from Statistical Models

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Wouldn't it be convenient to have some kind of 'computer model' that could accurately forecast sales? After all, knowing what the future is likely to be would give us an obvious advantage over competitors, not only in minimizing the impact of a bad forecast, but in entering new markets or avoiding certain markets that offer little hope of profitable sales. Yet, like any other method that attempts to predict the future, a mechanical forecast based on history is only as accurate as the information you have, and can easily make false predictions when unexpected events force radical changes in your business environment.

Some models are better than others at predicting otherwise 'random' changes in the business environment.
But, with little historic data to derive any kind of statistical calculations off of, you might as well
throw darts at a dart board. There's really no difference in the forecast accuracy of results obtained
from one model over another, no matter what anyone tells you, when your historic data is limited to a
range that's shorter than the future period you are attempting to predict. It's just not mathematically
sound to think that there is.

Although there are a plethora of statistical models, and most likely others not yet discovered, the
statistical models to be discussed here fall under a few basic categories. They are:

- Average (mean value)
- Linear Regression (mean value plus trend)
- Moving Average (mean value adjusted for recent trend)
- Cyclic (Fourier analysis)
- Decomposition (Average + trend + cyclic + chaos + noise)
- Artificial Intelligence (rules system, neural network)

The 'average' model (mean value of historic data) is by far the simplest method by which
historic data can be used to generate a mechanical forecast. The forecast simply becomes
the average of all of the historic data points over time. When the amount of sales history is
limited, or when sales are generally consistent from year to year on established products, this
model is usually adequate. Further, if sales for a particular item (or for a particular customer)
is sporadic in nature, an average line is really as good as any other model, due to the high
standard deviation from the mean for the historic data.

A linear regression line indicates the average and slope of the historic data, and can be very useful for generating a forecast when steady growth or steadily declining sales are present in the history. Several months of history are required in order to provide a proper basis for forecasting; I would recommend at least as much history as the forecast period (in other words, to generate 1 year of forecast, at least 1 year of history is needed). The linear regression line is basically a "best fit" linear equation that passes nearest to the various data points (sales quantities and/or revenue) with respect to time. One technique that the Demand Planning Tool employs is a "weighted regression" line, similar to a linear regression with more weight being placed on the most recent data points and their delta from the mean, than on data points further back in history. In either case, a line with a particular slope is generated. Forecast is then assigned to the closest value to this line.

A linear regression line is usually the best model for sales data that is not seasonal in nature.
Often, however, business practices tend to insert 'artificial seasonality' into the actual sales
history. A statistical model that does not reject seasonal trends, the way the 'average' and 'linear regression'
models do, might further complicate the forecast and reinforce bad business habits. Linear regression takes
only the overall trend into account, and rejects any seasonality or 'cyclical trends' in the process.

In many ways, a 'moving average' combines some of the effects that a linear regression (trend) line
provides, with special consideration for recent changes in historic data. Essentially, a block
of 'n' data points are averaged, and used to predict 'n+1'. This 'average window' is then moved
forward 1 data point, and the predicted data becomes the last data point in the new window. Window
size can be determined from historic data to produce the best model (least standard error). For large
'average window' sizes and short forecast periods, this model can be very effective in predicting
a forecast when sudden changes in the business environment are only reflected in the most recent historic
data. The Cyclical trends are also represented in the forecast generated by such a model, but are
"smoothed out" by the averaging process. 'Exponential Smoothing' is one typical statistical model that
uses a 'moving average'.

Cyclical models work best for businesses that are seasonal in nature, or expect repeating business
cycles on a continuous basis. Typically, this will involve a Fourier analysis or some similar method
that involves cyclical functions. Fourier analysis derives the coefficients of a set of sine and cosine
functions of varying frequencies that, summed together, best represent the historic data. The cyclic
nature of the history is then used to generate a forecast. This model, however, is computationally
expensive unless certain assumptions are made about the data, such as what the expected cycle periods
are expected to be. Also, when history is limited, the forecast tends to become a duplication of the
historic data itself. As such, this model should only be used when several years of history are available,
and the forecast should be limited to 1 or 2 years, much less than the amount of history.

A decomposition model is essentially one that combines 2 or more of the basic components of the historic data. These are:

- Average - This is the mean value of the historic data.
- Trend - This is the slope of the historic data. Though the linear regression model is actually 'trend' plus 'average', it is not considered to be a 'decomposition' model in and of itself. The 'trend' component is essentially the slope of the linear regression line.
- Cyclic - This is 'seasonality', or cycles within the historic data. Typically, more product will be sold in certain months within the year, or weeks within the quarter, than in others. This component averages the cyclic trends within the historic data, and attempts to predict when the least and most sales will occur.
- Chaos - Essentially, this is a component that appears to be random, but can be predicted when the appropriate type of model is used. Chaotic data is typically 'hi frequency' in nature, and therefore appears to be noise at an initial glance at a chart or graph. Chaotic data may also be non-linear, or related to other parameters not normally measured. As such, the analysis of the 'chaotic' component may be important for some types of data, but is usually disregarded or treated as 'noise'.
- Noise - This is randomness in historic data that can not be predicted nor accounted for directly or indirectly. The 'noise' component is indicated by the 'standard error' between the model 'predicting the past' (i.e. attempting to generate historic data points using the model's parameters), and the actual historic data. The standard deviation from the mean is also a good indicator of 'noise', but only relates to an 'average' model, and also includes the effect from the other components (i.e. trend, cyclic, chaos).

Depending upon the type of model used, a decomposition model generally requires a significant amount of historic
data in order to be valid. This is because the cyclic component (normally present in a decomposition model
that is not merely a modified linear regression) requires the most history in order to be valid. Typically,
to predict 1 year of forecast you should have at least 2 years of history.

By far the Artificial Intelligence technique is the most accurate, and generally the most computationally expensive method that can be used, as long as the programming is well suited to the nature of the data it is analyzing. A 'neural network', which attempts to simulate the activity of the human brain, is the most flexible method for making the kinds of adaptive decisions necessary for a good artificial intelligence model. The Demand Planning Tool uses an artificial intelligence scheme within the 'Demand Analysis' module to generate exception reports from a pre-conceived set of rules that apply to the type of data being analyzed. A set of pre-conceived 'rules' produces the least computationally expensive artificial intelligence model, and if properly designed, it is quite adequate for that particular purpose. With such a model, it is up to the programmer to determine which rules apply, and how to implement them. A 'neural network', however, will generally build its own rules based on the patterns found in the historic data, but requires considerably more computing power than pre-conceived rules.

A neural network system, though, must be 'trained'. Here is where the greatest danger lies. An over-trained neural network
will simply predict the future the same as its historic data, as though it should happen again, exactly as it has before.
This is obviously a poor application of a neural network. But, with proper 'training', a neural network can begin taking
into account the interrelationship between product lines, product lifecycles, and so forth. This may also require additional
data, such as marketing trends for other (competing) businesses, overall economic information, and so forth. Obviously, such
a model can become QUITE complicated and require a tremendous amount of history to review. Still, when properly designed, a
neural network can outperform the most popular statistical models with the least amount of historic data. And, when combined
with a set of pre-conceived rules specific to the type of analysis being performed, the neural network can easily outperform
any other statistical model with respect to accuracy and the amount of available historic data.

Although it may be an attractive idea to fully forecast a business based entirely on a computerized statistical model, it has been our experience at S.F.T., Inc. that actual operating conditions are far too unpredictable by such means. There are just too many variables for even the best statistical model to generate a reliable forecast, especially at the product and customer level, where data is often spurious, with very high variability. No mechanical model could possibly provide this kind of detail with any amount of reliability, due to the erratic nature of the data. It's just not statistically possible.

So why even bother with mechanical forecasting models in the first place? After all, if they're unreliable and inaccurate, what purpose could they possibly serve? Again, it has been our experience at S.F.T. Inc. that a mechanically generated forecast makes an excellent 'starting point' for generating a 'bottom up' forecast, from the product and customer detail level. Those customers who make frequent and consistent orders will likely continue to maintain the same ordering pattern for established products. For such customers, the mechanical model may generate a reasonable forecast. On the other hand, new product introduction may cause the ordering to shift from an existing product to the new product over time. For this reason, the sales and marketing people must review the mechanically generated forecast for accuracy, applying their product and customer knowledge, and their understanding of the marketplace as a whole, to modify the mechanical forecast as necessary to ensure the best possible accuracy. The mechanical forecast then becomes the starting point for developing a new forecast, making the overall forecasting process much faster and simpler.

In addition, once a forecast has been developed, a mechanical forecast at the aggregate level can be used to verify that the results
'make sense'. Here, I believe, is the best use of such a mechanical model. For at the aggregate level you have adequate data available,
much lower variability, and usually a lot more history. Once the sales force has generated a forecast for all customers for a given product,
the aggregate sum of the forecast can be compared to a mechanically generated forecast. Generally, a chart of some type would be used to
visually compare the results to several mechanical models simultaneously. These mechanical models form a guideline as to what the forecast
ought to look like, if business patterns remain consistent. Sales managers, marketing managers, and corporate executives can then review
the aggregate data, and use this information to make sound business decisions, including final approval of the forecast. For this purpose,
mechanical forecast models are ideal. And, the forecasting models themselves no longer need to be very sophisticated - a simple linear
regression, or a mean value line, is usually sufficient for this purpose.

Ideally, a statistical model ought to be able to predict the future based on the past, when adequate historic data is available,
and the business climate remains consistent. However, we do not live in an ideal world, nor do we run our businesses in an ideal business
environment. As such, a mechanically generated forecast can only be a guideline. We must constantly measure performance against
the forecast, and generate mechanical forecasts only when there is adequate history available to do so. Even so, those who best
understand the markets, the people in the sales force and marketing force, must apply human intelligence and understanding to
the forecast, at a customer and product level, from the bottom up, in order to make the forecast as accurate as possible. The
mechanical models themselves will get you close, but can almost never accurately predict new product introduction, product end of life,
or the effect of competing businesses on the market share.

At S.F.T., Inc., this has been our experience: For most businesses, it is a good idea to avoid complex statistical models in generating
a forecast. It is better to generate a mechanical forecast with a linear regression or mean line, then adjust the mechanical forecast based
on product and customer knowledge, and constantly measure sales performance to the forecast, making minor adjustments where necessary. Further,
when the forecast itself doesn't contain a lot of 'artificial seasonality' (based on previous business practices), the result from this process
is a 'smoothing out' of the deliveries of product to the various customers, reducing the 'noise' component and making the process more controlled,
and forecasts more accurate in the future, without the use of complex statistical models. This 'whole business process' approach is something
that even the most accurate mechanical forecast cannot accomplish alone. And, this is why our Demand Planning Tool application
does not focus on providing a whole array of statistical forecasting models, but instead focuses on forecast and plan verification, and measuring
the 'whole business' performance to this plan. For, without the 'error' feedback, and product/customer knowledge provided by the sales and marketing
force, one mechanical model is really as good as any other at trying to do something that really is impossible: predict the future.

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