To view this presentation, you'll need to enable Flash.

Show me how

After you enable Flash, refresh this webpage and the presentation should play.

Loading...

PPT – Data Modeling and Least Squares Fitting PowerPoint presentation | free to download - id: c0b30-ZDc1Z

The Adobe Flash plugin is needed to view this content

View by Category

Presentations

Products
Sold on our sister site CrystalGraphics.com

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Data Modeling andLeast Squares Fitting

- COS 323

Data Modeling

- Given data points, functional form,find

constants in function - Example given (xi, yi), find line through

themi.e., find a and b in y axb

yaxb

(x6,y6)

(x3,y3)

(x5,y5)

(x1,y1)

(x7,y7)

(x4,y4)

(x2,y2)

Data Modeling

- You might do this because you actually care about

those numbers - Example measure position of falling object,fit

parabola

time

p 1/2 gt2

position

? Estimate g from fit

Data Modeling

- or because some aspect of behavior is unknown

and you want to ignore it - Example measuringrelative resonantfrequency of

two ions,want to ignoremagnetic field drift

Least Squares

- Nearly universal formulation of fittingminimize

squares of differences betweendata and function - Example for fitting a line, minimizewith

respect to a and b - Most general solution technique take derivatives

w.r.t. unknown variables, set equal to zero

Least Squares

- Computational approaches
- General numerical algorithms for function

minimization - Take partial derivatives general numerical

algorithms for root finding - Specialized numerical algorithms that take

advantage of form of function - Important special case linear least squares

Linear Least Squares

- General pattern
- Note that dependence on unknowns is linear,not

necessarily function!

Solving Linear Least Squares Problem

- Take partial derivatives

Solving Linear Least Squares Problem

- For convenience, rewrite as matrix
- Factor

Linear Least Squares

- Theres a different derivation of

thisoverconstrained linear system - A has n rows and mltn columnsmore equations than

unknowns

Linear Least Squares

- Interpretation find x that comes closest to

satisfying Axb - i.e., minimize bAx
- i.e., minimize bAx
- Equivalently, minimize bAx2 or (bAx)?(bAx)

Linear Least Squares

- If fitting data to linear function
- Rows of A are functions of xi
- Entries in b are yi
- Minimizing sum of squared differences!

Linear Least Squares

- Compare two expressions weve derived equal!

Ways of Solving Linear Least Squares

- Option 1 for each xi,yi compute f(xi), g(xi),

etc. store in row i of A store yi in

b compute (ATA)-1 ATb - (ATA)-1 AT is known as pseudoinverse of A

Ways of Solving Linear Least Squares

- Option 2 for each xi,yi compute f(xi), g(xi),

etc. store in row i of A store yi in

b compute ATA, ATb solve ATAxATb - These are known as the normal equations of the

least squares problem

Ways of Solving Linear Least Squares

- These can be inefficient, since A typically much

larger than ATA and ATb - Option 3 for each xi,yi compute f(xi), g(xi),

etc. accumulate outer product in U accumulate

product with yi in v solve Uxv

Special Case Constant

- Lets try to model a function of the form

y a - In this case, f(xi)1 and we are solving
- Punchline mean is least-squares estimator for

best constant fit

Special Case Line

- Fit to yabx

Weighted Least Squares

- Common case the (xi,yi) have different

uncertainties associated with them - Want to give more weight to measurementsof which

you are more certain - Weighted least squares minimization
- If uncertainty is ?, best to take

Weighted Least Squares

- Define weight matrix W as
- Then solve weighted least squares via

Error Estimates from Linear Least Squares

- For many applications, finding values is useless

without estimate of their accuracy - Residual is b Ax
- Can compute ?2 (b Ax)?(b Ax)
- How do we tell whether answer is good?
- Lots of measurements
- ?2 is small
- ?2 increases quickly with perturbations to x

Error Estimates from Linear Least Squares

- Lets look at increase in ?2
- So, the bigger ATA is, the faster error

increasesas we move away from current x

Error Estimates from Linear Least Squares

- C(ATA)1 is called covariance of the data
- The standard variance in our estimate of x is
- This is a matrix
- Diagonal entries give variance of estimates of

components of x - Off-diagonal entries explain mutual dependence
- nm is ( of samples) minus ( of degrees of

freedom in the fit) consult a statistician

Special Case Constant

Things to Keep in Mind

- In general, uncertainty in estimated

parametersgoes down slowly like 1/sqrt(

samples) - Formulas for special cases (like fitting a line)

are messy simpler to think of ATAxATb form - All of these minimize vertical squared distance
- Square not always appropriate
- Vertical distance not always appropriate

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Home About Us Terms and Conditions Privacy Policy Presentation Removal Request Contact Us Send Us Feedback

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Data Modeling and Least Squares Fitting" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!