After watching the video on resampling, I have learned about Cross-validation, Validation Set and Bootstrapping.
Resampling involves taking samples from an existing set of observations and creating a new data set.
Bootstrapping, on the other hand, is a statistical process that involves resampling a single set of observations to create multiple simulated samples.
In the Validation Set approach, the data is divided into a training set and a testing set.
The training set is the one that can be used to estimate the error rate when a model is created using the training set.
The principal component analysis (PCA) was employed to narrow down the features of %Obese and %Inactive to two primary components,
which account for the greatest variability of the data. A scatter plot illustrates the distribution of the data points in this new 2-dimensional space,
allowing for the detection of any patterns or clusters between the two components.
Polynomial regression is an extension of linear regression where higher-degree terms (squared, cubed, etc.) of the predictor variables are included in the model to fit non-linear trends in the data. Let’s delve into how to plot a polynomial regression model:
1. Understanding Polynomial Regression:
In a simple linear regression, we try to fit a straight line to the data. For example:
Y=β0+β1X+ϵY=β0+β1X+ϵ
However, in polynomial regression, the equation could look something like:
Y=β0+β1X+β2X2+…+βnXn+ϵY=β0+β1X+β2X2+…+βnXn+ϵ
Where nn is the degree of the polynomial.
2. Fitting the Model:
To plot a polynomial regression model, you first need to:
Choose the degree of the polynomial based on your data. This involves a balance: a higher degree might fit the training data better but can lead to overfitting.
Use statistical software or libraries (e.g., scikit-learn in Python) to fit the polynomial regression model to your data.
3. Plotting:
Once the model is fitted, you can plot it. The procedure generally involves:
Plotting the actual data points, usually as scatter points.
Generating a range of predictor values (often a fine grid across the range of your data).
Using the polynomial regression model to predict the response for each of these predictor values.
Plotting the predicted values, usually as a smooth curve.
4. Visual Interpretation:
When viewing the plot, you’ll see the data points and the curve representing the polynomial regression. The curve should capture the underlying trend of the data points. Depending on the degree of the polynomial, this curve can be a simple curve (e.g., quadratic) or more complex, wavy shapes.
5. Potential Pitfalls:
While polynomial regression can capture complex non-linear trends, it also has potential pitfalls:
Overfitting: Higher-degree polynomials can fit the training data very closely, capturing noise and making poor predictions on new, unseen data.
Interpretability: As the degree of the polynomial increases, the model can become harder to interpret.
6. Visual Enhancements:
For a clearer visual representation:
Ensure the polynomial curve is smooth.
Use color or different markers to distinguish between actual data points and the polynomial curve.
If plotting multiple polynomial models (e.g., of different degrees), use different colors or line styles for each.
In summary, plotting a polynomial regression model involves fitting a curve to data points, allowing for the visualization of non-linear relationships. Proper care should be taken to choose an appropriate polynomial degree and to avoid overfitting.