quadratic forms and their application to whale localization · introduction quadratic forms the...

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

The PrincipleAxesTheorem

A QuickExample

ThreeComponentAnalysis

Quadratic Forms and their Application toWhale Localization

Nicole Bender

Marist CollegePoughkeepsie, NY 12601

Hudson River Undergraduate Mathematics Conference

April 6, 2013

Nicole Bender Quadratic Forms and their Application to Whale Localization

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Introduction

Topics:

1 Quadratic Forms

2 Principle Axis Theorem

3 An Example

4 Three Component Analysis of Ellipsoids

5 Ocean Bottom Seismometers

6 A Three Component Analysis Example

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Quadratic Forms as Polynomials and SquareMatrices

An expression in the form

ax2 + by2 + cxy =!x y

" # a c/2c/2 b

is called a quadratic form in 2 variables; x and y. Aquadratic form in n variables is a function f : Rn ! R ofthe form

f (x) = xTAx

where A is a symmetric n " n matrix and x is in Rn.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Ellipses

An ellipse is in standard position to the coordinate axesif its equation can be expressed as

" # 1a2 00 1

where a, b > 0 and either a > b, a < b, or a = b.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Ellipses

An ellipse is in standard position to the coordinate axesif its equation can be expressed as

" # 1a2 00 1

where a, b > 0 and either a > b, a < b, or a = b.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Definitions

An Orthogonal Matrix is an n " n matrix whose columnsform an orthogonal set of unit vectors.

An orthogonal matrix, Q, diagonalizes a symmetric n " nmatrix, A if

QTAQ = D

where D is a diagonal matrix.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Definitions

An Orthogonal Matrix is an n " n matrix whose columnsform an orthogonal set of unit vectors.

An orthogonal matrix, Q, diagonalizes a symmetric n " nmatrix, A if

QTAQ = D

where D is a diagonal matrix.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Principle Axes Theorem

A is the n " n symmetric matrix associated with thequadratic form xTAx. (This may have cross productterms.)

Q is an orthogonal matrix such that QTAQ = D, where Dis a diagonal matrix

The change of variable x = Qy transforms xTAx intoyTDy, which has no cross-product terms.

Note that if the eigenvalues of A are !1, . . . ,!n andy = [y1, . . . , yn]T , then

xTAx = yTDy = !1y21 + · · ·+ !ny

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

xTAx = yTDy = !1y21 + · · ·+ !ny

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

xTAx = yTDy = !1y21 + · · ·+ !ny

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

xTAx = yTDy = !1y21 + · · ·+ !ny

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

xTAx = yTDy = !1y21 + · · ·+ !ny

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

A Quick Example

Let’s look at the equation 5x2 + 2y2 + 4xy = 6. The lefthand side of this equation is a quadratic form that can bewritten as xTAx = 6 where

#5 22 2

The eigenvalues of A are found to be !1 = 1 and !2 = 6by solving for ! in det(A# !I ) = 0.

The diagonal matrix of eigenvalues is therefore

#1 00 6

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

A Quick Example

#5 22 2

#1 00 6

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

A Quick Example

#5 22 2

#1 00 6

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

A Quick Example

The corresponding eigenvectors are

&,q2 =

%2!51!5

We construct Q using these eigenvectors

to orthogonally diagonalize A as QTAQ = D.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

A Quick Example

The corresponding eigenvectors are

&,q2 =

%2!51!5

We construct Q using these eigenvectors

to orthogonally diagonalize A as QTAQ = D.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

A Quick Example

By substituting x = Qx# into xTAx = 6 we havex#TQTAQx# = 6.

Since QTAQ = D, it follows that x#TDx# = 6.

then (x #)2 + 6(y #)2 = 6 or(x !)2

6 + (y #)2 = 1

which is the desired equation for an ellipse with major axisof length

$6 and minor axis of length 1.

The eigenvectors of A determine the positions of the x#

and y# axes.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

A Quick Example

then (x #)2 + 6(y #)2 = 6 or(x !)2

6 + (y #)2 = 1

and y# axes.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

A Quick Example

then (x #)2 + 6(y #)2 = 6 or(x !)2

6 + (y #)2 = 1

and y# axes.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

A Quick Example

then (x #)2 + 6(y #)2 = 6 or(x !)2

6 + (y #)2 = 1

and y# axes.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

A Quick Example

The graph of 5x2 + 4xy + 2y2 = 6 is:

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Three-Dimensional Quadratic Forms

Quadratic forms can also be applied to three-dimensionalstructures such as an ellipsoid:

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Application to Whale Localization

Three component ocean-bottom seismometers recorddisplacements in the ocean bottom caused by acousticwaves travelling through the water.

These seismometers record the displacements from threedi!erent directions; north, east, and depth.

Current research utilizes a number of ocean-bottomseismometers o! of the coast of New Zealand.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Data wasrecorded on theseinstruments for24 hours onMarch 6, 2009.

We will focus onthe 16th hourrecorded onNZ29.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Covariance Matrix

The covariance matrix, S is 3" 3, real, and symmetric.

(Szz Szn SzeSzn Snn SneSze Sne See

The terms of S are auto- and cross-variances of the threecomponents (north, east, and depth).

S is evaluated as

Sjk =XXT

xijxik

where X = [xij ], i = 1, · · · ,N, and j = 1, 2, 3. The i th

sample of component j is xij , and N is the number ofsamples.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Covariance Matrix

S is evaluated as

Sjk =XXT

xijxik

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Covariance Matrix

S is evaluated as

Sjk =XXT

xijxik

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Covariance Matrix

S is the matrix of the quadratic form representing anellipsoid.

S has real non-negative eigenvalues (some may be zero).

The 3 eigenvalues !j and the associated eigenvectors uj ,where j = 1, 2, 3, are found by solving

(S# !2I)u = 0.

The three principle axes are given by !juj and tell us theorientation of the ellipse.

In many real applications, all three eigenvalues are nonzeroand unequal.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Covariance Matrix

(S# !2I)u = 0.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Covariance Matrix

(S# !2I)u = 0.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Covariance Matrix

(S# !2I)u = 0.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Covariance Matrix

(S# !2I)u = 0.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Azimuth of a Signal

The azimuth of a wave arrival can be estimated from thehorizontal orientation of rectilinear motion, given by u1corresponding to the largest eigenvalue.

We can find the azimth for the direction the signal camefrom using

"azimuth = tan"1

,u21u31

where uj1 are the j th components of eigenvector u1.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Azimuth of a Signal

The azimuth of a wave arrival can be estimated from thehorizontal orientation of rectilinear motion, given by u1corresponding to the largest eigenvalue.

We can find the azimth for the direction the signal camefrom using

"azimuth = tan"1

,u21u31

where uj1 are the j th components of eigenvector u1.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Three Component Directions of station NZ29, hour 16, call 7

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Three Component Analysis of Whale Calls

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Questions

Are there any questions?

Thank you for your time. You have been a great audience.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Questions

Are there any questions?

Thank you for your time. You have been a great audience.

QuadraticForms and

theirApplication to

WhaleLocalization

Nicole Bender

Introduction

QuadraticForms

A QuickExample

Bibliography

Jurkevics, Andy. ”Polarization Analysis ofThree-Component Array Data.” Seismological Society ofAmerica 78 (1988): 1725-743.

Poole, David. ”Section 5.5 Applications.” Linear Algebra:A Modern Introduction. 2nd ed. N.p.: ThomsonBrooks/Cole, 2006. 411-26.

quadratic forms and their application to whale localization · introduction quadratic forms the...

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