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## Applied Least Squares

by: Jordane Kemmer

8

0

19

# Applied Least Squares ST 708

Jordane Kemmer
NCS
GPA 3.79

Hughes-Oliver

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COURSE
PROF.
Hughes-Oliver
TYPE
Class Notes
PAGES
19
WORDS
KARMA
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## Popular in Statistics

This 19 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 708 at North Carolina State University taught by Hughes-Oliver in Fall. Since its upload, it has received 8 views. For similar materials see /class/223956/st-708-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15
JM Hughes Oliver ST708 Applied Least Squares Remedial Measures for Collinearity Impact of collinearity wrt inferential goals 0 causes variance inflation of estimated parameters collinearity creates serious problems if the purpose of the regression is to understand the process to identify important variables in the process or to obtain meaningful estimates of the regression coefficientsquot Rawlings Pantula Dickey 1998 p 458 o ineffective determination of relative importancequot of explanatory variables The best recourse to the collinearity problem when the objective is to assign relative importance is to recognize that the data are inadequate for the purpose and obtain better data perhaps from controlled experimentsquot Rawlings Pantula Dickey 1998 p 446 o no effect on precision of estimated responses and predictions at observed points in the Xspace If the regression analysis is intended solely for prediction of the dependent variable the presence of near singularities in the data does not create serious problems as long as certain very important conditions are met Rawlings Pantula Dickey 1998 p 457 Fall 2006 JM Hughes Oliver ST708 Applied Least Squares 0 causes variance inflation of estimated responses and predictions at unobserved points in the Xspace OLS estimator is BLUE 0 collinearity maintains unbiasedness o collinearity causes increase in variances Sacri ce unbiasedness to get smaller variance M Biased Regression 0 improved wrt MSE estimation of parameters 0 improved wrt MSE estimation of estimated responses at unobserved points in certain regions of the Xspace 0 still ineffective determination of relative importancequot of explanatory variables Some biased regression approaches 0 Principal C0mponents Regression o Ridge Regression 0 Partial Least Squares Fall 2006 2 JMHughes Ol139ver l l ST708 Applied Least Squares 0 Assuming X is n x p 1 with a column of ones Y 2 X3 e 1neo XB e where X is n x p X 1n X 3 30 e3939 1neo X XCB e where Yquot 1nY1 Life 1an is n x p XC X Y is centered version of X ln o TV XeD12 D126 e where D diagXch r WW 1n Z 6 1n Z5 e and Z is centered and scaled version of X 0mitting the intercept column 0 Z UL12V by singular value dec0mposition o W ZV converts the p columns of Z into p principal components corresponding to eigenvalues A1 2 2 AP columns of W are orthogonal reasonable to drop principal components with small eigenvalues o Y 1n7 Z66 ltgt Y 1717 ZVV 5e 1717 W7 e vv W 397 Fall 2006 JM Hughes Oliver ST708 Applied Least Squares Steps 1 Regress Y on W to get P 2 Eliminate all principal components that a have condition index gt 10 and b have nonsignificant coefficients 73 Endproduct is Vg and quot79 3 Convert back to centered and scaled variables A 1 59 Vg39yg wuth var5g VgLgVgMSE where MSE is from the reduced model this is what SAS does 4 Convert back to original variables A am 2 D 12Vg39Yy D 12535 Why is this called biased regression A 5a Vlt9gtVlt39g50LS gt E5g Vlt9gtV39g5 75 5 In fact bias 2 Ema 5 Vs39ys where V5 is the set of principal c0mponents dropped earlier and 75 is the corresponding set of coefficients By only omitting nonsignificant nyi s in step 2 we ensure a Small bias Fall 2006 4 JM Hughes Oliver ST708 Applied Least Squares Algae Example 13degree nonorthogonal case Summary 0 VIF flags all explanatory variables Drop them all 0 COLLINOINT flags 9 condition indices gt 10 with all explanatory variables appearing in the flagged principal c0mponents 0 Test if we can drop the bott0m 9 principal c0mponents RMSE9214 9 RMSE2149 RMSE2 pvalue PrF194 gt 26809 2 04781 F 26809 Cannot drop bott0m 9 principal c0mponents 0 Test if we can drop the bott0m 8 principal c0mponents F 03184 pvalue PrFf4 gt 03184 2 094571 Drop bott0m 8 principal c0mponents 0 After dropping bott0m 8 principal c0mponents get estimate as Fall 2006 5 JM Hughes Oliver ST708 Applied Least Squares 6a ampe03 I intercept day 375067 39956 05671 02522 options nodate ls85 ps25 nonumber data algae input day density 00 datalines 1 530 1 184 2 1183 2 664 3 1603 3 1553 4 1994 4 1910 5 2708 5 2585 6 3006 6 3009 7 3867 7 3403 8 4059 8 3892 9 4349 9 4367 10 4699 10 4551 11 4983 11 4656 12 5100 12 4754 13 5288 13 4842 14 4969 14 5374 title1 height run n00004 00002 15in quotAlgae Example Polynomial Regression 0 0 Fall 2006 JM Hughes Oliver ST708 Applied Least Squares data algae2 set algae dayday75 day2dayday day3day2day day4day3day day5day4day day6day5day day7day6day day8day7day day9day8day day10day9day day11day10day day12day11day day13day12day run 13degree polynomial based on quotday75 variable Principal components regression proc reg dataalgae2 outestfixcoll noprint title2 height15in quot13degree polynomial around day 75 title3 height15in quotPrincipal Components Regressionquot model densityday day2 day3 day4 day5 day6 day7 day8 day9 day10 day11 day12 day13 vif collin pcomit1 to 12 edf outseb run proc print datafixcoll run data testdrop set fixcoll if n1 or type IPCquot if n1 then do mseOrmsermse edfOedf end retain mseO edfO run data testdrop set testdrop jn1 edfedfOj ftestrmsermseedf mseOedfO jmseO pvalue1cdf F ftestjedf0 run proc print run Fall 2006 7 m 9 Algae 5mm 39Po norm39aIRegression 13degree po nomialamund day 75 Pn39ncigal Components Regression r nunnn7271 nunnn11 75 7975911 U unnn31 118151 1 15671 7 nunnn1aa1 8 22511911 8 5835911 8 13127911 1 162mm 71 5295911 71 7135911 m 9 n unnn31 n unnn11 n unnn31 Algae bmmple Paynomp39amegmssmn 13degree po nomialamund day 75 Pn39ncigal Components Regression runnnn7271 757576611 736562611 716256E7M 73 1562911 runnnnnnm 72 7115911 717135911 JM Hughes Oliver ST708 Applied Least Squares Ridge Regression 0 Assuming X is n x p 1 with a column of ones Y 2 X3 e 1neo XB e where X is n x p X 1n X 3 30 e3939 1n130 XBi XCB e where Y 1nY1 1nf2 blip is n X p XC X Y is centered version of X 1n130 XHk KGB U2 D126 6 where D diagX Xc r WW 1n Z 5 1n Z5 e and Z is centered and scaled version of X 0mitting the intercept column 0 Ridge Estimator fr0m Ridge Factor k 2 0 is 5kt Z Z kI1Z Y 150k 2 13 an mm L Bok YBk 5kD 125k o How to choose k3 k p MSE Z Z150j2 Hoerl Kennard Baldwin 1975 Fall 2006 1 JM Hughes Oliver ST708 Applied Least Squares Ridge Trace plot Bk3j versus k for each j 1 p Select k when estimates stop changing rapidly not when get a flat line When VIFj stabilizes wrt k for each j 1 1 Why is this called biased regression Soc 2 Z Z kI 1Z Z30LS gt bias 2 E5k 5 z z kI1Z Z I 5 kZ Z Ian 16 so that bias increases as Ridge Factor k increases Variance decreases as Ridge Factor k increases var5k 02Z Z kI1Z ZZ Z kaI1 Why is this called a shrinkage estimator 0 50c gt 0 as k increases 0 Bayesian interpretation prior is 6 N 0 2 2I large k indicates strong belief that 6 z 0 prior d0minates small k indicates little prior knowledge data d0minates Fall 2006 2 JM Hughes Oliver ST708 Applied Least Squares Algae Example 13degree nonorthogonal case Summary 0 VIF flags all explanatory variables Drop them all COLLINOINT flags 9 condition indices gt 10 with all explanatory variables appearing in the flagged principal c0mponents 0 PCR drop bott0m 8 principal c0mponents get estimate as I intercept day day2 day3 day4 zz 375067 39956 01968 00065 00023 00004 0 0 0 0 0 se6 05671 02522 00349 00075 00003 00002 0 0 0 0 0 o Ridge Reg n what k to use k 21077 x 10 9 Hoerl Kennard Baldwin 1975 Ridge Trace k 001 VIFj k 004 I choose this one Get estimate as I intercept day day2 day3 day4 7004 377660 39375 02762 00001 00000 00002 0 0 0 0 0 se 004 07919 03081 01152 00166 00030 00003 0 0 0 0 0 Fall 2006 JM Hughes Oliver ST708 Applied Least Squares options nodate ls85 ps25 nonumber data algae input day density datalines 1 530 1 184 2 1183 2 664 3 1603 3 1553 4 1994 4 1910 5 2708 5 2585 6 3006 6 3009 7 3867 7 3403 8 4059 8 3892 9 4349 9 4367 10 4699 10 4551 11 4983 11 4656 12 5100 12 4754 13 5288 13 4842 14 5374 14 4969 title1 height15in quotAlgae Example Polynomial Regressionquot run data algae2 set algae dayday75 day2dayday day3day2day day4day3day day5day4day day6day5day day7day6day day8day7day day9day8day day10day9day day11day10day day12day11day day13day12day run Fall 2006 JM Hughes Oliver ST708 Applied Least Squares 13degree polynomial based on quotday75 variable Ridge regression proc reg dataalgae2 outestfixcoll noprint title2 height15in quotisdegree polynomial around day 75 title3 height15in quotRidge Regressionquot model densityday day2 day3 day4 day5 day6 day7 day8 day9 day10 day11 day12 day13 ss1 ss2 vif collinoint ridge0 to 002 by 001 edf outseb outstb outvif plot ridgeplot vref0 nomodel nostat title4 height15in quotRidge Tracequot run data choosek set fixcoll run proc gplot datachoosek where type RIDGEVIFquot and day13lt30 plot dayday13 ridge overlay legend title4 quotRidge VIFs run proc print datafixcoll title4 run data chooseka set fixcoll if type PARMS run data choosekb set fixcolldropin p edf rsq if type RIDGESTBquot and ridge000 run data choosek merge chooseka choosekb kinrmsermse day2day22day32day42day52day62 day72day82day92day102day112day122day132 edfrmsermse1rsq run title4 quotHoerl Kennard Baldwin Recommendation for k proc print run Fall 2006 5 EsHmnH The REG Procedure A gae Enamp e Po ymrmal Regresswon Iiidegree Dnlynnmlal arnund day 75 Rm an as Regressl Ridge Yrace Hr rn a PM 00025 00050 00075 00100 00125 Ridge r ay r dayz r daya 7 day4 r days r day dayE 7 gt7 gt days gt6er dayll gtltgtltrgtlt daylz gtltgtltrgtlt dayla 00m 00175 00200 7 dayS 44 dayl day so A gae Emmp e Po ymrmal Regressm 00020099 0m00mm ar0un0 000 05 0009 9009550n 0009 000 0 X 0 0 0 K 0 0 0 r Y Q i 1 0 x g x g 7 0 a0 i i 000 iiiimi gt x x 0 0 0 0 0 0 0 0 0 002 0000 0000 0000 0000 0002 0000 0 000 0000 0020 0009 regress0n 00n0000 03009 PL Y day day2 day3 day4 day5 day6 day7 day L39days daylo gtltgtltgtltdayll XXXdayIZ XXXdayla 545mm Algae Enmple Polynomial Regressmn 1141211122 polynomial around day 75 nge Regression Obs MODEL 1va DEFVAR RIDGE FOOMIT RMSE Intertept any aayz day aayn days days uay7 daya days aaym any day12 day13 density IN F EDF R50 1 MODEM FARMS 321m D212D7 3D3499 D311 D13 D13 DD4 VEI D4 VEI DD DDD DDD VEI DD VEI DD DDD DDD VEI DD 71 13 14 14 D99D91 2 MODEM SEE 321m D212D7 D13527 D232 D1D D23 DD3 DD3 DD1 DD1 DDD DDD DDD DDD DDD DDD r1 1 MODEM D1DOEV1F mm D DDD 391 773 4332 2D 524275 D3 DD3D79 DD 54373D99 72 24912D55 73 111D72DDD4 9D 14D332757 42 52D5DD95D7 D7 13D3D2D2412 52931D3D3719 24351249 5D 742D323DD 1D 71 4 MODEM D1DOE 321m DDDD D212D7 3D3499 D311 D13 D13 DD4 VEI D4 VEI DD DDD DDD VEI DD VEI DD DDD DDD VEI DD 71 5 MODEM D1DOESTD 321m DDDD D212D7 DDDDDD D793 r117 1D41 1414 711931 779 33 53D 53 195 D3 427B 53 3212813 13D7 34 D3 95 7493 3D 71 s MODEM D1DOESED 321m DDDD D212D7 D13527 D232 D1D D23 DD3 DD3 DD1 DD1 DDD DDD DDD DDD DDD DDD r1 7 MODEM D1DOEV1F 321m DDD1 15D35 23 4D 1D7 3D 73 7D 33 32 3313 5714 33 D9 193D 511 512 47 4D 55 9D 71 5 MODEM D1DOE 321m DDD1 D22334 37D737 D413 VEI D3 VEI DD DDD VEI DD DDD DDD VEI DD DDD VEI DD DDD DDD VEI DD 71 5 MODEM D1DOESTD 321m DDD1 D22334 DDDDDD 1D31 VEI 2D D13 DD5 VEI D3 DD3 DD3 VEI D1 DD7 VEI D1 DD2 DDD VEI D3 71 1o MODEM D1DOESED 321m DDD1 D22334 DDD392 DD41 DD2 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 11 MODEM D1DOEV1F 321m DDD2 11 D53 2D34 5441 35D1 1373 19D5 273D 1D93 D5D 349 323 2417 2D13 V1 12 MODEM D1DOE 321m DDD2 D22495 37D3DD D4D3 VEI D3 VEI DD DDD VEI DD DDD DDD DDD DDD VEI DD DDD VEI DD VEI DD 71 11 MODEM D1DOESTD may DDD2 D22495 DDDDDD 1D33 VEI 27 VEI D3 DD2 VEI D3 DD3 DD1 DD1 DD4 VEI D1 DD2 VEI D1 DD1 V1 14 MODEM D1DOESED 321m DDD2 D22495 DDD179 DD35 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 15 MODEM D1DOEV1F 321m DDD3 9337 172D 3417 2379 1133 1445 1343 574 49D 2D7 25D 1744 1794 71 1s MODEM D1DOE may DDD3 D22313 377957 D399 VEI D3 VEI DD DDD VEI DD DDD VEI DD DDD DDD VEI DD DDD VEI DD DDD V1 17 MODEM R1DOESTD 321m DDD3 D22313 DDDDDD 1D19 VEI 23 VEI D2 DD1 VEI D7 DD3 DD1 DD1 DD2 VEI DD DD2 VEI D1 DD1 V1 15 MODEM D1DOESED 321m DDD3 D22313 DDDD31 DD32 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 15 MODEM D1DOEV1F may DDD4 D357 1493 2425 1D44 D7D 11 D5 11 D2 37D 323 247 223 1399 1293 71 2a MODEM R1DOE 321m DDD4 D2273D 37733D D394 VEI D3 DDD VEI DD VEI DD DDD VEI DD DDD DDD DDD DDD VEI DD DDD V1 21 MODEM D1DOESTD 321m DDD4 D2273D DDDDDD 1DD5 VEI 25 DDD VEI DD VEI D7 DD3 VEI D2 DD2 DD1 DDD DD2 VEI D1 DD2 V1 22 MODEM D1DOESED may DDD4 D2273D DD7919 DD31 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD 71 21 MODEM R1DOEV1F 321m DDD5 7394 1323 1D31 1515 729 1DD5 79D 23D 232 21D 2DD 1177 1D12 V1 24 MODEM D1DOE 321m DDD5 D22D33 377399 D3D9 VEI D3 DDD VEI DD VEI DD DDD VEI DD DDD DDD DDD DDD VEI DD DDD V1 25 MODEM D1DOESTD 321m DDD5 D22D33 DDDDDD D994 VEI 25 DD2 DD1 VEI D7 DD2 VEI D3 DD2 DDD DDD DD2 VEI D1 DD2 V1 25 MODEM D1DOESED 321m DDD5 D22D33 DD7D33 DD3D DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 27 MODEM D1DOEV1F 321m DDD3 7195 11D3 15D5 12D7 332 D7D 31D 2DD 177 193 1D4 1D13 D32 71 25 MODEM D1DOE 321m DDD3 D22995 377134 D3D5 VEI D3 DDD VEI DD VEI DD DDD VEI DD DDD VEI DD DDD DDD VEI DD DDD V1 25 MODEM D1DOESTD 321m DDD3 D22995 DDDDDD D9D3 VEI 24 DD4 VEI D2 VEI D3 DD2 VEI D3 DD2 VEI DD DD1 DD2 VEI DD DD2 V1 10 MODEM D1DOESED 321m DDD3 D22995 DD7739 DD29 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 11 MODEM D1DOEV1F 321m DDD7 13793 1D3D 12134 1115 533 7134 4D5 139 14D 177 172 D92 71D 71 12 MODEM D1DOE 321m DDD7 D23132 3713952 D3D1 VEI D3 DDD VEI DD VEI DD DDD VEI DD DDD VEI DD DDD DDD VEI DD DDD 71 11 MODEM D1DOESTD 321m DDD7 D23132 DDDDDD D974 VEI 24 DD5 VEI D2 VEI D3 DD2 VEI D4 DD2 VEI D1 DD1 DD1 VEI DD DD2 V1 14 MODEM D1DOESED 321m DDD7 D23132 DD7721 DD2D DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 15 MODEM D1DOEV1F 321m DDDD 13454 972 1D9D 9D1 511 1377 39D 142 115 132 131 794 321 V1 15 MODEM D1DOE 321m DDDD D23275 373757 D37D VEI D3 DDD VEI DD VEI DD DDD VEI DD DDD VEI DD DDD DDD DDD DDD V1 17 MODEM D1DOESTD 321m DDDD D23275 DDDDDD D935 VEI 23 DD3 VEI D2 VEI D3 DD2 VEI D4 DD2 VEI D1 DD1 DD1 DDD DD2 V1 15 MODEM D1DOESED 321m DDDD D23275 DD73D7 DD2D DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 15 MODEM D1DOEV1F 321m DDD9 3159 D91 931 D72 47D 3D5 335 122 D97 149 153 713 554 71 4o MODEM D1DOE 321m DDD9 D23423 373573 D375 VEI D3 DDD VEI DD VEI DD DDD VEI DD DDD VEI DD DDD DDD DDD DDD V1 41 MODEM D1DOESTD 321m DDD9 D23423 DDDDDD D957 VEI 23 DD7 VEI D3 VEI D3 DD1 VEI D4 DD2 VEI D1 DD1 DD1 DDD DD2 V1 42 MODEM D1DOESED may DDD9 D23423 DD7334 DD27 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD r1 515m 545mm 41 MODEM R1DGEV1F densny DD1D 5898 822 85D 782 435 544 287 1D7 D83 138 145 545 5D2 V1 44 MODEM R1DGE densny DD1D D23575 3754D8 D372 VD D2 DDD VD DD VD DD DDD VD DD DDD VD DD DDD DDD DDD DDD V1 45 MODEM D1DGESTD densny DD1D D23575 DDDDDD D949 D 23 DD8 VD D3 VD D5 DD1 VD D4 DD2 VD D1 DD1 DD1 DDD DD2 V1 46 MODEM R1DGESE8 densny DD1D D23575 DD755D DD27 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 47 MODEM R1DGEV1F densny DD11 5552 753 779 7D7 4D7 493 25D D95 D73 129 139 588 459 71 4a MODEM R1DGE densny DD11 D2373D 375251 D359 VD D2 DDD VD DD VD DD DDD VD DD DDD VD DD DDD DDD DDD DDD 71 4s MODEM D1DGESTD densny DD11 D2373D DDDDDD D942 D 22 DD9 VD D3 VD D5 DD1 VD D4 DD1 VD D1 DD1 DD1 DD1 DD2 V1 50 MODEM R1DGESE8 densny DD11 D2373D DD7543 DD25 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 51 MODEM R1DGEV1F densny DD12 5448 711 714 543 383 448 22D D85 D55 121 133 539 424 V1 52 MODEM R1DGE densny DD12 D23888 3751D2 D355 VD D2 DDD VD DD VD DD DDD VD DD DDD VD DD DDD DDD DDD DDD V1 55 MODEM D1DGESTD densny DD12 D23888 DDDDDD D935 D 22 DD9 VD D4 VD D5 DD1 VD D4 DD1 VD D2 DD1 DD1 DD1 DD2 V1 54 MODEM R1DGESE8 densny DD12 D23888 DD7543 DD25 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 55 MODEM R1DGEV1F densny DD13 5251 555 559 588 351 41D 195 D78 D58 114 128 495 395 71 5s MODEM R1DGE densny DD13 D24D49 375952 D354 VD D2 DDD VD DD VD DD DDD VD DD DDD VD DD DDD DDD DDD DDD V1 57 MODEM D1DGESTD densny DD13 D24D49 DDDDDD D928 D 22 D1D VD D4 VD D4 DD1 VD D4 DD1 VD D2 DD1 DDD DD1 DD2 V1 58 MODEM R1DGESE8 densny DD13 D24D49 DD7548 DD25 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 55 MODEM R1DGEV1F densny DD14 5D59 525 512 54D 342 375 177 D72 D53 1D8 123 459 359 71 so MODEM R1DGE densmy DD14 D24212 375829 D351 VD D2 DDD VD DD VD DD DDD VD DD DDD VD DD DDD DDD DDD DDD 71 a1 MODEM D1DGESTD densny DD14 D24212 DDDDDD D922 D 22 D11 VD D4 VD D4 DD1 VD D4 DD1 VD D2 DD1 DDD DD1 DD2 V1 52 MODEM R1DGESE8 densny DD14 D24212 DD7557 DD25 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD 71 s MODEM R1DGEV1F densmy DD15 49DD 59D 572 498 325 345 15D D55 D48 1D2 118 427 348 71 s4 MODEM R1DGE densny DD15 D24375 3757D1 D359 VD D2 DDD VD DD VD DD DDD VD DD DDD VD DD DDD DDD DDD DDD 71 s5 MODEM D1DGESTD densny DD15 D24375 DDDDDD D915 D 21 D11 VD D4 VD D4 DDD VD D4 DD1 VD D2 DD1 DDD DD1 DD1 71 es MODEM R1DGESE8 densmy DD15 D24375 DD757D DD25 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 67 MODEM R1DGEV1F densny DD15 4743 558 537 451 31D 32D 147 D52 D45 D97 114 398 328 71 ea MODEM R1DGE densny DD15 D24541 37558D D355 VD D2 DDD VD DD VD DD DDD VD DD DDD VD DD DDD VD DD DDD DDD V1 55 MODEM D1DGESTD densmy DD15 D24541 DDDDDD D91D D 21 D12 VD D4 VD D3 DDD VD D4 DD1 VD D2 DD1 VD DD DD1 DD1 71 7a MODEM R1DGESE8 densny DD15 D24541 DD7585 DD25 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 71 MODEM R1DGEV1F densny DD17 4595 53D 5D5 429 295 297 135 D58 D42 D93 11D 373 311 V1 72 MODEM R1DGE densny DD17 D247D8 375453 D354 VD D2 DDD VD DD VD DD DDD VD DD DDD VD DD DDD VD DD DDD DDD V1 75 MODEM R1DGESTD densny DD17 D247D8 DDDDDD D9D4 D 21 D12 VD D5 VD D3 DDD VD D4 DD1 VD D2 DD1 VD DD DD1 DD1 V1 74 MODEM R1DGESE8 densny DD17 D247D8 DD77D4 DD25 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 75 MODEM R1DGEV1F densny DD18 4457 5D5 478 4DD 283 275 125 D54 D39 D89 1D5 35D 295 71 7s MODEM R1DGE densny DD18 D24875 375351 D352 VD D2 DDD VD DD VD DD DDD VD DD DDD VD DD DDD VD DD DDD DDD V1 77 MODEM D1DGESTD densny DD18 D24875 DDDDDD D899 D 21 D13 VD D5 VD D3 DDD VD D4 DD1 VD D2 DD1 VD DD DD1 DD1 71 7a MODEM R1DGESE8 densny DD18 D24875 DD7725 DD25 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD V1 75 MODEM R1DGEV1F densny DD19 4327 482 454 374 271 258 115 D51 D37 D85 1D3 33D 282 71 Do MODEM R1DGE densny DD19 D25D42 375242 D35D VD D2 DDD VD DD VD DD VD DD VD DD DDD VD DD DDD VD DD DDD DDD 71 a1 MODEM D1DGESTD densny DD19 D25D42 DDDDDD D894 D 21 D13 VD D5 VD D2 VD DD VD D4 DD1 VD D2 DD1 VD D1 DD1 DD1 V1 32 MODEM R1DGESE8 densny DD19 D25D42 DD7747 DD24 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD 71 a MODEM R1DGEV1F densny DD2D 42D4 451 431 351 25D 241 1D8 D49 D35 D82 1DD 312 259 V1 84 MODEM R1DGE densny DD2D D2521D 375138 D348 VD D2 DDD VD DD VD DD VD DD VD DD DDD VD DD DDD VD DD DDD DDD V1 85 MODEM D1DGESTD densny DD2D D2521D DDDDDD D889 VD 2D D14 VD D5 VD D2 VD DD VD D4 DD1 VD D2 DD1 VD D1 DD2 DD1 71 as MODEM R1DGESE8 densny DD2D D2521D DD7771 DD24 DD1 DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD r1 Hummus 518m 545mm Algae Enmple Polynomial Regressmn 1J degreepolymrmal around day 75 nge Regression Hoerl Kennard Baldwin Recommendation for k lobsl MODEL I TYPE I DEPVAR I RIDGE I PCOMIT I RMSE Ilmerceptl day I aayz I days I day4 I days I days I day7 I days I days I uaym day I any12 I day1G ldensltyl IN I F I EDF I list I k I 1IMODEL1 I WDGESTE I densny I n I I 5121287 I n I 5179318 I71 15595 I1n4nsa I 141412 I711931n I7793252 I558555 I 195558 I7127n55 I7212 E53 1351754 I 839458 I74955n5 7139 13 I 14 I 14 Inaanm I21D7EE79 I 51 ml

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All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com