data("mtcars")
mtcars## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160.0 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160.0 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108.0 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360.0 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225.0 105 2.76 3.460 20.22 1 0 3 1
## Duster 360 14.3 8 360.0 245 3.21 3.570 15.84 0 0 3 4
## Merc 240D 24.4 4 146.7 62 3.69 3.190 20.00 1 0 4 2
## Merc 230 22.8 4 140.8 95 3.92 3.150 22.90 1 0 4 2
## Merc 280 19.2 6 167.6 123 3.92 3.440 18.30 1 0 4 4
## Merc 280C 17.8 6 167.6 123 3.92 3.440 18.90 1 0 4 4
## Merc 450SE 16.4 8 275.8 180 3.07 4.070 17.40 0 0 3 3
## Merc 450SL 17.3 8 275.8 180 3.07 3.730 17.60 0 0 3 3
## Merc 450SLC 15.2 8 275.8 180 3.07 3.780 18.00 0 0 3 3
## Cadillac Fleetwood 10.4 8 472.0 205 2.93 5.250 17.98 0 0 3 4
## Lincoln Continental 10.4 8 460.0 215 3.00 5.424 17.82 0 0 3 4
## Chrysler Imperial 14.7 8 440.0 230 3.23 5.345 17.42 0 0 3 4
## Fiat 128 32.4 4 78.7 66 4.08 2.200 19.47 1 1 4 1
## Honda Civic 30.4 4 75.7 52 4.93 1.615 18.52 1 1 4 2
## Toyota Corolla 33.9 4 71.1 65 4.22 1.835 19.90 1 1 4 1
## Toyota Corona 21.5 4 120.1 97 3.70 2.465 20.01 1 0 3 1
## Dodge Challenger 15.5 8 318.0 150 2.76 3.520 16.87 0 0 3 2
## AMC Javelin 15.2 8 304.0 150 3.15 3.435 17.30 0 0 3 2
## Camaro Z28 13.3 8 350.0 245 3.73 3.840 15.41 0 0 3 4
## Pontiac Firebird 19.2 8 400.0 175 3.08 3.845 17.05 0 0 3 2
## Fiat X1-9 27.3 4 79.0 66 4.08 1.935 18.90 1 1 4 1
## Porsche 914-2 26.0 4 120.3 91 4.43 2.140 16.70 0 1 5 2
## Lotus Europa 30.4 4 95.1 113 3.77 1.513 16.90 1 1 5 2
## Ford Pantera L 15.8 8 351.0 264 4.22 3.170 14.50 0 1 5 4
## Ferrari Dino 19.7 6 145.0 175 3.62 2.770 15.50 0 1 5 6
## Maserati Bora 15.0 8 301.0 335 3.54 3.570 14.60 0 1 5 8
## Volvo 142E 21.4 4 121.0 109 4.11 2.780 18.60 1 1 4 2help(mtcars)1.The data was extracted from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles.
2.This dataset contains 11 variables with 32 observations.
mpg: Miles/(US) gallon which represents the number of miles the car can travel on a gallon of fuel.
cyl: Number of cylinders. More cylinders result in greater power but possibly decreased fuel efficiency. Common numbers are 4, 6, and 8.
disp: Displacement is the measure of the total volume of all the cylinders in an engine. A higher displacement usually means more power but less fuel efficiency.
hp: Horsepower. This measures the power of engine produces.
drat: Rear axle ratio. This represents the gear ratio of the real axle, which can influence the carโs speed and acceleration.
wt: Weight(1000 lbs), total weight of the car.
qsec: 1/4 mile time. The time taken by the car to travel a quarter-mile from a standing start.
vs: Engine type (0 = V-shaped, 1 = straight)
am: Transmission type (0 = automatic, 1 = manual)
gear: Number of forward gears
carb: Number of carburetors
3.This dataset is cross-sectional data. Specifically, the data was collected at a single point in time and provide a snapshot of the characteristics of the automobiles for the 1973-1974 model years.
4.Our target variable (y) will be mpg (miles per gallon) based on other independent variables.
I tend to choose mpg as dependent variable, disp and wt independent variables.
\[ mpg = \beta_0\ + \beta_1disp\ + \beta_2 wt\ + \epsilon\ \]
Predict number of miles the car can travel on a gallon of fuel based on the values of displacement and weight.
lm_mpg = lm(mtcars$mpg ~ mtcars$disp + mtcars$wt)
print(lm_mpg)##
## Call:
## lm(formula = mtcars$mpg ~ mtcars$disp + mtcars$wt)
##
## Coefficients:
## (Intercept) mtcars$disp mtcars$wt
## 34.96055 -0.01772 -3.35083#checking the n/a
any(is.na(mtcars))## [1] FALSE# x variebles
X <- as.matrix(mtcars$disp)
#add a column of ones for beta 0
X <- cbind (1, X)
#add the wt column to design marix
X <- cbind(X, mtcars$wt)
print(X)## [,1] [,2] [,3]
## [1,] 1 160.0 2.620
## [2,] 1 160.0 2.875
## [3,] 1 108.0 2.320
## [4,] 1 258.0 3.215
## [5,] 1 360.0 3.440
## [6,] 1 225.0 3.460
## [7,] 1 360.0 3.570
## [8,] 1 146.7 3.190
## [9,] 1 140.8 3.150
## [10,] 1 167.6 3.440
## [11,] 1 167.6 3.440
## [12,] 1 275.8 4.070
## [13,] 1 275.8 3.730
## [14,] 1 275.8 3.780
## [15,] 1 472.0 5.250
## [16,] 1 460.0 5.424
## [17,] 1 440.0 5.345
## [18,] 1 78.7 2.200
## [19,] 1 75.7 1.615
## [20,] 1 71.1 1.835
## [21,] 1 120.1 2.465
## [22,] 1 318.0 3.520
## [23,] 1 304.0 3.435
## [24,] 1 350.0 3.840
## [25,] 1 400.0 3.845
## [26,] 1 79.0 1.935
## [27,] 1 120.3 2.140
## [28,] 1 95.1 1.513
## [29,] 1 351.0 3.170
## [30,] 1 145.0 2.770
## [31,] 1 301.0 3.570
## [32,] 1 121.0 2.780#y variable
y <- as.matrix(mtcars$mpg)
y## [,1]
## [1,] 21.0
## [2,] 21.0
## [3,] 22.8
## [4,] 21.4
## [5,] 18.7
## [6,] 18.1
## [7,] 14.3
## [8,] 24.4
## [9,] 22.8
## [10,] 19.2
## [11,] 17.8
## [12,] 16.4
## [13,] 17.3
## [14,] 15.2
## [15,] 10.4
## [16,] 10.4
## [17,] 14.7
## [18,] 32.4
## [19,] 30.4
## [20,] 33.9
## [21,] 21.5
## [22,] 15.5
## [23,] 15.2
## [24,] 13.3
## [25,] 19.2
## [26,] 27.3
## [27,] 26.0
## [28,] 30.4
## [29,] 15.8
## [30,] 19.7
## [31,] 15.0
## [32,] 21.4#checking the rows and columns for design and response matrix
dim(X)## [1] 32 3dim(y)## [1] 32 1Now that we have design (X) and response (y) Matrices, we can use them to slove for parameters by using closed form solution:
\[ \beta = (X'X)^{-1} X'y \]
betas <- solve(t(X) %*% X) %*% t(X) %*% y
betas## [,1]
## [1,] 34.96055404
## [2,] -0.01772474
## [3,] -3.35082533print(lm_mpg)##
## Call:
## lm(formula = mtcars$mpg ~ mtcars$disp + mtcars$wt)
##
## Coefficients:
## (Intercept) mtcars$disp mtcars$wt
## 34.96055 -0.01772 -3.35083Thus, we got same results by using package and OLS closed form function.
Based on the the results, we can get our linear regression function as:
\[ mpg = 34.96055 - 0.01772disp - 3.35083wt + \epsilon\ \]
With there is one unit increase in displacement, there is 0.01772 decrease in mpg, holding other variables remain constant.
With there is one unit increase in weight, there is 3.35083 decrease in mpg, holding other variables remain constant.