Non-linear Regression in Mathematics AI
Types of Regression Models
Non-linear regression extends beyond simple linear relationships to model more complex patterns in data. The most common types of non-linear regression models include:
- Quadratic: $y = ax^2 + bx + c$
- Cubic: $y = ax^3 + bx^2 + cx + d$
- Exponential: $y = ae^{bx}$
- Power: $y = ax^b$
- Sine: $y = a\sin(bx + c) + d$
For instance, population growth might follow an exponential model, while oscillating phenomena like sound waves could be modeled using a sine function.
Exam techniqueSelecting the Right Regression Model
Choosing the appropriate non-linear regression model depends on the nature of the data and the underlying relationship between variables. Here are some common guidelines:
- Quadratic Regression: Used when the data follows a parabolic shape (e.g., projectile motion, area-based relationships).
- Cubic Regression: Useful when data has more than one bend, showing an "S"-shaped or wave-like pattern.
- Exponential Regression: Appropriate when data grows or decays at an increasing rate, such as population growth or radioactive decay.
- Power Regression: Used when a variable changes at a proportional rate to another variable, often seen in physics and engineering.
- Sine Regression: Ideal for periodic or oscillatory data, such as temperature cycles or seasonal trends.
Least Squares Regression
- The method of least squares is used to find the best-fitting curve for a given set of data points.
- This method minimizes the sum of the squares of the residuals (the differences between observed and predicted values).
In the context of AI and machine learning, least squares regression is a fundamental technique for training models on numerical data.
HintModern technology, such as graphing calculators and statistical software, can quickly perform least squares regression for various model types. Students are expected to be proficient in using such tools to evaluate different regression models.
Sum of Square Residuals (SSres)
The Sum of Square Residuals (SSres) is a measure of the total deviation of the response values from the fit to the response values. It is calculated as:
$$ SSres = \sum_{i=1}^n (y_i - \hat{y}_i)^2 $$
Where:
- $y_i$ is the observed value
- $\hat{y}_i$ is the predicted value from the model
A smaller SSres indicates a better fit of the model to the data.
Coefficient of Determination (R²)
The coefficient of determination, denoted as R², is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s).
R² is calculated as:
$$ R^2 = 1 - \frac{SSres}{SStot} $$
Where SStot is the total sum of squares:
$$ SStot = \sum_{i=1}^n (y_i - \bar{y})^2 $$
And $\bar{y}$ is the mean of the observed data.
NoteR² values range from 0 to 1, with 1 indicating a perfect fit (SSres = 0) and 0 indicating that the model explains none of the variability in the data.
Interpreting R²
- An R² of 0.75 means that 75% of the variance in the dependent variable is predictable from the independent variable(s).
- For linear models, R² is the square of Pearson's correlation coefficient.
Students often assume that a higher R² always indicates a better model. However, this is not always the case, especially when comparing models of different complexities or types.
