 Before you go on to fitting the model for our more realistic examples, let us look at this analytical solution for the cumulative sales as well as sales rate, nothing but total adopters and the adoption rate and time of peak sales or the inflection point here. So, time of peak sales will coincide with the inflection point, so this is a different terminologies for same thing, just make a note. So, in Bas model we have written that AR is nothing but dA by dt that is change in the adopters which is A into P plus C into I into P into A by n. Since we said we will set C into delta is equal to Q and we already know that P is nothing but n minus A, we can rewrite this as A into n minus A plus Q into n minus A into A by n. It can be further rewritten as P plus Q into A by n into n minus A. Again note A is equal to 0 at t equal to 0 rather A t is equal to 0, that should be subscript t for all A, all A and P has a subscript t in it. We started off with similar dA by dt equations in the previous case and then we had a lengthy derivation. I will take the liberty to skip that part and directly jump to solving one yeah A small oh sorry should be small p I am sorry small p small p sorry. Solving one we can get adopters at time t as n into 1 minus e power minus of P plus Q into t divided by 1 plus Q by P d power minus P plus Q into t and call it equation 2. Based on this we can get our time of peak adoption in a similar fashion or your t star what is earlier is minus 1 by P plus Q to ln of P by Q. So, this t star for the cases when P is greater than Q becomes negative right. So, that means t is negative. So, that is why you are having a peak at t equal to 0 and the number of adopters at t star that is A R at t star is equal to 1 by 4 Q into P plus Q the whole square. This can be derived in a similar fashion as we saw last class it is just little time consuming, but not that difficult you can solve that because it is just one variable you just have to integrate it yeah to our BAS model. So, people have been using the BAS diffusion model for quite some time. The various studies that has been reported to look at the diffusion speed across various countries and products for example, that paper the link is given. So, they are given some best guesses of P's for various type of products. The P's and Q's are going to differ based on whether it is a FMCG good for the fast moving consumer goods or is a durable goods etcetera. For example, a baseline case of US consumer durable is as in 76 was given as the base case and based on that people have given various estimates for P's. Similarly, we have various estimates for Q's as you can see even in the base case. The base case for P is 0.02, the best guess for Q is 0.47 where word of mouth is clearly dominating here as compared to over advertising effects. Similarly, things can be compared across it is a non-durable goods for example, let us see. There are things like industrial goods which are quite comparable is 1.149, it is 1.05 things are somewhat comparable in case of some of the items as opposed to the other ones where it is quite far apart. So, with that let us move on and figure out how we are going to do for the same example we will take up what we did last class. So, suppose we are given the sales rate, again we have a nice hum shaped pattern of the sales rate or the adoption rate. Now, how will we fit our parameters or our estimator parameters for the bus model which in this case happens to be P, Q and N. When it was only Q and N that is C into I and N we use the use couple of ways that we solved it. Now, we have another factor called as P. So, let us see how we can do that. Let us take the equation as it is, let us take what is happening to our say AR instead let us just put AR at time t plus 1 is P plus Q by N into A t N minus A t which can be further expanded as AR at time t plus 1 is P into N plus Q minus P into A t minus Q by N into A t square. This particular equation is of the form, this is of the form y equal to A plus B x plus C x square where A is nothing but P into N, B is nothing but Q minus P and C is nothing but minus Q by N. So, now what you have is a nice polynomial already here which we have the output variable y in terms of input x which x and x square and I am just fitting a polynomial to that. So, now I need to estimate my A, B and C based on this and use it to compute my values of P, N and Q to go one step further we can write this for example, Q minus P is nothing but what is Q, let us take Q from here which is nothing but minus C into N minus A by N. So, Q is equal to minus C into N it is coming from equation 3 here and P is equal to A by N which is coming from here, Q minus P is equal to B which comes from here. This if you solve it you can get C into N squared plus B N plus A equal to 0 which just implies that N is nothing but the roots of these which is equal to minus B plus or minus root of. So, total population is nothing but the root of this equation and once you have N you can use it to compute your Q and P. You are not required, but just you can do it. We changed it, sorry. So, now, we can actually use this to build a regression model which is what you guys are going to do. So, you can just make note of the equation A, B and Q again once you get A, B and C you can always solve these equations directly or you can compute N and then you can solve it. So, now let us go back to our slides. So, in order to fit a best model we estimated these parameters P, Q and then we need to estimate and we wrote it, we rewrote the equation sort of option rate and we got this polynomial of this particular form. So, we need to fit the polynomial of the same form. So, we are going to use regression to estimate these parameters. In fact, you guys are going to do it. So, use spreadsheet do your computation, bus diffusion class dot excel s is presented there. Compute cumulative sales is already computed in the sheet it is available in module please download it. So, cumulative sales A is already computed, you compute the square of cumulative sales and then in excel add the data analysis tool pack and then you click tools, data analysis, regression and give these exact ranges that is given here. So, you are using the same spreadsheet that is what you will be giving and click ok. And then get your A, B and C and using the equation that we just gave you where in to solve for N, P and Q the same data I have already pre computed the values for A because you already know how to do that. You just have to include A square and use the regression and just follow the instructions because you need both the columns. So, you got it because the regression includes A column as well as A square column. So, you need to have both the columns selected so that the parameters can be fit and excel we pre compute second parameters and give it. Even if you did not get A, B and C at least using A, B and C tell me values of N, P and Q these are values of A, B and C what will be N, P and Q because to simulate it this is what we want. Any one value of N 8002 ok, P and Q already have N you can directly compute P and Q P is nothing, but A by N you already have A divided by N should give you P to be 0.0255 something and Q is 1.1852. So, now these values you can put it in your bus diffusion model in a Wensum file and simulate it to see where you are getting a bell shaped curve for adoption rate. You can substitute these values directly in your Wensum file right away and simulate the model. What are the Wensum, the Wensum model file you have for bus diffusion. Substitute the values of this N, P and Q. Since P is less than Q we are going to get a bell shaped pattern for the adoption rate. Please check it and put the values in Wensum of N, P and Q. Simulate it to get the values of adoption, adopters and adoption rate. Once you get it I have already shown how to calculate the values for the how to visually how to see the values in a table form. Once you open a table there is a feature to copy it can copy and paste the data value into your excel. Then you can compare the actual data with the simulated data. If you do that you will get this curve. So, the red curve is what I got through Wensum which is you see the exact same thing you will get it in your Wensum model also. There is a randomness you will get the exact same thing and the blue line is a real data that we actually have. Pretty reasonable fit using the bus model. So to get this graph we have to copy paste the data from Wensum. You need to plug the data in Wensum, simulate it, copy the data, paste it back in excel and then do this. There are ways in which we can get the input data itself the not input the original sales data the blue line that you see here into Wensum, but we will I will show that later. Now, to summarize this full sales example let us look at comparison of all the fits that we have been got for the adoption rate. First we, so the blue line represents the actual sales data remember that the ragged line and the green line was a logistic model that we first fit. Remember the logarithms that we took unfit a linear regression line that is represented in this green line which had a n value of 7900, a naught as 40 and c i as 1.55. The logistic model and then we did using solver where we directly use the expression for 80 and use the solver to determine the values of n, a naught and c i and this is the best estimate we had to minimize the means the square error. So these are the values we got n is 8,075, a naught is 89, c i is 1.24 and the bus model which is shown in red the logistic solver is in pink this is red in red color is the latest data that we just got with n is 8002 and p is 0.02 and q is 1.18. So if you see the c i and p are equivalent, if you can see here the c i and p are reducing from 1.55 to 1.24 to 1.18. At the same time if you can observe here, here n is very low and a naught is also quite low compared to the second case. So they you try to take a corresponding q value so that it can try to fit a curve as close as possible. In this case, a naught is much higher nearly 90 while the c i or the q is 1.24 but here we overcame the start up problem as well with a very small value for effect of advertisement and the p value again is kind of comparable to the logistic model solver scenario. So for the same, another thing you have to observe is for the same data set, we had used slightly different models and we got different results. Of course, we can just focus on logistic model solver and the bus model here. The curves are very similar to each other and both are correct. It is not like one is better over the other, it depends on how we want to start. If there is indeed advertisement happening, then we might want to prefer bus model. If there is no advertisement happening, we actually you know seeded the market by giving people freebies or initial set of people were directly approached and they immediately got this product after which word of mouth predominantly took place then you might want to represent it as a simple logistic model itself. This one green line is to illustrate that even with all these models, if you do not pay attention and be careful, you will get some plot and then you will say ok, this is also a shape let me use it, but we can definitely get a better fit if we use solvers or we use exact analytical solutions to compute the values, ok. Please remember that and observe how good a fit is and how we can improve it further based on the data given. So, this bus model can be extended similar to other diffusion model. One is to include discard and replacement purchase. Assume adopters move back to potential adopters after discarding or consuming their first purchase, they buy it and then they leave it, they abandon it. We can include it by adding additional rate which comes from adopters to the potential adopters which is shown as a discard rate. So, in this case the equation with discard rate will be adopters divided by product lifetime. You can simulate this above scenario. When the, when average, when discard rate is not considered, then all the potential adopters that is N is 100, everybody will buy the product. When there is discard rate, intuition says that some fraction will keep discarding the product every time unit. So, the steady state that we will achieve will be lower than 100. What that exact value is you can try to, we can simulate it right now. You already have the bus model, you have to open it and input these values N is 100, P is 0.01, Q is 2.5, average product lifetime is 5. Of course, you have to include this rate in your model and define the equations for that. What is the value it is saturating? The rest of the model is same as your bus model. So, in the bus model you have to add this. So, do not end up just creating two stocks and one flow. There is another flow from potential adopters, adopters which are adoption rate, word of mouth, auditing effect. Everything is the same. All you are doing is adding one more flow within your existing model, ok. I did not show that part. The equation for discard rate is adopters divided by lifetime. Other extensions are also possible. For example, if you want to model repeat purchases, there is no discard, but suppose we have an adoption rate, but we want to model repeat purchases. That is they already purchase a product and they are already adopted, but now I am just going to keep you know purchasing again and again. So, here we can distinguish between sales rate and adoption rate. So, adoption is you become kind of what can I say, you adopt that brand and sales rate is how much units are being purchased every time period. So, that can be simply modeled as for example, adoption rate. We can create a separate variable called sales rate and initial purchase rate will be initial sales per adopter. So, this suppose assume initial sales is 1. So, initial purchase rate will be adoption rate into sales rate, sales per adopter. So, sales rate will be equal to the adoption rate only from this side. Now, for repeat purchases I can simply add a variable called as repeat purchase rate and we can come up with an average consumption per adopter. Suppose, you are going to repeat it how often or what quantity they repeat it in and I can add it to the sales rate. So, me effective sales rate is nothing, but the sum of repeat purchases as well as initial purchases. So, adopters are people who are actually owning the brand and I am going to do it again and again. They are already into the product. Now, how often they are going to keep buying it again and again. Again, we can there is no discard rate in this remember. So, you can modify the bus model we can you know set n is 100, p is 0.01, q is 2.5, initial purchase is 1 per unit per person, 1 unit per person and replacement just 20 percent units per person per year. We can see what happens to the actual sales rate. The sales rate is at 100 will be more than 100 eventually it has to be little more than 100 because everybody would adopt the product, but then how does that change is something you can try it at your home. Thank you.