University:
Whitman College
Course:
MATH-125 | Calculus I
Academic year:

2021
Views:

333

Pages:

2
Author:

Eden Robbins

Real-World Application: The Line Of Best Fit Given a set of p data points, (x1 , y1 ), (x2 , y2 ), . . . , (xp , yp ), we would like to find the equation of the line y = mx + b that “best” describes this data. If the data were exactly linear, we would only need two data points to determine the line- The problem is, there is some “error”- No line will exactly go through all the data. Algebraically, we are saying that for any choice of m, b, it will probably be the case that: yi 6= mxi + b Therefore, the error taken at xi is typically taken to be (yi − (mxi + b))2 = (yi − mxi − b)2 Sum these errors together, and this is our error function: E(m, b) = (y1 − mx1 − b)2 + (y2 − mx2 − b)2 + . . . + (yp − mxp − b)2 This function depends on two inputs, m, b. We can simplify this to a function of one variable by the following: Define x to be the mean (or average) of the x’s, and y to be the mean of the y’s. It can be shown that the intercept will be: b = y − mx Now take x̂i = xi − x and ŷ = yi − y (this is called mean-subtracting the data). Then our new error function is: E(m) = (ŷ1 − mx̂1 )2 + (ŷ2 − mx̂2 )2 + . . . + (ŷp − mx̂p )2 The minimum error is found by differentiating the error, setting the result to zero and solve for the slope: dE = 2(ŷ1 − mx̂1 )(−x̂1 ) + 2(ŷ2 − mx̂2 )(−x̂2 ) + . . . + 2(ŷp − mx̂p )(−x̂p ) = 0 dm Divide by 2, expand the result and isolate the slope: m x̂21 + xˆ2 2 + . . . x̂2p = x̂1 ŷ1 + x̂2 ŷ2 + . . . + x̂p yˆp m= x̂1 ŷ1 + x̂2 ŷ2 + . . . + x̂p yˆp x̂21 + xˆ2 2 + . . . x̂2p Example: Find the line of best fit through the points: x −1 0 1 2 3 y −4.9 −2.0 1.4 4.3 6.8 SOLUTION: 1 First, x= −1 + 0 + 1 + 2 + 3 =1 5 y= −4.9 − 2 + 1.4 + 4.3 + 6.8 = 1.12 5 so once we find the slope m, the intercept b = 1.12 − m(1). Mean-subtract the data, then compute the slope: x x̂ −1 −2 0 −1 1 0 2 1 3 2 ŷ y −4.9 −6.02 −2.0 −3.12 1.4 0.28 4.3 3.18 6.8 5.68 x̂ŷ x̂2 12.04 4 3.12 1 0 0 3.18 1 11.36 4 The sum of the column x̂ŷ is 29.7 and the sum of x̂2 is 10. This gives the slope and then the intercept: 29.7 m= = 2.97 b = 1.12 − 2.97 = −1.85 10 The line of best fit is y = 2.97x − 1.85. We might also note that the actual error function in this case is: E(m) = (−6.02 + 2m)2 + (−3.12 + m)2 + (0.28 − 0)2 + (3.18 − m)2 + (5.68 − 2m)2 If we were to expand and simplify, the graph of E is a parabola. Figure 1: The data and the line of best fit. 2

Real-World Application The Line Of Best Fit

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