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4.1.1 Introductory Example A classic application of the method of least squares is illustrated by the following example: Example 4.1. Consider the three points (1;1), (3;2) and (4;5). As we can see these do not lie on a straight line: But suppose we want want to nd a line that’s really close to the points, what-ever that might mean.
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- Definition
- Fact I
- Fact 2
- Intermediate Value Theorem
Note that this definition is also implicitly assuming that both f(a)f(a) and limx→af(x)limx→af(x) exist. If either of these do not exist the function will not be continuous at x=ax=a. This definition can be turned around into the following fact.
This is exactly the same fact that we first put down backwhen we started looking at limits with the exception that we have replaced the phrase “nice enough” with continuous. It’s nice to finally know what we mean by “nice enough”, however, the definition doesn’t really tell us just what it means for a function to be continuous. Let’s take a look at...
To see a proof of this fact see the Proof of Various Limit Propertiessection in the Extras chapter. With this fact we can now do limits like the following example. Another very nice consequence of continuity is the Intermediate Value Theorem.
All the Intermediate Value Theorem is really saying is that a continuous function will take on all values between f(a)f(a) and f(b)f(b). Below is a graph of a continuous function that illustrates the Intermediate Value Theorem. As we can see from this image if we pick any value, MM, that is between the value of f(a)f(a) and the value of f(b)f(b) an...
The equation of the line of best fit is y = ax + b. The slope is a = .458 and the y-intercept is b = 1.52. Substituting a = 0.458 and b = 1.52 into the equation y = ax + b gives us the equation of the line of best fit. \[y=0.458 x+1.52 \nonumber \] We can superimpose the plot of the line of best fit on our data set in two easy steps.
PART I: Least Square Regression 1 Simple Linear Regression Fitting a straight line to a set of paired observations (x1;y1);(x2;y2);:::;(xn;yn).Mathematical expression for the straight line (model)
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Polynomials become more ‘squiggly’ as their order increases. A ‘squiggly’ appearance comes from inflections in function. Consideration #1: 3rd order - 1 inflection point 4th order - 2 inflection points nth order - n-2 inflection points. overfit. Consideration #2: 2 data points - linear touches each point.
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Chapter 4. itting & Correlation4.1 IntroductionThe process of constructing an approximate curve , which fit best to a given discrete. et of points is called curve fitting. Curve fitting and interpolat. on are closely associated procedures. In interpolation, the fitted function should pass through all given data points; whereas curve fitting ...
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A pattern of shapes that fit perfectly together! A Tessellation (or Tiling ) is when we cover a surface with a pattern of flat shapes so that there are no overlaps or gaps. Examples:
