Weighted least-squares

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Problem 9.33

Find “best-fit” straight lines to the data in Table 9.33a.

Table 9.33a. data.
0.21 0.49 0.71 1.00 1.42 1.73 2.03 2.47
0.51 1.31 1.54 2.58 1.79 2.20 2.76 2.72
3.05 3.09 3.28 3.64 3.70 3.84 4.07 4.24
4.42 3.25 3.07 3.50 3.73 3.63 3.87 3.88
  1. First plot the data and determine by eye a best-fit line,
  2. Second, find the unweighted best-fit line by least-squares (i.e., weights of 1)
  3. Find the least-squares best-fit line by weighting according to the vertical distances from the line in (a), and finally
  4. By discarding the three wildest points (weighting them zero)

Background

To fit a straight line to a data set such as that in Table 9.33a, we can find the constants and such that the sum of the squares of the “errors” is minimized (see also problem 9.22). An error is the difference between an observed point and that predicted by the equation. If we wish to give added weight to some data points, usually because we consider them more reliable than other values, we give the error squared the weight as in equation (9.33a). Then we write the sum of the errors squared as


(9.33a)

and minimize by varying and . This gives these equations:


(9.33b)


(9.33c)

We rewrite these as simultaneous equations to be solved for and :


(9.33d)


(9.33e)

where .

Curves other than a straight line can be fit to data sets in a similar manner. Other definitions of “best fit” can also be used. Additional constraints, for example, that the curve should pass through the origin, can also be added.

Figure 9.33a.  Straight-line fits to the data.

Solution

The data are plotted in Figure 9.33a and the calculations given in Table 9.33c. The best-fit line determined by eye is shown by the dashed line; its equation is.

The line for equal weighting shown by the solid line has the equation.


The line giving increased weighting to data that lie closer to the equal-weighting line is shown by short dashes; its equation is

If we simply throw away the three points that lie farthest away, ( in Table 9.33b) we get the equation (not plotted)

Table 9.33b. Least-squares calculations.
0.21 0.51 0.04 0.11 1 1 0
0.49 1.31 0.24 0.64 1 5 1
0.71 1.54 0.50 1.09 1 5 1
1.00 2.58 1.00 2.58 1 1 0
1.42 1.79 2.02 2.54 1 3 1
1.73 2.20 2.99 3.81 1 5 1
2.03 2.76 4.12 5.60 1 2 1
2.47 2.72 6.10 6.72 1 5 1
3.05 4.42 9.30 13.48 1 1 0
3.09 3.25 9.55 10.04 1 4 1
3.28 3.07 10.76 10.07 1 3 1
3.64 3.50 13.25 12.74 1 4 1
3.70 3.73 13.69 13.80 1 3 1
3.84 3.63 14.75 13.94 1 4 1
4.07 3.87 16.56 15.75 1 5 1
4.24 3.88 17.98 16.45 1 4 1
38.97 44.76 123.13 129.37 16 13
140.11 152.41 452.21 454.91 55

The changes in values are (standard deviation 3%) and the different weighting schemes make relatively little difference in this example.

Eye-ball fit


Equations for equal weighting line:
.
Weighting by proximity to above line:
.
Throwing away 3 wild points:
.

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