Considering linear, quadratic, and exponential lines of best fit:
If you're studying a set of data, how can you decide which of those three types of model would be most appropriate for the data?
linear: constant difference
quadratic: constant 2nd differences
exponential: constant ratio over equal intervals
Steve, thank you, but what do you mean constant 2nd differences? And constant ratio over equal intervals? For me, the difference between linear and exponential is that linear go up/down at a constant amount by adding that same constant amount. Exponential goes up/down by multiplying that same constant amount.
x _y____y(n)-y(n-1) ___ next diff
1__0______
2__5______ 5
3__10_____ 5____________0
4__15_____ 5____________0
5__20_____ 5____________0
6__25______5____________0
THAT is constant diff
now
x _y____y(n)-y(n-1) ___ next diff
1__1______
2__4______ 3
3__9_______5____________ 2
4__16 ____ 7____________ 2
5__25_____ 9_____________2
6__36______11____________2
THAT is constant difference of difference - Get it?
To determine which type of model (linear, quadratic, or exponential) is the most appropriate for a set of data, you can follow these steps:
1. Plot the data: Start by creating a scatter plot of the data points. This will help you visualize the relationship between the dependent variable (y-axis) and the independent variable (x-axis).
2. Analyze the pattern: Examine the general pattern formed by the data points on the scatter plot. Look for any apparent trends, relationships, or patterns such as increase, decrease, or a combination of both.
3. Linear model: If the plot shows a roughly straight line pattern, such that the data points seem to be following a linear trend, a linear model may be appropriate. A linear model represents a constant rate of change over time or across the x-axis.
4. Quadratic model: If the scatter plot suggests a U-shaped or an inverted U-shaped pattern, where the data points curve upward or downward, a quadratic model may be appropriate. Quadratic models fit data that exhibit a curved pattern, indicating a nonlinear relationship.
5. Exponential model: If the scatter plot shows a pattern where the data points increase or decrease exponentially, meaning the rate of growth or decay is proportional to the current value, an exponential model may be appropriate. These models demonstrate rapid growth or decay over time.
6. Evaluate residuals: After fitting a line of best fit for each model type, evaluate the residuals. Residuals measure the difference between the observed data points and the predicted values from the model. A smaller average residual indicates a better fit.
7. Compare R-squared values: Calculate and compare the R-squared values for each model. The R-squared value indicates the proportion of the variance in the dependent variable that is explained by the model. Generally, the model with the highest R-squared value is the most appropriate for the data.
Using these steps, you can determine which model type---linear, quadratic, or exponential---best fits the data based on the observed pattern, residuals, and R-squared values. Keep in mind that visual examination alone may not always lead to a definitive answer, and further statistical analysis may be required.