23 Oct Least Squares Method 1
Least Squares Method
1
Goals: To determine the best linear approximation to a set of data pairs (x,y) using the Least Squares Method.
Requirements: Basic algebra and some knowledge of graphing, plotting data, and straight lines and slopes.
Questions: Total of 10 questions in this Lab. Background: Often in physics, data follows a linear trend that if all the data were measured
correctly, the data would lie on a straight line. In real life, however, measuring data can involve uncertainties due to the measuring process causing a scattering in the data. A line of best fit can be roughly determined using an eyeball method by drawing a straight line through as many points as possible, as shown below1.
The red line is said to model the data – it is the best fit that represents all the data, that is, the trend of the data. A straight line follows the linear equation
? = ?? + ? [1]
Where x is the independent variable, m is the slope of the line, and b is the y- intercept (where the line intercepts the y axis when x = 0). For the line above, we see clearly that b = 5.0, and the slope m is approximately 0.14, calculated from
? = ?2 − ?1 ?2 − ?1
For example, selecting two points on the line, say (0,5) and (35,10), we see the slope is
? = 10 − 5
35 − 0 = 0.14
Equation 1 for the data becomes
? = 0.14? + 5
Least Squares Method
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Rather than use the eyeball method, a better approximation can be obtained using analytical methods. Analytical methods are useful when the dataset is large, and the methods provide a more accurate and consistent approximation. The following procedure is called the Least Square Method and can be applied to any number of n ordered (x,y) data pairs:
? = ∑ ?? −
(∑ ?)(∑ ?) ?
∑ ?2 − (∑ ?)2
?
[2]
? = �̅� − ?�̅� = ∑ ?
? − ?
∑ ?
? [3]
For example, consider the following n = 4 (x,y) data pairs:
x y
1 1.2
2 2.5
3 3.4
4 4.1
n = 4 ∑ ? = 1 + 2 + 3 + 4 = 10 ∑ ? = 1.2 + 2.5 + 3.4 + 4.1 = 11.2 ∑ ?? = (1)(1.2) + (2)(2.5) + (3)(3.4) + (4)(4.1) = 32.8 ∑ ?2 = 12 + 22 + 32 + 42 = 30
(∑ ?) 2
= 102 = 100
And we see from Equation 2 that
? = 32.8 −
(10)(11.2) 4
30 − 100
4
= 0.96
And from Equation 3 that
? = 11.2
4 − (0.96)
10
4 = 0.4
We can now model our data using Equation 1 that is the best fit for the data
? = 0.96? + 0.4
Below is a graph of the data along with fitted line
Least Squares Method
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We see from the graph that when x = 0, y =0.4. Our model allows us to predict values beyond our dataset if we know the data will follow the trend of the first four data pairs.
1. Setup:
Consider the following dataset consisting of n = 10 (x,y) pairs:
x y xy x2
8 3
2 10
11 3
6 6
5 8
4 12
12 1
9 4
6 9
1 14
?? = ?? = ??? = ??? =
a. Fill in the table (white cells) calculating for each data pair xy and x2 values. b. Finally, sum each column and fill in the yellow cells.
Least Squares Method
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2. Exercises: a. Question # 1: What is your answer for ?? ? b. Question # 2: What is your answer for ?? ? c. Question # 3: What is your answer for ??? ? d. Question # 4: What is your answer for ??? ?
Given the results in questions #1 through #4 and n = 10, we can now model the data
e. Question # 5: Using Equation 2, what is the slope m ? f. Question # 6: Using Equation 3, what is the y-intercept b ?
At this point knowing m and b, we have modeled the data and have determined Equation 1:
? = ?? + ?
Let’s now, plot the data and compare/check our results
Click here to bring up the laboratory Activity in another window
If the above link does not work, please copy the following URL into a browser: https://phet.colorado.edu/sims/html/least-squares-regression/latest/least-squares-regression_en.html
You should see this:
3. Setup: a. Ensure all options are selected and checked as shown below (Custom and additional grid
lines checked).
b. Drag all 10 data points from the bucket of points (bottom left) on to the graph, estimating
as best as possible the position of the (x,y) pair on the graph.
Least Squares Method
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Your plot of the data should look similar to the following:
c. Click on Best-Fit Line button (left top). d. Select Best-Fit Line as show below to see the slope m and y-intercept b
4. Exercises: a. Question # 7: What value of slope m did the graphing program return? b. Question # 8: What value of y-intercept b did the graphing program return?
Calculate the percent difference in the slope from your value found in Question 5 to the value computed by the graphing program:
??????? ?????????? = ???????? #7 ????? − ???????? #5 ?????
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