Now everything that we've been talking about so far has focused on looking at forecasts as a particular estimate, a pinpoint estimate or the point estimate. Well, really when we're making predictions, we're not just talking about a single point estimate, we're talking about distributions. And so, imagine your regression line. If this is our regression line, imagine at a particular point overlaying a normal distribution around that point. And so, that normal distribution reflecting the range of values potentially that we might expect to observe around that. So if I were to get multiple observations with a value of X2, I have my best guess, but I've got that entire range of uncertainty around that and that's important to keep in mind. That we're not just predicting a point, we're predicting overall how much certainty do we have that it's going to fall within a particular range and that's something that's going to come up in the example that we work through a little bit later. Now, how well we built a nice regression model? Did we really need it? Could we have done something that easier? Well, what if we used a Simple Moving Average model? Or what if we used formulated it as in other regressive models saying, let's use the new most recent weeks of observations and those are going to be our predictors and we'll use those two data points to predict. So if I look back one week, look back two weeks, use that to predict what's demand going to be like in the current week. It turns out, you're not going to do too poorly with that particular measure. So our regression line for forecasting is in black, the purple line here giving us that simple moving average based model, just using those two weeks of data. And it looks reasonably similar, but the piece that I wanted to point out and you can see it clearly in this case here as well that the smoothing based approach where the auto-regressive model, where we're relying just on those recent observations, it tends to take longer to adjust than the regression-based model. So any time there's that discontinuity, the end of a quarter, the end of a month. Because I'm only looking back at the most recent two weeks, it's not as quick to adjust as our regression-based model. The other thing that we've gotta keep in mind is how far out into the future am I trying to forecast? Our regression includes a trend and month specific effect. I can forecast a year out, no problem. But with our simple moving average model, what do I do if I want to go beyond a two-week period? I actually need to keep on forecasting to produce those X variables, and I'm actually going to have to resort to a simulation-based procedure to take into account the amount of variation that I'm going to observe and be able to forecast out that four-year period. If we look at measures such as the average absolute error, our regression based model's doing better there. The farther out into the future that we're trying to forecast, keep in mind the more uncertainty that we're potentially going to observe. So this will, I pulled off stock forecasts. For Google and just comparing the amount of uncertainty, the bounds on this simulation procedure, looking out one month, two months, one year. Notice how much variation we observe if we try to predict out a year, whereas if we're just going out a couple of months, the amount of uncertainty that we have is much smaller. And so whenever you're using methods that rely on the most recent observations as inputs into your forecasts, you're going to have, there's this uncertainty that's going to keep on growing the further out you try to make those predictions. So, let's discuss the Excel exercise that we're going to work through next. We're going to do a demand planning scenario. So David's Retail, Art and Trade Shop. I've got to make some orders for the holiday season, determine how much I'm going to order based on historic consumer demand. Now suppose that our forecasts are too high, well, let's order a lot of products. Well, now we've got inventory that's gone unsold. We're going to have to get rid of it, maybe we can return it, maybe we've got to liquidate it. What if I don't order enough? Now I'm leaving money on the table, because I have consumer demand and I don't have the inventory to meet that demand. So, what we're going to be looking at is building a tool to assist with this demand planning exercise. So based on the historic demand, we're going to figure out what is the right quantity to be ordering where here is some of the constraints that we are dealing with. Order quantity is going to be placed in, quantities are going to be ordered in 25 unit increments. We're going to assume that we're going to sell it at a price of $15, so we haven't changed the price over time. How much does it cost us? Well, how much it costs us depends on how much we order. And if we're ordering less than 100 units, it's going to cost us $12. If we're ordering between 101 and 200 units, the first 100 units are going to cost us $12, the next 100 units would cost us $10 a piece. And ordering above 200 units, $12 for the first 100 units. $10 for the next 100 units. $8 for each unit above an beyond that and we do have a value that we can salvage, any merchandise that's left over we'll be able to get recouped $5 a month for it. So we don't completely waste our money, but we're not going to get the $15 if we have unsold merchandise. And so, here's the problem that we have is what's the right quantity for us to order where our objective is profit maximization. And so if I order too little, potentially if the money on the table. If I order too much, there is a chance that I don't sell it and that I end up spending too much money, because I don't get all of my money back for the excess, for the merchandise in excess of the demand quantity. So for each month, we want to know what's the number that we need to order and we want to know how much confidence do we have of profit falling into a particular range? So what is the range that contains centered around our best guess, centered around our estimate for profit? What's the range that we're 50% confident, 75% confident and 95% confident that profit's going to fall into? As far as deconstructing this problem and how do we approach it, we're going to start by building a forecasting model, just like the regression models that we've been looking at so far. Once we have our forecasting model, we know what the baseline level of demand is. We're going to know what the variation looks like say, from month to month. And we're going to be able to predict, what is demand going into the future and how much uncertainty do we have in those estimates? Well, then we're going to look at suppose we knew what the level of demand was, we're going to establish how does that connect to revenue and costs based on our order quantity with demand known. So then for whatever quantity we decide to order, we're going to simulate out what the different levels of demand are using the results from our forecasting model. And based on that stimulation for each time we run that simulation, we're going to be able to construct what's the revenue? What are the costs that we incur? And ultimately, what's that profit if we knew exactly what the demand was going to be? Well, we're going to simulate out a bunch of different levels of demand based on our forecasting model and then we're going to take the average across them that's going to give us our expectation for profit. We're also going to use those simulations to characterize those ranges we were looking for. So what's the range on 50% sure of profit falls in the range of 75% sure, 95% sure? So that Excel spreadsheet with the data we're going to be working with is up on the course website. So, we'll turn to that next.