Today. Least squares
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- Claribel Arnold
- 5 years ago
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1 Today Least squares 1
2 Least squares model fitting 2
3 Least squares model fitting How do we find the best line to fit through the data? 2
4 Least squares model fitting How do we find the best line to fit through the data? 2
5 Least squares model fitting 3
6 Least squares model fitting 3
7 Least squares model fitting y=ax (x n,y n ) (x i,y i ) i=1,2,3,...,n (x 1,y 1 ) 3
8 Least squares model fitting y=ax (x n,y n ) (x i,y i ) i=1,2,3,...,n (x 1,y 1 ) 3
9 Least squares model fitting y=ax (x n,y n ) (x i,y i ) i=1,2,3,...,n (x 1,y 1 ) Each red bar is called a residual. We want all the residuals to be as small as possible. 3
10 The residuals are... (A) r i = y i 2 + x i 2 (B) r i = a 2 (y i 2 + x i2 ) y=ax (C) r i = y i - ax i (x i,y i ) (D) r i = y i - x i (E) r i = x i - y i 4
11 To minimize the residuals, we define the objective function... (A) f(a) = y 1 -ax 1 + y 2 -ax y n -ax n (B) f(a) = (y 1 -ax 1 ) 2 + (y 2 -ax 2 ) (y n -ax n ) 2 (C) f(a) = (y 1 -ax 1 )(y 2 -ax 2 )...(y n -ax n ) (D) f(a) = (ay 1 -x 1 ) 2 + (ay 2 -x 2 ) (ay n -x n ) 2 5
12 To minimize the residuals, we define the objective function... (A) f(a) = y 1 -ax 1 + y 2 -ax y n -ax n (B) f(a) = (y 1 -ax 1 ) 2 + (y 2 -ax 2 ) (y n -ax n ) 2 (C) f(a) = (y 1 -ax 1 )(y 2 -ax 2 )...(y n -ax n ) (D) f(a) = (ay 1 -x 1 ) 2 + (ay 2 -x 2 ) (ay n -x n ) 2 (B) is called the sum of squared residuals. (A) is also reasonable but not as good (take a stats class to find out more). 6
13 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Define f(a): (A) SSR(a) = 5-4a + 7-6a (B) SSR(a) = (4-5a) 2 + (6-7a) 2 (C) SSR(a) = (5-4a) 2 + (7-6a) 2 (D) SSR(a) = (5-4-a) 2 + (7-6-a) 2 7
14 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Define f(a): (A) SSR(a) = 5-4a + 7-6a (B) SSR(a) = (4-5a) 2 + (6-7a) 2 (C) SSR(a) = (5-4a) 2 + (7-6a) 2 (D) SSR(a) = (5-4-a) 2 + (7-6-a) 2 Recall: f(a) = (y 1 -ax 1 ) 2 + (y 2 -ax 2 ) 2 8
15 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): (A)a = 7/6 (B) a = 5/4 (C) a = (7/6 + 5/4) / 2 (D) a = 31/26 9
16 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): (A)a = 7/6 (B)a = 5/4 (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 10
17 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 10
18 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 = a a 2 (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 10
19 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 = a a a a 2 (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 10
20 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 = a a a a 2 SSR (a) = (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 10
21 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 = a a a a 2 SSR (a) = a (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 10
22 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 = a a a a 2 SSR (a) = a (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 10
23 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 = a a a a 2 SSR (a) = a a (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 10
24 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 = a a a a 2 SSR (a) = a a = 0 (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 10
25 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 = a a a a 2 SSR (a) = a a = 0 (C)a = (7/6 + 5/4) / 2 a = ( ) / (D)a = 31/26 10
26 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 = a a a a 2 SSR (a) = a a = 0 (C)a = (7/6 + 5/4) / 2 a = ( ) / ( ) (D)a = 31/26 10
27 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 = a a a a 2 SSR (a) = a a = 0 a = ( ) / ( ) = ( ) / ( ) 10
28 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 = a a a a 2 SSR (a) = a a = 0 (C)a = (7/6 + 5/4) / 2 a = ( ) / ( ) (D)a = 31/26 = ( ) / ( ) = 62/52 10
29 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 = a a a a 2 SSR (a) = a a = 0 a = ( ) / ( ) = ( ) / ( ) = 62/52 = ( x 1 y 1 + x 2 y 2 ) / ( x x 2 2 ) 10
30 Find a so that y=ax fits (4,5), (6,7) in the least squares sense. Find the a that minimizes SSR(a): SSR(a) = (5-4a) 2 + (7-6a) 2 (A)a = 7/6 (B)a = 5/4 (C)a = (7/6 + 5/4) / 2 (D)a = 31/26 = a a a a 2 SSR (a) = a a = 0 a = ( ) / ( ) = ( ) / ( ) = 62/52 = ( x 1 y 1 + x 2 y 2 ) / ( x x 2 2 ) Desmos + mean + L 1 on doc cam (if time). 10
31 Definitions 11
32 Definitions A model is a function that you use to summarize or fit data. For example, some common ones: f(x)=ax, f(x)=ax+b, f(x)=ce -kx. 11
33 Definitions A model is a function that you use to summarize or fit data. For example, some common ones: f(x)=ax, f(x)=ax+b, f(x)=ce -kx. Residuals are a measure of how far each model value is from the data value: r i =y i -f(x i ). 11
34 Definitions A model is a function that you use to summarize or fit data. For example, some common ones: f(x)=ax, f(x)=ax+b, f(x)=ce -kx. Residuals are a measure of how far each model value is from the data value: r i =y i -f(x i ). The Sum of Squared Residuals (SSR) is a measure of how well the model fits all the data: SSR = (y i -f(x i )) 2. Small is better. 11
35 Definitions A model is a function that you use to summarize or fit data. For example, some common ones: f(x)=ax, f(x)=ax+b, f(x)=ce -kx. Residuals are a measure of how far each model value is from the data value: r i =y i -f(x i ). The Sum of Squared Residuals (SSR) is a measure of how well the model fits all the data: SSR = (y i -f(x i )) 2. Small is better. The best fit model is the model with parameter value(s) (a, a&b, C&k) that gives the smallest SSR. 11
36 Spreadsheet warnings Excel and other spreadsheet don t give y=ax fits, just y=ax+b. Don t use a y=ax+b function to get a for y=ax. Trendline in Excel s chart gives y=ax+b and usually not to enough decimals for WW. Do these manually using cells.
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