# Stanford University Statistical Learning Quiz Answer | Moving Beyond Linearity

**Stanford University Statistical Learning Quiz Answer |**

**Moving Beyond Linearity**

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**Polynomials and Step Functions Quiz**

### 7.1.R1

**Which of the following can we add to linear models to capture nonlinear effects?**

- Spline terms
- Polynomial terms
- Interactions
- Arbitrary linear combinations of the variables
- Step functions

**Piecewise-Polynomials and Splines Quiz**

### 7.2.R1

**Why are natural cubic splines typically preferred over global polynomials of degree d?**

- Polynomials have too many degrees of freedom
- Polynomials tend to extrapolate very badly
- Polynomials are not as continuous as splines

### 7.2.R2

**Let 1{x leq t} denote a function which is 1 if x leq t and 0 otherwise.**

**Which of the following is a basis for linear splines with a knot at t? Select all that apply:**

- 1, x, (x – t)1{x > t}
- 1, x, (x – t)1{x leq t}
- 1{x > t}, 1{x leq t}, (x – t)1{x > t}
- 1, (x – t)1{x leq t}, (x – t)1{x > t}

**Smoothing Splines**

### 7.3.R1

**In terms of model complexity, which is more similar to a smoothing spline with 100 knots and 5 effective degrees of freedom?**

- A natural cubic spline with 5 knots
- A natural cubic spline with 100 knots

**Generalized Additive Models and Local Regression**

### 7.4.R1

**True or False: In the GAM y sim f_1(X_1) + f_2(X_2) + e, as we make f_1 and f_2 more and more complex we can approximate any regression function to arbitrary precision.**

- True
- False

**Nonlinear Functions in R**

### 7.R.R1

**Load the data from the file 7.R.RData, and plot it using plot(x,y). What is the slope coefficient in a linear regression of y on x (to within 10%)?**

- -0.6748

### 7.R.R2

**For the model y ~ 1+x+x^2, what is the coefficient of x (to within 10%)?**

- 77.7

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**Chapter 7 Quiz**

### 7.Q.1

**Suppose we want to fit a generalized additive model (with a continuous response) for y against X_1 and X_2. Suppose that we are using a cubic spline with four knots for each variable (so our model can be expressed as a linear regression after the right basis expansion).**

Suppose that we fit our model by the following three steps:

- First fit our cubic spline model for y against X_1, obtaining the fit hat f_1(x) and residuals r_i = y_i – hat f_1(X_{i,1}).
- Then, fit a cubic spline model for r against X_2 to obtain hat f_2(x).
- Finally construct fitted values hat y_i = hat f_1(X_{i,1}) + hat f_2(X_{i,2}).

**Will we get the same fitted values as we would if we fit the additive model for y against X_1 and X_2 jointly?**

- yes, no matter what
- only if X_1 and X_2 are uncorrelated
- not necessarily, even if X_1 and X_2 are uncorrelated.