Thoughts, ideas and detailed information on Forecasting.
This graph is from a client and while it is only one series, it is so illustrative. The lesson here is to model and not fit. There might not be strong enough seasonality to identify it when you only have a few months that are seasonal, unless you are LOOKING for exactly that. Hint: The residuals are gold to be mined.
This will be our shortest BLOG ever, but perhaps the most compelling? Green is FPro. Red is Autobox. Actuals are in Blue.
Let's take a look at Microsoft's Azure platform where they offer machine learning. I am not real impressed. Well, I should state that it's not really a Microsoft product as they are just using an R package. There is no learning here with the models being actually built. It is fitting and not intelligent modeling. Not machine learning.
The assumptions when you do any kind of modeling/forecasting is that the residuals are random with a constant mean and variance. Many aren't aware of this unless you have taken a course in time series.
Azure is using the R package auto.arima to do it's forecasting. Auto.arima doesn't look for outliers or level shifts or changes in trend, seasonality, parameters or variance.
Here is the monthly data used. 3.479,3.68,3.832,3.941,3.797,3.586,3.508,3.731,3.915,3.844,3.634,3.549,3.557,3.785,3.782,3.601,3.544,3.556,3.65,3.709,3.682,3.511, 3.429,3.51,3.523,3.525,3.626,3.695,3.711,3.711,3.693,3.571,3.509
It is important to note that when presenting examples many will choose a "good example" so that the results can show off a good product. This data set is "safe" as it is on the easier side to model/forecast, but we need to delve into the details that distinguish the difference between real "machine learning" vs. fitting approaches. It's important to note that the data looks like it has been scaled down from a large multiple. Alternatively, if the data isn't scaled and really is 3 digits out then you also are looking for extreme accuracy in your forecast. The point I am going to make now is that there is a small difference in the actual forecasts, but the level(lower) that Autobox delivers makes more sense and that it delivers residuals that are more random. The important term here is "is it robust?" and that is what Box-Jenkins stressed and coined the term "robustness".
Here is the model when running this using auto.arima. It's not too different than Autobox's except one major item which we will discuss.
. 
The residuals from the model are not random. This is a "red flag". They clearly show the first half of the data above 0 and the second half below zero signaling a "level shift" that is missing in the model.
Now, you could argue that there is an outlier R package with some buzz about it called "tsoutliers" that might help. If you run this using tsoutliers, a SPURIOUS Temporary Change(TC) up (for a bit and then back to the same level is identified at period #4 and another bad outlier at period #13 (AO). It doesn't identify the level shift down and made 2 bad calls so that is "0 for 3". Periods 22 to 33 are at a new level, which is lower. Small but significant. I wonder if MSFT chose not to test use the tsoutliers package here.
Autobox's model is just about the same, but there is a level shift down beginning at period 11 of a magnitude of .107.
Y(T) = 3.7258 azure
+[X1(T)][(- .107)] :LEVEL SHIFT 1/ 11 11
+ [(1- .864B** 1+ .728B** 2)]**-1 [A(T)]
Here are both forecasts. That gap between green and red is what you pay for.
Note that the Autobox upper confidence limits are much lower in level.
Autobox's residuals are random
Guess What We Uncovered ? The 2001 M3 Competition's Monthly calculations for SMAPE were off for most of the entries. How did we find this? We are very detailed.
14 off the 24 to be exact. The accuracy rate was underestimated. Some entries were completely right. ARARMA was almost off by 2%. Theta-SM was off by almost 1%. Theta-SM's 1 to 18 SMAPE goes from 14.66 to 15.40. Holt and also Winter were both off by 1/2%.
The underlying data wasn't released for many years so this made this check impossible when this was released. Does it change the rankings? Of course. The 1 period out forecast and the averaged of 1 to 18 are the two that I look at. The averaged rankings had the most disruption. Theta went from 13.85 to 13.94. It's not much of a change.
The three other seasonalities accuracies were correctly computed.
if you saw our release of Autobox for R, you would know that Autobox would place 2nd for the 1 period out forecast. You can use our spreadsheet and the forecasts from each of the competitors and prove it yourself.
See Autobox's performance in the NN3 competition here. SAS sponsored the competition, but didn't submit any forecasts.
IBM released version SPSS Modeler 18 recently and with it a 30 day trial version.
We tested it and have more questions than answers. We would be glad to hear any opinions(as always) differing or adding to ours.
There are 2 sets of time series examples included with the 30 day trial.
We went through the first 5 "broadband" examples that come with the trial that are set to run by default. The 5 examples have no variability and would be categorized as "easy" to model and forecast with no visible outliers. This makes us wonder why there is no challenging data to stress the system here?
For series 4 and 5 both are find to have seasonality. The online tutorial section called "Examining the data" talks about how Modeler can find the best seasonal models or nonseasonal models. They then tell you that it will run faster if you know there is no seasonality. I think this is just trying to avoid bad answers and under the guise of it being "faster". You shouldn't need to prescreen your data. The tool should be able to identify seasonality or if there is none to be found. The ACF/PACF statistics helps algorithms(and people) to help identify seasonality. On the flipside, a user may think there is no seasonality in there data when there actually is so let's take the humans out of the equation.
The broadband example has the raw data and we will use that as we can benchmark it. If we pretend that the system is a black box and just focused on the forecast, most would visually say that it looks ok, but what happens if we dig deeper and consider the model that was built? Using simple and easy data avoids the difficult process of admitting you might not able complicated data.
The default is to forecast out 3 periods. Why? With 60 months of data, why not forecast out at least one cycle(12)? The default is NOT to search and adjust for outliers. Why? They certainly have many varieties of offerings with respect to outliers, but makes me wonder if they don't like the results? If you enable outliers only "additive" and "level shift" are used unless you go ahead a click to enable "innovational", "transient", "seasonal additive", "local trends", and "additive patch". Why are these not part of the typical outlier scheme?
When you execute there is no audit trail of how the model go to its result. Why?
You have the option to click on a button to report "residuals"(they call them noise residuals), but they won't generate in the output table for the broadband example. We like to take the residuals from other tools and run them in autobox. If a mean model is found then the signal has been extracted from the noise, but if Autobox finds a pattern then the model was insufficient...given Autobox is correct. :)
There is no ability to report out the original ACF/PACF being reported. This skips the first step for any statistician to see and follow why SPSS would select a seasonal model for example 4 and 5. Why?
There are no summary statistics showing mean or even number of observations. Most statistical tools provide these so that you can be sure the tool is in fact taking in all of the data correctly.
SPSS logs all 5 time series. You can see here how we don't like the kneejerk movement to use logs.
We don't understand why differencing isn't being used by SPSS here. Let's focus on Market 5. Here is a graph and forecast from Autobox 
Let's assume that logs are necessary(they aren't) and estimate the model using Autobox and auto.arima and both software uses differencing. Why is there no differencing used by SPSS for a non-stationary series? This approach is most unusual. Now, let's walk that back and run Autoboc and NOT use logs and differencing is used with two outliers and a seasonal pulse in the 9th month(and only the 9th month!). So, let's review. SPSS finds seasonality while Autobox & Auto.arima don't.
How did SPSS get there? There is no audit of the model building process. Why?
We don't understand the Y scale on the plots as it has no relationship to the original data or the logged data.
The other time series example is called "catalog forecast". The data is called "men". They skip the "Expert modeler" option and choose "Exponential Smoothing". Why?
This example has some variability and will really show if SPSS can model the data. We aren't going to spend much time with this example. The graph should say it all. Autobox vs SPSS
The ACF/PACF shows a spike at lag 12 which should indicate seasonality. SPSS doesn't identify any seasonality. Autobox also doesn't declare seasonality, but it does identify that October and December's do have seasonality (ie seasonal pulse) so there is some months which are clearly seasonal. Autobox identifies a few outliers and level shift signifying a change in the intercept(ie interpret that as a change in the average).
If we allow the "Expert Modeler", the model identified is a Winter's additive Exponential smoothing model.
We took the SPSS residuals and plotted them. You want random residuals and these are not it. If you mismodel you can actually inject structure and bias into the residuals which are supposed to be random. In this case, the residuals have more seasonality(and two separate trends?) due to the mismodeling then they did with the original data. Autobox found 7 months to be seasonal which is a red flag.
I think we know "why" now.
SAP has a webpage with a tutorial on using their Predictive Analytics 2.3 tool(formerly KXEN Modeler)using daily data. They released this back in December, but didn't see until browsing Twitter. It provides an unusual public record of what comes out of SAP. They didn't publish the model with p-values and all of the output, but this is good enough to compare against. We ran numerous scenarios with different modeling options to understand what the outcome would be using these modeling(ie variable) techniques. Autobox has some default variables it brings in with daily data. We will have to suppress some of those features so that when we use the SAP variables they don't collide with them and make a multicollinear regression.
The Tutorial is well written and allows you to easily download the 1,724 days of data and model this yourself. While SAP had a .13 MAPE(in sample), they had a challenge at the end for those who get a MAPE less than .12 to contact them. Can you predict what Autobox did? .0724. Guess who is going to contact them? I will also add, that if you can do better contact us as we might have something to learn too. I also suggest that you post how other tools handle this as well as that would be interesting to see as well. Autobox thrives(1st among automated) on daily data as it did in a daily forecasting competition and is much more difficult to model and something we have dedicated 25 years to perfecting.
After reading the SAP user's guide let's make the distinction that Autobox uses all of the data to build the model, while SAP (like all other tools) withholds data to "train" on.
Autobox adjusts for outliers. One could argue that by using adjusting for outliers the MAPE will only go down which is true, but it be aware that it allow for a clearer identification of the relationships in the data( ie coefficients / separating signal from noise).
The first approach in the SAP tutorial is running with only historical data and they add in the causals later. Outliers are identified and has a MAPE of .197.
A bunch of very curious variables(66??----PenultimateWednesday) are included that we have never seen before which made us scratch our heads (with delight???). They seem to try and capture the day of the week so we will turn that off some of Autobox's searches to avoid collinearity when we run with these in the first pass. They seem to use a day of year variable which I have never seen before. What book are they getting ideas to use these kind of variables from? Not one that I have ever seen, but perhaps someone can enlighten me? There are two variables that are measuring the number of working days that have occurred in the month and the number left in the month. We did find that some of these variables do have importance in the tests we ran so SAP has some ideas generating useful variables, but many are collinear and this could be called "kitchen sink" modeling. We will do more research into these. There is a holiday variable which also flags working days so the two variables would seem to be collinear. These two end up as the second and third most powerful variables in the SAP model. When we tried these in Autobox, both runs found them significant. Perhaps they measure (implicitly) holidays too? We are not sure, but they help.
There are weather variables which are useful and actually represent seasonality so using both monthly dummies/weekly dummies and the weather variables could be problematic. The holidays have been all combined into one catch all variable. This assumes that each holiday behaves similarly. It should be noted that a major difference is that SAP does not search for lead or lag relationships around the causals while Autobox can do that. Just try running this example in Autobox and then SAP. We ran with all of these curious variables. We then reduced these variables and kept only Holiday, gust, rain, tmean, hmean, dmean, pmean, wmean, fmean, TubeStrike and Olympics and removed the curious other variables. The question which might arise "how much can you trust the weather predictions?", but here we are looking at only the MAPE of the fit so that is not a topic of concern.
SAP ended up with a .13 MAPE when using there long list of causals. The key here is that no outliers are identified in the analysis. This is a distinction and why Autobox is so different. If you ignore outliers they do still exist and yes they exist in causal problems. By ignoring something that doesn't mean it goes away, but ends up impacting you elsewhere such as the model and you likely aren't even aware of its impact. By not being able to deal with outliers your model with causals will be skewed, but no one talks about this in any school or text book so sorry to ruin this illusion for you. Alice in Wonderland(search on alice) thought everything was perfect too, until.....
Autobox does stepdown regression, but also does "stepup" where it will search for changes in seasonality(ie day of the week), trend/level/parameters/variance as things sometimes drastically change. If you're not looking for it then you will never find it! s. The MAPE we are presenting can be found in the detail.htm audit report from the Autobox run(hint:near the bottom). We suppressed the search for special days of the month which are useful in ATM data, but not theoretically plausible for this data. Autobox allows for holidays in the Top 15 GDP's, but in general assumes the data is from the US so we will need to suppress that search. We suppressed the search for special days of the month which are useful in ATM daily data as payday's are important, but not theoretically plausible for this data.
To summarize: We can run this a few different ways, but we can't present all of these results down below as it would be too much information to present here. We included some output and the Autobox file (current.asc-rename that if you want to reproduce the results) so you can see for yourself. What we do know is that including ARIMA increases run time.
MAPE's
Let's consider the model that was used to develop the lowest MAPE of .0724.
There were 38 outliers identified over the 1,724 observations so the goal is not to have the best fit, but to model and be parsimonious.
So, what did we do to make things right? We started by deleting all kinds of variables. There were linearly redundant variables such as WorkingDay that is perfectly correlated (inverse here) to Holiday which by definition should never be done when using dummy variables. The variable "Special Event" is redundant with TubeStrike and Olympics as well. Special Event name isn't even a number, but rather text and also is redundant.
All other software withholds data whereas Autobox uses all of the data to build the model as we have adaptive technology that can detect change (seasonality/level/trend/parameters/variance plus outliers). We won best dedicated forecasting tool in J. Scott Armstrong's "Principles of Forecasting". For the record, we politely disagree against a few of the 139 "Principles" as well.
We report the in sample MAPE, in the file "details.htm" seen below...
Another way to compare the Autobox and SAP results are by comparing side by side the actual and fit and you will clearly see how Autobox does a better job. The tutorial shows the graph for univariate, but unfortunately not for the causal run! Here is the graph of the actual, fit and forecast.
We prefer the actual and residuals plot as you can see the data more clearly.
The sign of the coefficients make sense(for the UK which is cold). When it's warmer people will skip the car and use the bike, for example so when Temperature goes up (+ sign) then people rent more bikes. When its gusty people will not and just drive. The tutorial explains the variables names in the back. tmean is average temperature, w is wind, d is dewpoint, h is humidity, p is barometric pressure, d is real feel temperature. All 6 holidays were found to be important with all but one having lead or lag impacts. When you see a B**-2 that means two days before the Christmas volume was low by 5036. Autobox found all 6 days of the week to be important. The SAP Holiday variable was a mixture of Saturday and Sunday and causes some confusion with interpretation of the model. This approach is much cleaner. The first day of the data is a Saturday(1/1/2011) and the variable "FIXED_EFF_N10107" is measuring that impact that Saturday is low by 4114. Sunday is considered average as day 7 is the baseline. See below for more on the day of the week rough verification(ie pivot table/contribution %).
Note the "level shift' variables added to the model. This meant that the volume changed up or down for a period and Autobox identified and ADAPTED to it. We call this "step up regression"(nothing there right? Yes, we own that world!) as we are identifying on the fly deterministic variables and adding them to the model. The runs with the SAP variables fit 2012 much better. The first time trend began at period 1 with volume steadily increasing 10.5 units each day. This gets tampered down with the second time trend beginning at 177 making the net effect +4.3 increase per day. 38 outliers were identified which is the key to whole analysis. They are sorted by their entry into the model and really their importance.
Note the Seasonal pulse where the first day becomes much higher starting at period 1639 and forward with an average 3956.8 higher volume. Thats quite a lot and if you do some simple plotting of the data it will be very apparent. Day 1 and Day 2 were always low, but over time Day 1 has become more average, Note the AR1 and AR7 parameters.
Let's consider the day of the week data by building a pivot table.
And getting this % of the whole. We call this the contribution %. Day 7 in Excel is Saturday which is low and notice Sunday(baseline) is even lower(remember that the holiday variable had a negative sign? The sign for Saturday was +1351.5 meaning it was 1351 higher than Sunday which matches the plot below. This type of summarization ignores trend, changes in day of the week impacts, etc. so be careful. We call this a poor man's regression because those percentages would be the coefficient if you ran a regression just using day of the week. It is directional, but by not means accurate as Autobox. We use this type of analysis to "roughly verify" Autobox with day of the week dummies, monthly dummies, and day of the month effects using pivot tables. The goal is not to overfit, but rather be parsimonious. Auto.arima is not parsimonious.
Let's look at the monthly breakout. Jan,Feb,Dec are average and the other months are higher with a slope up to the Summer months and down back to Winter. The temperature data replaces the use of monthly or weekly dummies here.
It's been 6 months since ourlast BLOG. We have been very busy.
We engaged in a debate on a linkedin discussion group over the need to pre-screen your data so that your forecasting algorithm can either apply seasonal models or not consider seasonal models. A set of GUARANTEED random data was generated and given to us as a challenge four years ago. This time we looked a little closer at the data and found something interesting. 1)you don't need to pres-creen your data 2)be careful how you generate random data
Here is my first response:
As for your random data, we still have it when you send it 4 years ago. I am not sure what you and Dave looked at, but if you download run the 30 day trial now and we always have kept improving the software you will get a different answer and the results posted here on dropbox.com.https://www.dropbox.com/s/s63kxrkquzc6e00/output_miket.zip
I have provided your data(xls file),our model equation (equ), forecasts(pro), graph(png) and audit of the model building process(htm).
Out of the 18 examples, Autobox found 6 with a flat forecast, 7 with 1 monthly seasonal pulse or a 1 month fixed effect, 4 with 2 months that had a mix of either a seasonal pulse or a 1 month fixed effect, 2 with 3 months that had a mix of either a seasonal pulses or a 1 month fixed effect.
Note that no model was found with Seasonal Differencing, AR12, with all 11 seasonal dummies.
Now, in a perfect world, Autobox would have found 19 flat lines based on this theoretical data. If you look at the data, you will see that there were patterns found where Autobox found them that make sense. There are sometimes seasonality that is not persistent and just a couple of months through the year.
If we review the 12 series where Autobox detected seasonality, it is very clear that in the 11 of the 12 cases that it was justified in doing so. That would make 17 of the 18 properly modeled and forecasted.
Series 1 - Autobox found feb to be low. A All three years this was the case. Let's call this a win.
Series 2 - Autobox found apr to be low. All three years were low. Let's that call this a win.
Series 3- Autobox found sep and oct to be low. 4 of the 6 were low and the four most recent were all low supporting a change in the seasonality. Let's call this a win.
Series 4- Autobox found nov to be low. All three years were low. Let's call this a win.
Series 5- Autobox found mar, may and aug to be low. All three years were low. Let's call that a win.
Series 7- Autobox found jun low and aug high. All three years matched the pattern. Let's call that a win.
Series 10 - Autobox found apr and jun to be high. 5 of the 6 data points were high. Let's call this a win.
Series 12 - Autobox found oct to be high and dec to be low. All three years this was the case. Let's call this a win.
Series 13 - Autobox found aug to be high. Two of the three years were very very high. Let's call this a win.
Series 14 - Autobox found feb and apr to be high. All three years this was the case. Let's call this a win.
Series 15 - Autobox found may jun to be high and oct low. 8 of the 9 historical data points support this, Let's call this a win.
Series 16 - Autobox found jan to below. It was very low for two, but one was quite high and Autobox called that an outlier. Let's call this a loss.
A little sleep and then I posted this response:
After sleeping on that very fun excercise, there was something that still wasn't sitting right with me. The "guaranteed" no seasonality statement didn't match with the graph of the datasets. They didn't seem to have randomness and seemed more to have some pattern.
I generated 4 example datasets from the link below. I used the defaults and graphed them. They exhibited randomness. I ran them through Autobox and all had zero seasonality and flat forecasts.
http://www.random.org/sequences/
You should be. There is information to be MINED in the model. Macro conclusions can be made from looking at commonalities across different series(ie 10% of the SKUs had an outlier four months ago---ask why this happened and investigate to learn what you are doing wrong or perhaps confirm what you are doing right!...and perhaps the other 90% SKUs also had some impact as well, but the model didn't detect it as it was borderline. You could then create a causal variable for all of the SKUs and rerun and now 100% of the SKUs have the intervention(maybe constrain all of the causals to stay in the model or lower the statistical test to accept them into the model) modeled to arrive at a better model and forecast. Let's explore more ways to use this valuable information:
LEVEL SHIFTS
When hurricane Sandy hit last October, it caused a big drop for a number of weeks. Your model might have identified a "level shift" to react to the new average. The forecast would reflect this new average, but we all know that things will return, but the model and forecast aren't smart enough to address that. It would make sense to introduce a causal variable that reflected the drop due to the hurricane, BUT the future values of the causal would NOT reflect the impact so the forecast would return to the original level. So, the causal would have a lot of leading zeroes, and 1's when the impact of Sandy was felt and 0's when the impact would disappear. You could actually transition the 1 to a 0 gradually with some ramping techniques we learned from the famous modeler/forecaster Peg Young of the US DOT. The 0 dummy variable might increment like this 10,0,0,0,0,0,0,,1,1,1,1,1,1,1,.9,.8,.7,.6,.5,.4,.3,.2,.1,0,0,0,0,0,0,etc.
OUTLIERS
When you see outliers you should be reviewiing them to see if there is any pattern to them. For example, if you don't properly model the "Super Bowl" impact, you might see an outlier on those days. It takes a little time and effort to review and think "why" does this happen. The benefits of taking the time to do this can have a powerful impact. You can then add a causal with a 1 in the history when the Supewr Bowls took place and then the provide a 1 for the next one. For monthly data, you might see a low June as an outlier. Don't adjust it to the mean as that is throwing the baby away with the bath water. This means you might not be modeling the seasonality correctly. You might need an AR12, seasonal differencing or seasonal dummies.
SEASONAL PULSES
Let's continue with the low June example. This doesn't necessarily mean all months have seasonality and assuming a model instead of modeling the data might lead to a false conclusion for the need of seasonality. We are talking about a "seasonal pulse" where only June has an impact and the other months are near the average. This is where your causal dummy variable has 0's and a 1 on the low Junes and also the future Junes(ie 1,0,0,0,0,0,0,0,0,0,0,0,1).
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".
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. 
You have data that is decreasing. You have three areas where the data seems to level off. Is it a trend or is it two level shifts?
If you have any knowledge about what drives the data then by all means use a causal variable. What to do if you have none? It then becomes an interesting and very debatable topic.
How many periods determines a level shift might be a big factor here.
Simpson's Paradox is where you have a global significance, but not local. From a global perspective, sure there is a trend. From a local, there is no trend. Who is to say that the overall trend will continue? Who is to say that the trend won't? Maybe it will go up?
If you run this without making assumptions, you get two level shifts at period 14 and 25 and some outliers using the following data
20324 19856 19012 17247 18616 17786 20509 19097 19437 18562 17648 18672 17324 16765 16108 14742 16567 16041 15511 15403 16797 13977 15570 16249 14005 16645 14098 12310 15923 13422 13030
Y(T) = 18776. monthly
+[X1(T)][(- 2800.9 )] :LEVEL SHIFT 14 2011/ 10
+[X2(T)][(- 2602.3 )] :LEVEL SHIFT 25 2012/ 9
+[X3(T)][(+ 3272.0 )] :PULSE 26 2012/ 10
+[X4(T)][(- 1998.3 )] :PULSE 22 2012/ 6
+[X5(T)][(+ 2550.0 )] :PULSE 29 2013/ 1
+ + [A(T)]
Exploiting the Value of Information
Don’t tell me I can look anywhere in the database. Tell me where to look for something I didn’t know.
Do I need to plan using daily data?
I need to know when we will reach our capacity. When in the future will we exceed a user-specified high-side critical value and precisely when is this expected to happen with a confidence level?
Have we detected a statistically significant change in variability?
Have we detected a statistically significant change in the model such that the older data needs to be truncated as the pattern has completely changed?
What can we expect to happen if we alter our advertising/promotion/price activity?
Forecasting Weekly Beer Sales
