There are a couple of problems here. First, this is not a good way to use lm(...). lm(...) is meant to be used with a data frame, with the formula expressions referencing columns in the df. So, assuming your data is in two vectors x and y,
set.seed(1) # for reproducible example
x <- 1:11000
y <- 3+0.1*x + rnorm(11000,sd=1000)
df <- data.frame(x,y)
# training set
train <- sample(1:nrow(df),0.75*nrow(df)) # random sample of 75% of data
fit <- lm(y~x,data=df[train,])
Now fit has the model based on the training set. Using lm(...) this way allows you, for example to generate predictions without all the matrix multiplication.
The second problem is the definition of R-squared. The conventional definition is:
1 - SS.residuals/SS.total
For the training set, and the training set ONLY,
SS.total = SS.regression + SS.residual
so
SS.regression = SS.total - SS.residual,
and therefore
R.sq = SS.regression/SS.total
so R.sq is the fraction of variability in the dataset that is explained by the model, and will always be between 0 and 1.
You can see this below.
SS.total <- with(df[train,],sum((y-mean(y))^2))
SS.residual <- sum(residuals(fit)^2)
SS.regression <- sum((fitted(fit)-mean(df[train,]$y))^2)
SS.total - (SS.regression+SS.residual)
# [1] 1.907349e-06
SS.regression/SS.total # fraction of variation explained by the model
# [1] 0.08965502
1-SS.residual/SS.total # same thing, for model frame ONLY!!!
# [1] 0.08965502
summary(fit)$r.squared # both are = R.squared
# [1] 0.08965502
But this does not work with the test set (e.g., when you make predictions from a model).
test <- -train
test.pred <- predict(fit,newdata=df[test,])
test.y <- df[test,]$y
SS.total <- sum((test.y - mean(test.y))^2)
SS.residual <- sum((test.y - test.pred)^2)
SS.regression <- sum((test.pred - mean(test.y))^2)
SS.total - (SS.regression+SS.residual)
# [1] 8958890
# NOT the fraction of variability explained by the model
test.rsq <- 1 - SS.residual/SS.total
test.rsq
# [1] 0.0924713
# fraction of variability explained by the model
SS.regression/SS.total
# [1] 0.08956405
In this contrived example there is not much difference, but it is very possible to have an R-sq. value less than 0 (when defined this way).
If, for example, the model is a very poor predictor with the test set, then the residuals can actually be larger than the total variation in test set. This is equivalent to saying that the test set is modeled better using it's mean, than using the model derived from the training set.
I noticed that you use the first three quarters of your data as the training set, rather than taking a random sample (as in this example). If the dependance of y on x is non-linear, and the x's are in order, then you could get a negative R-sq with the test set.
Regarding OP's comment below, one way to assess the model with a test set is by comparing in-model to out-of-model mean squared error (MSE).
mse.train <- summary(fit)$sigma^2
mse.test <- sum((test.pred - test.y)^2)/(nrow(df)-length(train)-2)
If we assume that the training and test set are both normally distributed with the same variance and having means which follow the same model formula, then the ratio should have an F-distribution with (n.train-2) and (n.test-2) degrees of freedom. If the MSE's are significantly different based on an F-test, then the model does not fit the test data well.
Have you plotted your test.y and pred.y vs x?? This alone will tell you a lot.
errors <- (ytest - yhat)1 - sum(errors^2)/sum((ytest-mean(ytest))^2)and I get -0.59! – Sherlock