Statistics and Forecast Help
Introduction to Forecast
Let's revisit the ''USA hurricane scenario.'' The predicted path for Hurricane Emma shown in Fig. 6-3 is a hypothesis, not an absolute. The absolute truth will become known eventually, and a path for Emma will be drawn on a map with certainty, because it will represent history!
The "Purple Line"
If you live in a hurricane-prone region, perhaps you have logged on to the Internet to get an idea of whether a certain storm threatens you. You look on the tracking/forecast map, and see that the experts have drawn a ''purple line'' (also known as the ''line of doom'') going over your town on the map! Does this tell you that the forecasters think the storm is certain to hit you? No. It tells you that the ''purple line'' represents the mean predicted path based on computer models.
As time passes – that is, as the experiment plays itself out – you will get a better and better idea of how worried you ought to be. If you're the coolheaded scientific sort, you'll go to the Internet sites of the government weather agencies such as the National Hurricane Center, analyze the data for a while, and then make whatever preparations you think are wise. Perhaps you'll decide to take a short vacation to Nashville, Tennessee, and do your statistical analyses of Emma from there.
Confidence Intervals Revisited
Instead of drawing a single line on a map, indicating a predicted track for Hurricane Emma, it's better to draw path probability maps such as the ones in Figs. 6-1 and 6-2. These maps, in effect, show confidence intervals. As the storm draws closer to the mainland, the confidence intervals narrow. The forecasts are revised. The ''purple line'' – the mean path of the storm – might shift on the tracking map. (Then again, maybe it won't move at all.) Most hurricane Web sites have strike-probability maps that are more informative than the path-prediction maps.
Probability Depends on Range
Imagine that a couple of days pass, Emma has moved closer to the mainland, and the forecasters are still predicting a mean path that takes the center of Emma over Wilmington. The probability lines are more closely spaced than they were two days ago. We can generate a distribution curve that shows the relative danger at various points north and south of the predicted point of landfall (which is actually on the New Jersey coast east of Wilmington). Figure 6-4A is a path probability map, and Fig. 6-4B is an example of a statistical distribution showing the relative danger posed by Emma at various distances from the predicted point of landfall.
The vertical axis representing the landfall point (labeled 0 in Fig. 6-4B) does not depict the actual probability of strike. Strike probabilities can be ascertained only within various ranges – lengths of coastline – north and/or south of the predicted point of landfall. Examples are shown in Figs. 6-5A and B.
In the ''Canada ice-cream scenario,'' a similar situation exists. We can draw a distribution curve (Fig. 6-6) that shows the taste inclinations of people, based on the H0 that 25% of them like plain vanilla ice cream better than any other flavor. Given any range or ''margin of error'' that is a fixed number of percentage points wide, say ±2% either side of a particular point, H0 asserts that the greatest area under the curve will be obtained when that range is centered at 25%. Imagine that H0 happens to be true. If that is the case, then someone who says ''Our survey will show that 23% to 27% of the people prefer vanilla'' is more likely to be right than someone who says ''Our survey will show that 12% to 16% of the people prefer vanilla.'' In more general terms, let P be some percentage between 0% and 100%, and let x be a value much smaller than P. Then if someone says ''Our survey will show that P% ± x% of the people prefer vanilla,'' that person is most likely to be right if P% = 25%. That's where the distribution curve of Fig. 6-6 comes to a peak, and that's where the area under the curve, given any constant horizontal span, is the greatest.
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