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## Watson, It's not Elementary!

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In 2011, IBM Watson shook our world when it beat Ken Jennings on Jeopardy and "Computer beats Man" was the reality we needed to accept.

IBM's WatsonAnalytics is now avalilabe for a 30 day trial and it did not shake my world when it came to time series analysis.  They have a free trial to download and play with the tool. You just need to create a spreadsheet with a header record with a name and the data below in a column and then upload the data very easily into the web based tool.

It took two example time series for me to wring my hands and say in my head, "Man beats Computer".  Sherlock Holmes said, "It's Elementary my dear Watson".  I can say, "It is not Elementary Watson and requires more than pure number crunching using NN or whatever they have".

The first example is our classic time series 1,9,1,9,1,9,1,5 to see if Watson could identify the change in the pattern and mark it as an outlier(ie inlier) and continue to forecast 1,9,1,9, etc.  It did not.  In fact, it expected a causal variable to be present so I take it that Watson is not able to handle Univariate problems, but if anyone else knows differently please let me know.

The second example was originally presented in the 1970 Box-Jenkin's text book and is a causal problem referred to as "Gas Furnace" and is described in detail in the textbook and also on NIST.GOV's website.  Methane is the X variable and Y is the Carbon Dioxide output.  If you know or now closely examine the model on the NIST website, you will see a complicated relationship where there is a complicated relationship between X and Y that occurs with a delay between the impact of X and the effect on Y (see Yt-1 and Yt-2 and Xt-1 and Xt-2 in the equation).  Note that the R Squared is above 99.4%!  Autobox is able to model this complex relationship uniquely and automatically.  Try it out for yourself here! The GASX problem can be found in the "BOXJ" folder which comes with every installed version of Autobox for Windows.

Watson did not find this relationship and offered a predictive strength of only 27%(see the X on the left hand of the graph) compared to 96.4%.  Not very good. This is why we benchmark. Please try this yourself and let me know if you see something different here.

Autobox's model has lags in Y and lags in the X from 0 to 7 periods and finds an outlier(which can occur even in simulated data out of randomness).  We show you the model output here in a "regression" model format so it can be understood more easily. We will present the Box-Jenkins version down below.

Here is a more parsimonious version of the Autobox model in pure Box-Jenkins notation.  Another twist is that Autobox found that the variance increased at period 185 and used Weighted Least Squares to do the analysis hence you will see the words "General Linear Model" at the top of the report.

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## Great time series analysis example using the "Ages at Death of the Kings of England" Dataset

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This is a great example of how ignoring outliers can make you analysis can go very wrong.  We will show you the wrong way and then the right way. A quote comes to mind that said "A good forecaster is not smarter than everyone else, he merely has his ignorance better organized".

A fun dataset to explore is the "age of the death of kings of England".  The data comes form the 1977 book from McNeill called "Interactive Data Analysis" as is an example used by some to perform time series analysis.  We intend on showing you the right way and the wrong way(we have seen examples of this!). Here is the data so you can you can try this out yourself: 60,43,67,50,56,42,50,65,68,43,65,34,47,34,49,41,13,35,53,56,16,43,69,59,48,59,86,55,68,51,33,49,67,77,81,67,71,81,68,70,77,56

It begins at William the Conqueror from the year 1028 to present(excluding the current Queen Elizabeth II) and shows the ages at death for 42 kings.  It is an interesting example in that there is an underlying variable where life expectancy gets larger over time due to better health, eating, medicine, cyrogenic chambers???, etc and that is ignored in the "wrong way" example.  We have seen the wrong way example as they are not looking for deterministic approaches to modeling and forecasting. Box-Jenkins ignored deterministic aspects of modeling when they formulated the ARIMA modeling process in 1976.  The world has changed since then with research done by Tsay, Chatfield/Prothero (Box-Jenkins seasonal forecasting: Problems in a case study(with discussion)” J. Roy Statist soc., A, 136, 295-352), I. Chang, Fox that showed how important it is to consider deterministic options to achieve at a better model and forecast.

As for this dataset, there could be an argument that there would be no autocorrelation in the age between each king, but an argument could be made that heredity/genetics could have an autocorrelative impact or that if there were periods of stability or instability of the government would also matters. There could be an argument that there is an upper limit to how long we can live so there should be a cap on the maximum life span.

If you look at the dataset knew nothing about statistics, you might say that the first dozen obervations look stable and see that there is a trend up with some occasional real low values. If you ignored the outliers you might say there has been a change to a new higher mean, but that is when you ignore outliers and fall prey to Simpson's paradox or simply put "local vs global" inferences.

If you have some knowledge about time series analysis and were using your "rule book"on how to model, you might look at the ACF and PACF and say the series has no need for differencing and an AR1 model would suit it just fine.  We have seen examples on the web where these experts use their brain and see the need for differencing and an AR1 as they like the forecast.

You might (incorrectly), look at the Autocorrelation function and Partial Autocorrelation and see a spike at Lag 1 and conclude that there is autocorrelation at lag 1 and then should then include an AR1 component to the model.  Not shown here, but if you calculate the ACF on the first 10 observations the sign is negative and if you do the same on the last 32 observations they are positive supporting the "two trend" theory.

The PACF looks as follows:

Here is the forecast when using differencing and an AR1 model.

The ACF and PACF residuals look ok and here are the residuals.  This is where you start to see how the outliers have been ignored with big spikes at 11,17,23,27,31 with general underfitting with values in the high side in the second half of the data as the model is inadequate.  We want the residuals to be random around zero.

Now, to do it the right way....and with no human intervention whatsoever.

Autobox finds an AR1 to be significant and brings in a constant.  It then identifies to time trends and 4 outliers to be brought into the model. We all know what "step down" regression modeling is, but when you are adding variables to the model it is called "step up".  This is what is lacking in other forecasting software.

Note that the first trend is not significant at the 95% level.  Autobox uses a sliding scale based on the number of observations.  So, for large N .05 is the critical value, but this data set only has 42 observations so the critical value is adjusted.  When all of the variables are assembled in the model, the model looks like this:

If you consider deterministic variables like outliers, level shifts, time trends your model and forecast will look very different.  Do we expect people to live longer in a straight line?  No.  This is just a time series example showing you how to model data.  Is the current king (Queen Elizabeth II) 87 years old?  Yes.  Are people living longer?  Yes.  The trend variable is a surrogate for the general populations longer life expectancy.

Here are the residuals. They are pretty random.  There is some underfitting in the middle part of the dataset, but the model is more robust and sensible than the flat forecast kicked out by the difference, AR1 model.

Here is the actual and cleansed history of outliers. Its when you correct for outliers that you can really see why Autobox is doing what it is doing.

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## Maybe the grass IS greener? We accept the challenge! Here are some 30 day trials of software in "time series analysis" or use the software you own.

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We're trying to make easier for you to prove that Autobox isn't what we think it is.  Post your model, fit and forecast and we'll post Autobox's output. Anyone, feel free to post any other 30 day trial links here as well that are "time series analysis" related.

RATS

http://www.estima.com/ratsdemo.shtml

Minitab

http://www.minitab.com/en-US/products/minitab/free-trial.aspx

Salford Systems - They say they have time series in the new version of SPM 7.0, but we can't find it so this won't do you any good. Click on the top right of the screen if you want to try your luck.

http://www.salford-systems.com/products/spm/whats-new

SYSTAT

http://www.systat.com/SystatProducts.aspx

XL Stat

GMDH Shell - New to the market. Click on the bottom of the screen to download. They offer the International Airline Passenger Series as soon as you run it. If you run it, it makes no attempt to identify the outliers known to be the demise of any modeler plus it has a very high forecast which was ther subject of criticism of Box-Jenkins using LOGS and ignoring the outliers. See Chatfield and Prothero's criticsm in the paper "Box-Jenkins seasonal forecasting: Problems in a case-study"

http://www.gmdhshell.com/

Here is the Passenger Series (monthly data) 144 obs

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