distFrechet: ~ Function: Frechet distance ~

Description Usage Arguments Details Value Author References See Also Examples

Description

Compute Frechet distance between two trajectories.

Usage

1
distFrechet(Px,Py,Qx, Qy, timeScale=0.1, FrechetSumOrMax = "max")

Arguments

Px

[vector(numeric)] Times (abscisse) of the first trajectories.

Py

[vector(numeric)] Values of the first trajectories.

Qx

[vector(numeric)] Times of the second trajectories.

Qy

[vector(numeric)] Values of the second trajectories.

timeScale

[numeric]: allow to modify the time scale, increasing or decreasing the cost of the horizontal shift. If timeScale is very big, then the Frechet's distance is equal to the euclidienne distance. If timeScale is very small, then it is equal to the Dynamic Time Warping.

FrechetSumOrMax

[character]: The Frechet's distance can be define using the 'sum' function or the 'max' function. This option let the user to chose one or the other.

Details

Given two curve P and Q, Frechet distance between P and Q is define as inf_{a,b} max_{t} d(P(a(t)),Q(b(t))). It's computation is a NP-complex problem. When P and Q are trajectories (discrete curve), the problem is polynomial.

The Frechet distance can also be define using a sum instead of a max: inf_{a,b} sum_{t} d(P(a(t)),Q(b(t)))

The function distFrechet is C compiled, the function distFrechetR is in R, the function distFrechetRec is in recursive (the slowest) in R.

Value

A numeric value.

Author

Christophe Genolini
1. UMR U1027, INSERM, Universit<e9> Paul Sabatier / Toulouse III / France
2. CeRSM, EA 2931, UFR STAPS, Universit<e9> de Paris Ouest-Nanterre-La D<e9>fense / Nanterre / France

References

[1] Thomas Eiter & Heikki Mannila:
"Computing Discrete Fr<b4>echet Distance"

[2] C. Genolini and B. Falissard
"KmL: k-means for longitudinal data"
Computational Statistics, vol 25(2), pp 317-328, 2010

[3] C. Genolini and B. Falissard
"KmL: A package to cluster longitudinal data"
Computer Methods and Programs in Biomedicine, 104, pp e112-121, 2011

See Also

distTraj

Examples

1
2
3
4
5
6
7
8
9
   Px <- 1:20
   Py <- dnorm(1:20,12,2)
   Qx <- 1:20
   Qy <- dnorm(1:20,8,2)

   distFrechet(Px,Py,Qx,Qy)

   ### Frechet using sum instead of max.
   distFrechet(Px,Py,Qx,Qy,FrechetSumOrMax="sum")

Example output

Loading required package: clv
Loading required package: cluster
Loading required package: class
Loading required package: rgl
Loading required package: misc3d
Warning messages:
1: In rgl.init(initValue, onlyNULL) : RGL: unable to open X11 display
2: 'rgl_init' failed, running with rgl.useNULL = TRUE 
3: .onUnload failed in unloadNamespace() for 'rgl', details:
  call: fun(...)
  error: object 'rgl_quit' not found 
              Q1          Q2          Q3         Q4         Q5        Q6
P1  0.0004362875 0.100024547 0.200191931 0.30121214 0.40520821 0.5144292
P2  0.1000009487 0.002215181 0.100383252 0.20181357 0.30690977 0.4178963
P3  0.2000004587 0.100024372 0.008756158 0.10357763 0.21022052 0.3234741
P4  0.3000002275 0.200011545 0.100377497 0.02692857 0.11910096 0.2337119
P5  0.4000000000 0.300005278 0.200173306 0.10346685 0.06432246 0.1566272
P6  0.5000031669 0.400000000 0.300071457 0.20152922 0.11794749 0.1187694
P7  0.6000577909 0.500042877 0.400000000 0.30055346 0.20769064 0.1503117
P8  0.7005036674 0.600511471 0.500332271 0.40000000 0.30236744 0.2209844
P9  0.8025816958 0.702788454 0.602607169 0.50142404 0.40000000 0.3052236
P10 0.9080374807 0.808768310 0.708938362 0.60731713 0.50315149 0.4000000
P11 1.0152999893 0.916630928 0.817299673 0.71568993 0.61023100 0.5030211
P12 1.1178617317 1.019269160 0.919983237 0.81838124 0.71284459 0.6051116
P13 1.2127794805 1.113648176 1.013892872 0.91225659 0.80770160 0.7021611
P14 1.3055772924 1.205863251 1.105709546 1.00440734 0.90175464 0.8000000
P15 1.4014768562 1.301503596 1.201305707 1.10064802 1.00000000 0.9017546
P16 1.5002351109 1.400219278 1.300127833 1.20000000 1.10064802 1.0044073
P17 1.6000216725 1.500014293 1.400000000 1.30012783 1.20130571 1.1057095
P18 1.7000009314 1.600000000 1.500014293 1.40021928 1.30150360 1.2058633
P19 1.8000000000 1.700000931 1.600021672 1.50023511 1.40147686 1.3055773
P20 1.9000000359 1.800001283 1.700022247 1.60022659 1.50139437 1.4052122
           Q7        Q8        Q9       Q10       Q11       Q12       Q13
P1  0.6252899 0.7278659 0.8191383 0.9080955 1.0020947 1.1003312 1.2000320
P2  0.5300823 0.6322883 0.7217945 0.8090966 0.9023268 1.0003643 1.1000349
P3  0.4370179 0.5383173 0.6252877 0.7103770 0.8026161 0.9004045 1.0000383
P4  0.3477987 0.4469475 0.5300603 0.6120631 0.7029830 0.8004531 0.9000420
P5  0.2661467 0.3600206 0.4368456 0.5143268 0.6034380 0.7005037 0.8000433
P6  0.2005299 0.2809086 0.3467164 0.4172603 0.5038964 0.6005115 0.7000306
P7  0.1672685 0.2153350 0.2607274 0.3203024 0.4039002 0.5003323 0.6000000
P8  0.1794772 0.1724757 0.1794772 0.2209844 0.3023674 0.4000000 0.5003323
P9  0.2288709 0.1677719 0.1112739 0.1147233 0.2000000 0.3023674 0.4039002
P10 0.3050085 0.2148488 0.1141499 0.0000000 0.1147233 0.2209844 0.3203024
P11 0.4000000 0.3009142 0.2000000 0.1141499 0.1112739 0.1794772 0.2607274
P12 0.5005491 0.4000000 0.3009142 0.2148488 0.1677719 0.1724757 0.2153350
P13 0.6000000 0.5005491 0.4000000 0.3050085 0.2288709 0.1794772 0.1672685
P14 0.7021611 0.6051116 0.5030211 0.4000000 0.3052236 0.2209844 0.1503117
P15 0.8077016 0.7128446 0.6102310 0.5031515 0.4000000 0.3023674 0.2076906
P16 0.9122566 0.8183812 0.7156899 0.6073171 0.5014240 0.4000000 0.3005535
P17 1.0138929 0.9199832 0.8172997 0.7089384 0.6026072 0.5003323 0.4000000
P18 1.1136482 1.0192692 0.9166309 0.8087683 0.7027885 0.6005115 0.5000429
P19 1.2127795 1.1178617 1.0153000 0.9080375 0.8025817 0.7005037 0.6000578
P20 1.3118552 1.2164547 1.1139856 1.0072841 0.9023220 0.8004531 0.7000540
          Q14        Q15        Q16         Q17         Q18          Q19
P1  1.3000019 1.40000007 1.50000000 1.600000000 1.700000000 1.8000000000
P2  1.2000020 1.30000007 1.40000000 1.500000000 1.600000000 1.7000000000
P3  1.1000022 1.20000008 1.30000000 1.400000000 1.500000000 1.6000000000
P4  1.0000023 1.10000006 1.20000000 1.300000001 1.400000002 1.5000000015
P5  0.9000018 1.00000000 1.10000006 1.200000076 1.300000073 1.4000000680
P6  0.8000000 0.90000176 1.00000231 1.100002216 1.200002045 1.3000018885
P7  0.7000306 0.80004334 0.90004202 1.000038334 1.100034907 1.2000320035
P8  0.6005115 0.70050367 0.80045309 0.900404534 1.000364292 1.1003312016
P9  0.5038964 0.60343797 0.70298296 0.802616139 0.902326773 1.0020946537
P10 0.4172603 0.51432681 0.61206313 0.710377030 0.809096581 0.9080955043
P11 0.3467164 0.43684559 0.53006032 0.625287682 0.721794456 0.8191382543
P12 0.2809086 0.36002063 0.44694747 0.538317330 0.632288256 0.7278658628
P13 0.2005299 0.26614670 0.34779871 0.437017946 0.530082293 0.6252899165
P14 0.1187694 0.15662716 0.23371194 0.323474148 0.417896253 0.5144292418
P15 0.1179475 0.06432246 0.11910096 0.210220520 0.306909768 0.4052082118
P16 0.2015292 0.10346685 0.02692857 0.103577626 0.201813567 0.3012121399
P17 0.3000715 0.20017331 0.10037750 0.008756158 0.100383252 0.2001919314
P18 0.4000000 0.30000528 0.20001155 0.100024372 0.002215181 0.1000245474
P19 0.5000032 0.40000000 0.30000023 0.200000459 0.100000949 0.0004362875
P20 0.6000038 0.50000014 0.40000000 0.300000006 0.200000011 0.1000000224
             Q20
P1  1.900000e+00
P2  1.800000e+00
P3  1.700000e+00
P4  1.600000e+00
P5  1.500000e+00
P6  1.400002e+00
P7  1.300030e+00
P8  1.200304e+00
P9  1.101905e+00
P10 1.007292e+00
P11 9.170537e-01
P12 8.244930e-01
P13 7.217946e-01
P14 6.120763e-01
P15 5.041763e-01
P16 4.009099e-01
P17 3.001280e-01
P18 2.000123e-01
P19 1.000010e-01
P20 6.691207e-05
              Q1          Q2          Q3         Q4         Q5        Q6
P1  0.0004362875 0.100024547 0.200191931 0.30121214 0.40520821 0.5144292
P2  0.1000009487 0.002215181 0.100383252 0.20181357 0.30690977 0.4178963
P3  0.2000004587 0.100024372 0.008756158 0.10357763 0.21022052 0.3234741
P4  0.3000002275 0.200011545 0.100377497 0.02692857 0.11910096 0.2337119
P5  0.4000000000 0.300005278 0.200173306 0.10346685 0.06432246 0.1566272
P6  0.5000031669 0.400000000 0.300071457 0.20152922 0.11794749 0.1187694
P7  0.6000577909 0.500042877 0.400000000 0.30055346 0.20769064 0.1503117
P8  0.7005036674 0.600511471 0.500332271 0.40000000 0.30236744 0.2209844
P9  0.8025816958 0.702788454 0.602607169 0.50142404 0.40000000 0.3052236
P10 0.9080374807 0.808768310 0.708938362 0.60731713 0.50315149 0.4000000
P11 1.0152999893 0.916630928 0.817299673 0.71568993 0.61023100 0.5030211
P12 1.1178617317 1.019269160 0.919983237 0.81838124 0.71284459 0.6051116
P13 1.2127794805 1.113648176 1.013892872 0.91225659 0.80770160 0.7021611
P14 1.3055772924 1.205863251 1.105709546 1.00440734 0.90175464 0.8000000
P15 1.4014768562 1.301503596 1.201305707 1.10064802 1.00000000 0.9017546
P16 1.5002351109 1.400219278 1.300127833 1.20000000 1.10064802 1.0044073
P17 1.6000216725 1.500014293 1.400000000 1.30012783 1.20130571 1.1057095
P18 1.7000009314 1.600000000 1.500014293 1.40021928 1.30150360 1.2058633
P19 1.8000000000 1.700000931 1.600021672 1.50023511 1.40147686 1.3055773
P20 1.9000000359 1.800001283 1.700022247 1.60022659 1.50139437 1.4052122
           Q7        Q8        Q9       Q10       Q11       Q12       Q13
P1  0.6252899 0.7278659 0.8191383 0.9080955 1.0020947 1.1003312 1.2000320
P2  0.5300823 0.6322883 0.7217945 0.8090966 0.9023268 1.0003643 1.1000349
P3  0.4370179 0.5383173 0.6252877 0.7103770 0.8026161 0.9004045 1.0000383
P4  0.3477987 0.4469475 0.5300603 0.6120631 0.7029830 0.8004531 0.9000420
P5  0.2661467 0.3600206 0.4368456 0.5143268 0.6034380 0.7005037 0.8000433
P6  0.2005299 0.2809086 0.3467164 0.4172603 0.5038964 0.6005115 0.7000306
P7  0.1672685 0.2153350 0.2607274 0.3203024 0.4039002 0.5003323 0.6000000
P8  0.1794772 0.1724757 0.1794772 0.2209844 0.3023674 0.4000000 0.5003323
P9  0.2288709 0.1724757 0.1724757 0.1724757 0.2000000 0.3023674 0.4039002
P10 0.3050085 0.2148488 0.1724757 0.1724757 0.1724757 0.2209844 0.3203024
P11 0.4000000 0.3009142 0.2000000 0.1724757 0.1724757 0.1794772 0.2607274
P12 0.5005491 0.4000000 0.3009142 0.2148488 0.1724757 0.1724757 0.2153350
P13 0.6000000 0.5005491 0.4000000 0.3050085 0.2288709 0.1794772 0.1724757
P14 0.7021611 0.6051116 0.5030211 0.4000000 0.3052236 0.2209844 0.1724757
P15 0.8077016 0.7128446 0.6102310 0.5031515 0.4000000 0.3023674 0.2076906
P16 0.9122566 0.8183812 0.7156899 0.6073171 0.5014240 0.4000000 0.3005535
P17 1.0138929 0.9199832 0.8172997 0.7089384 0.6026072 0.5003323 0.4000000
P18 1.1136482 1.0192692 0.9166309 0.8087683 0.7027885 0.6005115 0.5000429
P19 1.2127795 1.1178617 1.0153000 0.9080375 0.8025817 0.7005037 0.6000578
P20 1.3118552 1.2164547 1.1139856 1.0072841 0.9023220 0.8004531 0.7000540
          Q14       Q15       Q16       Q17       Q18       Q19       Q20
P1  1.3000019 1.4000001 1.5000000 1.6000000 1.7000000 1.8000000 1.9000000
P2  1.2000020 1.3000001 1.4000000 1.5000000 1.6000000 1.7000000 1.8000000
P3  1.1000022 1.2000001 1.3000000 1.4000000 1.5000000 1.6000000 1.7000000
P4  1.0000023 1.1000001 1.2000000 1.3000000 1.4000000 1.5000000 1.6000000
P5  0.9000018 1.0000000 1.1000001 1.2000001 1.3000001 1.4000001 1.5000001
P6  0.8000000 0.9000018 1.0000023 1.1000022 1.2000020 1.3000019 1.4000018
P7  0.7000306 0.8000433 0.9000420 1.0000383 1.1000349 1.2000320 1.3000295
P8  0.6005115 0.7005037 0.8004531 0.9004045 1.0003643 1.1003312 1.2003036
P9  0.5038964 0.6034380 0.7029830 0.8026161 0.9023268 1.0020947 1.1019046
P10 0.4172603 0.5143268 0.6120631 0.7103770 0.8090966 0.9080955 1.0072921
P11 0.3467164 0.4368456 0.5300603 0.6252877 0.7217945 0.8191383 0.9170537
P12 0.2809086 0.3600206 0.4469475 0.5383173 0.6322883 0.7278659 0.8244930
P13 0.2005299 0.2661467 0.3477987 0.4370179 0.5300823 0.6252899 0.7217946
P14 0.1724757 0.1724757 0.2337119 0.3234741 0.4178963 0.5144292 0.6120763
P15 0.1724757 0.1724757 0.1724757 0.2102205 0.3069098 0.4052082 0.5041763
P16 0.2015292 0.1724757 0.1724757 0.1724757 0.2018136 0.3012121 0.4009099
P17 0.3000715 0.2001733 0.1724757 0.1724757 0.1724757 0.2001919 0.3001280
P18 0.4000000 0.3000053 0.2000115 0.1724757 0.1724757 0.1724757 0.2000123
P19 0.5000032 0.4000000 0.3000002 0.2000005 0.1724757 0.1724757 0.1724757
P20 0.6000038 0.5000001 0.4000000 0.3000000 0.2000000 0.1724757 0.1724757


 Result = 0.1724757 
[1] 0.1724757
              Q1          Q2          Q3         Q4         Q5        Q6
P1  0.0004362875 0.100024547 0.200191931 0.30121214 0.40520821 0.5144292
P2  0.1000009487 0.002215181 0.100383252 0.20181357 0.30690977 0.4178963
P3  0.2000004587 0.100024372 0.008756158 0.10357763 0.21022052 0.3234741
P4  0.3000002275 0.200011545 0.100377497 0.02692857 0.11910096 0.2337119
P5  0.4000000000 0.300005278 0.200173306 0.10346685 0.06432246 0.1566272
P6  0.5000031669 0.400000000 0.300071457 0.20152922 0.11794749 0.1187694
P7  0.6000577909 0.500042877 0.400000000 0.30055346 0.20769064 0.1503117
P8  0.7005036674 0.600511471 0.500332271 0.40000000 0.30236744 0.2209844
P9  0.8025816958 0.702788454 0.602607169 0.50142404 0.40000000 0.3052236
P10 0.9080374807 0.808768310 0.708938362 0.60731713 0.50315149 0.4000000
P11 1.0152999893 0.916630928 0.817299673 0.71568993 0.61023100 0.5030211
P12 1.1178617317 1.019269160 0.919983237 0.81838124 0.71284459 0.6051116
P13 1.2127794805 1.113648176 1.013892872 0.91225659 0.80770160 0.7021611
P14 1.3055772924 1.205863251 1.105709546 1.00440734 0.90175464 0.8000000
P15 1.4014768562 1.301503596 1.201305707 1.10064802 1.00000000 0.9017546
P16 1.5002351109 1.400219278 1.300127833 1.20000000 1.10064802 1.0044073
P17 1.6000216725 1.500014293 1.400000000 1.30012783 1.20130571 1.1057095
P18 1.7000009314 1.600000000 1.500014293 1.40021928 1.30150360 1.2058633
P19 1.8000000000 1.700000931 1.600021672 1.50023511 1.40147686 1.3055773
P20 1.9000000359 1.800001283 1.700022247 1.60022659 1.50139437 1.4052122
           Q7        Q8        Q9       Q10       Q11       Q12       Q13
P1  0.6252899 0.7278659 0.8191383 0.9080955 1.0020947 1.1003312 1.2000320
P2  0.5300823 0.6322883 0.7217945 0.8090966 0.9023268 1.0003643 1.1000349
P3  0.4370179 0.5383173 0.6252877 0.7103770 0.8026161 0.9004045 1.0000383
P4  0.3477987 0.4469475 0.5300603 0.6120631 0.7029830 0.8004531 0.9000420
P5  0.2661467 0.3600206 0.4368456 0.5143268 0.6034380 0.7005037 0.8000433
P6  0.2005299 0.2809086 0.3467164 0.4172603 0.5038964 0.6005115 0.7000306
P7  0.1672685 0.2153350 0.2607274 0.3203024 0.4039002 0.5003323 0.6000000
P8  0.1794772 0.1724757 0.1794772 0.2209844 0.3023674 0.4000000 0.5003323
P9  0.2288709 0.1677719 0.1112739 0.1147233 0.2000000 0.3023674 0.4039002
P10 0.3050085 0.2148488 0.1141499 0.0000000 0.1147233 0.2209844 0.3203024
P11 0.4000000 0.3009142 0.2000000 0.1141499 0.1112739 0.1794772 0.2607274
P12 0.5005491 0.4000000 0.3009142 0.2148488 0.1677719 0.1724757 0.2153350
P13 0.6000000 0.5005491 0.4000000 0.3050085 0.2288709 0.1794772 0.1672685
P14 0.7021611 0.6051116 0.5030211 0.4000000 0.3052236 0.2209844 0.1503117
P15 0.8077016 0.7128446 0.6102310 0.5031515 0.4000000 0.3023674 0.2076906
P16 0.9122566 0.8183812 0.7156899 0.6073171 0.5014240 0.4000000 0.3005535
P17 1.0138929 0.9199832 0.8172997 0.7089384 0.6026072 0.5003323 0.4000000
P18 1.1136482 1.0192692 0.9166309 0.8087683 0.7027885 0.6005115 0.5000429
P19 1.2127795 1.1178617 1.0153000 0.9080375 0.8025817 0.7005037 0.6000578
P20 1.3118552 1.2164547 1.1139856 1.0072841 0.9023220 0.8004531 0.7000540
          Q14        Q15        Q16         Q17         Q18          Q19
P1  1.3000019 1.40000007 1.50000000 1.600000000 1.700000000 1.8000000000
P2  1.2000020 1.30000007 1.40000000 1.500000000 1.600000000 1.7000000000
P3  1.1000022 1.20000008 1.30000000 1.400000000 1.500000000 1.6000000000
P4  1.0000023 1.10000006 1.20000000 1.300000001 1.400000002 1.5000000015
P5  0.9000018 1.00000000 1.10000006 1.200000076 1.300000073 1.4000000680
P6  0.8000000 0.90000176 1.00000231 1.100002216 1.200002045 1.3000018885
P7  0.7000306 0.80004334 0.90004202 1.000038334 1.100034907 1.2000320035
P8  0.6005115 0.70050367 0.80045309 0.900404534 1.000364292 1.1003312016
P9  0.5038964 0.60343797 0.70298296 0.802616139 0.902326773 1.0020946537
P10 0.4172603 0.51432681 0.61206313 0.710377030 0.809096581 0.9080955043
P11 0.3467164 0.43684559 0.53006032 0.625287682 0.721794456 0.8191382543
P12 0.2809086 0.36002063 0.44694747 0.538317330 0.632288256 0.7278658628
P13 0.2005299 0.26614670 0.34779871 0.437017946 0.530082293 0.6252899165
P14 0.1187694 0.15662716 0.23371194 0.323474148 0.417896253 0.5144292418
P15 0.1179475 0.06432246 0.11910096 0.210220520 0.306909768 0.4052082118
P16 0.2015292 0.10346685 0.02692857 0.103577626 0.201813567 0.3012121399
P17 0.3000715 0.20017331 0.10037750 0.008756158 0.100383252 0.2001919314
P18 0.4000000 0.30000528 0.20001155 0.100024372 0.002215181 0.1000245474
P19 0.5000032 0.40000000 0.30000023 0.200000459 0.100000949 0.0004362875
P20 0.6000038 0.50000014 0.40000000 0.300000006 0.200000011 0.1000000224
             Q20
P1  1.900000e+00
P2  1.800000e+00
P3  1.700000e+00
P4  1.600000e+00
P5  1.500000e+00
P6  1.400002e+00
P7  1.300030e+00
P8  1.200304e+00
P9  1.101905e+00
P10 1.007292e+00
P11 9.170537e-01
P12 8.244930e-01
P13 7.217946e-01
P14 6.120763e-01
P15 5.041763e-01
P16 4.009099e-01
P17 3.001280e-01
P18 2.000123e-01
P19 1.000010e-01
P20 6.691207e-05
              Q1           Q2          Q3          Q4         Q5         Q6
P1  4.362875e-04  0.100460835  0.30065277  0.60186491  1.0070731  1.5215024
P2  1.004372e-01  0.002651468  0.10303472  0.30484829  0.6117581  1.0296543
P3  3.004377e-01  0.102675840  0.01140763  0.11498525  0.3252058  0.6486799
P4  6.004379e-01  0.302687385  0.11178512  0.03833619  0.1574372  0.3911491
P5  1.000438e+00  0.602692664  0.31195843  0.14180304  0.1026587  0.2592858
P6  1.500441e+00  1.002692664  0.61202989  0.34333226  0.2206061  0.2214281
P7  2.100499e+00  1.502735541  1.01202989  0.64388572  0.4282968  0.3709178
P8  2.801003e+00  2.103247012  1.51236216  1.04388572  0.7306642  0.5919022
P9  3.603584e+00  2.806035466  2.11496933  1.54530976  1.1306642  0.8971258
P10 4.511622e+00  3.614803776  2.82390769  2.15262689  1.6338157  1.2971258
P11 5.526922e+00  4.531434704  3.64120736  2.86831682  2.2440467  1.8001469
P12 6.644783e+00  5.550703864  4.56119060  3.68669806  2.9568913  2.4052584
P13 7.857563e+00  6.664352040  5.57508347  4.59895465  3.7645929  3.1074195
P14 9.163140e+00  7.870215290  6.68079302  5.60336199  4.6663475  3.9074195
P15 1.056462e+01  9.171718887  7.88209872  6.70401001  5.6663475  4.8091742
P16 1.206485e+01 10.571938165  9.18222656  7.90401001  6.7669956  5.8135815
P17 1.366487e+01 12.071952458 10.58222656  9.20413784  7.9683013  6.9192910
P18 1.536487e+01 13.671952458 12.08224085 10.60435712  9.2698049  8.1251543
P19 1.716487e+01 15.371953389 13.68226252 12.10459223 10.6712817  9.4307316
P20 1.906487e+01 17.171954672 15.38228477 13.70481882 12.1726761 10.8359438
           Q7        Q8        Q9       Q10       Q11       Q12      Q13
P1  2.1467923 2.8746581 3.6937964 4.6018919 5.6039866 6.7043178 7.904350
P2  1.5597366 2.1920249 2.9138193 3.7229159 4.6252427 5.6256070 6.725642
P3  1.0856979 1.6240152 2.2493029 2.9596799 3.7622960 4.6627006 5.662739
P4  0.7389478 1.1858953 1.7159556 2.3280187 3.0310017 3.8314548 4.731497
P5  0.5254325 0.8854531 1.3222987 1.8366255 2.4400635 3.1405672 3.940611
P6  0.4219580 0.7028666 1.0495830 1.4668433 1.9707397 2.5712512 3.271282
P7  0.3886966 0.6040316 0.8647589 1.1850613 1.5889615 2.0892938 2.689294
P8  0.5503951 0.5611723 0.7406495 0.9616339 1.2640013 1.6640013 2.164334
P9  0.7792659 0.7181670 0.6724461 0.7871694 0.9871694 1.2895368 1.693437
P10 1.0842745 0.9330158 0.7865960 0.6724461 0.7871694 1.0081538 1.328456
P11 1.4842745 1.2339300 0.9865960 0.7865960 0.7837200 0.9631972 1.223925
P12 1.9848235 1.6339300 1.2875103 1.0014449 0.9514919 0.9561956 1.171531
P13 2.5848235 2.1344791 1.6875103 1.3064534 1.1803628 1.1309691 1.123464
P14 3.2869846 2.7395906 2.1905313 1.7064534 1.4855863 1.3519535 1.273776
P15 4.0946862 3.4524352 2.8007623 2.2096049 1.8855863 1.6543210 1.481466
P16 5.0069428 4.2708165 3.5164523 2.8169220 2.3870104 2.0543210 1.782020
P17 6.0208357 5.1907997 4.3337519 3.5258604 2.9896175 2.5546532 2.182020
P18 7.1344839 6.2100689 5.2503829 4.3346287 3.6924060 3.1551647 2.682063
P19 8.3472633 7.3279306 6.2656829 5.2426662 4.4949877 3.8556684 3.282121
P20 9.6591185 8.5443853 7.3796685 6.2499503 5.3973097 4.6561215 3.982175
         Q14       Q15       Q16       Q17       Q18       Q19       Q20
P1  9.204352 10.604352 12.104352 13.704352 15.404352 17.204352 19.104352
P2  7.925644  9.225644 10.625644 12.125644 13.725644 15.425644 17.225644
P3  6.762741  7.962741  9.262741 10.662741 12.162741 13.762741 15.462741
P4  5.731499  6.831499  8.031499  9.331499 10.731499 12.231499 13.831499
P5  4.840612  5.840612  6.940612  8.140612  9.440612 10.840613 12.340613
P6  4.071282  4.971284  5.971286  7.071288  8.271290  9.571292 10.971294
P7  3.389324  4.189368  5.089410  6.089448  7.189483  8.389515  9.689545
P8  2.764845  3.465349  4.265802  5.166206  6.166571  7.266902  8.467205
P9  2.197333  2.800771  3.503754  4.306371  5.208697  6.210792  7.312697
P10 1.745716  2.260043  2.872106  3.582483  4.391580  5.299676  6.306968
P11 1.570641  2.007487  2.537547  3.162835  3.884629  4.703767  5.620821
P12 1.452439  1.812460  2.259407  2.797725  3.430013  4.157879  4.982372
P13 1.323994  1.590141  1.937940  2.374957  2.905040  3.530330  4.252124
P14 1.242234  1.398861  1.632573  1.956047  2.373943  2.888372  3.500449
P15 1.360181  1.306556  1.425657  1.635878  1.942787  2.347996  2.852172
P16 1.561710  1.410023  1.333485  1.437062  1.638876  1.940088  2.340998
P17 1.861782  1.610196  1.433862  1.342241  1.442624  1.642816  1.942944
P18 2.261782  1.910201  1.633874  1.442265  1.344456  1.444481  1.644493
P19 2.761785  2.310201  1.933874  1.642266  1.444457  1.344892  1.444893
P20 3.361789  2.810202  2.333874  1.942266  1.644457  1.444892  1.344959


 Result = 1.344959 
[1] 1.344959

longitudinalData documentation built on May 2, 2019, 8:53 a.m.