Math Modeling
Why can't we just add more points to improve our estimate using interpolation?
If the number of points is larger than the degree of polynomial that you want to use, then the linear system for determining the coefficients will be over-determined
Discuss and give examples from several parts of the course why numerical methods often involve subjective judgments.
Initial guesses When is the error low enough? (How many iterations?)
Provide two general approaches to curve fitting on the basis of the amount of error with the data. When should we interpolate and when might a least squares regression be a better approach?
Interpolation vs. Least Squares Regression Interpolation -not suitable for substantial error -curves can oscillate widely in the intervals between points -determines an nth order polynomial that fits n+1 data points -estimate intermediate values between precise data points -not great for high degrees Least Squares regression -find line to fit shape or general trend of data without necessarily matching individual points -If the number of points is larger than the degree of polynomial that you want to use, then the linear system for determining the coefficients will be over-determined so regression would be better doesn't go through the points
Distinguish between using an existing model or constructing a model.
Model selection vs model construction constructing a model is complicated but you could make it however you wanted modeling tornado behavior from scratch selecting a model would be easier but it might not exactly answer the question taking a model of wind speed and using it to model a tornado
Discuss the (engineering) problem solving process p.12 and Figure 1.1 and mathematical modeling process (Meyer)
Problem Definition - Mathematical Model - Numeric/Graphic results - implementation
Give and describe the modeling process according to the handout and Meyer.
Step 1: Identify the problem Step 2: Make assumptions Step 3: Solve or interpret the model Step 4: Verify the model Step 5: Implement the model Step 6: Maintain the model
Compare and contrast the statement "Correlation is not causation." Give four or five possible explanations as to how the variables might be related if there is strong correlation. Give an example for each.
a correlation between two variables does not mean that one caused the other, there can be other explanations there can be a positive or negative correlation explanations for correlation: simple causality - x determines y more cars on the highway causes slower speeds (traffic) reverse causality - y determines x there were more storks in years with more births more births means more houses which leaves more area for nests mutual causality - x and y cause each other advertising raises sales but sales raise ability to advertise hidden variable causes both - third lurking variable determines x and y there are more liquor stores where there are more churches, both are caused by third variable: population complete accident - there is no cause or reason why they are correlated heights and social security numbers of random people could have relationship by luck
According to Meyer in chapter 3, provide and define several characters and principles used in determining the strengths and weaknesses in a model.
accuracy - output of model is correct or near correct precision - prediction is a definite number (not a range) descriptive realism - based on correct assumptions robustness - immune to errors in input data (not sensitive to error of input) generality - applies to a wide variety of situations fruitfulness - the conclusion is useful or it inspires other good models
Give some common stumbling blocks of nearly all models including the "Catch-22" of modeling.
catch 22 of modeling - we want to know how far we are from the truth (our error) but we don't know the truth because that's what we are trying to predict all models have to be based upon assumptions
Descriptive v prescriptive models (Meyer p 61)
descriptive - describes or predicts how something works or how it will work modeling how a particular engine works prescriptive - meant to help us choose the best way for something to work modeling an airplane engine
Discuss the benefits of an incorrect but fruitful model.
even if the model is wrong, it could spur another model that is helpful whether for the same problem or for a new one
Be able to give a reason as to why all cubic splines may not be the same on all programs.
n+1 conditions going through the points n-1 conditions to match the first derivative n-1 conditions to match the second derivative so matlab may do it a different way than mathematica
Compare and contrast well-conditioned and robustness.
robustness - characterization of model, not data model is not sensitive to change or error in the input well conditioned - problem with a low condition number a change in the initial conditions will not greatly affect your output
Discuss the role of belief in adjusting data.
sometimes you don't have repeated measurements of a quantity so you can't take the mean and you can't calculate error methods: grouping - do categories instead of exact values to smooth out data moving averages least squares interpolation - inserting values between more values you either believe the data or you don't, this decides whether you should adjust or not
Be able to draw a linear spline and a cubic spline.
spline - applying lower polynomials to subset of data points
Discuss what r squared means. Discuss the ramifications of adding another variable to increase r squared .
statistical measure of how close the data are to the fitted regression line. 100% indicates that the model explains all the variability of the response data around its mean. If a model has too many predictors and higher order polynomials, it begins to model the random noise in the data. This condition is known as overfitting the model and it produces misleadingly high R-squared values and a lessened ability to make predictions. the more variables you have, the more variance can be explained by the model
What is the fundamental difference between regression and interpolation? What serious problems could follow if the two are confused?
the main difference between interpolation and regression, is the definition of the problem they solve: Given n data points, when you interpolate, you look for a function that is of some predefined form that has the values in that points exactly as specified. When you do regression, you look for a function that minimizes sum of squares of errors. You don't require the function to have the exact values at given points, you just want a good approximation.
Analyze and provide examples discussing the question "What is real anyhow?" Compare and contrast the idea of a model which "works" with one which is descriptively realistic.
what is reality anyway? anything could be descriptively realistic if we tweak it to be so is this just our interpretation of reality?
Be able to describe some sources of error
Blunders (human imperfection) unlikely with computers Formulation errors - bias in forming model Data uncertainty round off error/error propagation truncation error
Why might the construction of a mathematical model be problematic?
Constructing your own model might result in complex math and unsolvable systems
What are the tradeoffs involved in numerical methods and modeling?
Cost Convenience Speed lots of computer calculations / iterations Repeatability Precision gives you a range instead of a number Should you put in the work for something that may not converge?
According to the Giordano handout, what are some considerations in a mathematical model?
Fidelity - How precisely does the model represent reality? representing the volume of a box Cost - total cost of modeling process high cost model requiring lots of calculations and time Flexibility - the ability to change and control conditions affecting the model modeling climate change (put in initial conditions)
What is a mathematical model?
A mathematical model can be broadly defined as a formulation or equation that expresses the essential features of a physical system or process in mathematical terms
Characteristics of mathematical models of the physical world.
Describes natural process or system in mathematical terms Represents an idealization and simplification of reality Yields reproducible results and can be used for predictive purposes
