In previous post, I have discussed on how to create a sample data-set in R. So let’s use the created data-set from previous post and start playing with Regression predictions.

With each prediction we want to measure, which one scores better the new values and where over-fitting start.

First we will create training and test subset for model prediction. We will use training dataset for model to train and later we will use test data set to actually test our model.

indices <- sample(1000,400)
train <- dat_set[indices, ]
test <-dat_set[-indices, ]

Once we have our subsets loaded, we can visualize a simple scatter plot, to get idea of the point dispersion and what kind of MSE we will be handling. For scatterplot we will use ggplot2 with simple geom_point function.

ggplot()+geom_point(data=train, aes(x=x, y=y))+ggtitle("TRAINING SET")
ggplot()+geom_point(data=test, aes(x=x, y=y))+ggtitle("TEST SET")

Now we can deploy lm function on training dataset to learn and score the model. we will be predicting variable Y (dependent) with independent variable X.

model <- lm(y~x, data=train)
model

Lm function shows (run: model or run: summary(model))

lm(formula = y ~ x, data = train)
Coefficients:
(Intercept) x
5.267 2.065

which can easily be tested. let us define x = 5 and observe the value:

x=5
predict(model, data.frame(x))
#results:
1
15.59432

This means that trained model will predict value Y with 15.59 when new value X =5 will be introduced into the model.

I will add some endpoints to draw a linear regression function. This function will serve as the idea how linear, quadratic, cube, etc. functions will behave when predicting better and accurate results.

x <- c(-5,30)
x_predict <- predict(model, data.frame(x))
endpoints <- data.frame(x, x_predict)

I hard coded the arbitrary endpoints based on the TRAIN graph (see above). in this case -5,30 as two points on the graphs (min and max endpoints). I let the prediction find the other points.

Now let’s plot the scatterplot together with the linear regression line and see the points vs. the line.

ggplot()+
geom_point(data=train, aes(x=x, y=y))+
geom_line(data=endpoints, aes(x=x,y=x_predict), color='brown',
size=1)+
ggtitle("TRAINING SET")

Seeing this one might get the quick idea how the “problem” or the dispersion of the dots is not nearly linear.

Now let’s play with prediction model and find the best polynomial line fitting the dots. All the time we will also look into the problem of over-fitting the regression function.

We will introduce MSE measure. MSE is mean square error measures the average of the squares of the “errors”, that is, the difference between the estimator and what is estimated.

For this purpose we will calculate use our x values against the predictions on our model.

#Calculating MSE
x <- test$x
p <- predict(model, data.frame(x)) # or: predict(model, test)

Based on this, we can see the predicted values for each x, which we will use for calculating the MSE.

Continuing we take the mean squares of difference between vector of predictions (that is: actual Y values) and vector of observed values corresponding to the inputs to the function which generated the predictions (that is: predicted model of X).

#MSE
sum((test$y - predict(model,data.frame(x)))^2)
#AVG - test mean square error
mse_test_value <- mean((test$y - predict(model,data.frame(x)))^2)

Now what we have a variable (mse_value) we can actually search for minimal MSE value among any next polynomial regression function.

So in first step we created linear function, now let’s create a quadratic function for train dataset.

# Quadratic
model_Q <- lm(y~x+I(x^2), data=train)
model_Q

For this quadratic function we create function for ggplot:

#function for quadratic
f_Q <- function(x) {
return(predict(model_Q,data.frame(x)))
}

This function is added to previous scatter plot with quadratic function.

#plotting training set with quadratic model for quadratic function
ggplot()+
geom_point(data=train, aes(x=x, y=y))+
geom_line(data=endpoints, aes(x=x,y=x_predict), color='brown',
size=1)+
stat_function(data=data.frame(x=c(-5,30)), aes(x=x), fun=f_Q,
color='blue', size=1)+
ggtitle("TRAINING SET")

Plot shows on training data that quadratic function sways from linear function.

But before we get ahead of ourselves, let’s repeat the calculation of MSE for linear vs. quadratic function and see where MSE measure is better.

#calculate test MSE for quadratic
mean((test$y-predict(model_Q, test))^2)

MSE Calculation is (for my case of dataset)

94.35723

where as for linear function:

mse_test <- mean((test$y - predict(model,data.frame(x)))^2)
#result
93.16901

MSE fits slightly better for linear function as for quadratic model. For cubic function (or degree 3) we use following function:

model_3 <- lm(formula=y~poly(x,3, raw=T), data=train)
mse_3 <- mean((test$y-predict(model_3,test))^2)

with result of:

95.07303

we see the cubic fit function is not as good as quadratic or linear and we may start already over-fitting the model. So let’s loop through the linear to degree of 12 of polynomial function and see the fit scores (MSE). Using following code:

for(i in 1:13) {
model <- lm(formula=y~poly(x,i, raw=T), data=train)
mse <- mean((test$y-predict(model,test))^2)
print(mse)
}

Returns following results:

[1] 93.16901
[1] 94.35723
[1] 95.07303
[1] 95.23098
[1] 94.79259
[1] 96.55435
[1] 108.6518
[1] 132.8873
[1] 130.5214
[1] 212.9898
[1] 169.7865
[1] 7321.596
[1] 226.708

And we see that at degree 5 the function fit is 94.79 as of degree of 2 with 94.35, but still linear model outfits / outperforms all other functions. What happens with function of degree 7 or higher is what we call over-fitting.

For better visualization I have reduced the function to only 10 iterations, mainly because at the degree of 11 and higher we get extreme outliers and high values.

mse_v<-numeric()
for(i in 1:10) {
model <- lm(formula=y~poly(x,i, raw=T), data=train)
mse_v[i] <- mean((test$y-predict(model,test))^2)
}
mse_v
#visualize MSE
y_m <- mse_v
x_m <- 1:10
mse_p <- data.frame(x_m, y_m)
ggplot()+
geom_point(data=mse_p, aes(x=x_m, y=y_m), size=2)+
geom_line(data=mse_p, aes(x=x_m, y=y_m), size=1)

Graph shows how the MSE is growing with each higher degree.

For the last part, let’s compare trained data to test data with all the functions from linear to function of degree 10.

Therefore we introduce following function:

mse_calc <- function(train,test){
for(i in 1:10) {
model <- lm(formula=y~poly(x,i, raw=T), data=train)
mse[i] <- mean((test$y-predict(model,test))^2)
}
return(mse)
}

making following visualization even better with for loop:

x<- 1:10
plot<-ggplot()
for(i in 1:10){
ind <- sample(1000,500)
train <- dat_set[ind,]
test <- dat_set[-ind,]
y<-mse_calc(train,test)
mse_poly <- data.frame(x,y)
plot<-plot+geom_point(data=mse_poly, aes(x,y), size=3)
plot<-plot+geom_line(data=mse_poly, aes(x,y))
}
plot

Making following outstanding plot:

Couple of words on for loop. In each loop we generate random subset of training and test data, which is predicted for each step and compared with each other, resulting in calculation of MSE. 10 times we calculate linear MSE fit measure for random values, 10 times we calculate quadratic MSE fit measure for random values, 10 times …. for degree 3 to degree 10. We see that over fitting start already at degree 4 and at degree 5 it just explodes.