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Today Least Square Method Linear LSM theory Least Squares method Hypothesis: m experimental data points (couples xj-yj) n-th grade model given from the summation of n functions φ of x n 1 y x i i x generic function of x i 0 Objective: Find the set of α parameters minimizing the deviation between the experimental values and the ones foreseen by the model, in order to have a model to estimate y, given x Least Squares method Terms to minimize: the square of the deviation between experimental data and the ones foreseen by the model y x j y j Deviation of the j-th couple 2 2 2 n 1 y j ii x j j m 2 2 m 2 j j 1 i 0 j 1 Least Squares method the term to minimize is a positive function of α, therefore, to find a set of α for the which this function has a minimum, we can look for a unique set of α for the which all the partial derivatives are null (point of minimum) 2 m n 1 0k 0, n 1 2 y j k x j 2 i i x j k x j m k j 1 j 1 i 0 m n 1 i x j k x j y j k x j m 1 i j 1 i 0 j 1 Least Squares method to simplify the i x1 formulation we define φi as a column vector in i ... the which each j-th i xm element is the result of the φi function applied to the experimental point xj. 0 x1 ... n 1 x1 grouping togheter the n B ... ... ... φi columns we have 0 xm ... n 1 xm matrix B of m row and n columns Least Squares method y1 0 if we define the column d ... ... vector d as the set of all ym n 1 the yj. experimental points and the column 0 x1 ... 0 xm vector α as the set of all B T ... ... ... the parameters, n 1 x1 ... n 1 xm equation (1) becomes: m n 1 i i x j k x j y j k x j k 0, n 1 B B B d m T T j 1 i 0 j 1 Please Note: OUR TASK IS TO FIND α GIVEN B and d Least Squares method if matrix B were orthogonal the product BTB would give a diagonal matrix, filled up with the norms of its basis, therefore we could write 0 0 0 0 0 0 d 0 ... 0 ... ... 0 0 n 1 n 1 n 1 n 1 d i d i i 0, n 1 but B i i is not orthogonal we can however operate a basis transformation as to make B orthogonal, allowing us to compute α Least Squares method to orthogonalize we use an iterative procedure: -the first new base is the same as the old -each further base is the same as the old, MINUS the dot product between the old base and the newfound basis (this is done to get rid of the interdependencies) 0 x 0 x n 1 B B B d y ii x T T i 1 i 0 i x i x i p p T p 0 B B B d n 1 y ii x T p p i 0 Gram-Schmidt orthogonalization i d i i 0, n 1 i i in the new system we have Least Squares method following the Gram-Schmidt orthogonalization we used an iterative procedure: 0 x 0 x introducing a further term i 1 i x i x i p i p p 0 p p p i, p p p βi,p we can rewrite the direct Gram-Schmidt This is not α !! orthogonalization as: i 0 x 0 x i d i i 0, n 1 i 1 i x i x i , p p i i p 0 Least Squares method This is not αi!! i d i i 0, n 1 i i using an inverse iterative procedure we can trace back the original α which we sought for: n 1 n 1 n 1 i i p p ,i p i 1 Least Squares method First grade linear systems 0 x 1 Model y ax b 1 x x 0 x 1 m Orthogonalization 1 0 x 1 1 x x x m j 1 xx 0 0 1 1 j 1 x m m 1 y j j x yj Orth. Model parameters 0 j 1 1 j 1 x m m 1 1 2 j x j 1 j 1 Orthogonal model y 0 1 1 x x Least Squares method First grade linear systems 0 x 1 Model y ax b 1 x x Orthogonal model y 0 1 1 x x 1 1 Model parameters 0 0 11,0 Exercise 6: Linear LS A rubber to metal device has been tested in order to assess its longitudinal stiffness obtaining the following results: Length Force applied mm N 50.56 0 50.87 50 51.06 75 50.90 100 51.14 150 50.86 200 51.37 300 51.69 400 52.75 500 52.40 600 53.52 700 53.98 800 Exercise 7: Linear LS (2nd grade) A bullet is shot at a unknown angle and then its distance (along the firing axis) is measured using an high-speed camera. Its initial velocity has to be determined, as well as its deceleration value. time position (x) 25.000 s m 0.000 1.404 20.000 0.001 1.100 Force Applied [N] 0.002 1.985 15.000 0.005 2.307 0.010 2.781 10.000 0.020 3.112 0.030 4.815 5.000 0.040 5.670 0.050 6.002 0.000 0.000 0.050 0.100 0.150 0.200 0.250 0.100 11.269 Length [mm] 0.150 16.623 0.200 21.146

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posted: | 5/2/2013 |

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