This lecture is about forecasting, which is a very specific kind of prediction problem. And it's typically applied to things like time series data. So, for example, this is the stock of information for Google on the NASDAQ, and so is this symbol GOOG. And you can see over time that there's a price for this stock and it goes up and down. So this introduces some very specific kinds of dependent structure and some additional challenges that must be taken into account when performing prediction. And so, first of all, the data are dependent over time, and so, that alone makes prediction a little bit more challenging than it is when you have independent examples. There's also some specific pattern types that should be paid attention to. Trends, such as long term increases or decreases, seasonal patterns are very common in this kind of data. For example, seasonal patterns over weeks, months, years, etc. Cycles, patterns that rise and fall periodically over a period that's longer than a year, for example. Here, the subsampling and the training and test can be a little bit more complicated because you can't just randomly assign samples into training and test. You have to take advantage of the fact that there's actually specific times that are being sampled and that points are dependent in time. Similar issues arise in predictions of spatial den, spatial data. For example, there's dependency between nearby observations and there may be location-specific effects that have to be modeled when doing prediction. Typically, the goal here is to predict one or more observations into the future and all standard prediction algorithms can be used, but you have to be a little bit cautious about how you use them. So, one thing to be aware of is that you have to be careful of spurious correlations. So, time series can often be correlate for reasons that do not make them good for predicting one from the other. So, if you look at, you can go to Google Correlate to correlate different words over time, the frequency of different words over time. And so, for example, here you can see a correlation between the Google stock price, shown in blue, and solitaire network, which is in red. And so, those don't necessarily have anything to do with each other at all, but they have a very high correlation, and you might think you might be able to predict one from the other, even though in the future, they might diverge substantially because they aren't necessarily related to each other at all. It's also very common in geographic analysis. This is actually a cartoon from xkcd that shows that heat maps particularly population-based heat maps had very similar shapes because of the place where many people live. So for example, the users of a particular site or the subscribers to a particular magazine or the consume, consumers of a particular type of website may all appear in the very similar places because the highest density in population in the United States is over here on the Eastern seaboard. And so, you see very similar heat maps of a large number of individuals at all of those different places. You should also beware of extrapolation. So this is a kind of a funny example that shows what happens if you extrapolate time series out without being careful about what could happen. So this shows on a long scale the winning time of a large number of oh sorry, of races that occurred at the Olympics. The blue times are men and the red times are women, and these authors of this paper extrapolated out into the future and said that in 2156 that would be when women would run faster than men in the sprint. And while we don't know when that, when or when that may or may not occur, one thing that was pointed out is that this kind of extrapolation is very dangerous. Eventually at some time in the future, both men and women will be predicted to run negative times for the 100 meters. And so, you have to be very careful about how far out you extrapolate from your data. So, I'm going to show a quick example of some forecasting using the quantmod package and some Google data. So, if I load this quantmod package and I can, I can load in a bunch of data from the Google stock symbol. And I can get it from the Google finance data set. And so if I look at this Google variable, I get the open, high, low, close, and volume information for a particular Google stock from the 1st of January, 2008 to December 31st, 2013. So I can summarize this monthly and store it as a time series. So I can use the two monthly variable or function to convert that to a monthly time series. And I can just take the opening information, and then I can create a time series object using the ts function in R. And if I plot that, I can see here's the monthly opening prices for Google over a period of seven years. So, an example time series decomposition would decompose this time series into a trend, any kind of consistent pattern, a seasonal pattern over time, and cyclic patterns where the data rises and falls over non fixed periods. And so, one way that we can do this is with the decompose function in R. So if I decompose this in an additive way, then I can see that there's a trend variable that appears to be an upward trend of the Google stock price. There also appears to be a seasonal pattern, as well as a more of a random cyclical pattern in the data set. So this is decomposing this series here into a series of different types of patterns in the data. So here for training and test sets, I have to build training and test sets that have consecutive time points. So here I am building a training set that starts at time point 1 and ends at time point 5. And then a test set that is the next consecutive sets of points after that. So that way, I can always build a training set and apply it to a test set that have consecutive time points that show the same sort of trends that I've observed in my data. So there's a couple different ways for doing forecasting. One is to do a simple moving average, which in another words, it basically averages up all of the values of, for a particular time point. And the prediction will be the average of the previous time points out to a particular time. You can also do exponential smoothing. In other words, basically we weight near-by time points as higher values or by more heavily than time points that are farther away. So there's a large number of different classes of smoothing models that you can choose. And for exponential smoothing, you can get an, you can fit a model where you have a different choices for the different types of trends that you might want to fit. And then when you forecast, you can get a prediction that comes out of your forecasting model. And you can also get sort of a prediction bounds for what are the possible values that you could get from that prediction. And you can get the accuracy using this accuracy function, so you can basically get the accuracy of your forecast using your test set, and it will give you root mean square to error and other metrics that are more appropriate for forecasting. I've obviously gone through this very fast and so, if you want more information, there's actually an entire field dedicated to forecasting and time series prediction. And I would highly recommend Rob Hyndman's Forecasting: principles and practice. This is a free book that's online and it's really, really good, and has a lot of information about how to get started at, in forecasting. So the cautions are to be wary of spurious correlations. Be very careful about how far you predic, predict out into the future with express, extrapolation, and be wary of dependencies like seasonal effects over time. If you would like information on you, for financial prediction and financial forecasting, the quantmod and quandl packages are also very useful in that area.
