Model Fitting with R (demo)

Data Science for Studying Language and the Mind

Katie Schuler

2023-10-17

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Data science with R

• Hello, world!
• R basics
• Data importing
• Data visualization
• Data wrangling

Stats & Model buidling

• Sampling distribution
• Hypothesis testing
• Model specification
• Model fitting
• Model accuracy
• Model reliability

More advanced

• Classification
• Feature engineering (preprocessing)
• Inference for regression
• Mixed-effect models

Roadmap

• Fit a model in R
• Goodness of fit: quantifying our intuition
• How do we estimate the free parameters?
  1. Gradient descent - iterative optimization algorithm
  2. Ordinary least squares - analytical solution for linear regression
• If time: another full example

Fit a model

	model specification
R syntax	rt ~ 1 + experience
R syntax	rt ~ experience
Equation	\(y=w_0+w_1x_1\)

flowchart TD
    spec(Model specification) --> fit(Estimate free parameters) 
    fit(Estimate free parameters) --> fitted(Fitted model) 

\(y = 211.271 + -1.695x\)

Fit a model in R

	model specification
R syntax	rt ~ 1 + experience
R syntax	rt ~ experience
Equation	\(y=w_0+w_1x_1\)

lm(rt ~ experience, data = data)

Call:
lm(formula = rt ~ experience, data = data)

Coefficients:
(Intercept)   experience  
    211.271       -1.695  

data %>% 
    specify(rt ~ experience) %>%
    fit()

# A tibble: 2 × 2
  term       estimate
  <chr>         <dbl>
1 intercept    211.  
2 experience    -1.69

linear_reg() %>%
    set_engine("lm") %>%
    fit(rt ~ experience, data = data)

parsnip model object


Call:
stats::lm(formula = rt ~ experience, data = data)

Coefficients:
(Intercept)   experience  
    211.271       -1.695  

Goodness of fit

Quantifying our intution with sum of squared error

experience	rt	prediction	error	squared_error
49	124	128.216	4.216	17.774656
69	95	94.316	-0.684	0.467856
89	71	60.416	-10.584	112.021056
99	45	43.466	-1.534	2.353156
109	18	26.516	8.516	72.522256

\(SSE=\sum_{i=i}^{n} (d_{i} - m_{i})^2 = 205.139\)

Sum of squared error

Given some data:

experience	rt	prediction	error	squared_error
49	124	128.216	4.216	17.774656
69	95	94.316	-0.684	0.467856
89	71	60.416	-10.584	112.021056
99	45	43.466	-1.534	2.353156
109	18	26.516	8.516	72.522256

Compute the sum of squared error:

\(SSE=\sum_{i=i}^{n} (d_{i} - m_{i})^2 = 205.139\)

  data %>%
    mutate(prediction = 211.271 + -1.695 * experience) %>%
    mutate(error = prediction - rt) %>%
    mutate(squared_error = error^2) %>%
    with(sum(squared_error))

[1] 205.139

Gradient descent

A search problem: we have a parameter space, a cost function, and our job is to search through the space to find the point that minimizes the cost function.

Gradient descent

Define our cost function:

SSE <- function(data, par) {
  data %>%
    mutate(prediction = par[1] + par[2]* experience) %>%
    mutate(error = prediction - rt) %>%
    mutate(squared_error = error^2) %>%
    with(sum(squared_error))
}

SSE(data = data, par = c(0, 0))

[1] 31791

Impliment gradient descent algorithm with optimg

optimg(
    data = data,  # our data
    par = c(0,0), # our parameters
    fn = SSE,     # our cost function
    method = "STGD") # our iterative optimization algorithm 

$par
[1] 211.26155  -1.69473

$value
[1] 205.138

$counts
[1] 12

$convergence
[1] 0

Ordinary least squares solution

\(y = w_0 + w_1x_1\)

We have a system of equations:

• \(124 = w_01 + w_149\)
• \(95 = w_01 + w_169\)
• …
• \(18 = w_01 + w_1109\)

We can express them as a matrix: \(Y = Xw + \epsilon\)

And solve with linear algebra: \(w = (X^TX)^{-1}X^TY\)

Ordinary least squares solution in R

 ols_matrix_way <- function(X, Y){
  solve(t(X) %*% X) %*% t(X) %*% Y
 }

We need to construct X and Y (must be matrices):

(response_matrix <- data %>% select(rt) %>% as.matrix())
(explanatory_matrix <- data %>% mutate(int = 1) %>% select(int, experience) %>% as.matrix())

      rt
[1,] 124
[2,]  95
[3,]  71
[4,]  45
[5,]  18

     int experience
[1,]   1         49
[2,]   1         69
[3,]   1         89
[4,]   1         99
[5,]   1        109

Then we can use our function to generate the OLS solution:

ols_matrix_way(explanatory_matrix, response_matrix)

                   rt
int        211.270690
experience  -1.694828

lm(rt ~ experience, data = data)

Call:
lm(formula = rt ~ experience, data = data)

Coefficients:
(Intercept)   experience  
    211.271       -1.695