##                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

From this data set, mtcars, we can set up a linear regression model through using the hp column as the predictor and mpg as the response. This is because the horsepower a car has will effect how many miles per gallon the car has.

Models with a positive slope are more accurately named as having a linear positive trend. When there is an increasing trend, the variables are said to have positive correlation while a decreasing trend has negative correlation.

There are two formulas relating to linear regression:

model: \(\text{y} = \beta_0 + \beta_1\cdot\text{x} + \varepsilon; \hspace{0.5cm} \varepsilon \sim \mathcal{N}(0; \sigma^2)\)
fitted line: \(\text{y} = \hat{\beta}_0 + \hat{\beta}_1\cdot\text{x};\)     \(\hat{\beta}_0 = b_0 - \text{ estimate of } \beta_0\); \(\hat{\beta}_1 = b_1 - \text{ estimate of }\beta_1\)

model: \(\text{price} = \beta_0 + \beta_1\cdot\text{carat } + \varepsilon; \hspace{0.5cm} \varepsilon \sim \mathcal{N}(0; \sigma^2)\)
fitted line: \(\text{price} = \hat{\beta}_0 + \hat{\beta}_1\cdot\text{carat}\)

While both cases were instances of simple linear regression, there are more complex depictions of this idea. This will be shown with the previously viewed data set mtcars, a data set smaller than diamonds and therefore easier to visualize. From the set, miles per gallon is dependent on both the horsepower and weight of the car. Unlike with the previous examples where a line was used to observe the linear regression function, a surface is used in a 3D space. That is so for each mpg value, there is a corresponding hp and wt value.

model <- lm(mpg ~ hp + wt, data = mtcars)

xx <- seq(min(hp), max(hp), by=0.05) 
yy <- seq(min(wt), max(wt), by=0.05) 
zsurface <- expand.grid(hp = xx, wt = yy, KEEP.OUT.ATTRS = F) 
zsurface$mpg <- predict.lm(model, newdata = zsurface) 
zsurface <-acast(zsurface, wt ~ hp, value.var = "mpg")

threeplot <- plot_ly(data = mtcars, x = ~hp, y = ~wt, z = ~mpg) %>% 
  add_markers(name = "data") %>%
  layout(title = "Miles Per Gallon vs Horsepower, Weight") %>% 
  add_trace(z = zsurface, x = xx, y = yy, type = "surface", 
            name = "Linear Regression") %>% hide_colorbar()
partial_bundle(threeplot)