Curve8.wk1 x y Exp y Ln y 1 1 2.71828 0 Curve Fitting--some general guidelines for interpretation. 2 1.1 3.00417 0.09531 3 1.2 3.32012 0.18232 Problem: I have some data--how might I decide what sort of curve to try to fit to it. 4 1.7 5.47395 0.53063 5 2.5 12.1825 0.91629 The nature of the data can suggest the type of curve. 6 2.9 18.1741 1.06471 7 4 54.5982 1.38629 If your concern is to forecast from your data: 8 6.1 445.858 1.80829 9 10.3 29732.6 2.33214 unbounded with no upper bound: try a linear, exponential, or logarithmic curve. linear--increases in the future occur at the same rate as increases in the present and past-- the slope of the line is constant. exponential--increases in the future occur at an increasing rate as time progresses the slope of the line increases as time progresses. logarithmic--increases in the future occur at a decreasing rate as time progresses-- the slope of the line is decreasing. A few types of interpretations: Linear--index numbers in time series analysis-- index numbers are used to relate a variable in one period of time to the same variable in another period of time (the base period). An index number is a relative number describing data that change over time (the data are a time series). Index numbers are often valuable in dealing with complicated data or data of great magnitude. In practice, they are often used to capture economic indicators--trends over time. Trend for the Index of Industrial Production Trend for the Wholesale Price Index -- source--Economic Report of the President Trend in equipment expenditures of the Curve Fitting Computer Company in millions of $$. Trend in overseas shipments of Eyepoint Needle Company in millions of needles. Trend in the Consumer Price Index, 1980-1990; 1985=100. For example: Year Index 1980 90 1981 93 1982 97 1983 102 1984 104 1985 100 1986 101 1987 99 1988 103 1989 107 1990 104 The year 1983, for example, is then referred to, relative to the base year of 1985, as year x=-2. Exponential--often used to model growth or decay Population, radio-carbon dating techniques, Compound interest problems. Base e arise naturally in calculus and in the compounding process--so, often employed. Logarithmic--inverse of exponential Not often used, but might be used when increase might cause slowing of growth-- as for example--when population of a city increases, area of course increases, but its rate of increase may slow, as for example when skyscrapers are introduced. Any exponential may be viewed as a logarithmic by inverting the role of dependent and independent variable. For example, supply and demand analysis traditionally sees price per unit as a function of quantity per unit of time-- but, in varying situations, one might have, instead of price=f(quantity), the alternate of quantity=f(price). An exponential in the traditional view becomes a logarithmic in the alternate view. Logistic--exponential type of growth initially--environmental resistance puts ceiling on growth. When a population gets sufficiently large, factors such as food supply and overcrowding tend to hold down the growth. Consumption is proportional to the amount remaining. Gompertz--flatter type of logistic--growth modelling. A few more references--classics in various fields. N. Keyfitz, Introduction to the Mathematics of Population, Reading, MA, Addison-Wesley, 1968. N. T. J. Bailey, The Mathematical Approach to Biology and Medicine, New York, John Wiley, 1967. J. I. Shonle, Environmetnal Applications of General Physics, Reading, MA, Addison-Wesley, 1975.