Nonlinear Regression Section 3 Quadratic Modeling

Transcription

1 Nonlinear Regression Section 3 Quadratic Modeling Another type of non-linear function seen in scatterplots is the Quadratic function. Quadratic functions have a distinctive shape. Whereas the exponential function has an L shape, quadratic functions have a U shape. This U shape is often called a Parabola. Because of their parabolic shape, Quadratic functions are commonly used to map airplane flights or missile launches. But a scatterplot does not have to have a U shape to use a Quadratic function. Quadratic functions can be used to map almost any curve in the scope of the data. But what is a Quadratic function? What does the function look like? Whereas linear functions have the form ŷ mx b and exponential functions are known for the variable exponents, Quadratic functions are known for their squared variables. The standard form for a quadratic function is 2 ŷ ax bx c where a, b and c are real numbers. Let us look at the following data set. This gives the value of a stock over a 6 month period. Stocks are notorious for going up and down in value depending on the state of the economy at the time. We let the explanatory variable be the number of months since January 1, and the response variable is the stock price per share in dollars. # of months since January Stock price $ per 1 share If we make a scatterplot of this data on Statcato, we get the following graph. Notice the distinctive upside down U shape.

2 So far, we have discussed the linear, exponential functions. Having Statcato find the least squares regression line and the best fit exponential function, we obtain the following graphs.

3 Notice that neither the regression line nor the exponential function seem to fit the data very well. In fact the exponential looks almost the same as the regression line. We will now have Statcato find the quadratic function that best fits the data. The quadratic function seems to fit the data very well. Let us look at the quadratic function that Statcato found. yˆ x 7.108x 2 2 Notice that the coefficient in front of the x term is negative. This is called the leading coefficient. When the leading coefficient is positive the quadratic will have a U shape, but as in this function, when the leading coefficient is negative the quadratic will have an upside down U shape. As I said earlier, don t make the mistake of thinking that quadratic functions are only useful when your scatterplot has a U shape. On the contrary, quadratic functions can be a good model for many curves.

4 For an example of this, let s look again at our world population data from section 4.1. Recall that this data gives the year and the world population from the year 500 BC to the year 1950 AD. Statcato found that the equation for that exponential function that best fit the data was yˆ the data and a scatterplot of the data with the exponential function. Year World Population in Billions x. Here is We said in section 4.1 that the exponential function fits the data set better than the regression line, but still not perfectly. We may think about whether another function might fit the data better than the exponential function. Plugging in this data into Statcato, we have the program find the quadratic function that best fits the data. Here is the scatterplot.

5 Notice that the quadratic function fits the data reasonably well. As with the exponential function, Statcato found the function that best fits the data. Looking at the graph, you may see that the leading coefficient says This does not mean that the leading coefficient is zero. (If that were the case, this function would not be quadratic.) It just means that the number when rounded to three decimal places is zero. Look under the Statistics menu in Statcato. Click on Correlation and Regression then on Nonlinear models. Under this menu, Statcato will give the function to better accuracy. We found the function to be yˆ x x 2. Notice the leading coefficient is positive, which corresponds to the graphs U shape. But now we have a quandary. In section 4.1, we found the linear and exponential functions that best fit this data. Now we also have the quadratic function to think about. So which is the best fit for the data? Assessing the fit of a quadratic function Let s start be looking again at the scatterplots. Which function looks like it fits the data the best, the line, the exponential curve, or the quadratic curve?

6

7 We definitely can see that the curves fit the data better than the line, but it is hard to tell which of the curves fit the data better. Recall that one way to answer the question of best fit is to look at the R-squared values for the line and each of the two curves. R-Squared (line) = R-Squared (Exponential curve) = R-Squared (Quadratic curve) = Remember that we do not like to use a more complicated model unless there is a significant improvement. We see from the R-squared values that the quadratic (73.7%) is significantly better than the line (58.8%) but not nearly as good as the exponential (90.8%). For this data set, it seems the exponential is the best model. We also like to look at the standard deviation of the residual errors S e. Recall that the curve with the smallest standard deviation is also an indication of best fit. We had Statcato calculate the standard deviations for the line and the two curves. S e (line) = Billion people S e (exponential) = Billion people S e (quadratic) = Billion people. As with the R-squared, the exponential model seems to be the best fit for this data. It has not only the highest R-squared value, but also the lowest standard deviation of the residual errors. Residual Plots Let s take a look at the residual plot for the quadratic population function above. Recall that a Residual plot is just a scatterplot with the original x values and the residuals as the y values.

8 What does the Residual Plot tell us? Recall one of the main reasons to look at a residual plot is to find which x values might be best for making a prediciton with this model. These x values are often called prediction intervals. Do you notice that for 500 x 1000 the points are reasonably close to the curve, but for 1750 x 1950 the points are much farther away from curve. This tells us that if we use the quadratic function to predict the population, then the predictions will be more accurate if we use years between 500 BC and 1000 AD and less accurate for years between 1750 and 1950 AD. You may have noticed that the residual plot for the Quadratic function looks very similar to the residual plot for the exponential function that we looked at in section 4.1. The residual plots for both the Exponential and Quadratic function have an S shape to them. We said that this could indicate that there may be a function that fits this data even better than the Exponential or Quadratic models. Can you think of a model that has an S shape to it? If you said cubic function, you may be right. Just to satisfy your curriosity here is a scatterplot of the population data with a cubic function. So what do you think of the fit?