Single variable linear regression, calculating baseline prediction, SSE, SST, R^2 of the model.

linear-regression.Rmd

---
title: "Single Variable Linear Regression R^2"
output: html_document
---

The following figure shows three data points and the best fit line:
y = 3x + 2.

The x-coordinate, or "x", is our independent variable and the y-coordinate, or "y", is our dependent variable.

Coordinates:

```{r, echo=FALSE}
data <- data.frame(x=c(0, 1, 1), y = c(2, 2, 8))
data
```

```{r}
data <- data.frame(x=c(0, 1, 1), y = c(2, 2, 8))
plot(data, pch=15, xlab='Independent', ylab='Dependent', ylim=c(0,10), xlim=c(-3, 3))

fit <- lm(y ~ x, data = data)
abline(fit)
```

What is the baseline prediction?

```{r}
# (2 + 2 + 8) / 3 = 4

baseline <- sum(data$y) / nrow(data)
plot(data, pch=15, xlab='Independent', ylab='Dependent', ylim=c(0,10), xlim=c(-3, 3))
abline(a=baseline, b=0, col='blue')
```

What is the Sum of Squared Errors (SSE) ?
```{r}
# (2 - 2)^2 + (2 - 5)^2 + (8 - 5)^2
# 0^2 + -3^2 + 3^2
# 0 + 9 + 9
# 18

squares <- apply(data, 1, function(row) {
  y <- row[2]
  yhat <- round(predict(fit, newdata = data.frame(x=row[1], y=row[2])))
  
  (y - yhat)^2
})

SSE <- sum(squares)
SSE
```

What is the Total Sum of Squares (SST) ?
```{r}
# (2 - 4)^2 + (2 - 4)^2 + (8 - 4)^2
# 4 + 4 + 16
# 24

squares <- apply(data, 1, function(row) {
  y <- row[2]
  yhat <- baseline
  
  (y - yhat)^2
})

SST <- sum(squares)
SST
```

What is the R² of the model?
```{r}
# 1 - (SSE / SST)
# 1 - (18 / 24)
# 1 - 0.75
# 0.25

1 - (SSE / SST)
```

plot1.png

plot2.png