Regression: Non-Linear Transformation
One kind of nonlinear function is called inverse variation. function that has direct variation between the variables, such as the proportional function formula of. One example of an inverse function is the speed required to travel between two. This page contains a short article on direct and inverse relationships. It is simple enough to solve this equation for a few values of X. For example, if we let X = 1 then Y = 4. Note that as X increases Y decreases in a non-linear fashion. In this section, you will learn most commonly used non-linear regression and how the following links to go directly to the detail and example of each non-linear.
Consider point D in Panel a of Figure We have drawn a tangent line that just touches the curve showing bread production at this point. It passes through points labeled M and N. The vertical change between these points equals loaves of bread; the horizontal change equals two bakers.
Linear & nonlinear functions: table
The slope of our bread production curve at point D equals the slope of the line tangent to the curve at this point. In Panel bwe have sketched lines tangent to the curve for loaves of bread produced at points B, D, and F.
Notice that these tangent lines get successively flatter, suggesting again that the slope of the curve is falling as we travel up and to the right along it. In Panel athe slope of the tangent line is computed for us: Generally, we will not have the information to compute slopes of tangent lines.
We will use them as in Panel bto observe what happens to the slope of a nonlinear curve as we travel along it. We see here that the slope falls the tangent lines become flatter as the number of bakers rises.
Notice that we have not been given the information we need to compute the slopes of the tangent lines that touch the curve for loaves of bread produced at points B and F. In this text, we will not have occasion to compute the slopes of tangent lines. Either they will be given or we will use them as we did here—to see what is happening to the slopes of nonlinear curves.
In the case of our curve for loaves of bread produced, the fact that the slope of the curve falls as we increase the number of bakers suggests a phenomenon that plays a central role in both microeconomic and macroeconomic analysis. As we add workers in this case bakersoutput in this case loaves of bread rises, but by smaller and smaller amounts.
Another way to describe the relationship between the number of workers and the quantity of bread produced is to say that as the number of workers increases, the output increases at a decreasing rate. In Panel b of Figure Indeed, much of our work with graphs will not require numbers at all. We turn next to look at how we can use graphs to express ideas even when we do not have specific numbers.
Graphs Without Numbers We know that a positive relationship between two variables can be shown with an upward-sloping curve in a graph. A negative or inverse relationship can be shown with a downward-sloping curve.
Some relationships are linear and some are nonlinear. We illustrate a linear relationship with a curve whose slope is constant; a nonlinear relationship is illustrated with a curve whose slope changes. Using these basic ideas, we can illustrate hypotheses graphically even in cases in which we do not have numbers with which to locate specific points. Consider first a hypothesis suggested by recent medical research: We can show this idea graphically. Daily fruit and vegetable consumption measured, say, in grams per day is the independent variable; life expectancy measured in years is the dependent variable.
Panel a of Figure Notice the vertical intercept on the curve we have drawn; it implies that even people who eat no fruit or vegetables can expect to live at least a while!
The graphs in the four panels correspond to the relationships described in the text. Panel b illustrates another hypothesis we hear often: Here the number of cigarettes smoked per day is the independent variable; life expectancy is the dependent variable.
The hypothesis suggests a negative relationship. Hence, we have a downward-sloping curve. As we saw in Figure We have drawn a curve in Panel c of Figure It is upward sloping, and its slope diminishes as employment rises.
Finally, consider a refined version of our smoking hypothesis. Suppose we assert that smoking cigarettes does reduce life expectancy and that increasing the number of cigarettes smoked per day reduces life expectancy by a larger and larger amount. Panel d shows this case. Again, our life expectancy curve slopes downward. But now it suggests that smoking only a few cigarettes per day reduces life expectancy only a little but that life expectancy falls by more and more as the number of cigarettes smoked per day increases.
We have sketched lines tangent to the curve in Panel d. The slopes of these tangent lines are negative, suggesting the negative relationship between smoking and life expectancy. They also get steeper as the number of cigarettes smoked per day rises. Whether a curve is linear or nonlinear, a steeper curve is one for which the absolute value of the slope rises as the value of the variable on the horizontal axis rises.
Thus far our work has focused on graphs that show a relationship between variables. We turn finally to an examination of graphs and charts that show values of one or more variables, either over a period of time or at a single point in time. Nonlinear relationship The data points in Plot 3 appear to be randomly distributed.
They do not fall close to the line indicating a very weak relationship if one exists.
If a relationship between two variables is not linear, the rate of increase or decrease can change as one variable changes, causing a "curved pattern" in the data. This curved trend might be better modeled by a nonlinear function, such as a quadratic or cubic function, or be transformed to make it linear. Plot 4 shows a strong relationship between two variables.
This relationship illustrates why it is important to plot the data in order to explore any relationships that might exist. Monotonic relationship In a monotonic relationship, the variables tend to move in the same relative direction, but not necessarily at a constant rate. In a linear relationship, the variables move in the same direction at a constant rate.
- NonLinear Transformation
- Nonlinear Relationships and Graphs without Numbers
- Linear, nonlinear, and monotonic relationships
Plot 5 shows both variables increasing concurrently, but not at the same rate. This relationship is monotonic, but not linear. The Pearson correlation coefficient for these data is 0.
Linear relationships are also monotonic. For example, the relationship shown in Plot 1 is both monotonic and linear.