From 3dcf7337253c6d72bf434d78ce8dbdf471aac496 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Gi=C3=B9=20Marcer?= Date: Sun, 24 May 2020 12:01:36 +0200 Subject: [PATCH] ex-7: revised and typo-fixed In addition, the folder ex-7/iters was created in order to plot the results of the Perceptron method as a function of the iterations parameter. --- ex-7/iters/iter.txt | 11 +++++ notes/images/7-iterations.pdf | Bin 0 -> 13088 bytes notes/images/7-percep-plane.pdf | Bin 38930 -> 37218 bytes notes/images/7-percep-proj.pdf | Bin 9723 -> 11599 bytes notes/sections/7.md | 70 +++++++++++++++++++------------- 5 files changed, 52 insertions(+), 29 deletions(-) create mode 100644 ex-7/iters/iter.txt create mode 100644 notes/images/7-iterations.pdf diff --git a/ex-7/iters/iter.txt b/ex-7/iters/iter.txt new file mode 100644 index 0000000..b6090c5 --- /dev/null +++ b/ex-7/iters/iter.txt @@ -0,0 +1,11 @@ +#iters w_x w_y b +1 0.4265959958 0.9044422902 1.7487522878 +2 0.5659895562 0.8244124103 1.4450529880 +3 0.6540716938 0.7564325610 1.2125805084 +4 0.6540716938 0.7564325610 1.2125805084 +5 0.6540716938 0.7564325610 1.2125805084 +6 0.6540716938 0.7564325610 1.2125805084 +7 0.6540716938 0.7564325610 1.2125805084 +8 0.6540716938 0.7564325610 1.2125805084 +9 0.6540716938 0.7564325610 1.2125805084 +10 0.6540716938 0.7564325610 1.2125805084 diff --git a/notes/images/7-iterations.pdf b/notes/images/7-iterations.pdf new file mode 100644 index 0000000000000000000000000000000000000000..91321a17f51fcefd2d3200334d6aacc3a80afda6 GIT binary patch literal 13088 zcmeHuc{r8b*T150D49d*Mo5M;Pa>Iy7sm2d!Ie5wf4IAT5Eq+vSvDZ%4iiFLiX?H z(6u`VG=zfa_HGC@H3(@&ai)4f7|>z?A*~L2P$8r?g-LOzJ42f{Bd9b-xFc!K09{|E zz9o~wq(ZpqL3(r=6T(hEn^BldDuV{$;a3FGj6rv>q%t8Ju#S!%n4jv)gpdaN!J1k# zwf0PH0PTj5mM#=WSDG_Km>zCzYVSsMU_wOJ-ZFS<9w0f-Qm(N7V=Qk^3|n4DR9^<3S7h#;h%JCLCc z)q(B^L~cZ-IWt`#G`uqw@IWsngG$+t@J+nFvxa6%Tva@x=*wK0`s~T1OKEQ4nx)lZ zP&jm@(fTHe0omM)zHd6#xwUTNGBl)?+cFSY5^L~}V$*?+KL77({Tidm6J2giZ4DGZ6VkQU)#YNcN6g^m0|odQ}I*YB2_l zD3F&Q_u+*1r><9$K`LRc{8NSFu1eRBagIxwufF=s?BtryndcIl3TEXtgG#-PYohb1dC`0ok*AtP(=&!v6>2xQ$nj0-)b>$&t#1EN z+Z)ibA1kmqH>a+~chWoUNZVdq|6W`|;OB=tvMcXdZ{+8+X^>L*s>2nVzBYz0UBkA& z_P~X8$!!<-+IIR)8Y`?b8Xib(%M>w8jLNzxEF-p&;HW1qa6y1nm1>DrU@nY`wyU+Wt$8}ulBYLOpjelsJ=N+oBe$DqNtW{v_ zwjgsgv9*0xh%*dZZN%<)?Yy<^4{AeHj)$(;xM5ivHCOin&d1}41818Ks2n-ltUyKO zF|CPpY%e*}^)78MwCURtzprt_Irl9fFF$VOrOk0=ssk_ki-Iz3wjA)X(W)Uup*A>F zzTSrLJktB{$-BFzreCX^CP&-aZp$G5QEhi0xwqR&TJrVHiUz87_r4pJIA!M=I$GiF z>Shk}JO>Xk$$?MV(jN8wh3D&6J7-k?{cOhuyU!je=@KPuWe~A*PvBU0*HX=RnF;=K zMu~#p{WcAme8JRxA$l6O#%yJ>u3oXbJMed?itQ67#YN}fwG^>^Zl@pPU)1AUMWx8Z zkW+Hy0cVV5Z1M-}HhB*hN5$yNdVSTBBOLWSRy$EN^z`eJF`4EL-r#RLcHevF`S5g_ zO-j6a{DyslL2oXKWT8Z!U%1an7HWx9)i^Tx{tGs|ui0qjHm?BVV{83urq;chv@E-D z?yXdq+m$5B3AsvJeZ9b^Ma*wZ+g<)5Q+$bn+8sp#yPDdys{UFjQ`P=Xw9!QGqz)=5 z?7+ztn zY1fjEu*>HE+@^R=-r4cod!-TEjRJ>XH6GM$I+6($`Vxc=nX$G%OC4YU%YFvQw0&J$)Uwhtsqp_YDHCiGY7kv($S zhELQ89JipRtq(o2Born|2gNHncFEWm+nDAFQvVrv5;L~m#Zsm0AN|zthl6t0`uaZN z54hUw{8H(KFS$0A+Gs6y>yBi>wqt9t0=oy+Ht%!ddM4S?CR!uQIiR>Nv~N4DWH3yV z=gN9v1>5Z43pZ$XoZH^Z_U<{*nWP+{zDAO#t^eiadhrQiTlxX!$0trHZiA#F&m`VO zk@DHZkB5GeY#3(T3%A#MF4<{wiG?`Ymg;&@AV)tI?fyafnY6uJ2ar(#x*-*{%4nHDqh>JhU;^WzF z)Vk(tLBH$azIPitA6b9BH*7?`%~&Rp^;rAko$OHyn#c0wiznhtOR}r|9l$g;lsxGPvndy zlIF=-J<3pkU5ZO*JSp`wq|rFCKCbo6k4{|OxzNvo8{Bg;dRN!gwnpt9I=f?a(gfin zPtx&gS!3brg)!eQuM|7!JS-%?UMO46VaxN})~9Knn1{S~mE@f4Zk)*0QOPu0+GjrO z=9Ko*?Q+n)!{s{?#zaK9beGC0IV=81W^e8nk>nLQH%P3_SNn`;W-2}Q4?D=GD-*E% z?Jk-f^?Gn}nB@8Kn=Tus-pKol_!idlbzRd+S%meuZqV<{I0k6wTpNxzyM3k{#ap|5 znfNoYqo>zDeSIPF^H)>8(Sz&~t7L=N;yL9QuXXj4v??q4sya}#?<-Y5b~v9qEH>8V z-u0M*eX!ozGLtTPqm*>XiuQN}x=-v;Rra__-&EJf!|m=yvs+ z2TSxG$GS-y`7ze+aqByk>6aKFcvx?Z{-)1gt04cY2>CINByPnb`$oU3kFtF{KeRsk zTjDKa$VrltT5#NI`wbxzfAZa_Q5C|Ol;zYX7A<=MIW~;7UM1*7rVb{DHm7p(Zh_a=*dpIDq4e~st}hw;sU3^WfRd|UkvEM{i#=^ zEet!^R(k8{?ampuI|fta~I) z$6k)pw@iG|A+l02Sm^FE5s^I<{`6tDb3+F$TOUMQ#J+v<^~eGBs%sK=7NK#jWBxlD z(b)M?R6m`&q5{RGr*H(Gr zcB0>F(vP-sbSS3?yX@Y!VdXB>elDF(k+Jujhi}CAzOl$ncS9E5Q?S3iY4lX%{#_5! 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With $N = 5$ iterations, the values of $w$ and $t_{\text{cut}}$ level off up to the third digit. The following results were obtained: +Different values of the learning rate were tested, all giving the same result, +converging for a number $N = 3$ of iterations. In @fig:iterations, results are +shown for $r = 0.8$: as can be seen, for $N = 3$, the values of $w$ and +$t^{\text{cut}}$ level off. +The following results were obtained: $$ w = (0.654, 0.756) \et t_{\text{cut}} = 1.213 $$ -where, once again, $t_{\text{cut}}$ is computed from the origin of the axes. In -this case, the projection line does not lies along the mains of the two -samples. Plots in @fig:percep_proj. +In this case, the projection line is not parallel with the line joining the +means of the two samples. Plots in @fig:percep_proj. + +

+ +## Efficiency test {#sec:7_results}
![View from above of the samples.](images/7-percep-plane.pdf){height=5.7cm} @@ -396,40 +411,37 @@ red. ## Efficiency test {#sec:7_results} -A program was implemented to check the validity of the two -classification methods. -A number $N_t$ of test samples, with the same parameters of the training set, -is generated using an RNG and their points are divided into noise/signal by -both methods. At each iteration, false positives and negatives are recorded -using a running statistics method implemented in the `gsl_rstat` library, to -avoid storing large datasets in memory. -In each sample, the numbers $N_{fn}$ and $N_{fp}$ of false positive and false -negative are obtained in this way: for every noise point $x_n$ compute the -activation function $f(x_n)$ with the weight vector $w$ and the -$t_{\text{cut}}$, then: +Using the same parameters of the training set, a number $N_t$ of test +samples was generated and the points were divided into noise and signal +applying both methods. To avoid storing large datasets in memory, at each +iteration, false positives and negatives were recorded using a running +statistics method implemented in the `gsl_rstat` library. For each sample, the +numbers $N_{fn}$ and $N_{fp}$ of false positive and false negative were obtained +this way: for every noise point $x_n$, the threshold function $f(x_n)$ was +computed, then: - - if $f(x) < 0 \thus$ $N_{fn} \to N_{fn}$ - - if $f(x) > 0 \thus$ $N_{fn} \to N_{fn} + 1$ + - if $f(x) = 0 \thus$ $N_{fn} \to N_{fn}$ + - if $f(x) \neq 0 \thus$ $N_{fn} \to N_{fn} + 1$ and similarly for the positive points. -Finally, the mean and standard deviation are computed from $N_{fn}$ and -$N_{fp}$ of every sample and used to estimate purity $\alpha$ -and efficiency $\beta$ of the classification: +Finally, the mean and standard deviation were computed from $N_{fn}$ and +$N_{fp}$ for every sample and used to estimate the purity $\alpha$ and +efficiency $\beta$ of the classification: $$ \alpha = 1 - \frac{\text{mean}(N_{fn})}{N_s} \et \beta = 1 - \frac{\text{mean}(N_{fp})}{N_n} $$ -Results for $N_t = 500$ are shown in @tbl:res_comp. As can be seen, the -Fisher discriminant gives a nearly perfect classification -with a symmetric distribution of false negative and false positive, -whereas the perceptron show a little more false-positive than -false-negative, being also more variable from dataset to dataset. -A possible explanation of this fact is that, for linearly separable and -normally distributed points, the Fisher linear discriminant is an exact -analytical solution, whereas the perceptron is only expected to converge to the -solution and thus more subjected to random fluctuations. +Results for $N_t = 500$ are shown in @tbl:res_comp. As can be seen, the Fisher +discriminant gives a nearly perfect classification with a symmetric distribution +of false negative and false positive, whereas the perceptron shows a little more +false-positive than false-negative, being also more variable from dataset to +dataset. +A possible explanation of this fact is that, for linearly separable and normally +distributed points, the Fisher linear discriminant is an exact analytical +solution, whereas the perceptron is only expected to converge to the solution +and is therefore more subject to random fluctuations. ------------------------------------------------------------------------------------------- @@ -442,4 +454,4 @@ Perceptron 0.9999 0.28 0.9995 0.64 Table: Results for Fisher and perceptron method. $\sigma_{\alpha}$ and $\sigma_{\beta}$ stand for the standard deviation of the false - negative and false positive respectively. {#tbl:res_comp} + negatives and false positives respectively. {#tbl:res_comp}