From 1b59ce5a0ca2751461ec29cf4523039fee4fb902 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Wed, 5 Sep 2018 16:46:51 +0200 Subject: [PATCH] added hint to exercise 5 --- doc/Projects/2018/hw2/html/hw2-bs.html | 7 +++++-- doc/Projects/2018/hw2/html/hw2.html | 7 +++++-- .../2018/hw2/ipynb/ipynb-hw2-src.tar.gz | Bin 207 -> 207 bytes doc/Projects/2018/hw2/pdf/hw2.p.tex | 6 ++++-- doc/Projects/2018/hw2/pdf/hw2.pdf | Bin 198097 -> 198515 bytes doc/Projects/2018/hw2/pdf/hw2.tex | 6 ++++-- doc/src/Projects/2018/Exercises/hw2.do.txt | 6 ++++-- 7 files changed, 22 insertions(+), 10 deletions(-) diff --git a/doc/Projects/2018/hw2/html/hw2-bs.html b/doc/Projects/2018/hw2/html/hw2-bs.html index 5ff410dfb..2cac701d8 100644 --- a/doc/Projects/2018/hw2/html/hw2-bs.html +++ b/doc/Projects/2018/hw2/html/hw2-bs.html @@ -171,12 +171,15 @@ Discuss these quantities as functions of the variable \( \lambda \) in the Ridge

Using the singular value decomposition, show that the variance of the direction vector -\( \hat{z}_i=\hat{X}\hat{v}_i \) is equal to (equation (3.49) of Hastie et al.) +\( \hat{z}_i=\hat{X}\hat{v}_i=\hat{u}_1d_1 \) is equal to (equation (3.49) of Hastie et al.) $$ \mathrm{Var}(\hat{z}_i)=\frac{d_i^2}{N}, $$ -where \( d_i \) are the singular values of the matrix \( \hat{X} \). Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. +where \( d_i \) are the singular values of the matrix \( \hat{X} \). In Hastie et al, the matrix elements of \( X \) are centered. The consequence is that the mean values of for example \( \hat{u}_i \) are zero. + +

+Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise.

diff --git a/doc/Projects/2018/hw2/html/hw2.html b/doc/Projects/2018/hw2/html/hw2.html index 66482871e..102ba1e39 100644 --- a/doc/Projects/2018/hw2/html/hw2.html +++ b/doc/Projects/2018/hw2/html/hw2.html @@ -136,12 +136,15 @@ Discuss these quantities as functions of the variable \( \lambda \) in the Ridge

Using the singular value decomposition, show that the variance of the direction vector -\( \hat{z}_i=\hat{X}\hat{v}_i \) is equal to (equation (3.49) of Hastie et al.) +\( \hat{z}_i=\hat{X}\hat{v}_i=\hat{u}_1d_1 \) is equal to (equation (3.49) of Hastie et al.) $$ \mathrm{Var}(\hat{z}_i)=\frac{d_i^2}{N}, $$ -where \( d_i \) are the singular values of the matrix \( \hat{X} \). Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. +where \( d_i \) are the singular values of the matrix \( \hat{X} \). In Hastie et al, the matrix elements of \( X \) are centered. The consequence is that the mean values of for example \( \hat{u}_i \) are zero. + +

+Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. diff --git a/doc/Projects/2018/hw2/ipynb/ipynb-hw2-src.tar.gz b/doc/Projects/2018/hw2/ipynb/ipynb-hw2-src.tar.gz index eddf88e9ac632bd4f11a05b825b072182e4eb6a8..dda671634a7ce1694cfc77fac4167cf1504b5782 100644 GIT binary patch literal 207 zcmV;=05Ja_iwFQo?2lUj1MSbjY63A3#&OR+#XP~1?5^v%P!Api5nmv((M?+uO(ME) z-&LVK)ti*k|L^iK!!XROW-qUcS&=V!^fXGTrmp3h(`Ir@Ro?3xtxM5WRaeud(OQcd ziisMDdfrLjPAQu>oYK8npRE6JZW!BcAMClmB47DOtUH%vzU*@sy!epqCWLqC&Sl{a zz28V1Y`XR(+T+PZE5`R?+SX+ky!jX(V%SBqPsRARKLG#$0000000000004mhHG005Z`V@Lo1 literal 207 zcmb2|=3wZ#(;v;i{Pw(W7L%huTjKS3i~J0}{Mp6#;!vx?Bb|KnAcGJ4YZ#+4>WmpQ G7#INHYheoj diff --git a/doc/Projects/2018/hw2/pdf/hw2.p.tex b/doc/Projects/2018/hw2/pdf/hw2.p.tex index 7ba297184..101bd04f8 100644 --- a/doc/Projects/2018/hw2/pdf/hw2.p.tex +++ b/doc/Projects/2018/hw2/pdf/hw2.p.tex @@ -211,11 +211,13 @@ Discuss these quantities as functions of the variable $\lambda$ in the Ridge and \subsection{Exercise 5} Using the singular value decomposition, show that the variance of the direction vector -$\hat{z}_i=\hat{X}\hat{v}_i$ is equal to (equation (3.49) of Hastie \emph{et al.}) +$\hat{z}_i=\hat{X}\hat{v}_i=\hat{u}_1d_1$ is equal to (equation (3.49) of Hastie \emph{et al.}) \[ \mathrm{Var}(\hat{z}_i)=\frac{d_i^2}{N}, \] -where $d_i$ are the singular values of the matrix $\hat{X}$. Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. +where $d_i$ are the singular values of the matrix $\hat{X}$. In Hastie \emph{et al}, the matrix elements of $X$ are centered. The consequence is that the mean values of for example $\hat{u}_i$ are zero. + +Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. % ------------------- end of main content --------------- diff --git a/doc/Projects/2018/hw2/pdf/hw2.pdf b/doc/Projects/2018/hw2/pdf/hw2.pdf index fd9334f70031d9bec14058f19f6f1f9682f17753..7fbb0fe019d42440a70eb6064227153ad0e46a69 100644 GIT binary patch delta 14814 zcmaibQ*_{M>}GA-wrx*2wQbwRuQsQ)?RKWNZF6d4YIncgv;V!_yCf$kZ!U6@oF~sa zAA@iokI>+T0M5n6n%uZa16=9oCTz7}2J9LZN+JF=_M%=UVV1b$kkI`-(;|&Z9VoEi z135dh_G{!~`bU7Tut7-@N6xyt9R|%D^)?bR^6VpjiPaNT@w%rbhrzd(|Kl=zS#vN( z*2U!;RdGqfi#U54t9cX`-QolalbVNr`e({6fk$@F`c21j7B=7RE>IKD^L4Mu9GY`$ zlmu!S%kszQHrfSqBquwaJ#N)UD7I!r1>^O(ljWwf=_-5Lc=*nKT}ZbUQr&x3C^X?y zNUwGJ+P=+K>mS)J@y4wh=l1cOyOxAkkK@}W&s__Vbe~U2aN9wVWUaJB8{z<`5k=_O z_kVe)Ml4%%Y3Ck_0~k;T_4J9Wf@7)ElZ-M}!L~)QgjSRKm|j1Amm#*+S0p+j*=P;ap?npFQwfh``Z)T4*4DPB?WFa44Kbl}kaY={$*B2eMn>AS90 znB)D&NuJ%#WBkf?w~+XLQ`JEBb^s*R^IreOs{fbly8HX>HBfi$_#99+f!h?gb0MwX zE1dFa613|Z8RjZqf7C3Q$#NKsCV!Mt$o;F-GM0m|nk9U8 zWI#Lh?`!KVdj*0528EAP)~&&5o{=!aW5dLQEHGA z?Zbjk3My^!Be3x%uPa6DX`ph=qKkPQg6VYMRJk*>8APNuh$6s$rgtcl`|WE^8H$|G zWnR+mm@VYecW$RgnD=yjCNwfUvc7t(o*PSH4o_x}dblG+EbR7p`W$qS=uKkWZtqh* z86m<(K`PYD#2%{5I;7lcy~FkCqnm!kWUL4|+Ui1L0EDAAN4i@a|0^L%pe}_ka&wEu zp03++Al67bxpl4NU0J)_DG?_Yh>;zLF_}3|3X>yT{5ddW(zaNSVPnGhjA7z-LUcH< zV<30@`R{MdCCy}Qj4rQNlCQgFdK63?o=Axep8Y0iPOhPB_?*LLx9pZmbk|VWVwZhK z>mIp^7_ezKusPimMjo#Adskh-G;f5``&lq@k0&7Z)n;*3{pHEHG-oJ+PoN_fCHzG^ z?LorKz4lOT$40s%dcWN!DT#}9^6wu*skU01^bAk;*rq1g6iJF>u~R!n9JpYF_=nZy zta-9;P?^Ew*KF&Blbm9|ABJFWj}O3!^Fv8< z(X0#vBGzR6`1-s&-&$j@c+w9{dT%Fr$CwQnzhxa+^Bcy>qAorNopWrSY6@o%Y7C5L z5P-S35@VPOzCHF3HK$fXn`cgtE#_Fbg^i=iNl49u1d}Pp3%x^J>N!pnq}v+}l`aS- zTFoJS6mOzpC8oy8Kk(a~OKA)mfIDK69FIaS?UrJ+<4=}BgFhIQI{e!eB5iRD`jez2 zUR+9@v0F)2aLRLKbLi%m6<0C|*qJX?5?}&r9)6p&V<|wW2tyeqPMd+^pF7elMZyTd zx?zE4F1{IwOWvR=6qGwUH>EUynV;aFi9+bDLFN`2j-xt1D|^I) ze2d!p?^+csBQo8+#0d_`5hi*MAPy>=+Snk z{cmFkf9x*XTe*;Gl%T(K$w(aCiZH(K45&x!T>~T?}Z|Wi@+CZpb zr?s#5$UcWKgmj}Q&(|D~Bmf7lX`PNwv=}m(sMLIGDyz6S2fPk(hG5oJ%86!qGG?Px zuz3g0#;})ID(6Ep1S|(;?x=Ga`Jy*NE3}hH6bd@7|5P47)DyPMY4UkurW-Xx77Ug^ z9CS~^9>mTECy#6H_|I?>hh1V6l&wSmMQ-;Gy^)JUw*f()kC_O2z>?VM=dn4w+Ez+^ zE4FBN_$mpU1z9d{XGV=U4rB!v@sT2QnF_*Yor2RYrK)X+p)AL{nhfimfz+cUzw-;kz+nkW^Ny{NkK{HAOuh(0J0Bv-0L} zqqRAU?+AMjr2k>oEskHXUf1OSqlwYUj-}7QeYpk`r-{@!Uyx+e!bDpiveuOd{cc71 zvs4UKfZ}W{?4*2E!oS^ESlx?nydM^a_kx&eB2@8{=4k0ETdKqvbMlV&wq0b(hJ%nq?(*1tA)0%y%9`}=W_qE6fF^47d_u{aCsiQ>;r&d#vMJmHncXt`| zjVD*v`}GFiVxE{UJ@d{>+%X4i7Tj@W1j4d`ZntgS*tpFG(%eS!-nJL%0twNRu*sB= zjx#bFpfP2*@K|&EV`F7~>wGNEmFC&i3k&%M0q?vPve?$C2O#L?Mx`7MtP8C`z&0~T z(CJ4Lt^dop(oOl%`fR&>J#M5)7ZFAR;EW{s&F@R2W8LYq+e1Wn)Weq5ZF+tw1_h{i ziDjaa(~Z~*)Bs%ZkE<`)a^e=F&kZX~CS`MqS>e41CcBxz%2O&S*g}03yiCceQ5I5P zb(PE6RD22yb^k7Ta9Prs3g%W0o^_uynIxBjMs;e&)#uG0kx$7)2VN!fG0yp)pTZ-X~kv{dLKhU9+p4|6b`!ygQrU8!_rE zV0iHA(`^X9xee{(*Q8W}Ch^K@i79jlSo6{1+Fgie-3pFNr z+o8dM>4SZRB_i2o@#8|_uE7g$h#)u8)57+qn7~JD_3YDgfw}vq&!+D)2?}b1qDx&; zo{dfbA}RuEBO*^C0)G~TIRKg?4bca64C34ZPI3kE#$p;6lm`^m5s7GPZvG7-!bXv; zIf)k|o&O$}Fi_kT$;2)osV=RD3BWC7K4--!2~=PuHhlT*3=p8!eT_cad0-u3all>OJMoi4=y z;&r(5`!EdnT7Rx_`g*7S6GHq8?PAKZin_|u;>lv)3{kKmdQ>eESmeKfkKa9q$?9_` z;9J<;&>+Ub>8@3gffMBVdJWp0NqT9Dpz*cnP*G3E-^Ae8p&~wHe&6XUNb!Li`W~R? z@%!8&eSrS9aTVP6zy4_HfXPc^oInteW4J)?b-r7}@GoB8bd$MK99=YFLku*?h7g|4 z{*S*NESK0(_4q^JP<;j4#sNJ5-B*;f$N&h<7FGzM&*gsNhnc3t8YyT|@6ejum&T_h z0Z+fkrI536I##qek1u!u3qM)_BLx&cLC!be0bCn4rflWr{vlyvnfdN)_-+Yr>&xx_ zMt%p@ar>F>kC()X^hsvE7y7qT{u;2Hg7b{5>}e(ehwTC9q@Sn;9IxIZci#kAVYzNhw1QTd@(JKvs@OSkyt|7pu7jDwu68Zo2!(ie_q1*%>`lL*N($ThwxGkFP1N~q1Zig4l>c+HTDJo$>N)IFD*A?IUL zq92@}xMP^+oy~|rPzj5iqX2O#2fJ%+h51Pq#x}m3W2B=I7>-;0L=0C8fmDbubKQ-A zDHpSK#sujt0}j;kjtIZ1cbw{ujS0?;<3PV2!ER(+@b3u)CuHrD@SRJI&S;-qF~J0M zbrOHPao&8UhFJU~BO>2xuNT9=T|gvyEBJ&wUjt0c?8SA@#71jF=QZD8Fk_W6oCM)=g?oc>UppsU@xL%N$Pz7V2@ zaOCrZy3WMf%tLH-1Sm4;#DJ$PhT&&sxa^6OUt7V0a#C9IMlfLUw#(na8&%ILCqiQ? z3bACaMc=E=d8){crw<++$>rnOnZ(fO5;nrbbl*CaBT|05hV$}P(wCYHVcJ!FK1o~H^OjqTy^!Zz{))lcBnk52@(l_l}FIM|ZAS&zsHi_PpyZv+3vi{3q9*rwiw~ z>(S2QwYiQ}$c`ufAybql)FMAM-fOF6k*O0cM7BoM9T~%c50r#h?v+jp&JZboIG*4^ z$tT^2M}|-P%dAy|5IXcwDD3FCGIGXpVLMC0MJS2aLZqpAiK#>+QKAL46$6B^-G%MKZKR(ms9t>9S!Dp*fW6>S zqFL(s5SV}4`BRuKr)PMdvRA4PyNG&(dPqR`1(_gK^KnjYf#mbS>hoW}t>L`L2uY;m zAz$4oqXwk&QArl@ewrY{W!HWDyUQFxcL_A{>Ecws8O2gJ`D-Nx`J=QYH*R2(Ex>1B zb(pyh8!0}#iKe@9bycrUqHEcCGId_%i~@=) z;Vy30^f4H_#M~xMKO~E89$-dFYue)l8)w4%H|3}KGnjROzOC}rQdgOePFWp`aRgp~ z;?Z;rq*RC_Ysvh+`nWRtrV8$+DFnB0xO359HtLUdGMVE7+;JH>Q9PFDuAeEg00i(I z1dJ!77}OKH3Ry6iAXmc9+7J>kHLYAWRklGLLt56ZKU6`%T@nEvO*_4Hl@a8NrTS?Z z-uy<7sSXEMaT@-XxTF3(%!JN}?v=RqJjQCBA(C~tIpde=_$dO&lM6~hV82*`CVb3G zKa8$(0)0Bl>)6{znH=dHgnN?Cy`tyn03-Z8Ee1k=NAJ9gKZ!joWXvUsnS0s-BV+Ca zU>^FL$*zWe^(TS_1Vn|*sItkd!M*sE)ALR@+GZ+dL{$nBX^TP264MWbZ6gL#S2h;k zCOLFMnF^zt^OQeQE6evu?XliFwfxHN! zzF2X(*0bmf!{Go;TflAY(A~JhC!M4#_y6hh&s;)L^^1?VFKp5Pi=U5!OGs8 z=F@$7>Z{hi$?f2~W{g{co*en>Yb#hg9nc7|KT7J^kEMEX$h~GObo0 z`J0q!Omp3l9&fMd{84eAy5^)dAXVeghB_i?V%Ao~w+>@x!=_M6s%8mft5JMX3S=p`9_bS%AK;^0}FB3;(wD*ME z{Tg<$N&13ao_L(7iykB@;q6wIwp>f{(dc^+E8W)*k#MD$J*_d^O{*Qr!ld(_;jG8< zBo;QjiIqz8_}vU!QnCyRoSU@JKfNpc?V86ouVv`}wH{bRa9+hLSN?Vptv%sC!kaFV zw(3RCl(*d7wp-Qyq_{G>a!criEyI>6pIiF5J54v;t^U>+cdnh{OM`n?Cu>kTw$jjk z-1O5|%p#!%#_-NW-+KB)iqsZu6>Aea!rSE$`oz|`rui(j?XWqAjLybg@yQU4#3On# zN266XlU6lId9^l=Z<+IP@~>mcvT?m|omY2d|Av8u_Jbxb;BpT2RiV#9aBuPLY$T5y zrz)@L93b7HsIW4(E$O*>OJ*DPmesR>9_XnG{oYVHSXBKIdkzoIB!DG_$7NIZJkR`V zxg>Rx^N=7^59gNP2w>gTLXm4f|cZ(a)7_Kb|uWIIj(c)2gs| z?Hb`M$yCLMeGq?6_7bxuYF|&s$Svr_I7l(Td1(}Q?AvEJ6L;J)Y23)5CyeZV@Xwp~6guFjI^gQB&&iTmWeT#v82 zRHD$4U~vhmEv4hHrcBhyLE}w(D?^NlEG20*yk9u&^**hPi*6fgSsV|hGD(UpNdPrr zv`Ew)zy(ce`;ily*r?0w7_zWqf)ElZZb-;B>JeLUKk8#D9AG|vkK-qhI4 z{&2@)%ff$YF?gIIjLGcn@v8GCC3v}5J7{d zk&-&XX_7r^{Cm{71Lt^V@z0Rmqv1`O3b*=eEJ&6_ak*Zz8Nw zi+wt`gty<8o<5EV`~9g63zA5Ce#GTb*%iZJ@6`*dUgO~wXY@DRpK7hU)gys%1MAsy zfWJmz-*hZF#FyvH?^^$bmaX>?mm0OJn|1I%Hr1w& zfZ-?A;V!AsV|Dy_WN33WKR^5of`M{E_4(&~dyMos)UYR6cDkvSK`OfX5k_V?>@5_# zM}(KtjE}$uNM|}B)p;W`5ysOJ!(8rj;DPKgkv#9fd2Krb!FH8M8E<8Fi23cz>s{Im zAKC49^QoL-ij3z4f*SKrCajY+_O;itDQuIwrqCGCxn}w`KOAQ=YE>&YrPUz7}$nK=DJBpG&Y0b=bz!p>R8Xj#m3h zPOH;b`EQHj??GqvFaq|Dtd_Q_1^u@wTP2xkF~y8p9vx!ga_6^h5f^apz@)PzMCtZc zh!crsT$yfeM{78$v|h?3iB6Dsr7)fA4QBPm+Jd9XclYi zaUOxq78#xiY09IEc6tx>CC+2YEeS{Sjlhm?pi<|c@BBx$ zTK{(gvG#>OuwUh$)Y+E(`XQT&_sg--T4Qy__N_MLdb4yXMy(VIvJ7`=WY8b(C|YyJ zKNse$da>?SBrGkIMU_u)8oPBO%!cElvm$9mv=0;`-S^sjj;~=n0BOL?lSGlV1AgnH zaSN+@P%ecK4@_m2Ha(TNB z6nA#GHZgAiqmWHCX;D5zegy|A_Tj zqEjFaXzHXif0m2@(uGgPS#NEccMSO!wuhT{c`$FE3`WcLd)L1Hd8I)+ zBmMK&BZ;bfe4V?5wvgbvT{nBPQ_j3$4(nS1RlmlD(ffrZaLG(%xw274#-6|zWfqZM zC-=xBCfvmkz4pVOC!@yC@1ETny2NU*9kY!XpPy=&=YtNH#7*!?-Ti3C zv%vhg2ybHqu-dN(2B%T5xV1IG-&L2aT#|>V{}@IGy=_oU`t4O(2)u+&>d_r84rBA~ znSk}Qs3LyMm8!yY?=#`Ad}j76Ut5Q!uf&!x8}bFXT2+@{KXhe<{O z+r6=&#*`CUy1mM%!BMwTPONGKoH(3nM6I+4FaM~#{c^w2Kr{}*=^-w>lz^_f{6Ux} zieYN!&J^m@sH*DM|C{cx;@$Tk{M7$~;18FjiGc#+!h%IUP7J@|5|*`a{z7LmF|w=c zvf)`NfJ*!sIWH?=uZ|!cTBFr&P>;c#KryLX=*LU5+~3^+rm#dQyS_!->g@GscCr_1 zP*lY~d@cu~+liuP%vVcsP^$K9N%yr;KR_O#^YnuVa__0e7KhcPXnSPrE=eR<03y+h zx-LJ~x~u5wZv_=Am%@hOXxR^>-r}*~nsc{LXd!=#l2a5jq z=aoFM6q;P^%Hw`JDXH+Z%~&;29asFr;PYhpW*%|e{JYw4-YtL5Vu+$aLNd9~ipOO- zA(Qvv^ikTw`dM=XM*5pw-3G@OPd$8LCp_swdcBu^IVv~qS)kF_0^YWk^T|yIc7tdU zpk2J)1kMP9-m%r5uL>_H8H??x69EGlJ zES$%YUsTa3LsopFii%5l@b%9A{7i}jYJbr{9BynQ3p17V)`$nk>sW2Gi&}U&z}vXc z5K0k!h8s0PeBr>DI}&1sf||c=Q%Uq`#i~YdPatgwZnC4C%{C7eqJ&7bV?LVhEIbVT z+Y>nc?0Vz|?^$5bUob!GRP%bQ$ac93dY zdi1O5f|gg)VU8ucKHgN^8fA%0y%Cjl&=tC3GmM%BQ4J=lll*0__38Ch!ji|qq{+4K z&xa4<{VDnrC|Y6)xBsZ2GFS{QK-RM5*P2IZfpXii!@EhWsM_(_q3XzcmObOKp)5C8 zvbt&v-2r`{|C%KycWd|)Ht0Hcyb)zfSv}bO(XX4`;}mP^I3OI5ABu}1)h*$#jScA| zQ_M1v>kR4qu(Jg7Ymlh$u?u1F5_rGXzw*9ENzz*4E}w2r&RHcCI2~L9KFfpN5Be+3 z3Qlf7~5V|@wpokP3-J*CJY`qPdj#K^M_ov6OAg7}o0)8Jfh zUphsuN@g+l_!5~Hiq)C-m|1-Ki8N*TXI_5N=kLlKJ2kLEe9dm^1 z$RoTg+_k>r($*3)k5vgX1cd>H4!;?M<*TOd`dt90d9%7l)1)tuJFIXSW>Pux^&d)P zIvsXQWX;5}1Sl`4%UGDUkA%k&^RAwS|iHbCuyD&uAGm8pMjS2rS(+4EA_fZd8oMQafVw+k3S1WJB)u2 z87JF`#A*3KEU^F{RJ`-hWr%t&G~I-Ysy{RE$f~9`pU-N3C+wYJ@L8u)6&IOBm$t}r zBd_}H|Cs35!Ke;$Vq@I%lSZ~o=0hL-NBU>AYrh@xk?om!eIKN{+?Y+Gn;49oEi(!Z ztte#2tf3H$Evmk;d*5x%8u!%?TboK#8EFzOkhMB&B-;V-qZ!vKJYoi=RL>6!pR!Qy zu-%DdP==8%E|Ky0i&Uhl;8MIRPZt}q>R^l}$~u_aPWK0!Kl+_Ao0;HYS>N=TNV$C{ zfKh%>NxSG+Y=!7@ZgX+oAiwWt$4JOM%MV?b{`}1mPvN*ry&P6%ttV}*c^Dy&_UQ;wtzpmJ!e+oSkz8(>qchdAf|*_nn6|o^f|$w=VPI4 z4;dd&R8HDLUEHXkLseoHnabI$$z*i3bcUcOr7=Byw_kfk;V7S8!x^0+e(NazyGS2V z6%LSXt^mx08)ZaC6;ghZ+LBT_ZR+#4_n3S|{RaZoN<4SpKlMRQQe|oKNc@H2?9Nsl z9=&TV)5(FOjjLxsH9Vbihp=vgf6rBU9Y&&DHJ|XA8;;9%Y#9l11ZlY=bfZ==MUEW6 zV!JQK%xf_4cff#F)PY3{RFRc@3vi>=AZ9xplCs zuOh+IU4Cr3ri8wCPae&6FmLO3k0cAVOK*tFBIIV)EKlV?^z0P_6A@c$Rs&_2Bba!u z>w!(QAFIHt6bufsanvTCV`)^Zrd9ZqW!~#Gf=|4sQUJC+MX;g4)E8 z$sm{iRnrOb_}mAFgprA7t*dGCyh9$hU!E>$@*q;#rNM;nr4uv~vvm5Y6-+ccu=J=y z_{godn+~4#Q&@9HT|lIQ-i22uHelBO875(6oX1c1#n}5#QnCkC-gNeoDD^Q$Wwt;*7|aSStb{K?8Kn@o%Qg>(+2FzFcB$%M(G1TaudUA#=3;hA9MjaJV%ZonP+ z@Vl`h#yoxehf2INMj$LpL-P@?+ynw+xhYSCJdb%wz6NWv7Y`%V41MkJ{WaB*2`gIu z_#whF(komkN#kHY%?Qepu^A(I$MzZ~(tT?>*<^P&vH7T}RD@PIBu0AN@3#S0OXPm3 zt!VX!O0g*#+Y|}V>iLMF0rT;nQJ%3%2iZ58NY#!gKBrrBYpy+u`^-xof~1z-(cBg@ zcpqXp0$bQOyz80Lb}=z28w;`Po^ok(u(x&Yms!etF2G5uT5C*ErD*DP3k0jLWk4vp zOr)Ca2Ab(y&j|-(R}fbs<{0S}g=CjSvo<`pKnTa1=FY#ht2{Ja*t^Q57P=4MdX&68 zWcQ_Y_XSku4U&t4i=8c<@g9um|B6cYVEv#h|A)vpt*fhw-`*1NTt9bkm@{CrU8<_u z8J);7j=-{>n3*W^&qxkj{{Y^Uo8S2R?X$MDrPH3}FRZ$yCZ!l7ql~TpMO90!IB6`) zPp!flbOoGhaf}&kGFU^aShxgO%&-LIQ!xk>TS~D8LFOVjG3r9NBP;aMXc7c0+6a5Z zFDiNfLBlrW4=PZMMKz~vfK%hTMg&b)EQ=e2m#_lFps8U_Mx@@*$6eCn=&F6Cfe(pc}RQ1C#=2tnt`cR%Ax15T>9g81vN_g?MfVLr`lXIJTUkm;EeD1!_Y& z@*K41M1iV0s4GkJYbZtZ_OS8@HW7&AGs{HeNo8Pbk!wojWTGbLiKivqDQ-8T#9@E{ z>RtKSXl0_~5H|QEFP2Mh)$EY)SI}ABhnJ@++^eyZ`qSZ#n@W1J4-?q@Em`>J zIzjjj*2Pmxb1_OFsh6;~{99OW^>O-J=JoMn^~;{~O0mqvVAnwHNG|OFcH>_`9)gXx zH+TAbzM@f`f<|)8-KGL>`?=xj!8|?8Ew}k?kD*9?OT#DF#4^=LfW%-ZQwj=H6QvcJk)@sU3Q$@y{3Qe1>n_60Xu7&Do9RiZ#t+?UF16 zD5N_Al~eNmI8Ev2J;-1IOG6;JoSUhbR)y*9&__OB&wWqh@X|pvc`PRTpU6z!I}95P z(sUaYwBuVm*^fnPQmluh0oiMEkn=8TtyE%*#XT9=i{SU3b?iIS*II+;CeB{2;SN)B zD2Kgu#tc5eO-T(Id~()GM?F8R+L*E%KQFA>`9k6!TP1Vy6X$g167zsQrGt`^W|TDb zh)}`9N+OZw7B&UesuWXG%T1!VJ8?fjQD$6ilemv0efM!Zqa3jeX-7|K&d-#sgpK0m z@kiQgoXece@G^GWoMEr)ll2?xd!hE}ayMPgiU7MeP!1W0a#36q36Wxq$3IY1LXCQf z8=NUr3vJCvBttII-{=PN;k11M5-KM_gq#N>tCTZ23Um*>aD!IFPWxz%DJT$d%{ zpBJO8;Pk-{xNiKq|L~mgR44AiPv z!X=Mc%kO_SiSadS2X4*U+oJ13@3qZLq7Z|jla6Dg@H5tl(YT|nmrW~gp=L38ZSbMl zx;=&{wIN&DVAH_$yR4^KBc!A?ld#cT1a^(h*FXa?znf3Ahr^`Iazz;wHy-q^YhEe2 zp98rNMc%@Qu2)t3-x&CWt2`$=AS&K~(&=oEBAG1hv~MH`;9@*%qLik$NTU5z-2P zSaSh=(m5L_c!F7ab9Bpd{Wz+J9|#7hdZ(Ny=}3BAIW|5M7xCtd<*=G|w6Oi3tAh+8`KdJ^PVaYBMjGgV=~ zAG&X&6Ydr^)U69HI>%H~Shw*Xdfzi4$P3daTWg36jBu?YZ58v_rd&;N-7aL5+KL3k z%eLrD2spw|I}JnflsnTp+*O1+V?V?xbt-H>Vl!0Hmx#(z*5U2+@tL*U!LnZ3HF;QJSCTAoZyE4=23sZ1m$$N-F zrHhW`ib;Nybh7iNBEQnzWX8f)bjka5yT}XUJ+XGUNlf4BmKBGj-gTTEuZ%Vv^4UtX z>NHm7Hv6Tnv_tIozV4VK?vAZvJfMXMWFgq*NTRKBcR9#ppVStsGSoLPV$K5Qp>4p5 z_p8)R_fE?FUQI`3dGiO+r4>tAX8yCn?$!5A|K2}0fs6{1lSel<(BS#MHcWP9e$$TN zo^}r=Qq)Fbe50ScIgf|ee$tMRsL?(yj!yS=HFb5_QhbTW9%=)UHy8aCc-IPblBG00 zcO^`ty{70>Ib1O)6^)~hMv{QzP(WhmUEEfB?-;l@jpCuyt2WZ1lEIjYfyTEOdw1?Q zy*Fe1jYHjJRLx<&3k9_cAMKIc%#j~QhT){n%4%~{if4Tj%{ykiXWcv8J#A=!kL!cO zvr$xH079V%jKrH61B+N;h|+I*nE3-&6uJe;(47fSk&+nbAeJ;$r~=@NFV*m<6|;Yk zEBNNy#qVWy73IT8$*}iRBOU!M2ZYK+&tN+qsyTvnE-e-!-Ov0?mFr8oIH(UxA0j&b zgXV*4?51-ce;E@9;3q;pRZWVq31G+D`UCF%-#kp>wD^49(@PQgEr8D~Uv9K?|mZ1A=L zdEV+W?AsvSSLRpwopHcZ^q&pBl83NA-`cx5WXdfMRaKCrdi97y?k>)I_3)rKr)+50ik;R*29mNu9va4G{vuo;k$z{_OP_8 z1V}vu@PAxt#X}FSyMcjctHHFWBDHhhg$ohyhqR|B*vs8oxbde0M-5BjhaGtODFh{B ze7zR$Dc?+%z#=B617_9kx6jMsnM$1ET6tXf@5px{GzRPWfL5$Q3r34bSX#8#ybN8) zWA5_z&UasU)&q1CzTHS`bo}b~0M<`QR!@A_Y3uh2R&}-oSGN8$7>{NL|McvdI6M38 zu_nP!cmCSRnEQF~Od(x0Q^gAw$5iNeg{OFA1#FgbfZC`9-nQsBbjE?_U=EjMAa`II zS-+zmy17V@5z=?;28NNjC;5pEEij-F1bm+$Lw{d}k4#-qna0SQanNA(j;`?;@7i;5 z|MK}goFEd<8|o_F%EferoCkIJEIlD1-IVqSUfvUC(b*S}`8nunWo8rK0|FV_hn>aTf` z87bSz#npx-&#Y&ZhyVRTNlBABa`JKI*5&*QZL+*M?&PB^U*}}w?yXHWxq!{xbJ~~L ziB?wd5jT0rCNfY$q%bv^6{PAKPa^-Kc1|8sEfr z!dK$Z9y2royLVL-3>rGQ6$nCAWT3|0mOH7~_BlCA7Di=wYo6dLCXuhieI$kC5MxuM zDwrb(U#nGPfFU0HT!%KhDPzsl6Q&_Q&8VkWFd9j=H`WRxLE%Y~nSQrWSpI;fZ!lL% zDq8^u@D^etY16&c*yFcHU(&&J({oYUO|cds639~o-AojpT-Ca(KG-)=YjNtL<*smM zt`HQZOIugz7Dk(`L@${~O>l|SLm$TW9A;DxH@_?Nestvg74=%V=neSliM(&u6GhG zrWjFObTrWep_$DrQXTv1>vYyIkl}!<%G|ehtmBzluT*nyehJ07k^a}#)wyjXy~olZ z`^DuA@?}G>uJ)h#&A+2gKib`WOoQP1azm)l1W$G1Uu1$df|8`|(yz8_m6 z5PaRdImFIw)=1guALt?9#!y!qIP>j~Kd|vV#?#6z0!?-V&PQ9UbgP@yY?ohvG-=w6$QSsJW8;AILvV0A zWgeCv;=H1q92~rCqCDcF;*#QG;$kckY^)q?lI*PF+(IM*|6dpV|Hmw6W8+FMgaD@p z+S$NCFv9kH9ngYN8i(*~FN0Br&`@JnZ?WJc&L1q;idDSP6j9*f#OzZz@Q!<@xxc4q z<1^Ruv)jjkEJ;`bv^n}_I|Lc&ex?*-;->=mDy0z9({OOgrX$e{{olZ?5F?$ delta 14400 zcmai)LyRsAu(U^KY}>YN+qP}{9h+xt+qP}nwspq$`;-5!@2Y#zNhP(aPCZX?unWtSpR4j@y)g8?BxA?N%f}U!QQicK_79j%si>YQqw$-8}aiu~;_HqqGNwr^6eq zm0vIavBRB%q%_S%vya6XrDDDUa>arp=TjELUy3y-h2F0Hhxf6m=sJCTI=zMAy^9Cwf>&J2Y+I2a@#3c>hr{2cX(?CpZPzSKXT5X;4oja6RjbRbGA`CiDNM(N3l%3q!)1fJ)($V{aM>5;Cja}WRou4 z1F=xV5=15SCY19;U4(ocQb*XL_lHXzoiYc&cxniJ1juFD^C+UqU8DeeOx$dH>oV39 zy<`F|J1dAsp~yzyt5Od5G=UlFt{%C2U!U_Y0R!p9By&sb-)oarkOcg8QIalu%X-HF zJ00yjF|Vwh@%{}&pPkLs=zJ6H*;Z)~9dAfS;jP$?1e~vl5Xk6Tog{kvV1M}`CnDX% zM!__C2wJi}^mc06)E2JMudRT5%X|(yc%T3f1{fR5dDBZjrna*&vH6~AcTqR5Mkl7w z*)}u{W8NM;q~7tg6&ce{R9O`4%;2RLWqn;&A;cbn&fr2eTnIbo-lfO=bNxEz&T&qzYM`ei;13q^vLG)AoQ@ zxByJy1%Hjdqr7l`ygu-U3F8P%x;I_buzLM8vHt#33?&i^xsOILjn))ZZ4-|_q5I6Y z;T$y!j%bcl0$=hcIB2P+WW1|9Hxuhhz(z`UF-vMemW1R6x~Qva+4uS&2?~=j5a&HU ziUD%m1fC!rx3+tFUi(k$-u)BTW}Z!?)KA8v&0e?}ejS(uuNe#7WtLM_WKMF|!rWFg+z*8lo~ zQIPg69xqd*+V)x0p*@`nvwJ)m*%M)7QInX9w1`;H(q!s{qlQ_=%5dJLpB@0vIyzjK ztJ;f}9DIMYv(1vk;x1^CpCnv}5zhNo3sDLLZ>}LXN|1du6C#)AkF}se@+DU5Ybr2@ z&pGut{QKLW6nYYdpRqs!OII{DZOfA=#R72zO4148z@W@76rki-L9(Ev=7Oqi_xU1U zDCd_dnv{a70=b708KbFa?nnT*VLG}v4pPjtDh@(gSetseZb(p%GFv6Xwi@o#WEmZt z{RiaU8HSE!u{eIwSBD`gk-g%GXHGP#`MjizH0C?mF4kNVq^Y5*8y<5-1C-_uXfYk) z-_WEK{hJ*pMFZNk_mNMmw0NnM?>)4RW}v zxJ2Q5@w(U(7!K0OpHvg)bB~c)eU^agN4Ioqp4>xsjLN~HUM71k?e!(jt8RH^jx^2j zO5@Ec*m;MALr$J83fljL`p^v`Ut=XH^U~#~_GIi-59>UWeX#@x!GXPsC#;E8BXfiw3f3{2K0{#um?#w zoaS*i=6T2DQ-Er=oHR~SsoCtG9P^;nVW6dXGGi2B#k6t-_DJS%HD}MRrs*T(MH__puGsgzCdY19{sc&=U&cFo~o>pGx~5i1C4@23+_VJ>8b4JRO%7 z6GsFH7-9fP@B(qQfq!+T+>KgB#`Cb4EPq?{G}7n1o;T9Z!f55rtI%XItJ^xdTzW1T zb?~@21t6l}*ujQ*E?RJHH$>&Icj6HPPNJ2ssFdD4Yn;Z_l;rPU+YQyM+XX&2(=r~A zi+fl0aeR>5(k%`dHpmmlC((1(_kD?9{bz(|cIyEFavH7C12K|9%-3>zCDqio4HG5E zM^%GEcvAto>O_|BWAkOjRDaFamysG?>v!_nUBH{Pt4CAp8{Fp+(xqbw%QWSe53d3o z_^O4yUA6H$Q;mRA8hHj|JWYTVJqk!DNQvEZH zYCXPWVt>}fX^;=%u8!i~88(p>I?__c!Zvxxa&k}J(dp}`^*r|7o?v-~L&C||E;?p4%OOEwmTKfwe9zGJ?D5i&YL;RX=9*Cme}Tb>gCNA_PiE zizD^6vcRGrh?>be=%j%Kl%^YWoe@M17H9@pGkXhHOJ+iLW=5v8fdybxFb-B0=KpdV zEnp@7t&fj|rP1KEH+cHGr%7XYq(`^br|xQUGL=sTN-UyUxB!v}4Qj(DWIQ!N`K$J$ zTcBs}ZtU&p?9Pe6gLlq9=ZW_W`YB6C%EsGpd*2(ARUaEVh!T{%w3M+sjR-$XxKo5^ zG1YMn2o)AWP>2~3GPb^+9Ugv&gvk;Q6M$a}D|qstAAp7%9sb!l-~==v8YzxBA6|68 z8MF)P2m>yA;F<2mhz|c}*sw2vYm|WD2KIy6q{b&abeUj*hx}>;mZnE%j#lD zvTRqIupjdDHW=#pO$das8Nv{Z`46ypx?m6@RD{ud0M0f{m@p6~m?2yngl)Ae9iRrt zZz`JoRQ-@p*suzS7Crb=fo@ZMmHk6GAg(l{g&wFa1?1VTx>`Y?=Dc0;?v+!!=$03 zp&{WoOFsd^;6AUA=5pOfbx*>NAb=433>2}~$r1HS!vtg=Kg9j8XTJ|LTONW@sB>uH z=U#<^Z`z@Rp&$i}A_(_)ZxAxJKW|j@n;qio;2u!?5cK~0$i~FM0qkoe$%-ErR zt5!I&x=g18eUv@}rabVQRMBe=XCGZlQ*05KvHEmP2;o0qYB#+9C2U8)Mu~v`HqZt` zz6s>a36cOpw+$Bm_|p&P2KYG=L0tGs^cs8s0t@&Az9;lxxVO4AnC3zgQ5XqH4GA@7 z(gXq)c^?`|faE-gCSN9S{{z481bkmV-$(NkX_*fl0I`UtK*(q0Wa%%)Axgp+u;{>4 z@V8I(w;(tjHlkkPFr?(AKVNAe00LJClznI)>KO6!x5?#qvIHYQ7?3gGdtm(QVe)GS z_as&r^o#Hb*w`3QxI$+c+V?do2V5llc#jjdhjSs{-M(~9^qusHSQYXqISOKX4OrAk70dW1{ zz>u(Ur>@hF(m#>gYzcgF`e4s9r)XR7<=s+OD_&%Ya(6zmMTp;luC`htlfW(0B4(|^-kThh#Pm&Nm zkjVij>FIw4f?@qQBp@D75>SfVBYfb5fb8%02@NiVeFgnyC%<11eYT_}2o!zJh5m-t z+;hetY{Op&{R4bvK^bLpn$D9!8dcn$B*|H_>NU-UFeM?4dgqC0#aRw#W7Mx2=#-WUxlE;gjqb361ujVU1YmC|vw(f@vGTr0wA!TkqB8luQF(nUo72qs<<& zeboh4JxP2`W>~+;AQ2{rSQh72$JjmxqR~8G6Dfs2H%oa**belOh8IjF!!3hsJz{w( zrQbx_)?|ox(Z7c$$^MH4!VFqB6gf6uk?1w+J%Bw&@y<2H&{JBZ2Y3ZI%Yqf&6PZu+ z6+1A1M6q7uYYmCzJ;ou8tY{D~F|5Om#&qdtXL+i&C3ZEPy+1I8;~6du_^jUfYEzc3 zC2(y>wS3ry%rulq46T9KHkzdRkw2T?(V49_qr%POo6A#dd<$w%5E*HgRWNz4R{~5= z0|4NAT$U{pp1{MB`ggZRr!rG`!|gEUA0s>9;Afd#IQd{BWdLQDCJ*2kmvl~-gQx82 z2(MM&C?sZnU$nOFq^m*hzkL>K*(H8)^a@HuMs_z-dyUky7A`Ccvfv9f<`sHIHW8y9 zt(I!978m39a$t`iv@7#~r|y|BT3OYtsS|mB?$EL8i4$V%=u3$#T6)rMRN*^ID4_+& z95#A4F0IJk3|P;^^M^wb_;ktdSXWZ1F}-eFHRc!})iI=3jTwiKCDFOSah z1DG^w{srr2DsG~_Qf5v-6e>umvcy=^*VqdJsW1+fLm>B7J1}9 zcl+@J9hN7v2fbc+JPRmrMdRUp%tEe7dj=ze-&w^u|5(AKw~h6XFyES$U4sDj3dG&! zuT7p;qCs*PWM%u*dp7jF^lZGgJbz5Xa!MK-7)I9*GKLP^D537&Lsx@2Dq1&Sl+Mz4y&7g|m%H7q7Yg5Ldt=W@QLGaN;C1Cgk#Kcs`pJ$`+)g&ocVb38N zZgXT}w%2)Bo)a{DYc&b9HvV&xy@ESZf0?kG)O(n$n#t-Xp99yy?eEL@%5#PwY-y0^ z@$^&ob(+@`B6m|kmN3&plR~dRcgbH#g!5CgPUg2BKO;#v?4F18s|m4?tl1%IMhG)G zFD#Xdear0p-hrWM{AiST1W@ux<5_yXxrxz1ZU_B1GnqNR9&qGaZ!`dBU#mHRi2Rsh zUUGJg-;4Y#EYu41%cQ6p=1$9&NkK73W(tg%y}$wjr4j@YH$HEUJ~P{Fk4=T(`)EY< z8`Rc>ghLjBRdWT#WuwZeK-JDWulVdMyFP5%d=x&la`y#Tjz-RK+HsKb`Bn08jl}rt z3|{ZvEzCzRrn8jd`z1hYGEbqWmL7)97m58onmTUAz~hP~Cdl`l{p6@~&eU40dcm)O z`ifp*c#~wxJa-1(COPb__jx)irqqJVgFEslqYrlTeFtd@y<*EWxm!CW8ZNMWkVpah z<>huv8daP%x9gU};CZI)l}UkBChiTPSu65B;@tVf^7G`VFdq=_MIh09QjK*7mKyIA z4d^02YU?dStKYUqunKoE%PY0Q2NyN*MS_vBoE6n$_wfpn@hjPurX~^ zg#l4tUYDX~;pCgL#rRxmHXde;JLj!Wk{6LFqGVTCft*ag&FtZHiU`1+*Y*tsaXJHv5O@m5Kdy9;uy z`^wbCTa0Z-)+=5ylU31A_TTZg%4}=$X;8FDpBq{z1nW+NKAtth;SY<*cMK18=Vdwu zR%1Rilt`eTUpe?Kq#TlJ#AjB<&d|Q9kqr=3603pr(JB&a>r$+rVdC~qz*&6jw%P%~ zhu_HO*#P!WT~JVtom3Aauc(A=giO$ne3=POwAMN#jFawi5j}zBPwA3>Y=t}A{+!Be zJ?F=1em~mx;V!#8w&DLsM03h1mR;SOp0jVVvh&Eg&A{WgGP8A($RscZKiYQWR`u#P zqWwn{q+Ta(b-t6AO$lzS11~dr0^MNvtt1J}!;qJyuN*`14Eh$W)^5if{!}8)FW14Ec%q?Zp1JqX!sm`StgZ+{) zEU*c8g>;~DG;;OMs}#kYCiXThN#$-0C(?PFxc%X$IH+L-;S>cXDw+Ci4NFDSh z!o|YwXv;${#Y@XwdX@U6U0AMA7L<9z6u^o4geg1o=pJ9ujo)djehMvjycUG?>D~b< zC&Q7()4VF2Lu+?vRN06QKCqo$)+yoX?)Iy-DZO>hx@266c;dH`PMD6}7S#t8*16F_ z1d@7$_<6jOy(>8Mc@d3W?_Ju6W838#jC{7q5FpSp8PlqZGC@*Na>M2sV?0kT0?4GE z>C#ON7~Av1QLt*rWDr=$de~SfzrV;jyLk=khLCDJ0fv3Esa0On8K(*btDQN#;>W1C z(c8-K!s=`Gs$oq8)%z@JF zb$nhQ)x@*mJ?DpIv{bP*HG|M`&ZD~*t^07hk61t#Md)K)n z*3(sF%G_y63<$A&6a<`F0+xG^wQ8xJO!-L=?qaS4Sm;3eJ*1CHi*a84@^|BS#N?EM z)p(fLyb9@UQIqAEkTm$^AG7PCilWTD0pFIR)oIoLgkOluA~2tB!F~No&)hyhib`cO zy#dcQRuvANjz$D4Sr(`6m+PIA=V@%k9_Ejy%h_R( zW!F!IbhLB+WVj)Q0HWB)x-`n*a;#Ub1k>Rk{)%hrNv`NM7&t9XInJ%gG4w9xMV{7J zmBK{V9P~Q5uLOO&7!VtazkFoXF|CyA!?zKwI{WZkap>3mV$oT$-7k$2$%D>rf^)Vt?_(lRpV%)ggFl8k(tRJt1 zn{gl7t=X-%fCcZ|NBt(?oMX*1o>KnM@EeV}4@T&zWWB+FexK((+Y8QZwpYyx+wra% z2`Y6VWQ%I^O#-xF#>uz1XMDHAVvAJK<6)Vt7osas#PQjZHau>Y%C(cvxuqT(uUpMN z^^$Y{Q>3HBNmXt+pz8C9cO!pJlhkhRme>lP>c-`K0vE;Uyni8= z`IdiIsnvJ>*>{?ps{Ex3;8qbjOA~7Y^&9>UqlVu?@Af5#d`?XYbJM*>52-`HQE~cK z?!!#8RWU!ro1~FXwk`(Fj zr>?^SH&2}};*;B&A>ULvKVwOQXg(_IIj$yRLpOf>FK6&S<++&wtt?l1`%Dg3;(+78 zB}a-CbZrW=lE01g_6TgDvyT400Mn8O0}9YWv~ay2_lI*i?X`-7bgFP zjlbl*O}Pp3MwW-yQKj6Lgt@sIx*XL=6b!tEyiSo(uuj4g;mU&d!RV{bdJ@#AJUvaJ z+q-qAh8B~`TWRi{y`JO9p_F1jJ0+B@qM%Mb9?7K=k6EY>C`ts*Qe8^gS}NCFX*Bt( z08^8UUNP;Gz1v&`Y5Oz-X0X^hTR&cBqb2NH-S_{#mqV)%$GCxR}r_six za5rjN3HA{(F`$tP-oYTB%cJc=CoT02gR=u52@iC&%njk;VZ_c2SzI(JBM^0y6x!Iy z7W_LntI6}PQMwbDtjg++=dX5;gjH(|YU5hY%Z+qx79kvk^W)&#G*=!p<-fZIsH z#I5ze`y(OBFoktAY_n1r5O>2naiJ#g3WR1ycb%B)t-1U<=XQ*)lZ?*0@8kX;OLqdp zoSN_^Rw;%*fuLh9S}L-4`hTwq8tq%(H#!*H7>7e5S^cG9F9+AV|eF-OKPO?qb%G zZdJBg={r&MCYmwTnDAp+4V8Ze18#;&rN4P}z9UjTC?p!;E!rkAcBA;_?Z2&*ar@Am zrwp}NM$?lCBwekEJi4vQc_|zi+la|`p4ugk%_z*IS|<)+)u=Jb*h7PkhNyIA{?;QCyty6f6n0$^bMaiHW5 zY7Xu#y=+jGd;|){-{{AF!hAlTbsZkAZ6=IU%#w8-Q2!_`Jd<%*Pw#(8Qac=^cB*WSh}v{z zsIz+bdH$&)r=WJ@WlgL71Gw>`|B=5o0z(jylP=4iD*hUy#5i^R$+C8ykNb%!9mFuu zIIgu%Xm@Y`y$&LL-XcEb{1 zy};|eazzz59dzd!f@$fz+%wQ*w0iUDr%LJtzsptVS<=3fc?pJj0BpD=z7M8nFK<%V z1t6T$uEfHV{Gdnoe3%?Tq+}NYmkaYdJ!xjMm!6t-rkb*&Wkt#)*|hc{n`i$i=Z{QS zDbe%ir+c}c?Rxd5oA)VOCYbP_2%dBcJp3~pdj|S`{tq)%%yd_OvStgVRCa)0Hg(JU zU-n$-nM=Z53w|a@41g%AREmb4xnxZK)?b5%hkMRlj`JD)wlTD@ak+q6nsdryc;1c) z??WhUlTMz+NNHJjG*CwKda{9DLZS8Zs{L)k-9U;DgnU+3%AmHF=lNhbfyZ<^V)%d4V^mj0iA^72S8#u9{xE3H*5M9bXh6I z68=YX506yria74v=JxH9{I-xS*ngZK=Ye+Rz*igWdm_7t#VLqtO7Rn%iOI)0NeJ+& zlw&&hFUfKa+1!OL!p5{og-XS(VC8e=_D<3}3L;$g?W$^_qX-V_Xl5vQiuv7Iz zjEfTMqt+;=Q1P_QLyW23(PyI`7EPSv(3q<_Q9!?^M6my`vF6$(|MdlxeM9XA5B^>| z*E3t|0N~qFwHY{<5-es@@He!6fwxr0g^}ZA;`g`QPIa1?peud@a zZ$EI!_>U2FqbOy>L2<{$Q7~kjHupArsASV})(~2%rk^ogU79LX zzhu9>r!UzOX;I=RLvx2LWDK2KtU0s-H|b6I9xt~+hGYcrl%&MWcO0mm?_5|4L&9)g+u_8qZzpYesdeBqOdY60re`PIRpbl+oSlaVBvR>?-e7`t_5Af{; zY=FWPdDLxo?!r0+Xv;s_$tr)VSu||l4F7@XL7aCwO08cp04MUU*wzj-3b2Q#0iba| zTE#AR7mMI-$fGqwuU*LRPyN5*Ud|m z%v51i#vGZrf3ZGBcW)Pml{K&uHGEp6eZJb1Z;0pOfq*=LR}{yq)Ca}izon*lK4e65 zmtH`p1n!@cKf% z&GliKa-1HX;gI6xq-(u_pmL6TX;>SNIWy6JxJwcGaBvd}zWOSRnb?i0BoU^tgq`&l z8<&M?1SFi7JmTdxFM_5^z+*y7#1hrq8k# z8hYAG)B|vSVIHJ#j%}c++$vHxPTL&(*OV4eIshuv0{(MN@8=+WX}#l~5WA5S&jP-G zsB|iXubnZc?7Sr|;icR1gDPN$FSVkl;gy?=2XeN6J?nOAQNr|A;`zn5E<8dCLKann z08>FltF?66Xc5GC24K0A5b!}d0%1dYY6L-LU!wuP;9MOVBGCDi$30(H8gt zEcNk+jfIPuC2j5*1poi9kIx_j!2hpdr3B3A=&0auq5Hqp%^#Vz8MoXnt}d%1K_*QD zC54)bu#W^Y!MVmV#O%lpetmq;DsAnsbM8Z_b*T|ntw=0#?R8Y{s23uNW0F-Xu0m17 zs1ZY-!6b$-u!@I{!J&o5E1e62A=;1$*YhzHLJLzAK%ZD5l|&K3l2e77!SN@dfdiUa z2DT&S%0r==Q!v1va8t*^Casp5u%~$Fsz)eZ+w?7Kdg|8)DRoUEfxvAC)#)KOi7n}H z(>YoZS=qeD4e!gMa+?PK1UoTwgQ&n0YtEZk-324#?-Uc3pauSW)(5K+#1)HZCxDKC z01HB2@LqvYAI^`3MC37QLKItUCjbDaknulU45U2}lcn{eI#LCIm_i5=K@K*|<2#ES z4{EOjYgRf!rHRJegpy}Mx}y$g0)dG=Q;U^@$oJ;EnqXcgn4Mh00j7b*j0NaYvma8NZGPy~*F6v8nz8%F2=K>(#m!7{> z$N3&M`kmuKx9cuh9_9dt?h3tqT~wjI9PQLT&K}$qQ&M~wf#=?j?b!+r7DLYTzWAK9 z|26Sd7a9*8{Q=?aL>Eo?rE`q&>86c26Q)sW{6m6P0%JE?4rUUi7ZVyw8)95XCs8c# z;=L(d?Ex_s&ENItc`32qAb?lkq1!UB6tAT=#9f*#g}8Hh`#GCxDul`j2j{af0xyUT zmg=et(cE!_F^7xa@T<<=_wZzcUf+v7_5Wx#>a&*{F|}D!uH1Y)gv*Ki)HDy5j^7Kg z(@Rsf4$8h>S_+NtD{o$JVBnuZ*xfgn^A+?~1Hq2XklxVCYG*%6TL2H%DWWka-rlzE z-b~d&H$ZJ=t?lgFni>IO=*n6dKRI}#I8%2b`a*GcJ2yD8N{cIRNtsmQ?BnF!SFd?h zR$ky6g1z=Lu)E6fdoX(9#?rMla*^>rkV_%;7fd-zd9RA1##;57=805U4jdH13!3UV zaetJi`12k_D7UF@0sdBqRRCBmG&brSgkYvwtVr%)j z6=M+ZW|?!qEr7Uikz01DdYXpyyWCQp`lEECWFt*SjgZy@qFYXes1t-sUA0szFbZR~ z^6E9V^}AkpDIjOeZvg=&q46@kR3^`gb`hV7k#1$ZC+p9opS=+x6EM8*has>VB$7+{ zU57WY7jrHcTvWn{U9v|#S<_-+L?oEPPrVlsqYzn6DS%EbaD%>qu^c)F23EOW7nNr{ z-8s*k5yY7*7Dr*QV48W@>);cTGcQMhcEG?YmfgkyI7cF|p2-h!4TYTvHsEg9X+XF#Pj>or8I!^_jP!GLFx$i9xxBw-6e%FPkZY?Y;r z30b+L(4sy^{m$*3g93fS2wZGvQ$(FZQbhPUXW_c=?Z5n zELy_ZsU$kc#A*O5tZgK5^puJYMW&@hm<1^6rmyH|d$V5BMl{<;k9vstY}HNh&_}i= zO^5HrVDE^S*3)Wm5m_Jt-LS-*D3yZBmlpEvx+`%IcR>k~1jpgz`Tnrj*QF2ZsAJ|5 z63{wzow(+KC6CLJw$&B7&>YZ0>=A1C;MpLKlhaGJZkUP+7vQDZX2V+PShbunNb0Pe zaP&T5{=VI!)W&2K`{z z+!9Hg$q0(ZQ)Yj+R5VpVA~arh^lnkT4>|8xFH*HqVp9Plj38Mf+h|m#EVrj@XPso$ zqTb)Q#9ca(CPKdDM-~%jZy<+6%Fm?Bj;^eQIBgXhX>{Y64wO#_-RBF_D82_O25bP4 zI9ZEWK3h5qRjGMv2Daao#jqCFckCuNI1h3>kz>WA>GeHb*}0*}u|xUdp1ppa-N$!IGs`AnT7 zUv;vPy1MC!L4LWP^U~uu10RQyCtJAQz zywMVrxX}mxW6A5PK0x8h3eqo3kVrC$U6MTf2s@**dh)xn#F%2S1;~uNF$-@CT=9Mt zo$33W_;Oa+UtZ7(K=5G4T+GgOl{vir`a62^;vtY%dcIq8cLEJqIJNI3LKCCD9N+Gr%4zB8aild9?f6t6qkHQhh_5(WII~fUEhJI6 zas6sLZX-~{vATlp*Wc4R( zy!<>|^yE#+07)cViAz#c7S8l%MFU^jtBt+6DJ9)LiDDad^*t^-66x_o&r7=tSvkRE z!XG-2e}pL)IZOhPpz%Tq^@Jkm^MVu;*@>UEdx44dcp*a?0NKnyFKHED#C^4p_^n93 zziCH9fS#cJ@ZgV+m$lC9`8P1e#G}Vcch+>in7~$3QqE(VE6SS-bPyd^BuehbqWdkv zzPl&~RgJIr8&BSR2Qm7}_ZdsaHTdO|HE*%o=Gbk5*H?IfZoA`ISw8aS1;k7k@~IQ$ zHKx^N``(gQK-w`v?EZ-OfWcJTeI$t&Y#(RJnHgWcL*OVV=NOx;AuDE{0qj0B$4Paj zem}%dg2747R})O6#+G2?)5FogEdxzLhyzI>wjin+=^TjUJPxoILInlQVnz+G@9+8f zy&eU)zY4PrbfY7)o9p}3>W{d(%@E+r%UR;m>;K3sj;!)#0uH{GHKjAl(fQuad zle?=Z-dPR!ce&d9mcNkFh;k>YMzV z$Eyd1@uqi<{ZXzr@fE;S5p&NV(kHoc)dAim!gV^vfNfpP3EjLRb-f z??5AJi~8jmE0YlDvufO$*!kznohhJwO8W4(3Kthq@x+x>to73dYSw=BUm~r<2`0@r zz}i_$uJS`Zt_rH~a;6$PjpnZd8neebWY7))0 zww`|Kc;oZts0ALUi^d>oa-UPOF1TbmV0kFgON+(PdKRe|symkFq+XiHHKWxz(om@6coKzQ#hd7SWcbrTN6wwc`h=VLrVZC>eq_=`@q&12xI2+EMEjZkSDL zz*)A;3$B|vIlwmf$n~;2oVu{sw{aP09Zl;QOS^k4@r~B-gzv=%q&J%bEZ|pU1RI<6h51-ah~CYWef? zKp%l?xo`%BOJHc9#68NN1G+d+zn38YxF!87jOVWG*Uj#Hg7fCJ*FWFnWj*c;=WSfw z1-#{>YCWxML6`Uf9EgGL4(w+#w)UAbr)0mw`wr9jdN?H9ZM_###sl;l&TFuW)6o*& zbYTRU@Nz-){D^Vx5D|c-Dt+~d$24-S$#pQ+P7lr=%9C)f_bWTTL8w2QpD(4^#{|j| zCn^cQApItWD`_ErKy8(T*_k-lSwzK|8O6BR*hHB{*f|+FnT0r+#aYCeIfaG!33>m| z6TSaMvI&`)S<{5TL1_RJ^h3a?p$|T{d99Ro6`5*EE;DQhVtYe8lC-vTyg@mdo`hD$ z64!Kzh9t*bSq8nwv~d|5d08D30tPY2oTT$qO}eml^YB;o;BM;ysMq~O*FxB>grIni zU|sbHS&l$s3E}iwue|h4^Kg9ia9rg=!0P!Jsrd*|^|0Ud&}4d;0YaETMwo$0n88k3 z73gPX8!KkZA=PRzmE*Y#h2&-ecSk y26sj1#||mwVwU|-*@IZDCj=g)r0d>9UPW@czIGBE{l=>Gvf>~`}2 diff --git a/doc/Projects/2018/hw2/pdf/hw2.tex b/doc/Projects/2018/hw2/pdf/hw2.tex index bb1c897c9..8670ad27a 100644 --- a/doc/Projects/2018/hw2/pdf/hw2.tex +++ b/doc/Projects/2018/hw2/pdf/hw2.tex @@ -181,11 +181,13 @@ Discuss these quantities as functions of the variable $\lambda$ in the Ridge and \subsection*{Exercise 5} Using the singular value decomposition, show that the variance of the direction vector -$\hat{z}_i=\hat{X}\hat{v}_i$ is equal to (equation (3.49) of Hastie \emph{et al.}) +$\hat{z}_i=\hat{X}\hat{v}_i=\hat{u}_1d_1$ is equal to (equation (3.49) of Hastie \emph{et al.}) \[ \mathrm{Var}(\hat{z}_i)=\frac{d_i^2}{N}, \] -where $d_i$ are the singular values of the matrix $\hat{X}$. Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. +where $d_i$ are the singular values of the matrix $\hat{X}$. In Hastie \emph{et al}, the matrix elements of $X$ are centered. The consequence is that the mean values of for example $\hat{u}_i$ are zero. + +Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. % ------------------- end of main content --------------- diff --git a/doc/src/Projects/2018/Exercises/hw2.do.txt b/doc/src/Projects/2018/Exercises/hw2.do.txt index 2b6e8071f..987cef3ec 100644 --- a/doc/src/Projects/2018/Exercises/hw2.do.txt +++ b/doc/src/Projects/2018/Exercises/hw2.do.txt @@ -54,10 +54,12 @@ Discuss these quantities as functions of the variable $\lambda$ in the Ridge and ===== Exercise 5 ===== Using the singular value decomposition, show that the variance of the direction vector -$\hat{z}_i=\hat{X}\hat{v}_i$ is equal to (equation (3.49) of Hastie *et al.*) +$\hat{z}_i=\hat{X}\hat{v}_i=\hat{u}_1d_1$ is equal to (equation (3.49) of Hastie *et al.*) !bt \[ \mathrm{Var}(\hat{z}_i)=\frac{d_i^2}{N}, \] !et -where $d_i$ are the singular values of the matrix $\hat{X}$. Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise. +where $d_i$ are the singular values of the matrix $\hat{X}$. In Hastie *et al*, the matrix elements of $X$ are centered. The consequence is that the mean values of for example $\hat{u}_i$ are zero. + +Give an interpretation of these results, in particular in connection with the variance of the coefficients you obtained in the previous exercise.