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.