For the `Wage` data, an additive model expressed with explicit spline basis functions can be fitted using `lm()`:

```R
gam1 <- lm(wage ~ ns(age, 5) + education, data = Wage)
```

To specify smooth terms with `s()`, use `gam()` from the `gam` library:

```R
library(gam)
gam2 <- gam(wage ~ s(year, 4) + s(age, 5) + education, data = Wage)
```

University of Michigan - Ann Arbor

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Generalized Additive Models (GAMs), similar to multiple linear regression models, attempt to predict a response value ($$Y$$) using many predictor variables ($$X_1, \dots, X_p$$). GAMs maintain their additivity by applying separate non-linear functions to each predictor variable ($$f_j(X_j)$$) and adding them together. They can be applied to both quantitative and qualitative responses.

Generalized Additive Models (GAMs)

Pros:
- With GAMS, one can fit non-linear functions to each predictor to automatically model non-linear relationships that standard linear regression will miss
- The non-linear fits *can* make more accurate predictions for the response Y
- The additive nature of GAMs means we can still examine the effect of each predictor variable on Y while holding other variables fixed.
- Degrees of freedom can summarize the smoothness of the GAM

Cons:
- The main limitation is that GAMs are restricted to be additive, meaning important interactions *can be missed*. However, as with regular linear regression, we can manually add interactions by including additional predictors.













Pros and Cons of GAMS

For Generalized Additive Models (GAMs) in regression problems, to depict a non-linear relationship between predictors and their response, we start with the multiple linear regression model $$y_i = \beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + \dots + \beta_px_{ip} + \epsilon_i$$. We then replace each linear component ($$\beta_jx_{ij}$$) with a non-linear function $$f_j(x_{ij})$$ so that the model becomes $$y_i = \beta_0 + f_1(x_{i1}) + f_2(x_{i2}) + \dots + f_p(x_{ip}) + \epsilon_i$$. A separate non-linear function $$f_j$$ is applied to every predictor variable, and these are then added together. This is why it is considered an additive model.

GAMs in Regression Problems

Similar to their use in regression, Generalized Additive Models (GAMs) can be applied to classification problems with qualitative responses. Assuming a binary response $$Y$$ that equals 0 or 1, we can extend the standard logistic regression model, $$\log\left(\frac{p(X)}{1-p(X)}\right) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_p X_p$$, by replacing each linear component ($$\beta_j X_j$$) with a non-linear function ($$f_j(X_j)$$). The resulting GAM for classification is: $$\log\left(\frac{p(X)}{1-p(X)}\right) = \beta_0 + f_1(X_1) + f_2(X_2) + \dots + f_p(X_p)$$.

GAMs for Classification Problems

https://codeburst.io/pygam-getting-started-with-generalized-additive-models-in-python-457df5b4705f



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