Linear Regression
Linear regression fits a straight line to the selected data using a method called the Sum Of Least Squares.
Sum Of Least Squares
The Sum Of Least Squares method provides an objective measure for comparing a number of straight lines to find the one that best fits the selected data.
- Plot each data point in a table
- Calculate the distance between each data point and the proposed straight line
- Square the distances (to remove negative values)
- Calculate the sum of the squares
- Repeat steps 2 to 4 for each possible line
- Select the line with the lowest sum of squares (from step 4).
Example
The table below demonstrates how the sum of squares is calculated for a line where
Price = 20.50 + 0.11 * day n
| Date | n | Closing Price | Price (20.50 + 0.11*n) |
Distance | Squared |
|---|---|---|---|---|---|
| 13-Feb | 1 | 20.55 | 20.61 | -0.06 | 0.0036 |
| 14-Feb | 2 | 20.80 | 20.72 | 0.08 | 0.0064 |
| 15-Feb | 3 | 20.95 | 20.83 | 0.12 | 0.0144 |
| 16-Feb | 4 | 20.78 | 20.94 | -0.16 | 0.0256 |
| 17-Feb | 5 | 21.10 | 21.05 | 0.05 | 0.0025 |
| Sum | 0.0525 |
The sum of squares is calculated for each possible line and the line with the lowest sum is selected.
Mathematical Formula
Manually calculating the sum of squares for each possible line would be enormously time-consuming. Fortunately there is a quicker way.
The formula for a straight line is
y = a + bx
For our purposes:
- y is the price
- x is the date
- a is the constant (the value when x equals zero)
- b is the slope of the line
The formula for calculating the line of best fit is
b = ( nΣxy - ΣxΣy ) / ( nΣx² - (Σx)² )
a = ( Σy - bΣx ) / n
Where n is the number of data points selected.