non-linear regression
DESCRIPTION
Non-linear regression. All regression analyses are for finding the relationship between a dependent variable (y) and one or more independent variables (x), by estimating the parameters that define the relationship. - PowerPoint PPT PresentationTRANSCRIPT
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Non-linear regressionAll regression analyses are for finding the relationship between a dependent variable (y) and one or more independent variables (x), by estimating the parameters that define the relationship.Non-linear relationships whose parameters can be estimated by linear regression: e.g, y = axb, y = abx, y = aebxNon-linear relationships whose parameters can be estimated by non-linear regression, e.g,
Non-linear relationships that cannot be represented by a function: loess
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Commonly Encountered Funtions0246810121416135XYy=x1.5y=x0.5y=x105101520250246810XYy=exy=100e-xy=1000.5x
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Growth curve of E. coliA researcher wishes to estimate the growth curve of E. coli. He put a very small number of E. coli cells into a large flask with rich growth medium, and take samples every half an hour to estimate the density (n/L).14 data points over 7 hours were obtained.What is the instantaneous rate of growth (r). What is the initial density (N0)?As the flask is very large, he assumed that the growth should be exponential, i.e., y = aebx (Which parameter correspond to r and which to N0?)
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SAS Program/* Fictitious data */data Ecoli;Input Density @@;Time = _N_;lnD = log(Density);datalines;20.023 39.833 80.571 161.102 317.923 635.672 1284.544 2569.430 5082.654 10220.777 20673.873 40591.439 81374.642 163963.873;proc reg; var Density lnD Time; model lnD = Time; plot Density*Time/ symbol='.'; plot lnD*Time/ symbol='.';run;Run, and ask students to compute parameters a and b from regression output. What is the initial density?This statement is necessary, otherwise Density will be an unknown variable for the reg procedure and the plot statement will fail.
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Body weight of wild elephantA researcher wishes to estimate the body weight of wild elephants. He measured the body weight of 13 captured elephants of different sizes as well as a number of predictor variables, such as leg length, trunk length, etc. Through stepwise regression, he found that the inter-leg distance (shown in figiure) is the best predictor of body weight.He learned from his former biology professor that the allometric law governing the body weight (W) and the length of a body part (L) states that W = aLb Use linear regression to find parameters a and b.
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SAS Program/*Fitcitious data */data Elephant;Input L W @@;lnL = log(L);lnW = log(W);datalines;0.3 1.657 0.4 2.500 0.5 4.680 0.6 7.075 0.7 10.0700.8 11.988 0.9 14.836 1 18.318 1.1 23.496 1.2 27.8971.3 36.796 1.4 44.611 1.5 50.183;proc reg; var L W lnL lnW; model lnW = lnL; plot W*L/ symbol='.'; plot lnW*lnL/ symbol='.';run;Run and ask students to calculate the parameters a and b.
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DNA and protein gel electrophoresisHow to estimate the molecular mass of a protein?A ladder: proteins with known molecular massDeriving a calibration curve relating molecular mass (M) to migration distance (D): D = F(M)Measure D and obtain MThe calibration curve is obtained by fitting a regression model
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Protein molecular massThe equation appears to describe the relationship between D and M quite well. This relationship is better than some published relationships, e.g., D = a b ln(M)The data are my measurement of D and M for a subset of secreted proteins from the gastric pathogen Helicobacter pylori (Bumann et al., 2002).Write a SAS program to use the data to find parameters a and bBumann, D., Aksu, S., Wendland, M., Janek, K., Zimny-Arndt, U., Sabarth, N., Meyer, T.F., and Jungblut, P.R., 2002, Proteome analysis of secreted proteins of the gastric pathogen Helicobacter pylori. Infect. Immun. 70: 3396-3403.
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Area and RadiusWhat is the functional relationship between the area and the radius?
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Toxicity study: pesticideWhat transformation to use?
Chart2
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Percentage
Dosage
Percentage killed
Sheet1
These are actually my species and the order that they should be in - and the real gtRNA data.. just done manually and not including the repeatmasked numbers
The initial files that I need to input are species specific - so this eg would be a compilation from 24 txt files
PheAsnAspHisSerTyrCysGly
GAAAAAGTTATTGTCATCGTGATGGCTACTGTAATAGCAACAGCCACC
H sapiensall12321191181413015
masked
P troglo.all1130113109213127111
masked
M musculusall7141610181057141
masked
C familiarisall102111329892110
masked
F catusall822157113619128191
masked
B taurusall2874048622318438103531576220
masked
G gallusall91987166105
masked
T rubripesall2024241202161411218
masked
D rerioall19710104524150138710424142365136983611
masked150099610750350540032002500400(eg)
D melan.all81214569714
masked
C elegansall1420271919191316
masked
S cerevisall111116828416
masked
I would also like an output from each species for all tRNA comparing repeat masked and not
Species: H sapiensSpecies: P troglodytes
number of tRNAnumber of tRNA
anticodongtRNAdbRM+RM-anticodongtRNAdbRM+RM-
AlaANN302010AlaANN302010
GNN25205GNN25205
CNN15610056CNN15610056
UNN12310023UNN12310023
GlyANNGlyANN
GNNGNN
CNNCNN
UNNUNN
ProANNProANN
GNNGNN
CNNCNN
UNNUNN
ThrANNThrANN
GNNGNN
CNNCNN
UNNUNN
ValANNValANN
GNNGNN
CNNCNN
UNNUNN
SerANNSerANN
GNNGNN
CNNCNN
UNNUNN
ANNANN
GNNGNN
ArgANNArgANN
GNNGNN
CNNCNN
UNNUNN
CNNCNN
GNNGNN
LeuANNLeuANN
GNNGNN
CNNCNN
UNNUNN
CNNCNN
GNNGNN
PheANNPheANN
GNNGNN
AsnANNAsnANN
GNNGNN
AspANNAspANN
GNNGNN
HisANNHisANN
GNNGNN
(etc)(etc)
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FavorOpposeMarginal
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166.504343892429.8188797975138.228274069623.0258509299483.01665351070.0000000
93.301726284968.002933973959.914645471152.0266916421394.0056396741
1009.4391114293601018030280
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IndXX2Dosagey1y2CDF1CDF2Probit1Probit2PercentageDosagePercentage
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Percentage
Dosage
Percentage killed
-
Probit and probit transformationProbit has two names/definitions, both associated with standard normal distribution: the inverse cumulative distribution function (CDF)quantile functionCDF is denoted (z), which is a continuous, monotone increasing sigmoid function in the range of (0,1), e.g., (z) = p (-1.96) = 0.025 = 1 - (1.96)The probit function gives the 'inverse' computation, formally denoted -1(p), i.e., probit(p) = -1(p) probit(0.025) = -1.96 = -probit(0.975)[probit(p)] = p, and probit[(z)] = z.
Chart3
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CDF2
z
CDF
Sheet1
These are actually my species and the order that they should be in - and the real gtRNA data.. just done manually and not including the repeatmasked numbers
The initial files that I need to input are species specific - so this eg would be a compilation from 24 txt files
PheAsnAspHisSerTyrCysGly
GAAAAAGTTATTGTCATCGTGATGGCTACTGTAATAGCAACAGCCACC
H sapiensall12321191181413015
masked
P troglo.all1130113109213127111
masked
M musculusall7141610181057141
masked
C familiarisall102111329892110
masked
F catusall822157113619128191
masked
B taurusall2874048622318438103531576220
masked
G gallusall91987166105
masked
T rubripesall2024241202161411218
masked
D rerioall19710104524150138710424142365136983611
masked150099610750350540032002500400(eg)
D melan.all81214569714
masked
C elegansall1420271919191316
masked
S cerevisall111116828416
masked
I would also like an output from each species for all tRNA comparing repeat masked and not
Species: H sapiensSpecies: P troglodytes
number of tRNAnumber of tRNA
anticodongtRNAdbRM+RM-anticodongtRNAdbRM+RM-
AlaANN302010AlaANN302010
GNN25205GNN25205
CNN15610056CNN15610056
UNN12310023UNN12310023
GlyANNGlyANN
GNNGNN
CNNCNN
UNNUNN
ProANNProANN
GNNGNN
CNNCNN
UNNUNN
ThrANNThrANN
GNNGNN
CNNCNN
UNNUNN
ValANNValANN
GNNGNN
CNNCNN
UNNUNN
SerANNSerANN
GNNGNN
CNNCNN
UNNUNN
ANNANN
GNNGNN
ArgANNArgANN
GNNGNN
CNNCNN
UNNUNN
CNNCNN
GNNGNN
LeuANNLeuANN
GNNGNN
CNNCNN
UNNUNN
CNNCNN
GNNGNN
PheANNPheANN
GNNGNN
AsnANNAsnANN
GNNGNN
AspANNAspANN
GNNGNN
HisANNHisANN
GNNGNN
(etc)(etc)
Sheet2
B
FavorOpposeMarginal
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6.88282647586.9265704209
DF
u3.83741955581
uA(F)0.01880236141601018030280
uA(M)-0.0188023614Disease PresentDisease absent1202036060560
uB(F)0.09861743421Loc1Loc2Loc1Loc21803054090840
uB(O)-0.0986174342Race144123810104
uAB(F,F)-0.19363923561Race22822201888245.660673733323.0258509299934.7322331602102.03592144991577.7410888874
uAB(F,O)0.193639235672345828192574.499009133859.91464547112118.9974513221245.66067373333543.6445988884
uAB(M,F)0.193639235613062934.7322331602102.03592144993397.4473353615404.98287032975656.0575891434
uAB(M,O)-0.193639235610686
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93.301726284968.002933973959.914645471152.0266916421394.0056396741
1009.4391114293601018030280
|BlackBlondBrownRedTotal632.7794785592255.88233187282012060360560
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Male3216439100
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Sheet3
IndXX2Dosagey1y2CDF1CDF2Probit1Probit2PercentageDosagePercentage
10.1-2.265902295527-1.6891650862-2.36496667180.04559389890.0090158468-1.6891650862-2.36496667180.90270.90
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121.2-0.680918492143-1.0593069185-0.74083128690.14473001260.2293978712-1.0593069185-0.740831286922.944322.94
131.3-0.5456633344-1.002047085-0.60223511050.15816042630.2735088237-1.002047085-0.602235110527.354427.35
141.4-0.542805095144-0.9447872516-0.59930627280.17238375080.2744843328-0.9447872516-0.599306272827.454427.45
151.5-0.522550578744-0.8875274182-0.57855143810.18739750570.28144594-0.8875274182-0.578551438128.144428.14
161.6-0.498983393645-0.8302675848-0.55440210660.20319375550.2896518424-0.8302675848-0.554402106628.974528.97
171.7-0.470768118945-0.7730077513-0.52548987050.21975889660.2996213466-0.7730077513-0.525489870529.964529.96
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191.9-0.352417668446-0.6584880845-0.40421597850.25511227560.3430269524-0.6584880845-0.404215978534.304634.30
202-0.323739911546-0.601228251-0.37482983610.27384398520.3538935116-0.601228251-0.374829836135.394635.39
212.1-0.316889801346-0.5439684176-0.36781051750.29323160310.3565072631-0.5439684176-0.367810517535.654635.65
222.2-0.267683959647-0.4867085842-0.31738921450.31323243050.375474145-0.4867085842-0.317389214537.554737.55
232.3-0.244231895547-0.4294487507-0.29335784760.33379834120.3846243207-0.4294487507-0.293357847638.464738.46
242.4-0.180825394748-0.3721889173-0.2283851070.35487609850.4096734332-0.3721889173-0.22838510740.974840.97
252.5-9.62E-0249-0.3149290839-0.14171514770.37640774930.4436525048-0.3149290839-0.141715147744.374944.37
262.6-6.30E-0249-0.2576692504-0.10763010790.39833108980.4571445565-0.2576692504-0.107630107945.714945.71
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282.8-2.20E-0249-0.1431495836-0.06566056230.44308602310.4738240357-0.1431495836-0.065660562347.384947.38
292.93.85E-0250-0.0858897501-0.00363194050.46577702980.4985510686-0.0858897501-0.003631940549.865049.86
3030.097458477850-0.02862991670.05677281290.48857987590.5226369144-0.02862991670.056772812952.265052.26
313.10.16755098510.02862991670.12859670960.51142012410.5511616150.02862991670.128596709655.125155.12
323.20.1922824353510.08588975010.15393907060.53422297020.56117111130.08588975010.153939070656.125156.12
333.30.2310412865520.14314958360.1936553260.55691397690.57677711290.14314958360.19365532657.685257.68
343.40.2891663705520.2004094170.25321619060.57941980240.59994942210.2004094170.253216190659.995259.99
353.50.2968843528520.25766925040.26112481910.60166891020.60300187370.25766925040.261124819160.305260.30
363.60.3020980212530.31492908390.26646727330.62359225070.60506031970.31492908390.266467273360.515360.51
373.70.3356066476530.37218891730.30080361520.64512390150.61821787430.37218891730.300803615261.825361.82
383.80.3401568773530.42944875070.30546624280.66620165880.61999446180.42944875070.305466242862.005362.00
393.90.3638908439530.48670858420.32978647550.68676756950.62921934640.48670858420.329786475562.925362.92
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414.10.4752570158540.6012282510.44390356620.72615601480.67144384690.6012282510.443903566267.145467.14
424.20.5702938518550.65848808450.54128796080.74488772440.70584544130.65848808450.541287960870.585570.58
434.30.5985572526550.71574791790.57024951190.76292649150.71574576050.71574791790.570249511971.575571.57
444.40.6732349345560.77300775130.64677184950.78024110340.7411101930.77300775130.646771849574.115674.11
454.50.6733942606560.83026758480.64693511120.79680624450.74116302970.83026758480.646935111274.125674.12
464.60.7558137059570.88752741820.73139044580.81260249430.76772965030.88752741820.731390445876.775776.77
474.70.7635332065570.94478725160.73930063020.82761624920.77013776660.94478725160.739300630277.015777.01
484.80.814393052581.0020470850.79141679350.84183957370.78564959331.0020470850.791416793578.565878.56
494.90.8295022315581.05930691850.80689919340.85526998740.79013771821.05930691850.806899193479.015879.01
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515.11.0110534606601.17382658530.99293502210.87976775350.83962918941.17382658530.992935022183.966083.96
525.21.0288352647601.23108641881.01115606460.89085472780.84402912981.23108641881.011156064684.406084.40
535.31.186456079611.28834625221.17267035980.90118727030.8795359891.28834625221.172670359887.956187.95
545.41.225763972621.34560608561.21294921930.9107852060.8874253821.34560608561.212949219388.746288.74
555.51.3584396169631.40286591911.34890216030.91967158730.91131580291.40286591911.348902160391.136391.13
565.61.456950587641.46012575251.44984650560.92787224190.92644933621.46012575251.449846505692.646492.64
575.71.4588461519641.51738558591.4517888940.93541531950.92671984351.51738558591.45178889492.676492.67
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CDF2
z
CDF
-
SAS programData Pesticide;Input Dosage Percent @@;NuProbit = probit(Percent/100);Cards;27 0.90 28 1.39 31 2.40 31 2.49 35 6.42 36 7.78 37 9.1638 10.21 38 11.71 40 16.24 41 16.90 43 22.94 44 27.35 44 27.45 44 28.14 45 28.97 45 29.96 45 30.50 46 34.30 46 35.39 46 35.65 47 37.55 47 38.46 48 40.97 49 44.37 49 45.71 49 46.66 49 47.38 50 49.86 50 52.26 51 55.12 51 56.12 52 57.68 52 59.99 52 60.30 53 60.51 53 61.8253 62.00 53 62.92 54 66.06 54 67.14 55 70.58 55 71.57 56 74.11 56 74.12 57 76.77 57 77.01 58 78.56 58 79.01 59 83.53 60 83.96 60 84.40 61 87.95 62 88.74 63 91.13 64 92.64 64 92.67 66 95.49 68 97.00 69 97.15; Proc reg; Model Percent = Dosage / R CLM alpha = 0.01 CLI; Plot Percent*Dosage / symbol = '.'; Model NuProbit = Dosage / R CLM alpha = 0.01 CLI; Plot NuProbit*Dosage / symbol = '.';run;Run and explainWhy divide Percent by 100?Graphic contrast between the original and the transformed DVCLM: CL of meanCLI: CL of individual observation
-
Commonly Encountered Funtions
-
Non-linear regressionIn rapidly replicating unicellular eukaryotes such as the yeast, highly expressed intron-containing genes requires more efficient splicing sites than lowly expressed genes.Natural selection will operate on the mutations at the slicing sites to optimize splicing efficiency.Designate splicing efficiency as SE and gene expression as GE.Certain biochemical reasoning suggests that SE and GE will follow the following relationships:
-
Guess initial valuesWhen GE=0 then SE = , so 0.4When GE increases from 2 to 8, SE increases from 0.47 to 0.75, so (0.75-0.47)/(8-2) 0.047With 0.4 and 0.047, then SE for GE = 12 should be 0.4+0.04712 = 0.96, but the actual SE is only about 0.77. This must be due to the quadratic term GE2, i.e., (0.77 - 0.96) = 122, so - 0.002
Chart2
0.46
0.47
0.57
0.61
0.62
0.68
0.69
0.78
0.7
0.74
0.77
0.78
0.74
0.8
0.8
0.78
SE
GE
SE
Sheet1
familyInvertebratesVertebrates
Caenorhabditis elegansFruitflyTunicatePuffer fishZebrafishFrogMouseRatHuman
NKX5111433222
DLX113856766
CDX112233323
HOX6810454035393939
PAX284973444
POU352161315141316
LIM777211412121112
Numbers of member genes of several homeobox gene families in animal species
Hypothesis: Gene family size is correlated with phenotypic/organismal complexity
Sheet2
AlaAGCGGCCGCTGC43AlaAGCCGCTGC
29592959
GlyACCGCCCCCTCC31GlyGCCCCCTCC
15791579
ProAGGGGGCGGTGG21ThrAGTCGTTGT
10471066
ThrAGTGGTCGTTGT22ValAACCACTAC
106611165
ValAACGACCACTAC32PheGAA
1116512
AspGTC
19
PheAAAGAA12GluCTCTTC
121313
AsnATTGTT33HisGTG
13211
LysCTTTTT34MetCAT
171720
AspATCGTC19TyrATAGTA
19114
GluCTCTTC26TrpCCA
13139
HisATGGTG11SerAGACGATGAGCT
1111458
GlnCTGTTG32ArgACGCCGTCGCCTTCT
211175656
LeuAAGCAGTAGCAATAA
SerAGAGGACGATGAACTGCT281210377
11458
ArgACGGCGCCGTCGCCTTCT29
75656
LeuAAGGAGCAGTAGCAATAA39
1210377
IleAATGATTAT27
1485
MetCAT20
20
TyrATAGTA15
114
SupresCTATTA3
12
CysACAGCA30
30
TrpCCA9
9
SelCysTCA3
3
Sheet3
3.365.046.728.410.0811.7613.4415.1216.818.4820.1621.8423.5225.226.8828.5630.2431.9233.6
Fit Method kdTree
BlendingLinear
Number of Observations168
Number of Fitting Points168168168168105105105656565656565414141333333
kd Tree Bucket Size1111222333444555666
Degree of Local Polynomials1111111111111111111
Smoothing Parameter0.020.030.040.050.060.070.080.090.10.110.120.130.140.150.160.170.180.190.2
Points in Local Neighborhood3568101113151618202123252628303133
Residual Sum of Squares1.58E-29293.099468.803603.620737.152736.747868.4591057.6091196.4861335.9921451.3301452.3251539.2971584.7431613.2661629.5741629.1161629.8851637.156
Trace[L]16872.32349.11037.10329.59929.44724.73521.14418.78616.82915.24415.17113.86412.64011.82511.06010.36310.3209.723
GCV.0.0320.0330.0350.0380.0380.0420.0490.0540.0580.0620.0620.0650.0660.0660.0660.0660.0660.065
AICC.3.1222.8842.8702.9282.9243.0073.1463.2323.3133.3723.3713.4113.4223.4283.4283.4183.4183.414
AICC1.567.834496.128487.424495.105494.511507.245529.958544.085557.343567.117567.045573.492575.368576.360576.196574.486574.464573.797
Delta1082.286110.055124.411132.582132.764138.596142.590145.565148.009149.973150.060151.668152.938153.969154.928155.724155.774156.525
Delta2078.612108.329123.533130.021130.209136.865140.298143.795146.623148.850148.941150.739151.587152.804153.917154.664154.721155.596
Equivalent Number of Parameters16858.93240.27530.61623.78023.65820.06716.87715.13713.66612.46012.40111.39610.2189.6189.0498.4498.4147.972
Lookup Degrees of Freedom.86.131111.809125.295135.193135.370140.349144.919147.358149.408151.104151.187152.603154.301155.142155.946156.790156.835157.459
Residual Standard Error.1.8872.0642.2032.3582.3562.5032.7232.8673.0043.1113.1113.1863.2193.2373.2433.2343.2353.234
n s/ 5
0.6721.0081.3441.682.0162.3522.6883.0243.363.6964.032
ab
xy58xyylnYSUMMARY OUTPUT
000109.90777500442.2933198029
0.10.277777777812020.01980352332.9967219598Regression Statistics
0.20.384615384624039.93730567413.6873108664Multiple R0.9999986922
0.30.441176470638080.22946203424.3848908044R Square0.9999973844
0.40.47619047624160159.13598182555.0697590683Adjusted R Square0.9999971832
0.50.55320320.60640764975.7702142264Standard Error0.0052043413
0.60.51724137936640644.21945828796.4680394418Observations15
0.70.5303030303712801273.2868469357.1493569046
0.80.5405405405825602562.13554287297.8485963862ANOVA
0.90.5487804878951205112.42618449178.539429362dfSSMSFSignificance F
10.5555555556101024010270.7434834399.237054694Regression1134.6197795551134.61977955514970239.749071220
1.10.5612244898112048020423.75318332139.9244538742Residual130.00035210720.0000270852
1.20.5660377358124096040864.389560084210.6180142919Total14134.6201316622
1.30.5701754386138192081612.379694820511.3097362414
1.40.573770491814163840164248.2057420212.0091340123CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
1.50.5769230769Intercept2.30329518080.0025584336900.275547984602.29776802122.30882234052.29776802122.3088223405
1.60.5797101449xylnYx0.69338656390.00031101892229.403451390400.69271464850.69405847930.69271464850.6940584793
1.70.5821917808120.0232.9969
1.80.5844155844239.8333.6847a =10.0071034002
1.90.5864197531380.5714.3891b =0.6933865639
20.58823529414161.1025.0820y=a*e^(bx)
2.10.58988764045317.9235.7618exp(b) =2.000478824
2.20.59139784956635.6726.45471.9989059382
2.30.592783505271284.5447.1582
2.40.594059405982569.4307.8514
2.50.595238095295082.6548.5336SUMMARY OUTPUT
2.60.59633027521010220.7779.2322
2.70.59734513271120673.8739.9366Regression Statistics
2.80.59829059831240591.43910.6113Multiple R0.9999977612
2.90.59917355371381374.64211.3190R Square0.9999955224
30.614163963.87312.0074Adjusted R Square0.9999951492
3.10.6007751938Standard Error0.0063862489
3.20.6015037594Observations14
3.30.602189781
3.40.6028368794ANOVA
3.50.6034482759dfSSMSFSignificance F
3.60.6040268456Regression1109.3003886925109.30038869252679970.583669860
3.70.6045751634Residual120.00048941010.0000407842
3.80.6050955414Total13109.3008781026
3.90.6055900621
40.6060606061CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
4.10.6065088757Intercept2.30281494620.0036051567638.755859529902.29495998462.31066990782.29495998462.3106699078
4.20.6069364162x0.69313870830.00042340421637.061569908100.69221618990.69406122680.69221618990.6940612268
4.30.6073446328
a =10.0022987961
b =0.6931387083
y=a*e^(bx)
exp(b) =1.9999830556
xylnxlny
0.30.803160565-1.2039728043-0.2192006287
0.41.1339796135-0.91629073190.1257332277
0.52.4318381914-0.69314718060.8886474288
0.63.0625604659-0.51082562381.1192513197
0.74.2531821711-0.35667494391.4476674489
0.86.005437243-0.22314355131.792665266
0.97.2742427818-0.10536051571.9843397227
110.314799622902.3335797205
1.112.25639649740.09531017982.5060479638
1.214.98507490390.18232155682.7070546993
1.317.00764518760.26236426452.8336629599
1.421.00333015250.33647223663.0446810038
1.525.32299803780.40546510813.2317129962
xylnxlnySUMMARY OUTPUT
10.31.657-1.2039728043-0.4117456216
20.41.921-0.91629073190.4916066294Regression Statistics
30.54.680-0.69314718060.9659051663Multiple R0.9966784784
40.67.075-0.51082562381.150692476R Square0.9933679893
50.710.070-0.35667494391.4543939753Adjusted R Square0.9927650793
60.811.988-0.22314355131.8675314511Standard Error0.0914006615
70.914.836-0.10536051572.0496882052Observations13
8118.31802.2052853306
91.123.4960.09531017982.4628797534ANOVA
101.227.8970.18232155682.7124034672dfSSMSFSignificance F
111.336.7960.26236426452.9224165743Regression113.764369002213.76436900221647.62217771160
121.444.6110.33647223663.0246265239Residual110.09189489010.0083540809
131.550.1830.40546510813.1356164515Total1213.8562638924
CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
Intercept2.29535271620.027636559983.054936206702.23452505812.35618037432.23452505812.3561803743
lnx2.12954106220.052463493240.590912501602.01406969232.24501243212.01406969232.2450124321
a =9.9279371388
GESEb =2.1295410622
10.46
20.47
30.57
40.61
50.62
60.68
70.69
80.78
90.7
100.74
110.770.0375
120.780.4
130.74-1100
130.8-1
150.8
160.78
xxia:max(1, ns/5)
xxia:int(n*s)
Sheet3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
SE
GE
SE
MBD0764C653.unknown
-
A few more twistsThe continuity condition requires thatThe smoothness condition requires that The two conditions implies that
-
SAS program/* Fictitious data */data Intron; input SE GE @@; datalines; .46 1 .47 2 .57 3 .61 4 .62 5 .68 6 .69 7 .78 8 .70 9 .74 10 .77 11 .78 12 .74 13 .80 13 .80 15 .78 16 ; title 'Quadratic Model with Plateau';proc nlin data=Intron; parms alpha=.4 beta=.047 gamma=-.002; GE0 = -.5*beta / gamma; if (GE < GE0) then Y = alpha + beta*GE + gamma*GE*GE; else Y = alpha + beta*GE0 + gamma*GE0*GE0; model SE = Y; if _obs_ =1 and _iter_ =. then do; plateau =alpha + beta*GE0 + gamma*GE0*GE0; put / GE0= plateau= ; end; output out=b predicted=SEp; run;
Initial guestimatesWrite out GE0 and plateau (which could be computed from the estimated , , and . However, the output of these parameters contain rounding errors.A conditional statement needed to model the two segmentscomputes predicted values for plotting and saves them to data set bRun and explain
-
Plotproc sgplot data=b noautolegend; yaxis label='Observed or Predicted'; refline 0.7775 / axis=y label="Plateau" labelpos=min; refline 12.7476 / axis=x label="GE0" labelpos=min; scatter y=SE x=GE; series y=SEp x=GE;run;
-
Fitting another function/* Fictitious data */data Intron; input SE GE @@; datalines; .46 1 .47 2 .57 3 .61 4 .62 5 .68 6 .69 7 .78 8 .70 9 .74 10 .77 11 .78 12 .74 13 .80 13 .80 15 .78 16 ; title 'Quadratic Model with Plateau';proc nlin data=Intron; parms alpha=.4 beta=.8 gamma=1; model SE = (alpha+beta*GE)/(1+gamma*GE); output out=b predicted=SEp; run;proc sgplot data=b noautolegend; yaxis label='Observed or Predicted'; scatter y=SE x=GE; series y=SEp x=GE;run;Run and explain the output
-
Plot output
-
Robust regressionLOWESS: robust local regression between Y and X, with linear fittingLOESS: robust local regression between Y and one or more Xs, with linear or quadratic fittingUsed with relations that cannot be expressed in functional formsSAS: proc loessData: Data set 1: monthly averaged atmospheric pressure differences between Easter Island and Darwin, Australia for a period of 168 months (NIST, 1998), suspected to exhibit 12-month (annual), 42-month (El Nino), and 25-month (Southern Oscillation) cycles (From Robert Cohen of SAS Institute)Data set 2: Two-channel microarray data. Background correction and two-channel loess normalization (from Wudu).
-
data ENSO;input Pressure @@;Month=_N_;datalines;12.9 11.3 10.6 11.2 10.9 7.5 7.7 11.7 12.9 14.3 10.9 13.7 17.1 14.0 15.3 8.55.7 5.5 7.6 8.6 7.3 7.6 12.7 11.0 12.7 12.9 13.0 10.9 10.4 10.2 8.0 10.913.6 10.5 9.2 12.4 12.7 13.3 10.1 7.8 4.8 3.0 2.5 6.3 9.7 11.6 8.6 12.410.5 13.3 10.4 8.1 3.7 10.7 5.1 10.4 10.9 11.7 11.4 13.7 14.1 14.0 12.5 6.39.6 11.7 5.0 10.8 12.7 10.8 11.8 12.6 15.7 12.6 14.8 7.8 7.1 11.2 8.1 6.45.2 12.0 10.2 12.7 10.2 14.7 12.2 7.1 5.7 6.7 3.9 8.5 8.3 10.8 16.7 12.612.5 12.5 9.8 7.2 4.1 10.6 10.1 10.1 11.9 13.6 16.3 17.6 15.5 16.0 15.2 11.214.3 14.5 8.5 12.0 12.7 11.3 14.5 15.1 10.4 11.5 13.4 7.5 0.6 0.3 5.5 5.04.6 8.2 9.9 9.2 12.5 10.9 9.9 8.9 7.6 9.5 8.4 10.7 13.6 13.7 13.7 16.516.8 17.1 15.4 9.5 6.1 10.1 9.3 5.3 11.2 16.6 15.6 12.0 11.5 8.6 13.8 8.78.6 8.6 8.7 12.8 13.2 14.0 13.4 14.8;ods output OutputStatistics=ENSOstats FitSummary=ENSOsummary;proc loess data=ENSO; model Pressure=Month / CLM smooth = 0.02 to 0.2 by 0.01 dfmethod=exact;run;symbol1 c=black i=join value=dot;symbol2 c=black i=join value=none;proc gplot data=ENSOstats; by SmoothingParameter; plot (DepVar Pred)*Month/overlay;run;
Three criteria for choosing the smooth parameter:GCV: generalized cross-validation (Craven and Wahba 1979)AIC: Akaike information criterion (Akaike 1973)AICC1: bias-corrected AIC (JUrvich and Simonoff 1988)The output data ENSOstats contains variables: SoothingParameter, Month, DepVar, Pred, etc.Step 1: characterize the most obvious trend.
-
Summary output: kd Tree, linearmax(1, n*s/5)int(n*s), where n is N. Fitting Points and s is Smotthing Param.
Sheet1
familyInvertebratesVertebrates
Caenorhabditis elegansFruitflyTunicatePuffer fishZebrafishFrogMouseRatHuman
NKX5111433222
DLX113856766
CDX112233323
HOX6810454035393939
PAX284973444
POU352161315141316
LIM777211412121112
Numbers of member genes of several homeobox gene families in animal species
Hypothesis: Gene family size is correlated with phenotypic/organismal complexity
Sheet2
AlaAGCGGCCGCTGC43AlaAGCCGCTGC
29592959
GlyACCGCCCCCTCC31GlyGCCCCCTCC
15791579
ProAGGGGGCGGTGG21ThrAGTCGTTGT
10471066
ThrAGTGGTCGTTGT22ValAACCACTAC
106611165
ValAACGACCACTAC32PheGAA
1116512
AspGTC
19
PheAAAGAA12GluCTCTTC
121313
AsnATTGTT33HisGTG
13211
LysCTTTTT34MetCAT
171720
AspATCGTC19TyrATAGTA
19114
GluCTCTTC26TrpCCA
13139
HisATGGTG11SerAGACGATGAGCT
1111458
GlnCTGTTG32ArgACGCCGTCGCCTTCT
211175656
LeuAAGCAGTAGCAATAA
SerAGAGGACGATGAACTGCT281210377
11458
ArgACGGCGCCGTCGCCTTCT29
75656
LeuAAGGAGCAGTAGCAATAA39
1210377
IleAATGATTAT27
1485
MetCAT20
20
TyrATAGTA15
114
SupresCTATTA3
12
CysACAGCA30
30
TrpCCA9
9
SelCysTCA3
3
Sheet3
3.365.046.728.410.0811.7613.4415.1216.80000000000
Fit Methodkd Tree
BlendingLinear
N168
N. Fitting Points1681681681681051051056565
kd Tree Bucket Size111122233
Deg. Local Polynom.111111111
Smoothing Param.0.020.030.040.050.060.070.080.090.1
Neighbor Points3.05.06.08.010.011.013.015.016.0
RSS0.0293.1468.8603.6737.2736.7868.51057.61196.5
Trace[L]168.072.349.137.129.629.424.721.118.8
GCV.0.00.00.00.00.00.00.00.1
AICC.3.12.92.92.92.93.03.13.2
AICC1.567.8496.1487.4495.1494.5507.2530.0544.1
Delta10.082.3110.1124.4132.6132.8138.6142.6145.6
Delta20.078.6108.3123.5130.0130.2136.9140.3143.8
Equivalent N. Param.168.058.940.330.623.823.720.116.915.1
Lookup DF.86.1111.8125.3135.2135.4140.3144.9147.4
Residual SE.1.92.12.22.42.42.52.72.9
N. Fitting Points
kd Tree Bucket Size
Deg. Local Polynom.
Smoothing Param.
Neighbor Points
RSS
Trace[L]
GCV
AICC
AICC1
Delta1
Delta2
Equivalent N. Param.
Lookup DF
Residual SE
Sheet3
1
#REF!
GE
SE
xxia:max(1, ns/5)
xxia:int(n*s)
xxia:max(1, ns/5)
xxia:int(n*s)
MBD07CC741B.unknown
MBD0764C653.unknown
-
More on Output
-
proc loess data=ENSO; model Pressure=Month / smooth = 0.01 to 0.2 by 0.01 dfmethod=exact degree = 2;run;
-
99% Confidence limitsods output OutputStatistics=ENSOstats FitSummary=ENSOsummary;proc loess data=ENSO; model Pressure=Month / r CLM smooth = 0.05 alpha=0.01 dfmethod=exact;run;symbol1 c=black i=none value=dot;symbol2 c=black i=join value=none;symbol3 c=red i=join value=none;symbol4 c=red i=join value=none;proc gplot data=ENSOstats; format DepVar 4.0; plot (DepVar Pred UpperCl LowerCL)*Month/ overlay hminor = 0 vminor = 0 vaxis = axis1 frame;run;
-
Pressure 0 4 8 12 16 20Month010203040506070809010011012013014015016017099% confidence limits
-
Step 2: Characterize other trends/* modify data set ENSO to filter out the 12-month cycle */data ENSO(drop=pi);set ENSO;pi = 4 * atan (1); /* or pi = 3.1415926 */cos1 = cos(2*pi*Month/12);sin1 = sin(2*pi*Month/12);proc reg data=ENSO;model Pressure = cos1 sin1;output out=ENSO1 r=FilteredPressure;run;proc print data=ENSO1; var Month Pressure FilteredPressure;run;ods output OutputStatistics=ENSO1stats;proc loess data=ENSO1;model FilteredPressure=Month/smooth = 0.12;run;proc gplot data=ENSO1stats; plot (DepVar Pred)*Month/overlay;run;r: residual, i.e., what is left after the monthly cycle has been removed.
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Cos1 and Sin1
Chart4
0.86602540380.5
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-0-1
0.5-0.8660254038
0.8660254038-0.5
1-0
Cos1
Sin1
Month
Cos1, Sin1
Sheet1
Array RowArray ColumnX LocationY Locationch1 Intensitych1 Backgroundch2 Intensitych2 BackgroundMonthCos1Sin1ObsMonthPressureFilteredP
1122808520469.22876166.962967231.89285354.88888910.86602540380.51112.9-0.62564
1125108500443.9776116.055557295.2830273.98148320.50.86602540382211.3-1.28396
1127508500421.165527133.40741227.32710374.9537053013310.6-0.52185
1129908510401.165405151.361115223.87736553.7222214-0.50.86602540384411.21.66893
1132208500668.037048184.287033265.13333175.5277795-0.86602540380.55510.92.66212
1134508500513.325867120.018517213.72093278.9814836-10667.5-0.08878
1136908500737.949158150.111115213.77777164.2222217-0.8660254038-0.5777.7-0.0577
1139358490715.324463112.185188261.1031892.2037058-0.5-0.86602540388811.73.00063
1141608500459.057983117.712959212.04586831.1481489-0-19912.92.73852
11439084951489.834839131.759262322.26403860.611111100.5-0.8660254038101014.32.54774
1146308510342.09649785.555557212.36558541.120369110.8660254038-0.5111110.9-2.14545
1148508510336.627258104.305557169.04705832.509258121-0121213.70.00545
1151008510370.43396123.166664183.93243445.379631130.86602540380.5131317.13.57436
1153308510331.234222102.537041177.47222941.490742140.50.86602540381414141.41604
1155708510364.921265160.444443156.55714426.9259261501151515.34.17815
1158008510360.279663126.333336153.9230849.25925816-0.50.866025403816168.5-1.03107
1160408510320.768524120.722221177.25300643.38888917-0.86602540380.517175.7-2.53788
1162708510312.330261166.666672172.25555428.22222118-1018185.5-2.08878
1165108520396.537201190.25165.72619644.519-0.8660254038-0.519197.6-0.1577
1122808800418.324554148.833328290.72827175.93518820-0.5-0.866025403820208.6-0.09937
1125108760342.302338114.018517196.82795763.81481621-0-121217.3-2.86148
1127408760489.361969137.083328243.17094457.462963220.5-0.866025403822227.6-4.15226
1129908770357.604309183.824081235.617386216.629623230.8660254038-0.5232312.7-0.34545
1132108750416.833344159.509262207.17346288.546295241-0242411-2.69455
1134408750484.618317102.379631247.52999973.564812250.86602540380.5252512.7-0.82564
1136808750447.929932156.71296721047.296295260.50.8660254038262612.90.31604
1139108740637.714966158.805557219.44274965.52777927012727131.87815
1141508740354.241669176.296295146.86315939.2314828-0.50.8660254038282810.91.36893
1143808740406.46093890.611115190.96939142.18518429-0.86602540380.5292910.42.16212
11462087402310.726807110.925926673.19897540.35185230-10303010.22.61122
1148408730371.04254297.629631170.96385252.34259431-0.8660254038-0.5313180.2423
1150908730379.864868146.074081159.01190233.37036932-0.5-0.8660254038323210.92.20063
1153208730352.589478105.601852162.92857448.45370533-0-1333313.63.43852
1155608730369.571442122.018517164.44085741.212963340.5-0.8660254038343410.5-1.25226
11578087357859.317871176.5740811655.24670456.027779350.8660254038-0.535359.2-3.84545
1160308720397.6138159.027771154.6326630.518518361-0363612.4-1.29455
11626087207629.537109161.7777711504.43090835.435184370.86602540380.5373712.7-0.82564
1165008710410.477478170.490738144.29545644.509258380.50.8660254038383813.30.71604
1122609010498.695648175.37037718492.9814833901393910.1-1.02185
1124908980379.150391139.833328159.32000764.83333640-0.50.866025403840407.8-1.73107
1127208980470.926483129.824081222.39814879.12963141-0.86602540380.541414.8-3.43788
1129708990366.175995183.703705191.3584987.06481242-1042423-4.58878
1131908970408.991302138.083328199.27192782.7543-0.8660254038-0.543432.5-5.2577
1134308970480.200012174.111115219.52000483.10185244-0.5-0.866025403844446.3-2.39937
1136608970413.056061162.481476218.55670235.52777945-0-145459.7-0.46148
1139208970457.100708110.314812183.91000438.212963460.5-0.8660254038464611.6-0.15226
1141308970526.518066145.472229214.42240959.333332470.8660254038-0.547478.6-4.44545
1143708970595.55676397.055557207.85714751.490742481-0484812.4-1.29455
11461589552107.196533168.638885456.04788233.083332490.86602540380.5494910.5-3.02564
1148308970362.287048100.40741178.94897535.277779500.50.8660254038505013.30.71604
1150708970379.63909981.398148157.63333141.5462955101515110.4-0.72185
1153108970364.72222991.166664144.76596132.93518452-0.50.866025403852528.1-1.43107
1155408970419.242981116.361115143.12345922.55555553-0.86602540380.553533.7-4.53788
1157808970510.121948207.388885141.65957642.28703754-10545410.73.11122
1160108960333.725677149.138885142.95698549.31481655-0.8660254038-0.555555.1-2.6577
1162508960333.829773174.046295160.43037452.00925856-0.5-0.8660254038565610.41.70063
1164808960329.64151107.04629514946.75925857-0-1575710.90.73852
112250924017495.826172148.0370334209.79687543.259258580.5-0.8660254038585811.7-0.05226
1124809210545.102295127.620369239.05479465.685188590.8660254038-0.5595911.4-1.64545
1127209210430.164642135.962967217.30000396.277779601-0606013.70.00545
1129609230396.960327146.046295184.05607678.259262610.86602540380.5616114.10.57436
1131909220429.756317131.425919263.33914254.259258620.50.86602540386262141.41604
1134209220440.186676139210.00869840.9351846301636312.51.37815
1136609220449.799988138.416672212.44564897.68518864-0.50.866025403864646.3-3.23107
11391092205970.16308696.4722211541.52880942.35185265-0.86602540380.565659.61.36212
1141309220398.245911114.666664142.86813439.28703766-10666611.74.11122
1143659200373.132751143.916672168.90425159.19444367-0.8660254038-0.567675-2.7577
1146009205389.592926106.074074158.28767447.59259468-0.5-0.8660254038686810.82.10063
1148209215401.941742119.194443150.01980643.99074269-0-1696912.72.53852
1150559195435.625106.851852173.8139549.601852700.5-0.8660254038707010.8-0.95226
1153009215344.738739171.675919133.38961856.064816710.8660254038-0.5717111.8-1.24545
1155309205418.148438158.638885181.69047528.342592721-0727212.6-1.09455
1157659200349.495575131.712967136.15583870.564812730.86602540380.5737315.72.17436
1160109195382.801575125.75184.15554842.925926740.50.8660254038747412.60.01604
1162409200435.63333159.888889193.85833745.7870377501757514.83.67815
1164809210467.66140791.296295193.78750626.4907476-0.50.866025403876767.8-1.73107
1122659465543.802734147.444443341.67031960.32407477-0.86602540380.577777.1-1.13788
1124909440528.645935113.129631247.30596965.37963178-10787811.23.61122
1127309450447.948059165.111115293.55139260.53703779-0.8660254038-0.579798.10.3423
1129709465452.482758127.935188207.65454140.0185280-0.5-0.866025403880806.4-2.29937
1132009435455.442963132.916672174.39999459.68518481-0-181815.2-4.96148
1134309455416.50930892.046295270.17947455.009258820.5-0.86602540388282120.24774
1136759450377.259247204.175919195.3018848.712963830.8660254038-0.5838310.2-2.84545
1139109410352.383942107.037041152.71084646.23148841-0848412.7-0.99455
1141409420317134.148148140.06976344.314816850.86602540380.5858510.2-3.32564
1143709420340.847839136.129623169.69999735.435184860.50.8660254038868614.72.11604
1146109420370.01962377.15741139.02221717.7962978701878712.21.07815
1148309420374.596161145.740738167.96807946.87036988-0.50.866025403888887.1-2.43107
1150809420362.582.138885155.54429635.65740689-0.86602540380.589895.7-2.53788
1153109420412.77520892.898148156.98924338.86111190-1090906.7-0.88878
11555094203523.5110.9074170520.67592691-0.8660254038-0.591913.9-3.8577
1157809420411.926239111.240738176.58407642.18518492-0.5-0.866025403892928.5-0.19937
1160209430424.669342131.981476151.27174444.96296393-0-193938.3-1.86148
11625094302128.712646166.416672405.38531531.861111940.5-0.8660254038949410.8-0.95226
1164909430399.08334478.129631141.92770429.185184950.8660254038-0.5959516.73.65455
1122709695797.730591149.027771316.8266660.666668961-0969612.6-1.09455
1124909680516.08783212.268524207.66336150.138889970.86602540380.5979712.5-1.02564
1127309680559.301392201.981476299.34646641.703705980.50.8660254038989812.5-0.08396
11297096903133.505859182.82408156453.76852990199999.8-1.32185
11320096701405.957397109.425926303.60388251.990742100-0.50.86602540381001007.2-2.33107
1134309670438.888885107.861115187.13461340.472221101-0.86602540380.51011014.1-4.13788
113670967019811.75176.8981484167.26953140.166668102-1010210210.63.01122
11391096706220.506348181.6944431320.21740755.51852103-0.8660254038-0.510310310.12.3423
1141409670423.322784144.055557174.95294252.296295104-0.5-0.866025403810410410.11.40063
1143709670372.85122781155.90525832.620369105-0-110510511.91.73852
1146109670800.5542691.574074271.07534840.1111111060.5-0.866025403810610613.61.84774
11483096704145.347656180.8055571129.28918534.8703691070.8660254038-0.510710716.33.25455
1150809670393.907684125.324074141.39361644.3333321081-010810817.63.90545
1153109670355.9829184.074074146.78161636.4907421090.86602540380.510910915.51.97436
1155509670402.657349135.009262169.536.8425941100.50.8660254038110110163.41604
1157809670409.741943128.861115186.71551558.2129631110111111115.24.07815
1160209660635.202454135.12962321943.472221112-0.50.866025403811211211.21.66893
1162509660646.97467162.601852229.4454527.129629113-0.86602540380.511311314.36.06212
1164909660440.5359578.138885197.91752631.629629114-1011411414.56.91122
1122609950496.792847116.759262312.71200663.277779115-0.8660254038-0.51151158.50.7423
11249099201107.978882139.648148241.55319239.379631116-0.5-0.8660254038116116123.30063
1127309920449.522125160.268524217.50981145.25117-0-111711712.72.53852
11297099301372.219727145.083328371.03466874.6759261180.5-0.866025403811811811.3-0.45226
11320099109013.628906148.0185241011.28973456.8055571190.8660254038-0.511911914.51.45455
1134309910835.783936133.064819188.60360739.51201-012012015.11.40545
1136709910813.408142143.546295200.39495844.6574061210.86602540380.512112110.4-3.12564
1139109910404.601501130.083328165.28712545.751220.50.866025403812212211.5-1.08396
1141409910301.935181136.870377164.27777140.3888891230112312313.42.27815
1143709910298.511627130.203705160.19047544.51852124-0.50.86602540381241247.5-2.03107
1146109910368.258057117.962959158.11340339.925926125-0.86602540380.51251250.6-7.63788
1148309910431.894379134.731476244.9239238.157406126-101261260.3-7.28878
1150809910467.813446158.185181148.28099132.712963127-0.8660254038-0.51271275.5-2.2577
1153109910384.48333784.648148186.67033430.175926128-0.5-0.86602540381281285-3.69937
11555099107582.845703151.6296231934.08972254.037037129-0-11291294.6-5.56148
1157809910336.125124.314812180.72042836.7962951300.5-0.86602540381301308.2-3.55226
1160209900360.90908878.351852155.48051546.4259261310.8660254038-0.51311319.9-3.14545
1162509900397.3773581.435188135.37878420.8425921321-01321329.2-4.49455
1164909900362.36029188.638885135.06849723.7129631330.86602540380.513313312.5-1.02564
11226010190424.102356131.333328184.41175868.842591340.50.866025403813413410.9-1.68396
11250010150447.953857143.620377236.93333498.194443135011351359.9-1.22185
11273010150404.873962123.879631221.46551561.824074136-0.50.86602540381361368.9-0.63107
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11438010150565.04937764.101852172.41584849.0648161430.8660254038-0.514314313.70.65455
11461010140354.445374103.90741178.19752533.6388891441-014414416.52.80545
11484010140339.105255172.398148170.54545629.2962971450.86602540380.514514516.83.27436
11508010140556.03228891.75221.05691530.6388891460.50.866025403814614617.14.51604
1153201014012989.09375101.4537054030.89160256.6759261470114714715.44.27815
1155501014011592.44921990.7222213354.70336933.287037148-0.50.86602540381481489.5-0.03107
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11602010140416.589.611115158.79762357.046295150-1015015010.12.51122
11626010140429.950806103.481483187.69879231.287037151-0.8660254038-0.51511519.31.5423
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11226010410896.98407151.84259280.92623959.212963153-0-115315311.21.03852
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11273010400537.239014142.268524305.96212858.0277791550.8660254038-0.515515515.62.55455
112980104102228.441406125.546295396.1727657.251561-015615612-1.69455
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11273010640560.564636157.870377229.51428272.101852
11298010650641.638306126.879631210.97761546.73148
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11344010630430.507568137.101852160.35105954.26852
11367010630683.111816120.666664198.59483350.638889
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11414010620854.43096986.65741176.39999425.342592
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11483010620575.802307129.805557185.86363225.027779
11508010610675.61175574.898148152.22222942.370369
115310106101302.454102130.34259208.93965169.953705
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11625010600441.594391150.814819137.27471928.787037
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112260108804746.080078122.15741715.77111845.898148
11250010850573.33764691.75272.97753946.712963
112730108501460.121338133.453705339.63372861.888889
11298010860846.54650987.129631263.721165.212959
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11367010850711.093567109.138885174.29203861.712963
11391010840431.875824126.222221121.38999955.527779
11414010850484.388885142.287033173.48979235.416668
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11461010840435.955353112.685188146.98684732.5
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11532010840383.59420898.240738136.519
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11626010840678.994568144.231476199.87049937.888889
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