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Conditional Probabilities: p \left( x | \omega_i \right) = \frac{1}{\pi b} \frac{1}{1+\left( \frac{x – a_i }{b} \right)^2 } for i = 1,2 . % Starting code here. a) Suppose the maximum error for classifying a patter that is actually in \omega_1 as if it were in \omega_2 is E_1 . Determine the decision boundary in terms of the variables given. E_1 = \int_{x^*}^{\infty} \frac{1}{\pi b} \frac{1}{1 + \left( \frac{x-a_i}{b} \right)^2 } \frac{1}{2} \mathrm{d} x = \frac{ \frac{\pi}{2} – \arctan\left( \frac{x^* – a_i}{b} \right) }{2 \pi} b) For this boundary, what is the error rate for classifying \omega_2 as \omega_1 ? The integral for E_2 is E_2 = \int_{-\infty}^{x^{*}} \frac{1}{\pi b} \frac{1}{1 + \left( \frac{x-a_i}{b} \right)^2 } \frac{1}{2} \mathrm{d} x = \frac{ \arctan\left( \frac{x^* – a_2 }{b} \right) + \frac{\pi}{2} }{2 \pi} If you use the vaue for x^* to get the E_1 value above, you get E_2 = \frac{ \arctan \left( \frac{ b \tan \left( \frac{\pi}{2} – 2 \pi E_1 \right) + a_1 – a_2 }{b} \right) + \frac{\pi}{2} }{2 \pi} c) What is the overall error rate under zero-one loss \mbox{Error} = E_1 + E_2 = E_1 + \frac{ \arctan \left( \frac{ b \tan \left( \frac{\pi}{2} – 2 \pi E_1 \right) + a_1 – a_2 }{b} \right) + \frac{\pi}{2} }{2 \pi} . d) Apply your results to the specific case b = 1 and a_1=-1 , a_2=1 and E_1=0.1 . e) Compare our calculation with the Bayes error rate E_{bayes} = \int_{-\infty}^{\infty} \min \left\{ p \left( x \left| \omega_1 \right. \right) p \left( \omega_1 \right) , p \left( x \left| \omega_2 \right. \right) p \left( \omega_2 \right) \right\} \mathrm{d} x pbayes = @(x) min( pxw1(x)*0.5 , pxw2(x)*0.5 );
error_bayes = integral( pbayes , -1500 , 1500 )
manual document ready 0.4 variable error_bayes 0.2498 1 1 5 true false pbayes = @(x) min( pxw1(x)*0.5 , pxw2(x)*0.5 ); 0 17 17 false true error_bayes = integral( pbayes , -1500 , 1500 ) 1 18 18 0
2021-03-01T06:57:37Z 2021-03-01T07:02:50Z
application/vnd.mathworks.matlab.code MATLAB Code R2019b
9.7.0.1183724 c5c2908e-cebb-4e62-b8de-a6860199f9f6
9.7.0.1471314
R2019b
Update 7
Sep 02 2020
3303242823