part1 full factorial designs for 2 level factors
play

Part1: Full factorial designs for 2-level factors Prepared by: - PowerPoint PPT Presentation

Jan 05, 2024 •38 likes •298 views

Part1: Full factorial designs for 2-level factors Prepared by: Paul Funkenbusch, Department of Mechanical Engineering, University of Rochester Terminology Experiment Model relationship Full factorial experiments Interactions

  1. Part1: Full factorial designs for 2-level factors Prepared by: Paul Funkenbusch, Department of Mechanical Engineering, University of Rochester

  2.  Terminology  Experiment  Model relationship  Full factorial experiments  Interactions and sparsity of effects DOE mini-course, part 1, Paul Funkenbusch, 2015 2

  3. Example Terms (measure the volume of a balloon as a function of temperature and pressure)  Temperature,  Factors  variables whose influence you want to study. Pressure  Levels  specific values given  50  C, 100  C to a factor during experiments 1Pa, 2Pa (initially limit ourselves to 2-levels)  Set T = 50  C, P = 1Pa  Treatment condition  one running of the experiment and measure volume  Response  result measured  measured volume for a treatment condition DOE mini-course, part 1, Paul Funkenbusch, 2015 3

  4. Leve vel Use X1, X2, etc. to designate factors. Factor -1 +1 Use -1, +1 to designate levels 50 100 X1. Temperature (  C) X2. Pressure (Pa) 1 2 X1 at level -1 means T = 50  C y  response y = volume of balloon TC TC X1 X1 X2 X2 y Use a table to show factor levels and response (a) for each 1 -1 -1 y 1 treatment condition. 2 +1 -1 y 2 For example, during TC2, 3 -1 +1 y 3 set T = 100  C and P = 1Pa, 4 +1 +1 y4 measure the balloon volume = y 2 etc. DOE mini-course, part 1, Paul Funkenbusch, 2015 4

  5.  “Effects” calculated from Leve vel responses (y 1, y 2 ) Factor -1 +1 50 100 X1. Temperature (  C)  m* TC TC X1 X1 y ◦ =(y 1 + y 2 )/2 1 -1 y 1 ◦ = overal all l average age 2 +1 y 2  D X1 ◦ = (avg. at +1) – (avg. at -1) y ◦ = m +1 – m -1 = y 2 – y 1 ◦ = “effect of X1” -1 1 X1 X1 DOE mini-course, part 1, Paul Funkenbusch, 2015 5

  6. TC TC X1 X1 y  Can fit a straight line 1 -1 y 1  y pred = a o + a 1 X 1 2 +1 y 2  a o ◦ intercept y ◦ related to m* ◦ (= m* for X1on ±1 scale) m* D  a 1 ◦ slope ◦ related to D X1 -1 1 X1 X1 ◦ (= D X1 /2 for X1on ±1 scale) DOE mini-course, part 1, Paul Funkenbusch, 2015 6

  7.  DOF or DF TC TC X1 X1 y ◦ Counter for information 1 -1 y 1 2 +1 y 2  Experimental data ◦ 2 data points (y1 & y2) ◦ = 2 DOF y  Analysis ◦ 2 results (m*, D X1  a o , a 1 ) ◦ = 2 DOF -1 1 X1 X1  Information is conserved ◦ 2 DOF  2 DOF DOE mini-course, part 1, Paul Funkenbusch, 2015 7

  8. Leve vel  One At a Time experiment Factor -1 +1 ◦ “OAT” 50 100 X1. Temperature (  C) ◦ what most people are taught X2. Pressure (Pa) 1 2 ◦ pick a baseline (X1=-1, X2=-1) ◦ Change one factor at a time TC TC X1 X1 X2 X2 y  Results 1 -1 -1 y 1 ◦ baseline = y 1 2 +1 -1 y 2 ◦ D X1 = y 2 – y 1 3 -1 +1 y 3 ◦ D x2 = y 3 – y 1 DOE mini-course, part 1, Paul Funkenbusch, 2015 8

  9. Leve vel  Choice of baseline is Factor -1 +1 critical (next slide) 50 100 X1. Temp (  C) X2. Pressure (Pa) 1 2  Inefficient use of data ◦ baseline = y 1 ◦ D X1 = y 2 – y 1 TC TC X1 X1 X2 X2 y ◦ D x2 = y 3 – y 1 1 -1 -1 y 1 ◦ Only use part of data to 2 +1 -1 y 2 calculate each effect 3 -1 +1 y 3 DOE mini-course, part 1, Paul Funkenbusch, 2015 9

  10. Leve vel  Study plant growth as a Factor -1 +1 function of watering and X1. watering never daily sunlight X2. daily sunlight none 10 hrs  Results ◦ baseline = y 1 = 0 TC TC X1 X1 X2 X2 growth th ◦ D X1 = y 2 – y 1 = 0 1 -1 -1 0 ◦ D x2 = y 3 – y 1 = 0 2 +1 -1 0 3 -1 +1 0  Conclusion ◦ Watering and sunlight do not affect plant growth DOE mini-course, part 1, Paul Funkenbusch, 2015 10

  11.  Test all combinations Leve vel Factor -1 +1  Results 50 100 X1. Temp (  C) ◦ m* = (y 1 +y 2 +y 3 +y 4 )/4 X2. Pressure (Pa) 1 2 ◦ D X1 = (y 3 +y 4 )/2 - (y 1 +y 2 )/2 ◦ D x2 = (y 2 +y 4 )/2 - (y 1 +y 3 )/2 TC TC X1 X1 X2 X2 y  Effect of X1 1 -1 -1 y 1 ◦ Averaged over both X2 levels ◦ No baseline 2 -1 +1 y 2 3 +1 -1 y 3  All results  all effects 4 +1 +1 y 4 ◦ Use all data in each calculation ◦ Efficient use of data DOE mini-course, part 1, Paul Funkenbusch, 2015 11

  12. TC TC X1 X1 X2 X2 X1*X2 *X2 y  Four degrees of freedom 1 -1 -1 +1 y 1  Model can include an extra (interaction) term 2 -1 +1 -1 y 2 3 +1 -1 -1 y 3  y pred = a o +a 1 X 1 +a 2 X 2 +a 12 12 X 1 X 2 4 +1 +1 +1 y 4  a 12  D 12 ◦ = (avg. for X 1 X 2 =+1) – (avg. for X 1 X 2 =-1) ◦ = (y 1 +y 4 )/2 - (y 2 +y 3 )/2 = “effect of X1*X2 interaction”  Can show interaction as a column to help calculations ◦ (Note: This simple algebra works only for 2-level factors, with levels set to ±1, but the underlying ideas also apply to other designs.) DOE mini-course, part 1, Paul Funkenbusch, 2015 12

  13.  Data on the removal Leve vel rate of osteotomy drills Factor -1 +1 is collected as a function of the load X1. applied load (kg) 2 3 applied and the number of previous X2. previous cuts 0 20 cuts made. (new)  Find the effects of the two factors and the TC TC X1 X1 X2 X2 X1*X2 *X2 Remov oval al rate interaction. Which (mm3/s) /s) is/are most important? 1 -1 -1 +1 3  Build a simple model to 2 -1 +1 -1 2 predict the removal rate. 3 +1 -1 -1 5 4 +1 +1 +1 2 DOE mini-course, part 1, Paul Funkenbusch, 2015 13

  14. TC TC X1 X1 X2 X2 X1*X2 *X2 Remov oval al rate (mm3/s) /s) 1 -1 -1 +1 3 2 -1 +1 -1 2 3 +1 -1 -1 5 4 +1 +1 +1 2  Effects ◦ m* = ? ◦ D X1 = ? ◦ D x2 =? ◦ D x1x2 = ? DOE mini-course, part 1, Paul Funkenbusch, 2015 14

  15. TC TC X1 X1 X2 X2 X1*X2 *X2 Remov oval al rate (mm3/s) /s)  Number of previous cuts (i.e. X2) is the 1 -1 -1 +1 3 more important factor 2 -1 +1 -1 2 (given the range of 3 +1 -1 -1 5 values tested). 4 +1 +1 +1 2  Effects ◦ m* = (y 1 +y 2 +y 3 +y 4 )/4 =(3+2+5+2)/4 = 3 ◦ D X1 = (y 3 +y 4 )/2 - (y 1 +y 2 )/2 = (5+2)/2 – (3+2)/2 = 1 ◦ D x2 = (y 2 +y 4 )/2 - (y 1 +y 3 )/2 = (2+2)/2 – (3+5)/2 = -2 ◦ D x1x2 = (y 1 +y 4 )/2 - (y 2 +y 3 )/2 = (3+2)/2 – (2+5)/2 = -1 DOE mini-course, part 1, Paul Funkenbusch, 2015 15

  16.  Use the normalized (±1) scale for the levels. Effects  y ◦ m* = 3 ◦ D X1 = 1 y m* ◦ D x2 = -2 D a 0 2 ◦ D x1x2 = -1 0 -1 1 -1 1 X1 X1 X1 X1  Then the midpoint of the design corresponds to the intercept (0) for both factors  a 0 = m* = 3.0  And the slope (a) for each effect is given by D /2 ◦ a 1 = D X1 /2 = 0.5; a 2 = D X2 /2 = -1.0; a 12 = D X12 /2 = -0.5  y pred = 3.0 + (0.5)X 1 - (1.0)X 2 - (0.5)X 1 X 2 (y in units of mm 3 /s) DOE mini-course, part 1, Paul Funkenbusch, 2015 16

  17.  L = load (kg) Leve vel  C = # of cuts Factor -1 +1 X1. applied load (kg) 2 3  X1 = -5 + 2L X2. previous cuts 0 20  X2 = -1 + 0.1C  y pred = 3.0 + (0.5)X 1 - (1.0)X 2 - (0.5)X 1 X 2 (y in mm 3 /s, for L in kg) DOE mini-course, part 1, Paul Funkenbusch, 2015 17

  18.  4 data points, 4 constants ◦  will hit all of the original data exactly  New trials under same conditions? ◦  How good was the original data? (error)  Other conditions (interpolation, extrapolation)? ◦  How good is the linear fit to the real dependence? DOE mini-course, part 1, Paul Funkenbusch, 2015 18

  19.  require more TC to include all combinations ◦ # of TC = 2 n , where n= # of factors  model includes more terms  # of interaction terms increases rapidly  includes “higher - order” interactions ◦ “order” = number of factors in the interaction term ◦ X 1 X 2  second order; X 1 X 2 X 3  third order, etc. DOE mini-course, part 1, Paul Funkenbusch, 2015 19

  20. TC TC X1 X1 X2 X2 X3 X3 y  2 3 = 8 1 -1 -1 -1 y 1  8 responses (8 y’s) 2 -1 -1 +1 y 2 3 -1 +1 -1 y 3  m* , D x1 , D x2 , D x3 , 4 -1 +1 +1 y 4 D x1x2 , D x1x3 , D x2x3 , 5 +1 -1 -1 y 5 D x1x2x3 6 +1 -1 +1 y 6 7 +1 +1 -1 y 7 8 +1 +1 +1 y 8  y pred = a o +a 1 X 1 +a 2 X 2 + a 2 X 3  8 DOF + a 12 X 1 X 2 +a 13 X 1 X 3 +a 23 X 2 X 3 ◦ m*  1 DOF +a 123 X 1 X 2 X 3 ◦ factors  3 DOF ◦ 2-factor inter.  3 DOF ◦ 3-factor inter.  1 DOF DOE mini-course, part 1, Paul Funkenbusch, 2015 20

  21.  4 factors at 2 levels  5 factors at 2 levels  2 4 = 16  2 5 = 32  16 DOF  32 DOF ◦ m*  1 DOF ◦ m*  1 DOF ◦ factors  4 DOF ◦ factors  5 DOF ◦ 2-factor inter.  6 DOF ◦ 2-factor inter.  10 DOF ◦ 3-factor inter.  4 DOF ◦ 3-factor inter.  10 DOF ◦ 4-factor inter.  1 DOF ◦ 4-factor inter.  5 DOF ◦ 5-factor inter.  1 DOF More factors  much more effort to measure interactions DOE mini-course, part 1, Paul Funkenbusch, 2015 21

  22.  System is usually dominated by the effects of factors and lower-order interactions.  Usually don’t really need 3 -factor, 4-factor, etc. interactions ~ 0.  But have just seen, that we may spend a lot of effort (i.e. lots of DOF) on them.  Can we make better use of these DOF? ◦  will discuss in Part II. DOE mini-course, part 1, Paul Funkenbusch, 2015 22

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Factorial Designs Two (or more) factors, say A and B , with a and b levels, respectively. A

Factorial Designs Two (or more) factors, say A and B , with a and b levels, respectively. A

ST 516 Experimental Statistics for Engineers II Factorial Designs Two (or more) factors, say A and B , with a and b levels, respectively. A factorial design uses all ab combinations of levels of A and B , for a total of ab treatments . When both

481 views • 22 slides
Two-Level Factors: The 2 k Factorial Design When several factors may affect a response, often each

Two-Level Factors: The 2 k Factorial Design When several factors may affect a response, often each

ST 516 Experimental Statistics for Engineers II Two-Level Factors: The 2 k Factorial Design When several factors may affect a response, often each has just two levels; e.g.: comparing two methods for one step in a process; presence or absence

364 views • 13 slides
The 2 k Factorial Design A design with k factors each with just 2 levels has 2 k treatments, and is

The 2 k Factorial Design A design with k factors each with just 2 levels has 2 k treatments, and is

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control The 2 k Factorial Design A design with k factors each with just 2 levels has 2 k treatments, and is known as a 2 k design. 2 k designs have the

472 views • 23 slides
Process Optimization with Designed Experiments Factorial experimental designs, including

Process Optimization with Designed Experiments Factorial experimental designs, including

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Process Optimization with Designed Experiments Factorial experimental designs, including fractionally replicated designs, are widely used to

520 views • 12 slides
From Factorial Designs to Hilbert Schemes Lorenzo Robbiano Universit di Genova Dipartimento di

From Factorial Designs to Hilbert Schemes Lorenzo Robbiano Universit di Genova Dipartimento di

From Factorial Designs to Hilbert Schemes Lorenzo Robbiano Universit di Genova Dipartimento di Matematica Lorenzo Robbiano (Universit di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 1 / 31 Abstract This talk is meant

1.81k views • 118 slides
Factorial Designs Tyson S. Barrett, PhD PSY 3500 Experiments Manipulating a factor (some

Factorial Designs Tyson S. Barrett, PhD PSY 3500 Experiments Manipulating a factor (some

Factorial Designs Tyson S. Barrett, PhD PSY 3500 Experiments Manipulating a factor (some attribute of a participant) to discover a causal relationship How do we know our manipulation of the independent variable worked? If we were studying

1.06k views • 8 slides
Extending a 2 k design The 2 k factorial design can detect main effects of factors and interactions

Extending a 2 k design The 2 k factorial design can detect main effects of factors and interactions

ST 516 Experimental Statistics for Engineers II Extending a 2 k design The 2 k factorial design can detect main effects of factors and interactions among factors, at the selected levels of the factors . To predict what will happen at other levels

290 views • 13 slides
Fractional Factorial Designs Each replicate of a 2 k design requires 2 k runs. E.g. 64 runs for k =

Fractional Factorial Designs Each replicate of a 2 k design requires 2 k runs. E.g. 64 runs for k =

ST 516 Experimental Statistics for Engineers II Fractional Factorial Designs Each replicate of a 2 k design requires 2 k runs. E.g. 64 runs for k = 6, or 1024 runs for k = 10. When this is infeasible, we use a fraction of the runs. As a result, we

580 views • 18 slides
Announcements Tree Recursion factorial (!) if n == 0 Recursive Factorial n! = 1 if n > 0

Announcements Tree Recursion factorial (!) if n == 0 Recursive Factorial n! = 1 if n > 0

Announcements Tree Recursion factorial (!) if n == 0 Recursive Factorial n! = 1 if n > 0 n! = n x (n-1) x (n-2) x ... x 1 factorial (!) factorial(5) def factorial(n): fact = 1 base case if n == 0 i = 1 n! = 1 while i <= n: 1

62 views • 4 slides
CS 147: Computer Systems Performance Analysis Fractional Factorial Designs 1 / 26 Overview

CS 147: Computer Systems Performance Analysis Fractional Factorial Designs 1 / 26 Overview

CS147 2015-06-15 CS 147: Computer Systems Performance Analysis Fractional Factorial Designs CS 147: Computer Systems Performance Analysis Fractional Factorial Designs 1 / 26 Overview CS147 Overview 2015-06-15 2 k p Designs Example

349 views • 30 slides
The General 2 k p Design To construct a 2 p fraction of the 2 k full factorial design, we

The General 2 k p Design To construct a 2 p fraction of the 2 k full factorial design, we

ST 516 Experimental Statistics for Engineers II The General 2 k p Design To construct a 2 p fraction of the 2 k full factorial design, we require p independent generators (such as D = AB ). Each generator contributes a word to the defining

384 views • 25 slides
Factorial Designs Merlise Clyde Design of Experiments: Chapter 6 Duke University March 2, 2016

Factorial Designs Merlise Clyde Design of Experiments: Chapter 6 Duke University March 2, 2016

Factorial Designs Merlise Clyde Design of Experiments: Chapter 6 Duke University March 2, 2016 STA 643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TIME: timing of

1.16k views • 101 slides
Computing Factorial Task: write a function that computes factorial n ( n 1)! if n >

Computing Factorial Task: write a function that computes factorial n ( n 1)! if n >

Computing Factorial Task: write a function that computes factorial n ( n 1)! if n > 1 Computer Programming: Skills & Concepts (CP) n ! = 1 if n = 1 Recursion (including mergesort ) Ajitha Rajan Tuesday 14 November 2017 CP

152 views • 4 slides
Part3: Other designs Prepared by: Paul Funkenbusch, Department of Mechanical Engineering,

Part3: Other designs Prepared by: Paul Funkenbusch, Department of Mechanical Engineering,

Part3: Other designs Prepared by: Paul Funkenbusch, Department of Mechanical Engineering, University of Rochester Extending what youve learned 3+ level factors Continuous Discrete Fractional factorials DOE mini-course,

665 views • 30 slides
Wisp - SRFI-119 define : factorial n if : zero? n . 1 * n : factorial {n - 1} I love the

Wisp - SRFI-119 define : factorial n if : zero? n . 1 * n : factorial {n - 1} I love the

Wisp - SRFI-119 define : factorial n if : zero? n . 1 * n : factorial {n - 1} I love the syntax of Python, but crave the simplicity and power of Lisp. Arne Babenhauserheide wisp January 27, 2016 1 / 10 Why Wisp? (Hello World!) Arne

184 views • 15 slides
Factorial private int factorial (int n) { assert n >= 0; if (n ==

Factorial private int factorial (int n) { assert n >= 0; if (n ==

Factorial private int factorial (int n) { assert n >= 0; if (n == 0) { return 1; } return n * factorial(n-1); } factorial(3) => 3 * factorial(2) = 6 2 factorial(2)

275 views • 8 slides
Investigation of Crouzeixs Conjecture via Nonsmooth Optimization Michael L. Overton Courant

Investigation of Crouzeixs Conjecture via Nonsmooth Optimization Michael L. Overton Courant

Investigation of Crouzeixs Conjecture via Nonsmooth Optimization Michael L. Overton Courant Institute of Mathematical Sciences New York University Joint work with Anne Greenbaum, University of Washington and Adrian Lewis, Cornell Workshop

1.65k views • 132 slides
How CLEVER is is your neural network? Robustness evaluation against adversarial examples Pin-Yu

How CLEVER is is your neural network? Robustness evaluation against adversarial examples Pin-Yu

How CLEVER is is your neural network? Robustness evaluation against adversarial examples Pin-Yu Chen IBM Research AI OReilly AI Conference @ London 2018 IBM Research AI Label it! IBM Research AI Label it! AI model says: ostrich IBM

852 views • 36 slides
Extreme precipitation event analysis based on GCM daily data Wei Ye International Global Change

Extreme precipitation event analysis based on GCM daily data Wei Ye International Global Change

Extreme precipitation event analysis based on GCM daily data Wei Ye International Global Change Centre Introduction Extreme weather events such as heavy rainfall can be serious treat agriculture production, infrastructure and peoples life;

624 views • 19 slides
Content Comparison of two nearness-to- collision surrogate indicators at a Problem statement

Content Comparison of two nearness-to- collision surrogate indicators at a Problem statement

Content Comparison of two nearness-to- collision surrogate indicators at a Problem statement signalized intersection in Minsk Literature review Surrogate measures of safety using Extreme Value Theory Application of EVT

457 views • 6 slides
Chris Parkin, MS Carmel Indiana Carmel, Indiana Argument The Art The Art of the Who has done

Chris Parkin, MS Carmel Indiana Carmel, Indiana Argument The Art The Art of the Who has done

Chris Parkin, MS Carmel Indiana Carmel, Indiana Argument The Art The Art of the Who has done a Who has done a presentation before? Who has seen a presentation before? Who knows someone who has seen a presentation before? before?

1.19k views • 76 slides
2019-2020 State Work Release Expansion WELCOME AND INTRODUCTIONS MIKE SCHINDLER Establishing

2019-2020 State Work Release Expansion WELCOME AND INTRODUCTIONS MIKE SCHINDLER Establishing

2019-2020 State Work Release Expansion WELCOME AND INTRODUCTIONS MIKE SCHINDLER Establishing Effective Meeting Protocols Welcome Who am I The role of the facilitator Why we asked you to attend / Local Advisory Committee (LAC)

671 views • 19 slides
Iteration and For Loops [Andersen, Gries, Lee, Marschner, Van Loan, White] Announcements: Prelim

Iteration and For Loops [Andersen, Gries, Lee, Marschner, Van Loan, White] Announcements: Prelim

CS 1110: Introduction to Computing Using Python Lecture 11 Iteration and For Loops [Andersen, Gries, Lee, Marschner, Van Loan, White] Announcements: Prelim 1 Rooms: aa200 jjm200 Baker Laboratory 200 jjm201 sge200

686 views • 44 slides
A PRAYER FOR PARENTS OF TEENS LORD, I ASK FOR THE PATIENCE TO BE A GOOD EXAMPLE TO MY CHILD

A PRAYER FOR PARENTS OF TEENS LORD, I ASK FOR THE PATIENCE TO BE A GOOD EXAMPLE TO MY CHILD

A PRAYER FOR PARENTS OF TEENS LORD, I ASK FOR THE PATIENCE TO BE A GOOD EXAMPLE TO MY CHILD DURING A TIME WHEN THEY WILL TEST EVERY PART OF ME. LORD, HELP ME NOT LOSE MY TEMPER, GIVE ME THE STRENGTH TO BOTH STAND FIRM WHEN I NEED TO AND LET GO

390 views • 18 slides

More recommend