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damsslides-2011

damsslides-2011 - DAMS Epiphany term Lecturer Abigail...

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DAMS Epiphany term Lecturer: Abigail Wacher [email protected] 1

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Purpose of course We shall introduce some of the simpler tech- niques that are used to construct and solve mathematical models of physical problems. These problems can come from many areas, including Engineering, Biology, Finance etc... but there exists a collection of basic techniques that are appropriate to all of these areas. There will necessarily be some abstraction - this is the power of mathematics - as the model that describes heat diffusion in a metal bar can also be shown to predict values of some finan- cial products. However we shall always try to bring the discussion back to ‘physical’ prob- lems. 2
Least Squares Approximation Many models involve parameters that must be estimated from experiment. That is, we know how the system behaves but not the values of the various ingredients of the model. For example, we know from Newton that an object falling under the force of gravity (ignor- ing air resistance) undergoes a constant accel- eration, but what value is this acceleration? To put this in another way, if we drop an object in a vacuum with initial displacement d 0 and initial velocity u , then the displacement at time t is given by d ( t ) = d 0 + ut + 1 2 gt 2 but what is g ? We could set up an experiment and make some measurements of d ( t ) and try to deduce g but it’s unavoidable that there will be errors in our experiment, so every measure- ment would give a slightly different g . 3

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Let’s look at this problem in a more general way: Suppose we have a set of n pieces of data ( x i , y i ), i = 1 , n which we believe should lie on a straight line (but probably doesn’t due to measurement errors) how do we evaluate this line? We take the form y = mx + c for the line and calculate the error at each point: e i = y i - ( mx i + c ) . We want to make these errors as small as pos- sible (if the data really were on a straight line we could make them zero). In order to do this we square the errors, add them up: S = n i =1 ( y i - mx i - c ) 2 , then try to make this sum as small as possible 4
Like any (smooth) function we can find the minimum (at least in principle) by setting the derivatives equal to zero. The unknown values in S on the previous slide were m and c , so we set 0 = ∂S ∂c = n i =1 2 ( y i - mx i - c ) ( - 1) 0 = ∂S ∂m = n i =1 2 ( y i - mx i - c ) ( - x i ) , or [ n i =1 1 n i =1 x i n i =1 x i n i =1 x 2 i ] [ c m ] = [ n i =1 y i n i =1 y i x i ] 5

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Example Fit a straight line to the data i 1 2 3 4 5 6 x i 0 1 2 3 4 5 y i 0 . 0 9 . 9 29 . 6 59 . 1 98 . 4 147 . 5 Now 6 i =1 1 = 6 (be careful!), 6 i =1 x i = 15, etc. and [ 6 15 15 55 ] [ c m ] = [ 344 . 5 1377 . 5 ] , and we solve to get m = 29 . 5, and c = - 16 . 33. If we plot this curve, together with the data it doesn’t look like we’ve made a very good job - our initial assumption that the data could be fitted by a straight line was wrong.
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damsslides-2011 - DAMS Epiphany term Lecturer Abigail...

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