Intro
Simple Linear Regression (SLR) has been tickled to death. One interesting tidbit about SLR is that of the different Sum of Squares formulations that exist and how they tie into just about everything. This posts tries to deconstruct the sum of squares formulations into alternative equations.
Definitions
In the least technical terms possible….
Sum of Squares provides a measurement of the total variability of a data set by squaring each point and then summing them.
$$ \sum\limits_{i = 1}^n {x_i^2} $$
More often, we use the Corrected Sum of Squares, which compares each data point to the mean of the data set to obtain a deviation and then square it.
$$ \sum\limits_{i = 1}^n { { {\left( { {x_i} - \bar x} \right)}^2} } $$
, where the mean is defined as: $$ \bar x = \frac{1}{n}\sum\limits_{i = 1}^n { {x_i} } $$
When we talk about Sum of Squares it will always be the later definition. Why? Well, using the initial definition is sure to cause a data overflow when working with large number (e.g. 1000000000000^2 vs. (1000000000 - 1000000)^2).
Arrangements
There are three key equations:
- Sum of Squares over
\(x\): $$ {S_{xx} } = \sum\limits_{i = 1}^n { { {\left( { {x_i} - \bar x} \right)}^2} } $$ - Sum of Squares over
\(y\): $$ {S_{yy} } = \sum\limits_{i = 1}^n { { {\left( { {y_i} - \bar y} \right)}^2} } $$ - Sum of
\(x\)times\(y\): $$ {S_{xy} } = \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right)\left( { {y_i} - \bar y} \right)} $$
Psst… The last one isn’t a square! In fact, it’s part of what’s called covariance. It’s listed here because of the similarities in manipulations that you will see later on.
These initial arrangements can be modified to take on different forms such as:
$$
\begin{align*} {S_{xx} } &= \sum\limits_{i = 1}^n { { {\left( { {x_i} - \bar x} \right)}^2} } \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - n{ {\bar x}^2} \notag \\ &= \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right){x_i} } \notag \\ \end{align*}
$$
and
$$
\begin{align*} {S_{xy} } &= \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right)\left( { {y_i} - \bar y} \right)} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - n\bar x\bar y \notag \\ &= \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right){y_i} } \notag \\ &= \sum\limits_{i = 1}^n {\left( { {y_i} - \bar y} \right){x_i} } \notag \\ \end{align*}
$$
The next two sections go into depth on how to manipulate these equations. The main point behind manipulating these equations is the use of the mean definition and some series properties.
Providing different forms of the Sum of Squares for \(S_{xx}\) and \(S_{yy}\)
These arrangements can be modified rather nicely to alternative expressions.
For instance, both 1 and 2 can be modified to be:
$$
\begin{align} {S_{xx} } &= \sum\limits_{i = 1}^n { { {\left( { {x_i} - \bar x} \right)}^2} } & \text{Definition} \notag \\ &= \sum\limits_{i = 1}^n {\left( {x_i^2 - 2{x_i}\bar x + { {\bar x}^2} } \right)} & \text{Expand the square} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - 2\bar x\sum\limits_{i = 1}^n { {x_i} } + { {\bar x}^2}\sum\limits_{i = 1}^n 1 & \text{Split Summation} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - 2\bar x\sum\limits_{i = 1}^n { {x_i} } + \underbrace {n{ {\bar x}^2} }_{\sum\limits_{i = 1}^n c = n \cdot c} & \text{Separate the summation} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - 2\bar x\left[ {n \cdot \frac{1}{n} } \right]\sum\limits_{i = 1}^n { {x_i} } + n{ {\bar x}^2} & \text{Multiple by 1} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - 2\bar xn \cdot \underbrace {\left[ {\frac{1}{n}\sum\limits_{i = 1}^n { {x_i} } } \right]}_{ = \bar x} + n{ {\bar x}^2}& \text{Group terms for mean} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - 2\bar xn\bar x + n{ {\bar x}^2} & \text{Substitute the mean} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - 2n{ {\bar x}^2} + n{ {\bar x}^2} & \text{Rearrange terms} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - n{ {\bar x}^2} & \text{Simplify} \\ \end{align}
$$
We’ll call this result the alternative definition.
We can further manipulate this expression…
$$
\begin{align} {S_{xx} } &= \sum\limits_{i = 1}^n { { {\left( { {x_i} - \bar x} \right)}^2} } & \text{Definition} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - n{ {\bar x}^2} & \text{Previous result} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - n{\left( {\frac{1}{n}\sum\limits_{i = 1}^n { {x_i} } } \right)^2} & \text{Substitute mean} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - n \cdot \frac{1}{ { {n^2} } } \cdot \sum\limits_{i = 1}^n { {x_i} } \cdot \sum\limits_{i = 1}^n { {x_i} } & \text{Square terms} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - \frac{1}{n}\sum\limits_{i = 1}^n { {x_i} } \cdot \sum\limits_{i = 1}^n { {x_i} } & \text{Simplify} \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - \bar x\sum\limits_{i = 1}^n { {x_i} } & \text{Substitute mean} \notag \\ &= \sum\limits_{i = 1}^n {\left( {x_i^2 - \bar x{x_i} } \right)} & \text{One summation} \notag \\ &= \sum\limits_{i = 1}^n {\left( { {x_i} \cdot {x_i} - \bar x \cdot {x_i} } \right)} & \text{Expand} \notag \\ &= \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right){x_i} } & \text{Factor} \\ \end{align}
$$
The last result we’ll refer to as the exterior definition.
Therefore, as stated previously, we have:
$$
\begin{align*} {S_{xx} } &= \sum\limits_{i = 1}^n { { {\left( { {x_i} - \bar x} \right)}^2} } \notag \\ &= \sum\limits_{i = 1}^n {x_i^2} - n{ {\bar x}^2} \notag \\ &= \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right){x_i} } \notag \\ \end{align*}
$$
Psst… For \(S_{yy}\), simply replace every \(x\) you see above with a \(y\).
e.g.
$$
\begin{align*} {S_{yy} } &= \sum\limits_{i = 1}^n { { {\left( { {y_i} - \bar y} \right)}^2} } \notag \\ &= \sum\limits_{i = 1}^n {y_i^2} - n{ {\bar y}^2} \notag \\ &= \sum\limits_{i = 1}^n {\left( { {y_i} - \bar y} \right){y_i} } \notag \\ \end{align*}
$$
Exploring the different forms of \(S_{xy}\)
Based on the previous section, what comes next should not be very surprising. The only real difference between these two sections is the inclusion of a different variable AND the fact that the number of observations between \(x\) and \(y\) are the same (e.g. \(n_{x} = n_{y} = n\)).
$$
\begin{align} {S_{xy} } &= \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right)\left( { {y_i} - \bar y} \right)} & \text{Definition} \notag \\ &= \sum\limits_{i = 1}^n {\left( { {x_i}{y_i} - {x_i}\bar y - \bar x{y_i} + \bar x\bar y} \right)} & \text{Expand Square} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - \bar y\sum\limits_{i = 1}^n { {x_i} } - \bar x\sum\limits_{i = 1}^n { {y_i} } + \bar x\bar y\sum\limits_{i = 1}^n 1 & \text{Split Summation} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - \bar y\sum\limits_{i = 1}^n { {x_i} } - \bar x\sum\limits_{i = 1}^n { {y_i} } + n\bar x\bar y & \text{Simplify} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - \bar y \cdot \left[ {n \cdot \frac{1}{n} } \right] \cdot \sum\limits_{i = 1}^n { {x_i} } - \bar x \cdot \left[ {n \cdot \frac{1}{n} } \right] \cdot \sum\limits_{i = 1}^n { {y_i} } + n\bar x\bar y & \text{Multiply by 1} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - \bar yn\left[ {\frac{1}{n} \cdot \sum\limits_{i = 1}^n { {x_i} } } \right] - \bar xn\left[ {\frac{1}{n} \cdot \sum\limits_{i = 1}^n { {y_i} } } \right] + n\bar x\bar y & \text{Identify Means} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - \bar yn\bar x - \bar xn\bar y + n\bar x\bar y & \text{Substitute Means} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - n\bar x\bar y - n\bar x\bar y + n\bar x\bar y & \text{Rearrange Terms} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - n\bar x\bar y & \text{Simplify} \\ \end{align}
$$
The result is a modified verison of the alternative definition.
We can obtain the similar form as the previous section, except this time we must choose to either have \(y_i\) or \(x_i\) on the exterior… Let’s start by opting for \(y_i\) on the exterior:
$$
\begin{align} {S_{xy} } &= \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right)\left( { {y_i} - \bar y} \right)} & \text{Definition} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - n\bar x\bar y & \text{Previous Result} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - n\bar x\left[ {\frac{1}{n}\sum\limits_{i = 1}^n { {y_i} } } \right] & \text{Substitute Mean} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - \bar x\sum\limits_{i = 1}^n { {y_i} } & \text{Simplify} \notag \\ &= \sum\limits_{i = 1}^n {\left( { {x_i} \cdot {y_i} - \bar x \cdot {y_i} } \right)} & \text{One summation} \notag \\ &= \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right){y_i} } & \text{Factor} \notag \\ \end{align}
$$
Alternatively, we can go the opposite route and have \(x_i\) on the exterior:
$$
\begin{align} {S_{xy} } &= \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right)\left( { {y_i} - \bar y} \right)} & \text{Definition} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - n\bar x\bar y & \text{Previous Result} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - n\left[ {\frac{1}{n}\sum\limits_{i = 1}^n { {x_i} } } \right]\bar y & \text{Substitute Mean} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - \bar y\sum\limits_{i = 1}^n { {x_i} } & \text{Simplify} \notag \\ &= \sum\limits_{i = 1}^n {\left( { {x_i} \cdot {y_i} - {x_i} \cdot \bar y} \right)} & \text{One Summation} \notag \\ &= \sum\limits_{i = 1}^n {\left( { {y_i} - \bar y} \right){x_i} } & \text{Factor} \notag \\ \end{align}
$$
Both are results from the exterior definition.
Therefore, we have the following equations:
$$
\begin{align*} {S_{xy} } &= \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right)\left( { {y_i} - \bar y} \right)} \notag \\ &= \sum\limits_{i = 1}^n { {x_i}{y_i} } - n\bar x\bar y \notag \\ &= \sum\limits_{i = 1}^n {\left( { {x_i} - \bar x} \right){y_i} } \notag \\ &= \sum\limits_{i = 1}^n {\left( { {y_i} - \bar y} \right){x_i} } \notag \\ \end{align*}
$$
A simple test
The above manipulation can be further scrutinized by seeing if it is accurate. To do so, let’s quickly right a few R functions to check the output.
# For reproducibility
set.seed(1337)
# Generate some random data
x = rnorm(10000,3,2)
y = rnorm(10000,1,4)
Let’s formulize the definitions.
# Sxx and Syy definition
s.xx = function(x){
sum((x-mean(x))^2)
}
# Sxx and Syy Definition definition
s.xx.alt = function(x){
n = length(x)
sum(x^2) - n*mean(x)^2
}
# Sxx and Syy Exterior definition
s.xx.ext = function(x){
sum((x-mean(x))*x)
}
# Sxy Definition
s.xy = function(x,y){
sum((x-mean(x))*(y-mean(y)))
}
# Sxy Alternative Definition
s.xy.alt = function(x,y){
n = length(x)
sum(x*y) - n*mean(x)*mean(y)
}
# Sxy Exterior Definition
s.xy.ext = function(x,y){
sum((x-mean(x))*y)
}
Now, let’s see the results of each function:
### Sxx and Syy
# All give the same value for Sxx Definition?
all.equal(s.xx(x), s.xx.alt(x), s.xx.ext(x))
## [1] TRUE
# What is the value?
s.xx.ext(x)
## [1] 40066.65
### Sxy
# All give the same value for Sxy Definition?
all.equal(s.xy(x,y), s.xy.alt(x,y), s.xy.ext(x,y))
## [1] TRUE
# What is the value?
s.xy.ext(x,y)
## [1] 330.3306
Timing
Aside from the derivations and the simple tests, there is one other item to consider… The amount of time it takes to calculate each equation.
# install.packages("microbenchmark")
# Load microbenchmark
library(microbenchmark)
# Benchmark Sxx definition against x data
microbenchmark(s.xx(x), s.xx.alt(x), s.xx.ext(x))
## Unit: microseconds
## expr min lq mean median uq max neval
## s.xx(x) 57.225 89.4490 123.5456 96.6575 105.0975 2487.600 100
## s.xx.alt(x) 50.978 88.8320 130.7216 95.4955 107.9100 3345.354 100
## s.xx.ext(x) 52.127 85.5605 93.9597 90.3075 98.3600 165.446 100
# Benchmark Syy definition against y data
microbenchmark(s.xx(y), s.xx.alt(y), s.xx.ext(y))
## Unit: microseconds
## expr min lq mean median uq max neval
## s.xx(y) 50.112 52.6700 66.75574 55.4875 86.5110 141.958 100
## s.xx.alt(y) 45.295 47.7435 63.25197 51.2505 82.0705 125.767 100
## s.xx.ext(y) 45.675 48.6085 62.67307 52.1655 81.7845 119.187 100
# Benchmark Sxy Definition
microbenchmark(s.xy(x,y), s.xy.alt(x,y), s.xy.ext(x,y))
## Unit: microseconds
## expr min lq mean median uq max neval
## s.xy(x, y) 73.759 77.2680 148.03748 93.6125 160.5925 2887.859 100
## s.xy.alt(x, y) 62.861 66.4635 127.68697 81.9395 116.7150 3353.675 100
## s.xy.ext(x, y) 45.143 49.0630 63.25256 51.5350 78.9100 160.691 100
In this case, we see that for the \(S_{xx}\) and \(S_{yy}\) the alternative definition is best whereas if we have \(S_{xy}\) then the best speed is from the exterior definition.