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They look like the same thing to me but I'm not sure.

Update: in retrospect, this was not a very good question. OLS refers to fitting a line to data and RSS is the cost function that OLS uses. It finds the parameters that gives the least residual sum of square errors. It is called ordinary in OLS refers to the fact that we are doing a linear fit.

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Here is a definition from Wikipedia:

In statistics, the residual sum of squares (RSS) is the sum of the squares of residuals. It is a measure of the discrepancy between the data and an estimation model; Ordinary least squares (OLS) is a method for estimating the unknown parameters in a linear regression model, with the goal of minimizing the differences between the observed responses in some arbitrary dataset and the responses predicted by the linear approximation of the data.

So RSS is a measure of how good the model approximates the data while OLS is a method of constructing a good model.

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Ordinary least squares (OLS)

Ordinary least squares (OLS) is the workhorse of statistics. It gives a way of taking complicated outcomes and explaining behaviour (such as trends) using linearity. The simplest application of OLS is fitting a line.

Residuals

Residuals are the observable errors from the estimated coefficients. In a sense, the residuals are estimates of the errors.

Let's explain the things using R code:

First fit a ordinary least square line of diamond datasets in UsingR library:

library(UsingR)
data("diamond")
y <- diamond$price
	x <- diamond$carat
n <- length(y)
olsline <- lm(y ~ x)
plot(x, y,
     main ="Odinary Least square line",
     xlab = "Mass (carats)", 
     ylab = "Price (SIN $)", 
     bg = "lightblue", 
     col = "black", cex = 2, pch = 21,frame = FALSE)
abline(olsline, lwd = 2)

enter image description here

Now, Let's calculate the residual i.e residual sum of squares: In R you can easily calculate the residual as resid(olsline), for visualisation let's calculate it manually:

# The residuals from R method
e <- resid(olsline)
## Obtain the residuals manually, get the predicated Ys first
yhat <- predict(olsline)
# The residuals are y -yhat, Let's check by comparing this with R's build in resid function
ce <- y - yhat
max(abs(e-ce))
## Let's do it again hard coding the calculation of Yhat
max(abs(e- (y - coef(olsline)[1] - coef(olsline)[2] * x)))
# Residuals arethe signed length of the red lines
plot(diamond$carat, diamond$price,
    main ="Residuals sum of (actual Y - predicted Y)^2",
     xlab = "Mass (carats)", 
     ylab = "Price (SIN $)", 
     bg = "lightblue", 
     col = "black", cex = 2, pch = 21,frame = FALSE)
abline(olsline, lwd = 2)
for (i in 1 : n) 
  lines(c(x[i], x[i]), c(y[i], yhat[i]), col = "red" , lwd = 2)

enter image description here

Hope these visualization will clear your doubts between RSS & OLS

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In a way, OLS is a model to estimate the regression line based on training data. While, RSS is a parameter to know the accuracy of model for both testing and training data.

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