Statistical Data Analysis Help
Introduction to Statistical Data Analysis - Tweaks, Trends, and Correlation
Graphs can be approximated or modified by “tweaking.” Certain characteristics can also be noted, such as trends and correlation. Here are a few examples.
The term interpolate means “to put between.” When a graph is incomplete, estimated data can be put in the gap(s) in order to make the graph look complete. An example is shown in Fig. 2-10. This is a graph of the price of the hypothetical Stock Y from Fig. 2-2, but there’s a gap during the noon hour. We don’t know exactly what happened to the stock price during that hour, but we can fill in the graph using linear interpolation . A straight line is placed between the end points of the gap, and then the graph looks complete.
Fig. 2-10. An example of linear interpolation. The thin solid line represents the interpolation of the values for the gap in the actual available data (heavy dashed curve).
Linear interpolation almost always produces a somewhat inaccurate result. But sometimes it is better to have an approximation than to have no data at all. Compare Fig. 2-10 with Fig. 2-2, and you can see that the linear interpolation error is considerable in this case.
Curve fitting is an intuitive scheme for approximating a point-to-point graph, or filling in a graph containing one or more gaps, to make it look like a continuous curve. Figure 2-11 is an approximate graph of the price of hypothetical Stock Y, based on points determined at intervals of half an hour, as generated by curve fitting. This does not precisely represent the actual curve of Fig. 2-2, but it comes close most of the time.
Curve fitting becomes increasingly accurate as the values are determined at more and more frequent intervals. When the values are determined infrequently, this scheme can be subject to large errors, as is shown by the example of Fig. 2-12.
The term extrapolate means “to put outside of.” When a function has a continuous-curve graph where time is the independent variable, extrapolation is the same thing as short-term forecasting. Two examples are shown in Fig. 2-13.
Fig. 2-11. Approximation of hypothetical stock price as a continuous function of time, making use of curve fitting. The solid curve represents the approximation; the dashed curve represents the actual price as a function of time.
Fig. 2-12. An example of curve fitting in which not enough data samples are taken, causing significant errors. The solid line represents the approximation; the dashed curve represents the actual price as a function of time.
In Fig. 2-13A, the price of the hypothetical Stock X from Fig. 2-2 is plotted until 2:00 P.M., and then an attempt is made to forecast its price for an hour into the future, based on its past performance. In this case, linear extrapolation , the simplest form, is used. The curve is simply projected ahead as a straight line. Compare this graph with Fig. 2-2. In this case, linear extrapolation works fairly well.
Figure 2-13B shows the price of the hypothetical Stock Y (from Fig. 2-2) plotted until 2:00 P.M., and then linear extrapolation is used in an attempt to predict its behavior for the next hour. As you can see by comparing this graph with Fig. 2-2, linear extrapolation does not work well in this scenario.
Fig. 2-13. Examples of linear extrapolation. The solid lines represent the forecasts; the dashed curves represent the actual data. In the case shown at A, the prediction is fairly good. In the case shown at B, the linear extrapolation is way off.
Extrapolation is best done by computers. Machines can notice subtle characteristics of functions that humans miss. Some graphs are easy to extrapolate, and others are not. In general, as a curve becomes more complicated, extrapolation becomes subject to more error. Also, as the extent (or distance) of the extrapolation increases for a given curve, the accuracy decreases.
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