Many
thanks to Steve Jones for this simplified explanation of Least Squares,
originally presented on a napkin at a Cracker Barrel Restaurant.
To make any least squares
analysis we need an estimate of what we think is the most probable value. We
then statistically analyze using the Least Squares Method to show that the sum
of the square of the residuals is at its least hence:
|
Meas |
R |
R2 |
|
1 |
-1 |
1 |
|
-> 2 |
0 |
0 |
|
3 |
+1 |
1 |
|
4 |
+2 |
4 |
|
5 |
+3 |
9 |
|
|
ĺ |
15 |
In the table above, we have
5 measurements. We select 2 as the most
probable value or best guess answer and
then subtract it from the other measurements to get their residuals. The
residuals are then squared to remove any negative
signs and summed giving a value of 15.
|
Meas |
R |
R2 |
|
1 |
-2 |
4 |
|
2 |
-1 |
1 |
|
-> 3 |
0 |
0 |
|
4 |
+1 |
1 |
|
5 |
+2 |
4 |
|
|
ĺ |
10 |
In the second table above,
we have selected 3 as the best guess answer and have repeated the
calculation of the residuals, their squares and the sum of the squares. In this
case the sum is 10.
|
Meas |
R |
R2 |
|
1 |
-3 |
9 |
|
2 |
-2 |
4 |
|
3 |
-1 |
1 |
|
-> 4 |
0 |
0 |
|
5 |
+1 |
1 |
|
|
ĺ |
15 |
In the final table above, 4
was selected as the best guess answer. The sum of the squares of the residuals
has been computed again, and its value is 15.
The final, most probable,
answer in this example is 3 because it has the least value for the sum
of the squares of the residuals.