable with if you're going to

Â be successful in implementing these quantitative models.

Â And the four functions that will be appearing through

Â the other modules in this course.

Â The linear the straight line function, the power functions thinks like quadratics,

Â the exponential function and the logarithm so

Â we're going to do a review of those as well.

Â 1:02

Now, in the business context,

Â the models that we talk about are not physical models.

Â So an architect might well create an architectural model of the building

Â that they plan on creating, it's not that sort of model that we're talking about.

Â What we're talking about is a formal description of

Â a business process and so that's what we think of as a model.

Â 1:29

Now, that description is invariably going to involve a set of

Â mathematical equations and incorporate what we term random variables.

Â We'll discuss this in more detail later on and later module exactly what random

Â variables are but these are the elements typically of a quantitative model.

Â Now it's important to realize that it's almost always

Â a simplification of the more complex business process.

Â And so, it's an art, as well as a science,

Â to achieve a suitable level of simplification.

Â We don't want to over simplify but on the other hand,

Â if our models are overly complex they will not be so useful.

Â And so, one needs to realize that they're not even striving,

Â typically, to be an exact representation of what's going on.

Â 2:30

There's always a set of assumptions that underlie the model and it's important

Â to be able to articulate those assumptions and the legitimacy of those assumptions.

Â And in terms of implementation within a business setting you'll

Â find that most of the quantitative models are implemented using

Â a spreadsheet tool like Excel, or Sheets.

Â Or potentially a custom computer program that is designed

Â to specifically implement an individual model.

Â So, that's what we think about when we talk

Â about a quantitative model in a business setting.

Â Now to provide some more concrete examples I'm going to show you some models and

Â illustrate the sorts of questions they're able to answer.

Â So, one of the things you might be interested in

Â thinking about if we were into the jewelry business is how a price

Â of a diamond varies as a function of its weight.

Â We typically have a sense that heavier

Â diamonds cost more money but what exactly does that relationship look like?

Â We could use a quantitative model to help us

Â understand the form of that relationship.

Â 3:55

Now, if you're into public policy and

Â you're dealing perhaps with some outbreak of a disease, an epidemic.

Â It's fundamental to be able to forecast or

Â anticipate the spread of that epidemic over time.

Â Most importantly, you probably want to do some resource planning

Â in the based on that epidemic, how many clinics do we need?

Â How many physicians need to be available within the next six months etc.?

Â And so, that sort of question understanding the spread of an epidemic

Â overtime, that's a place a quantitative model can be very useful.

Â 4:33

Going to the discipline of economics,

Â one of the most fundamental ideas there is to look at

Â the association between the price of a product and the demand for that product.

Â As I increase the price of my product, what happens its demand?

Â And ultimately, what's the best price to charge for

Â my product if I want to maximize my profit?

Â That's a question that we're going to come back to, so

Â there's a relationship we'll be interested in modelling.

Â The relationship between the price and demand., if I'm more into marketing realm.

Â I might be thinking about what's likely to happen in a market

Â as I introduce a new product, what's the uptake of that product likely to be?

Â Can I forecast the total number of units sold?

Â And so, understanding how a new product defuses through a market

Â is an idea that lends itself to qualitative modeling.

Â So those are some examples in disparate areas but

Â all can be addressed to the use of a quantitative model.

Â 6:15

the weight of a diamond and the price that it's going to to go for.

Â And so, often times we're going to think of representing the model

Â that we have through some graphical approach.

Â And so, in this course I'm going to be using a lot of graphics because

Â they are perhaps the most elegant way to produce and

Â represent and share your models with other people.

Â And so, what you're looking at here is a graph where on the horizontal axis,

Â we often call that the X axis,

Â you have the weight of the diamond that is measured in carats.

Â And on the vertical axis you'll have the expected price of the diamond and

Â what I'm looking at here is a potential model.

Â It's a very straight forward model, it's what we termed a linear model,

Â because it's a straight line.

Â And I have the equation associated with the model at the bottom of the slide here.

Â And what I'll do later on is discuss it much more detail such the liner equation.

Â But right now, I just want to show you that given such a model.

Â You would be able to use it to help forecast the expected price of a diamond

Â and so if for example I'm looking at a diamond ring that weights 0.3 of a carat.

Â All that I need to do is go into this graph

Â 7:43

Identify the point 3 on the horizontal axis.

Â Go up to the graph itself of the line, read off the value on the vertical axis

Â that we often call the Y-axis and there, I have an expected price for a diamond.

Â And so, in this particular case we have got a linear model,

Â it's not clear that that's going to work for all diamonds.

Â But, if you have a look at the range of the x axis, here, it's somewhat limited.

Â These are diamonds between 0.15, and 0.35 of a carat,

Â is the realm that I'm going to apply this model.

Â I'm not saying that it, necessarily,

Â applies to a diamond that weighs one carat or two carats way outside of the range.

Â But it might be reasonable that within this limited range,

Â one would see a linear relationship.

Â So that's an example of what we call a linear model.

Â 8:39

What about the spread of an epidemic?

Â Now, depending on the nature of the disease, and

Â the time at which we are following the epidemic.

Â One of the basic models, at least, to get started with,

Â to think about a spread of an epidemic, is what we term, and exponential model.

Â And here, I have a graph of an exponential function.

Â On the bottom axis we have week and

Â on the vertical axis we have the number of cases that have been reported.

Â And notice now that this graph, it's no longer linear,

Â it's what we would term a nonlinear relationship.

Â It is growing very quickly with term exponential growth and

Â it might be more appropriate for the spread of an epidemic in its early phases.

Â Now we would really hope that the exponential graph does not continue on for

Â long because the thing about these exponential graphs.

Â They're sometimes called hockey sticks when one refers to them in the business

Â context, is that they shoot up very, very quickly.

Â And I would not hear a claim that this would be a reasonable model over

Â a long period of time.

Â But in the initial phase is of an epidemic,

Â it might well serve as reasonable approximation.

Â And again, with such a structure, by which I mean the graph itself,

Â let's say, with sitting at week 30.

Â And we want to make a comment about what we think is going to happen at week 35.

Â We can use the graph,

Â we can use the equation to help us predict how many cases there are going to be.

Â So, that's an example of a non-linear relationship and

Â in particular it's called an exponential function and

Â I have presented the function at the bottom of the slide.

Â We'll talk about it in more detail later on.

Â 10:49

Now, in this situation,

Â we are looking at what is often turned the negative association.

Â The previous two examples, the graph one was a straight line the other

Â was an exponential function, would both going from bottom left to top right.

Â We term that positive association.

Â This time around, we're looking at something that has negative association

Â because typically for most goods, as the price increases.

Â Then, so the quantity solved is actually going to decrease then so

Â that's why we got a graph that goes from top left to bottom right.

Â 11:23

Now, I'm using a different sort of Mathematical

Â function to capture this association.

Â And the type of function that you're looking at here is called

Â a power function.

Â In terms of the model that we're using,

Â we have the quantity demanded is equal to some multiplicative constant

Â that's a 60,000 times the price to the power minus 2.5.

Â And for the particular data that sat behind this example,

Â this was a reasonable model to use.

Â This is different from the exponential function,

Â the power function that we're looking at here.

Â And it has some very special features,

Â this power function, again, to be described.

Â But it's an example of another place where these quantitative models can be very,

Â very useful, and in particular one of the uses that one would be able to find.

Â For this model is to think about what an optimal price should be.

Â Clearly, viewing increase the price the,

Â one unit of this product is going to bring in more money, but

Â you're going to be selling less units if you increase the price.

Â So, there's a trade off going on there and

Â the question is, how do we optimize that trade off?

Â How do we find the best price?

Â And so, economics is a discipline that is full of quantitative models and

Â this is a basic quantitative model for demand.

Â 12:56

So my final example here is a model for the uptake of a new product and

Â it's different from the previous examples that we've seen.

Â Because this graph has a feature is,

Â that it is increasing, but, then, it starts to tail off.

Â But the reason for that is, because the variable, the outcome that I'm looking at,

Â is the proportion of a market that has being exposed to the product.

Â But it's the product the proportion can never be greater than one so

Â therefore the graph cannot keep going up and up.

Â This particular function that we're looking at here is termed

Â a logistic function and it has the potential to map a process.

Â Where at the initial stages there's a slow start that

Â would be the early adapters picking up the product.

Â Then, there's a rapid take up of the product as more and

Â more people get to know about it.

Â Then, at some point you can't have a proportion greater than one.

Â So the proportion of the market that has actually purchased

Â the product has to start to tail off, cannot go above one.

Â And so, there's a special curve that is able to capture these intrinsic

Â features of the outcome variable that I am interested in here, the proportion.

Â Proportions go between zero and one, so I need a model that can reflect that.

Â This logistic function has the ability to do that and I just presented at the bottom

Â of the slide here what that logistic model looks like mathematically.

Â So those are four examples of models and

Â you can see that from a qualitative perspective.

Â They're able to pick up different features in an underlying process.

Â Linear modelling exponential model, we saw the power function and

Â here we've finished off by having a look at the logistic model.

Â So these would all be quantitative models

Â that would suddenly have a role in the business setting.

Â