From Insurance to Aerospace: A Mathematical Journey. Tyler Hayes June 29, 2016

Transcription

1 From Insurance to Aerospace: A Mathematical Journey Tyler Hayes June 29, 2016 tlh6792@rit.edu

2 Overview About me My experience with mathematics Internship at Liberty Mutual Insurance Internship at UTC Aerospace Systems Application Problem Summary

3 About Me Origin: Buffalo, NY Hobbies: Ukulele, Skiing, & Cats Education: BS: Applied Math (RIT) MS: Applied & Computational Math (RIT) PhD: Imaging Science (RIT) Starting Fall 2016 Internship Experience: Liberty Mutual Insurance UTC Aerospace Systems

4 Early Experience with Mathematics Everyday Mathematics Program Honors math classes from 5 th grade on AP Calculus AB in high school Technology classes throughout high school (PLTW): Introduction to Engineering Design Principles of Engineering Electronics Image:

5 College Experience with Mathematics Started at RIT as Mechanical Engineering Major Fell in love with beauty of mathematics Patterns Understanding how and why things worked Versatility in application Switched major summer after freshman year to math and never looked back Image: mathematics-in-nature-- uncategorized-presentationgnngdd8vz0

6 Things I Wish I had Known About Math in High School There exist multiple career paths for math majors: Teacher/Professor Analyst (IT, Operations, Research, etc.) Actuary Software Developer Multiple types of companies hire math majors: Insurance (Liberty Mutual, Excellis) Software Development (Google, Microsoft) Defense (United Technologies, Harris, MIT Lincoln Labs)

7 Internship #1: IT Intern at Liberty Mutual Insurance Summer after sophomore year at RIT Interview as a result of RIT Career Fair Led to full-time employment opportunity at Liberty Mutual

8 Liberty Mutual Insurance (LMI) Global insurer for personal and business insurance needs Provide wide range of insurance products: Personal automobile Homeowners Worker s compensation Commercial automobile Global specialty Image:

9 Global Specialty at LMI Liberty International Underwriters insure specialized risks in multiple countries throughout the world Examples: Marine Construction Aviation Crisis Management Surety Image:

10 Role as an Information Technology (IT) Intern Questions posed at start of internship: 1. What is the role of Predictive Analytics in business and IT at LMI? 2. What are the best predictive models for IT purposes in Global Specialty at LMI? 3. What is the best software to deploy such models? Image:

11 Predictive Analytics Definition: Predictive Analytics utilizes current and previous company data to make predictions about future data trends. Choose predictors Choose subjects Collect Data Form Statistical Model Based on techniques from regression and machine learning High School Math Concepts: Choosing sample sets Updating models As more data becomes available Validate and Revise Model Make Prediction Act on predictions

12 Different Statistical Models Generalized Linear Model (GLM) Similar to Linear Least Squares: f(x) = mx + b High School Math Concepts: Curve fitting Bayes Theorem Line formula Bayesian Non-Linear Model Based on Bayes Theorem: P(A B) = P B A P(A) P(B) Multivariate Adaptive Regression Splines (MARS) Seen as extension to linear models k MARS Model: f(x) = σ i=1 c i B i (x) Image:

13 Software Used for Statistical Models SAS, SSAS, R Educate students about the existence of math-based software Encourage students to use such software for special in-class assignments

14 Internship #2: Image Science Intern at UTC Aerospace Systems Summer after first year of Masters degree at RIT Interview as a result of on-campus recruiting event Led to research in imaging science for Master s Thesis and eventual enrollment in Imaging Science PhD program Image:

15 UTC Aerospace Systems World supplier of aerospace and defense products Design, manufacture, and service systems and components and provide integrated solutions for commercial, regional, business, and military aircraft and (previously) helicopters Major supplier to international space programs Image:

16 Role as an Image Science Intern Determine how well airborne sensors capture image resolution and sharpness (using quantitative estimates) Determine these quantities by fitting different non-linear functions to data Test confidence in fits using resampling techniques Image:

17 Modulation Transfer Function How well a sensor transfers contrast of an object to an image Image:

18 Relative Edge Response Calculated from the Edge Spread Function (ESF) which determines how well sensor transfers edges from original object to image High School Math Concepts: Function evaluations RER = ESF 0.5 ESF 0.5 Amplitude Image:

19 Method to Determine MTF Given initial chip, estimate edge location Fit Edge Spread Function (ESF) to edge data Take the derivative of the ESF to obtain the Line Spread Function (LSF) Take the Fourier Transform of the LSF to obtain the MTF High School Math Concepts: Curve fitting Derivatives Images:

20 Fitting Edge Spread Function (ESF) to Data Sigmoid Fit Function:f r = c + A 1+e br High School Math Concepts: Different representations of data Image: Heidingsfelder et. al.

21 Fitting Edge Spread Function (ESF) to Data Sigmoid Fit Function:f r, A, b, c = c + A changes the amplitude of the sigmoid A 1+e br High School Math Concepts: Mathematical transformations b changes the slope of the sigmoid c translates the sigmoid up and down

22 Calculating Uncertainty via Bootstrap Resampling Resample chip data and calculate new RER and MTF for resampled data Estimate mean and standard deviation of estimates to place error bars on calculations High School Math Concepts: Calculating mean and standard deviation Calculating error Image:

23 Application Problem A company designs cameras that attach to airplanes to capture images. The company wants to determine how well these cameras capture edges and lines and have assigned you to the task. They have asked you to do the following: 1. Plot the edge data captured from the camera displayed in the table below. x y Answer:

24 Application Problem Cont. 2. The data you plot in the previous part should look like a familiar trigonometric function; the cosine function! If I told you the cosine function should be of the form y = c + Acos(bx), what would be your best estimate for the parameters A, b, and c? Why? (Hint: For b, notice the domain of x-values is approximately pi, pi. Does this give us any information about the period of 2 2 the cosine wave?) Answer: A = 1, b = 2, c = 3. I determined these parameters 2 because A is the amplitude of the curve, 2π is the period of the b curve, and c is the vertical shift of the curve. This process is called curve fitting.

25 Application Problem Cont. 3. Using a graphing device, determine the values of the cosine curve you found in the previous part (y = c + Acos bx ) to fill in the table below rounded to two decimal places. Answer: x y Notice that the values you found in the previous part have the same value independent of the sign on x. We call functions of this type even functions.

26 Application Problem Cont. 4. Plot the values you found in part 3 on the same plot you created in part 1. Be sure to make a legend to indicate the difference between the plot from part 1 and the plot from part 3. Answer:

27 Application Problem Cont. 5. Use the values you found in part 3 to determine the error of each data point from part 1. Recall, Error = Actual Value Predicted Value. Answer: x y Based on the error values you found in part 5, would you trust your model to estimate the data from the camera? Why or why not? Answer: I would trust my model to estimate the camera data since all of the error values were relatively small.

28 Application Problem: Other Ideas Ask calculus students to also: Determine the derivative of the function from part 2 Determine the values of the derivative at the x-values from part 1 Plot the derivative (Line Spread Function) based on the points from the previous part

29 Summary of High School Level Concepts Choosing sample sets for statistics Curve fitting Bayes Theorem Function evaluations Mathematical transformations (scaling, translation, reflection, etc.) Derivatives Mean, standard deviation, etc. Computing error Image: 7B7316CD21EE54EBCF9B99F357CC05D

30 Thank you! Image: