# Practice quiz 3

Not graded, just practice

Author

Katie Schuler

Published

October 26, 2023

## Model fitting

1. True or false, gradient descent and orinary least squares are both iterative optimization algorithms.
1. What cost function have we been using to perform our gradient descent?
1. True or false, when performing gradient descent on the model given by the equation $$y = w_0 + w_1x_1 + w_2x_2$$, we might arrive at a local minimum and miss the global one.
1. Which of the following would work to estimate the free parameters of a nonlinear model?
1. True or false, in gradient descent, we search through all possible parameters in the parameter space.

## Model fitting in R

Questions 6-9 refer to the code and output below, performing gradient descent with optimg:

optimg(data = data, par = c(0,0), fn=SSE, method = "STGD")
$par [1] 3.37930046 0.06683237$value
[1] 959.4293

$counts [1] 6$convergence
[1] 0
1. How many steps did the gradient descent algorithm take?

2. What was the sum of squared error of the optimal paramters?

3. What coefficients does the algorithm converge on?

1. What parameters were used to initialized the algorithm?

Questions 10-12 refer to the output below from lm():


Call:
lm(formula = y ~ x, data = data)

Coefficients:
(Intercept)            x
3.37822      0.06688  
1. Use R notation to write the model specification.
answer
y ~ x  # this works (implicit intercept)

y ~ 1 + x # this also works (explicit intercept)
1. Given the model is specified by the equation $$y = w_0+w_1x_1$$, what is the parameter estimate for $$w_0$$ = and $$w_1$$ = .

2. True or false, for this model, optimg() with gradient descent would converge on the same parameter estimates?

## Model accuracy

Question 13 refers to the following figure:

1. In the figure above, which of the following corresponds to the residuals?
1. Suppose the $$R^2$$ value on the model in the figure above is about 0.88. Given this value, which of the following best describes the accuracy of the model?
1. Suppose 0.88 reflects the $$R^2$$ for our fitted model on our sample. Which of the following is true about the $$R^2$$ for our fitted model on the population?
1. Which of the following best describes an overfit model?
1. How can we estimate $$R^2$$ on the population? Choose all that apply.
1. Fill in the blanks below to best describe cross validation:
• Leave some out
• Fit a model on the data
• Evaluate the mdoel on the data

## Model accuracy in R

Questions 19-20 refer to the following code and output:

model <- lm(y ~ 1 + x, data)
summary(model)

Call:
lm(formula = y ~ 1 + x, data = data)

Residuals:
Min       1Q   Median       3Q      Max
-1.38959 -0.32626 -0.04605  0.31967  1.65977

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  2.08280    0.07418   28.08   <2e-16 ***
x            0.47844    0.01242   38.51   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.5139 on 198 degrees of freedom
Multiple R-squared:  0.8822,    Adjusted R-squared:  0.8816
F-statistic:  1483 on 1 and 198 DF,  p-value: < 2.2e-16
1. What is the $$R^2$$ value for the model fit above?

1. Does the value you entered in 19 reflect $$R^2$$ on the population or on the sample?

Questions 21-23 refer to the following code and output:

# we divide the data into v folds for cross-validation
set.seed(2)
splits <- vfold_cv(data)

# model secification
model_spec <-
linear_reg() %>%
set_engine(engine = "lm")

# add a workflow
our_workflow <-
workflow() %>%
add_model(model_spec) %>%
add_formula(y ~ x)

# fit models to our folds
fitted_models <-
fit_resamples(
object = our_workflow,
resamples = splits
)

fitted_models %>%
collect_metrics()
# A tibble: 2 × 6
.metric .estimator  mean     n std_err .config
<chr>   <chr>      <dbl> <int>   <dbl> <chr>
1 rmse    standard   0.507    10  0.0397 Preprocessor1_Model1
2 rsq     standard   0.890    10  0.0146 Preprocessor1_Model1
1. In the cross-validation performed above, how many folds were the data split into?
1. What $$R^2$$ do we estimate for the population?

1. What model has been fit?

Questions 24-26 refer to the following code and output:

# we bootstrap the data for cross-validation
set.seed(2)
bootstrap <- bootstraps(data)

# fit models to our folds
fitted_models_boot <-
fit_resamples(
object = our_workflow,
resamples = bootstrap
)

fitted_models_boot %>%
collect_metrics()
# A tibble: 2 × 6
.metric .estimator  mean     n std_err .config
<chr>   <chr>      <dbl> <int>   <dbl> <chr>
1 rmse    standard   0.507    25 0.00946 Preprocessor1_Model1
2 rsq     standard   0.887    25 0.00377 Preprocessor1_Model1
1. How many bootstrap samples did we generate?

1. True or false, we fit the same model to the bootstrap data as we did in the cross-validation code.
1. True or false, the $$R^2$$ estimated by bootstrapping is equal to the $$R^2$$ estimated by cross-validation.