The goal of regression analysis is to determine the values of parameters
for a function that cause the function to best fit a set of data
observations that you provide. In linear regression, the function
is a linear (straight-line) equation. For example, if we assume the value
of an automobile decreases by a constant amount each year after its
purchase, and for each mile it is driven, the following linear function
would predict its value (the dependent variable on the left side of the
equal sign) as a function of the two independent variables which are age
and miles:
value = price + depage*age + depmiles*miles
where value, the dependent variable, is the value of the car,
age is the age of the car, and miles is the number of miles
that the car has been driven. The regression analysis performed by NLREG
will determine the best values of the three parameters, price, the
estimated value when age is 0 (i.e., when the car was new), depage,
the depreciation that takes place each year, and depmiles, the
depreciation for each mile driven. The values of depage and
depmiles will be negative because the car loses value as age and
miles increase.
For an analysis such as this car depreciation example, you must provide a data
file containing the values of the dependent and independent variables for a set of
observations. In this example each observation data record would contain three
numbers: value, age, and miles, collected from used car ads for the same model
car. The more observations you provide, the more accurate will be the estimate
of the parameters. The NLREG statements to perform this regression are shown
below:
Variables value,age,miles;
Parameters price,depage,depmiles;
Function value = price + depage*age + depmiles*miles;
Data;
{data values go here}
Once the values of the parameters are determined by NLREG, you can
use the formula to predict the value of a car based on its age and miles
driven. For example, if NLREG computed a value of 16000 for price,
-1000 for depage, and -0.15 for depmiles, then the function
value = 16000 - 1000*age - 0.15*miles
could be used to estimate the value of a car with a known age and number of
miles.
If a perfect fit existed between the function and the actual data, the
actual value of each car in your data file would exactly equal the
predicted value. Typically, however, this is not the case, and the
difference between the actual value of the dependent variable and its
predicted value for a particular observation is the error of the estimate
which is known as the "deviation'' or "residual''. The goal
of regression analysis is to determine the values of the parameters that
minimize the sum of the squared residual values for the set of
observations. This is known as a "least squares'' regression fit.
Here is a plot of a linear function fitted to a set of data values. The
actual data points are marked with ''x''. The red line between a
point and the fitted line represents the residual for the observation.
NLREG is a very powerful regression analysis program. Using it you can
perform multivariate, linear, polynomial, exponential, logistic, and
general nonlinear regression. What this means is that you specify the form
of the function to be fitted to the data, and the function may include
nonlinear terms such as variables raised to powers and library functions
such as log, exponential, sine, etc. For complex analyses, NLREG allows
you to specify function models using conditional statements (if,
else), looping (for, do, while), work
variables, and arrays. NLREG uses a state-of-the-art regression algorithm
that works as well, or better, than any you are likely to find in any
other, more expensive, commercial statistical packages.
As an example of nonlinear regression, consider another depreciation
problem. The value of a used airplane decreases for each year of its age.
Assuming the value of a plane falls by the same amount each year, a linear
function relating value to age is:
value = p0 + p1*Age
Where p0 and p1 are the parameters whose values are to be
determined. However, it is a well-known fact that planes (and automobiles)
lose more value the first year than the second, and more the second than
the third, etc. This means that a linear (straight-line) function cannot
accurately model this situation. A better, nonlinear, function is:
value = p0 + p1*exp(-p2*Age)
Where the ''exp'' function is the value of e (2.7182818...) raised to
a power. This type of function is known as "negative exponential" and is
appropriate for modeling a value whose rate of decrease is proportional to
the difference between the value and some base value. Here is a plot of a
negative exponential function fitted to a set of data values.
Much of the convenience of NLREG comes from the fact that you can enter
complicated functions using ordinary algebraic notation. Examples of functions
that can be handled with NLREG include:
Linear: Y = p0 + p1*X
Quadratic: Y = p0 + p1*X + p2*X^2
Multivariate: Y = p0 + p1*X + p2*Z + p3*X*Z
Exponential: Y = p0 + p1*exp(X)
Periodic: Y = p0 + p1*sin(p2*X)
Misc: Y = p0 + p1*Y + p2*exp(Y) + p3*sin(Z)
In other words, the function is a general expression involving one dependent
variable (on the left of the equal sign), one or more independent variables, and
one or more parameters whose values are to be estimated. NLREG can handle up
to 500 variables and 500 parameters.
Because of its generality, NLREG can perform all of the regressions handled by
ordinary linear or multivariate regression programs as well as nonlinear
regression.
Some other regression programs claim to perform nonlinear regression but
actually do it by transforming the values of the variables such that the
function is converted to linear form. They then perform a linear
regression on the transformed function. This technique has a major flaw:
it determines the values of the parameters that minimize the squared
residuals for the transformed, linearized function rather than the original
function. This is different than minimizing the squared residuals for the
actual function and the estimated values of the parameters may not produce
the best fit of the original function to the data. NLREG uses a true
nonlinear regression technique that minimizes the squared residuals for the
actual function. Also, NLREG can handle functions that cannot be
transformed to a linear form.
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