# Statistical Analysis and Modeling Demo

## Modeling techniques include nonlinear multiple regression, binary or multinomial logistic regression, and canonical analysis.

### Exploratory statistical modeling is used to discover which variables are associated with sales

Impact of Weather Forecast Accuracy on Predictive Models of Behavior
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The Impact of Weather Forecast Accuracy on Predictive Models

Weather conditions influence human behavior and the choices people make.  Ice-cream sales, success of a garage sale, number of hot-dogs sold by a New York vendor, crime rates, school truancy rates, church attendance, and even stock market returns, are all affected to a varying extent by weather conditions.  In many cases, the ways in which weather conditions affect behavior have been researched, documented, and analyzed.     The end result of this research and data analysis is often a theoretic model that describes and quantifies how certain weather elements (such as wind, rain, temperature, or humidity) influence human behavior.  BrainTech, LLC uses similar statistical techniques to transform patterns in past sales into a descriptive model of behavior that can help a company better understand their customers.  These behavioral models include variables such as day-of-week, time of year, proximity to holidays, general demand trend, and multiple weather elements.  Using modeling techniques, we can identify variables that may have influenced customer purchasing behavior in the past, and quantify the strength of association between each variable and behavior.  Almost without exception, the models of customer behavior contain at least one weather element with a quantifiable and predictable association with daily sales figures.

Although a primary reason to develop models is to understand customer behavior, many of the models can reproduce the number of sales made on any given day with a high degree of accuracy.  Using statistical techniques such as cross-validation, we can refine models so that they reflect current customer behavior and preference with a high degree of certainty.  These cross-validated models can be extremely useful for predicting short-term, future sales.

The primary challenge of using predictive models that contain weather-related variables to estimate future sales is dealing with the variable accuracy of weather forecasts.  In essence, the accuracy of a weather forecast limits the number of days the model can be expected to produce reasonable estimates of future sales.  As an example, if customers buy fewer products on days when it is raining, even a perfect model of customer behavior will fail to accurately estimate sales seven days in advance if the forecasted rainfall seven days from now turns out to be inaccurate.

This informal report addresses two important factors that predictably influence the accuracy of weather forecasts.  These two factors are: (1) properties of the geographic location for which the forecast is being made (i.e. region, elevation, climate), and (2) the weather element being forecast (i.e. temperature, wind speed, precipitation).  By understanding how forecast accuracy varies based on geographic location and the weather element being forecast, we are able to estimate how many days in advance a specific model is likely to predict sales with a high degree of accuracy.

1.     Geographic Location
For 3-day weather forecasts in Key West, Florida, the predicted temperature has been within 3 degrees of the actual temperature 90% of the time.  In Seattle, that number is around 75%.  The same holds for precipitation predictions; forecasts of precipitation in Eastern Washington or Los Angeles tend to be more accurate than forecasts of precipitation in Seattle or tropical regions of Florida.  In general, the more stable the climate, the better forecasts tend to be.  Therefore, clients in charge of inventory management for stores in Southern California and Eastern Washington may be able to extend sales estimates further into the future than those who manage inventory for stores in Western Washington and Oregon.

2.     The forecasting of certain elements (such as rain) in a 1-to-3-day period out is considered to be extremely accurate, exceeding 90% in 2002.  We have been unable to locate a comprehensive, easily accessible collection of weather forecast accuracy data by region, per weather element.  Therefore, models that rely heavily on certain weather elements require that we perform our own analysis of each weather element for each weather region.  This is why we track changes in weather forecasts.

There are cases when weather forecasts change at the last minute due to unexpected movement of storm fronts, or slower-than-anticipated air movement in zones of high or low pressure.  We have received several questions about how unexpected, last-minute change in a weather forecast affects a predictive sales models.  While the answer to this question depends largely on customer behavior and the predictive model, using statistical models to generate estimates of future sales carries little or no risk of increasing negative impact when a dramatic forecast change occurs.  In other words, if a client is using weather forecasts to estimate demand, the estimates generated by the model are equally likely to be negatively impacted when an unexpected weather change occurs.  However, because models are updated each night based on actual weather collected during the day, forecast changes are immediately incorporated into the modeling process and future estimates are adjusted accordingly.  In the rare event a weather system stalls or arrives early during an important transition period (from weekend to weekday, or vice versa), estimated sales for the next week remain valid, and shipments can be reduced or increased by an appropriate amount based on predicted sales for the upcoming week.  In summary, although the negative impact of last-minute, unexpected changes in weather forecasts on sales estimates cannot be ignored or brushed aside, using modeling to estimate future sales carries little risk of increasing or prolonging the impact of such an event.

The two following figures illustrate the effect of weather forecast accuracy on sales estimated by the same predictive model.  In each figure, the right-most blue bar indicates the date on which the prediction is made.  The diverging lines show the anticipated range of change in prediction as time progresses, given the accuracy of weather forecasts.  In other words, if the model were used to estimate how many units will be sold six days from now, the bands indicate how estimates of sales might change over the upcoming six days due to adjustments made to weather forecasts.  The first figure shows a 95% confidence band for estimates produced by the model 6 days into the future assuming accuracy of forecasts typical of a Southern-California-like climate.  In other words, given the specific model with typical Southern California weather forecast accuracy, over the next six days, sales estimates will remain within the upper and lower bands 95% of the time.  Figure 2 shows the same model’s 95% band of prediction assuming weather forecast accuracy typical of a Western-Washington-like climate.

Figure 1. Assuming forecast accuracy typical of a region with stable climate, 95% of the time the model will continue to estimate sales within the confidence bands over the next six days.

Figure 2. Assuming forecast accuracy typical of a region with variable climate, 95% of the time the model will continue to estimate sales within the confidence bands over the next six days.

The 95% confidence bands diverge at an increasing rate in the second figure due to reduced accuracy of weather forecasts for the region.  However, it is also important to note that the bands remain fairly narrow within a 3 day forecast window, regardless of Southern-California-like or Western-Washington-like weather forecast accuracy.  In summary, models that use weather forecasts to estimate future sales for the following three days remain extremely robust despite variable accuracy of weather forecasts between regions.  However, as can be seen in figure two, the model’s estimate of sales 4-7 days in the future is likely to change a great deal as the weather forecast is updated.

In conclusion, weather forecasts are sufficiently accurate so that companies able to place and receive orders within a three-day window can be reasonably confident that sales predicted by the model up to three days in advance will remain fairly stable, despite updates to weather forecasts.     Companies that produce or stock perishable goods are often able to improve inventory management and better estimate short-term demand using predictive models in conjunction with weather forecasts. In regions with more stable climates, it may be reasonable to use predictive models in conjunction with weather forecasts to estimate sales up to 4-5 days in advance.