If you must be wrong, how little wrong can you be?
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1 MATH Harrell If you must be wrong, how little wrong can you be? Lecture 13 Copyright 2013 by Evans M. Harrell II.
2 About the test Median was 35, range 25 to 40. As it is written:
3 About the test Percentiles: 90th: 40 75th: 38 50th: 35 25th: 32 Range: 25 to 40
4 This week s learning plan Some applications of optimization. Max and min problems in many constraints. Spherical and cylindrical coordinates Vector fields
5 How does the Army beta test correlate with the SAT? Source: army.mil
6 Error analysis and optimization You have a bucket of data {(x i, y i )}. They don t really fit on a line, but what is the best fit in the least-squares sense? You want to minimize i (y i -(mx i +b)) 2 But what are the %(@!! variables? ANSWER: m and b!
7 Error analysis and optimization So we take the gradient of the objective function f(m,b) := i (y i -(mx i +b)) 2 with respect to variables m and b!
8 Error analysis and optimization So we take the gradient of the objective function f(m,b) := i (y i -(mx i +b)) 2 with respect to variables m and b! Critical points when 0 = -2 i x i (y i -(mx i +b)) and 0 = -2 i (y i -(mx i +b))
9 Error analysis and optimization Rewrite 0 = -2 i x i (y i -(mx i +b)) and 0 = -2 i (y i -(mx i +b)) In the form of a linear system of equations like m + b = : ( i x i 2 )m + ( i x i ) b = i x i y i ( i x i )m + N b = i y i
10
11 Example Best linear fit to (0,1), (1, 3), (2,4) Calculate: N = 3 i x i = 3 i y i = 8 i x 2 i = 5 i x i y i = 11 m = 9/6, b = 7/6
12 x x x
13 Other best fits Best quadratic, cubic, etc. Best combination of sines and cosines ( Fourier series ). Other functions representing your preconceptions about the data.
14 Clicker quiz What is the maximum value of 3x + 4y when x 2 + y 2 = 25? A 5 B 24 C 25 D (24) 1/2 E none of the above.
15 The Lagrange condition Assuming f and C are smooth, at a boundary maximum point x 0 where g(x 0 ) 0, f(x 0 ) = λ g(x 0 ) for some scalar value λ.
16 What if we have more than one constraint?
17 Example The intersection of two planes, such as x + 2 y - 3 z = 6 and x + y + z = 1 is a line. What is the closest point on the line to the origin?
18 Lagrange with two constraints Assuming f is smooth, and constrained by two smooth functions, g(x) = 0, and h(x) = 0. Lagrange s condition for a doubly constrained critical point is f(x 0 ) = λ g(x 0 ) + µ h(x 0 ) for some scalar values λ and µ.
19 Example The intersection of two planes, such as x + 2 y - 3 z = 6 and x + y + z = 1 is a line. What is the closest point on the line to the origin?
20
21 Example Objective function: f(x,y,z) = x 2 +y 2 +z 2 Lagrange says: 2xi + 2yj + 2zk = λ (1i+2j-3k) + µ (1i+1j+1k) Five unknowns (x,y,z,λ,µ). This vector equation represents three scalar eqns. We need two more
22 Lagrange conditions constraints
23 Coordinate systems if you are not square Cylindrical = polar plus z Spherical = geographic coordinates plus radius
24
25
26 Coordinate systems if you are not square Cylindrical = polar plus z r = distance from vertical axis, 0 r θ = angle, any range of length 2π z = height, - < z <
27 Coordinate systems if you are not square Cylindrical to Cartesian: x = r cos θ y = r sin θ z = z
28 Coordinate systems if you are not square Cartesian to Cylindrical : r = (x 2 + y 2 ) 1/2 θ = arctan(y/x) z = z
29 Coordinate systems if you are not square Spherical = geographic plus ρ ρ = distance from origin θ = polar angle in xy plane = longitude φ = angle from pole, colatitude
30 Coordinate systems if you are not square Cartesian to spherical: ρ = (x 2 + y 2 + z 2 ) 1/2 tan θ = y/x; or cot θ = x/y; or cos θ = x/(x 2 +y 2 ) 1/2 cos φ = z/(x 2 + y 2 + z 2 ) 1/2
31 Coordinate systems if you are not square Spherical to cylindrical: r = ρ sin φ θ = θ z = ρ cos φ
32 Coordinate systems if you are not square Spherical to Cartesian: x = ρ sin φ cos θ y = ρ sin φ sinθ z = ρ cos φ
33 Nice application How far is it by the shortest air route from Atlanta to Moscow? Atlanta latitude= longitude= Moscow latitude= Longitude=
34 Nice application How far is it by the shortest air route from Atlanta to Moscow? Note: ρ = km Atlanta φ = colatitude= sin φ =.8325, cos φ =.5540 θ = longitude= sin θ = , cos θ =.0973
35 Nice application How far is it by the shortest air route from Atlanta to Moscow? Note: ρ = km Moscow φ = colatitude= sin φ =.5653, cos φ =.8249 θ = longitude= sin θ =.6125, cos θ =.7905
36 Nice application How far is it by the shortest air route from Atlanta to Moscow? Note: ρ = km Atlanta x = ρ (.8325)(.0973) =.0810 y = ρ (.8325)(-.9953) = z = ρ (.5540)
37 Nice application How far is it by the shortest air route from Atlanta to Moscow? Note: ρ = 6378 km Moscow x = ρ (.5653)(.7905) =.4469 y = ρ (.5653)(.6125) =.3462 z = ρ (.8249) Cos(α) =.4469* *(-.8286) *.5540 α = , dist = 6378 α = 8693 km.
38 Unit vectors for curvilinear coordinate systems
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