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BTS Home > Publications > Journal of Transportation and Statistics > Volume 7 Number 2/3 > Paper 5
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Figure 5 - Illustration of Covariance Function and Correlation for a Very Congested Two-Link Example: Day 7

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5A: LINK 1

Time,
t (seconds)		 ln (travel time)
lnY g
102 6.21 6.25
109 6.12 6.25
137 6.27 6.26
204 6.36 6.26
205 6.44 6.26
214 6.27 6.26
227 6.46 6.27
233 6.27 6.27
259 6.48 6.29
271 6.46 6.30
327 6.43 6.33
362 6.42 6.37
379 6.40 6.40
401 6.38 6.43
403 6.50 6.43
419 6.39 6.44
430 6.46 6.44
516 6.61 6.49
519 6.60 6.50
522 6.45 6.50
549 6.59 6.52
572 6.61 6.54
582 6.68 6.55
606 6.70 6.56
625 6.59 6.58
634 6.59 6.58
641 6.43 6.59
669 6.64 6.61
730 6.90 6.66
751 6.71 6.68
772 6.90 6.71
773 6.74 6.71
814 6.79 6.75
821 6.73 6.76
850 6.87 6.77
855 6.69 6.77
881 6.93 6.77
906 6.92 6.78
910 7.00 6.78
913 6.81 6.78
915 6.68 6.78
974 6.94 6.78
994 6.65 6.78
1,012 7.01 6.79
1,036 6.67 6.79
1,038 6.67 6.79
1,040 6.70 6.79
1,042 6.70 6.79
1,101 6.68 6.77
1,166 6.65 6.74
1,185 6.65 6.73
1,198 6.77 6.72
1,237 6.66 6.70
1,332 6.61 6.59
1,396 6.58 6.48
1,978 5.51 6.28
2,033 6.77 6.34
2,058 6.55 6.36
2,066 6.39 6.37
2,286 6.48 6.50
2,296 6.47 6.50
2,335 6.47 6.49
2,340 6.46 6.49
2,346 6.46 6.49
2,355 6.46 6.49
2,552 6.31 6.31
2,805 5.70 6.14
3,122 6.69 6.31
3,136 6.58 6.32
3,212 5.56 6.34
3,274 6.32 6.35
3,326 6.73 6.36
3,411 6.27 6.33
3,520 6.17 6.27
3,888 6.36 6.13
3,959 5.46 6.11
3,972 6.35 6.10
4,028 6.18 6.07
4,038 5.97 6.07
4,081 5.95 6.07
4,086 6.26 6.07
4,185 6.02 6.03
4,207 6.09 6.02
4,243 5.95 6.01
4,275 6.04 5.99
4,281 6.04 5.99
4,301 5.92 5.98
4,309 6.12 5.97
4,316 6.00 5.97
4,318 5.43 5.97
4,467 5.72 5.93
4,486 5.97 5.94
4,510 6.07 5.94
4,517 6.08 5.94
4,527 5.88 5.95
4,557 5.89 5.96
4,563 5.91 5.96
4,566 6.10 5.96
4,673 5.67 5.96
4,897 5.94 5.95
4,932 5.99 5.95
4,938 6.03 5.96
5,006 5.87 5.96
5,243 6.03 5.97
5,272 5.95 5.95
5,403 5.83 5.86
5,410 5.83 5.85
5,642 4.92 5.40
5,751 4.74 5.24
5,833 5.38 5.23
5,851 5.05 5.23
5,924 4.94 5.20
5,929 5.59 5.20
5,950 5.64 5.19
5,982 5.50 5.18
5,984 5.57 5.18
6,048 5.02 5.16
6,074 5.38 5.16
6,142 5.20 5.13
6,174 5.16 5.10
6,201 5.09 5.06
6,202 4.95 5.06
6,212 5.04 5.04
6,214 5.10 5.04
6,220 5.11 5.03
6,244 5.04 5.00
6,251 5.03 4.99
6,255 5.20 4.99
6,294 5.21 4.94
6,317 4.84 4.92
6,438 4.76 4.88
6,482 4.89 4.85
6,526 4.88 4.82
6,558 4.82 4.79
6,565 4.86 4.79
6,566 4.93 4.79
6,603 4.89 4.74
6,626 4.88 4.72
6,653 4.66 4.70
6,664 4.76 4.70
6,671 4.65 4.70
6,726 4.80 4.70
6,742 4.65 4.69
6,795 4.66 4.71
6,879 4.50 4.75
6,916 4.50 4.72
6,920 4.43 4.72

5B: LINK 2

Time,
t (seconds)		 ln (travel time)
lnY g
102 6.71 6.69
109 6.72 6.68
137 6.68 6.68
204 6.61 6.67
205 6.59 6.67
214 6.61 6.67
227 6.57 6.67
233 6.66 6.67
259 6.62 6.66
271 6.60 6.66
327 6.62 6.66
362 6.63 6.66
379 6.62 6.66
401 6.62 6.67
403 6.84 6.67
419 6.62 6.67
430 6.59 6.68
516 6.61 6.68
519 6.73 6.68
522 6.61 6.68
549 6.59 6.67
572 6.73 6.67
582 6.78 6.66
606 6.74 6.65
625 6.62 6.64
634 6.63 6.64
641 6.66 6.64
669 6.62 6.63
730 6.62 6.61
751 6.54 6.60
772 6.63 6.60
773 6.51 6.60
814 6.57 6.59
821 6.52 6.59
850 6.53 6.58
855 6.52 6.58
881 6.55 6.58
906 7.01 6.57
910 6.65 6.57
913 6.62 6.57
915 6.59 6.57
974 6.59 6.56
994 6.56 6.55
1,012 6.64 6.55
1,036 6.52 6.55
1,038 6.53 6.55
1,040 6.50 6.55
1,042 6.49 6.55
1,101 6.50 6.55
1,166 6.51 6.54
1,185 6.50 6.54
1,198 6.50 6.54
1,237 6.48 6.54
1,332 6.49 6.53
1,396 6.48 6.53
1,978 6.79 6.54
2,033 6.38 6.54
2,058 6.51 6.54
2,066 6.53 6.53
2,286 6.37 6.44
2,296 6.38 6.44
2,335 6.37 6.42
2,340 6.38 6.42
2,346 6.35 6.41
2,355 6.35 6.41
2,552 6.38 6.34
2,805 6.42 6.28
3,122 6.17 6.26
3,136 6.15 6.25
3,212 6.30 6.22
3,274 6.08 6.19
3,326 6.12 6.17
3,411 6.17 6.14
3,520 6.12 6.11
3,888 6.06 6.07
3,959 6.22 6.06
3,972 6.07 6.06
4,028 6.00 6.06
4,038 6.04 6.06
4,081 6.04 6.06
4,086 6.06 6.06
4,185 6.05 6.06
4,207 5.95 6.06
4,243 5.98 6.05
4,275 6.08 6.04
4,281 6.08 6.04
4,301 6.06 6.03
4,309 6.00 6.03
4,316 6.06 6.03
4,318 6.07 6.02
4,467 6.01 5.95
4,486 5.95 5.94
4,510 5.86 5.93
4,517 5.87 5.93
4,527 5.94 5.92
4,557 5.85 5.91
4,563 5.87 5.91
4,566 6.01 5.91
4,673 5.91 5.88
4,897 5.82 5.84
4,932 5.83 5.83
4,938 5.82 5.82
5,006 5.78 5.79
5,243 5.58 5.69
5,272 5.50 5.67
5,403 5.51 5.58
5,410 5.53 5.58
5,642 5.55 5.53
5,751 5.56 5.53
5,833 5.63 5.54
5,851 5.77 5.55
5,924 5.60 5.56
5,929 5.51 5.56
5,950 5.59 5.56
5,982 5.49 5.55
5,984 5.58 5.56
6,048 5.76 5.55
6,074 5.47 5.55
6,142 5.49 5.54
6,174 5.51 5.53
6,201 5.50 5.53
6,202 5.57 5.53
6,212 5.51 5.52
6,214 5.50 5.52
6,220 5.47 5.52
6,244 5.48 5.52
6,251 5.48 5.51
6,255 5.61 5.51
6,294 5.59 5.50
6,317 5.48 5.50
6,438 5.54 5.49
6,482 5.52 5.48
6,526 5.53 5.48
6,558 5.49 5.48
6,565 5.49 5.47
6,566 5.51 5.48
6,603 5.48 5.47
6,626 5.48 5.47
6,653 5.49 5.46
6,664 5.51 5.46
6,671 5.51 5.46
6,726 5.46 5.44
6,742 5.45 5.43
6,795 5.38 5.41
6,879 5.29 5.36
6,916 5.26 5.34
6,920 5.22 5.33

5C: LINKS 1 and 2

Time,
t (seconds)		 Covariance, untransformed scale
5th percentile 95th percentile
102 −2,324.34 −1,014.81
109 −2,354.33 −1,011.67
137 −2,440.88 −1,009.05
204 −2,682.59 −1,018.79
205 −2,685.41 −1,020.02
214 −2,712.88 −1,023.91
227 −2,747.73 −1,035.53
233 −2,759.29 −1,039.30
259 −2,826.37 −1,058.60
271 −2,853.28 −1,065.96
327 −2,986.16 −1,111.99
362 −3,066.18 −1,141.66
379 −3,097.59 −1,150.47
401 −3,148.64 −1,173.57
403 −3,158.22 −1,173.88
419 −3,201.78 −1,188.86
430 −3,229.12 −1,195.75
516 −3,465.64 −1,269.29
519 −3,479.25 −1,271.86
522 −3,491.86 −1,275.82
549 −3,581.20 −1,302.93
572 −3,646.20 −1,326.05
582 −3,670.02 −1,337.35
606 −3,731.31 −1,361.35
625 −3,759.18 −1,376.50
634 −3,782.78 −1,386.89
641 −3,790.97 −1,393.76
669 −3,859.30 −1,425.78
730 −3,992.19 −1,470.35
751 −4,029.73 −1,488.37
772 −4,078.59 −1,506.39
773 −4,087.30 −1,507.45
814 −4,140.83 −1,515.79
821 −4,139.59 −1,518.21
850 −4,158.80 −1,516.91
855 −4,158.16 −1,516.40
881 −4,170.82 −1,515.84
906 −4,158.27 −1,504.73
910 −4,154.19 −1,505.62
913 −4,151.60 −1,505.77
915 −4,151.23 −1,504.60
974 −4,037.40 −1,468.04
994 −3,993.51 −1,454.06
1,012 −3,950.28 −1,439.94
1,036 −3,886.02 −1,423.37
1,038 −3,882.83 −1,422.65
1,040 −3,880.78 −1,422.27
1,042 −3,871.31 −1,420.58
1,101 −3,677.82 −1,374.15
1,166 −3,442.04 −1,322.07
1,185 −3,374.32 −1,313.50
1,198 −3,339.00 −1,308.59
1,237 −3,231.50 −1,288.74
1,332 −2,923.14 −1,244.79
1,396 −2,719.45 −1,229.89
1,978 −2,253.95 −1,005.90
2,033 −2,298.06 −981.64
2,058 −2,298.71 −971.74
2,066 −2,301.48 −970.13
2,286 −2,255.43 −913.97
2,296 −2,250.53 −912.15
2,335 −2,231.90 −900.98
2,340 −2,220.98 −896.34
2,346 −2,208.66 −893.58
2,355 −2,194.53 −891.77
2,552 −1,818.41 −823.61
2,805 −1,506.82 −729.20
3,122 −1,562.91 −677.77
3,136 −1,555.28 −672.86
3,212 −1,552.15 −657.18
3,274 −1,534.31 −649.32
3,326 −1,526.08 −644.71
3,411 −1,456.70 −638.55
3,520 −1,358.60 −637.29
3,888 −1,173.04 −514.36
3,959 −1,164.09 −478.00
3,972 −1,160.15 −472.83
4,028 −1,135.95 −450.03
4,038 −1,133.57 −447.13
4,081 −1,113.49 −432.35
4,086 −1,112.52 −430.72
4,185 −1,063.61 −406.24
4,207 −1,048.22 −399.99
4,243 −1,032.07 −390.45
4,275 −1,013.64 −382.92
4,281 −1,013.81 −381.79
4,301 −998.51 −376.35
4,309 −993.52 −374.90
4,316 −988.14 −373.02
4,318 −987.26 −372.23
4,467 −904.44 −346.70
4,486 −895.07 −344.67
4,510 −889.39 −343.43
4,517 −888.25 −342.42
4,527 −885.91 −343.00
4,557 −874.24 −342.94
4,563 −869.20 −343.03
4,566 −867.06 −342.92
4,673 −812.94 −338.56
4,897 −751.94 −317.65
4,932 −737.80 −313.88
4,938 −735.98 −313.06
5,006 −703.41 −300.16
5,243 −558.24 −242.17
5,272 −539.01 −233.17
5,403 −444.74 −194.25
5,410 −439.94 −192.41
5,642 −326.86 −144.29
5,751 −314.00 −130.90
5,833 −313.02 −125.48
5,851 −312.28 −124.89
5,924 −313.81 −121.73
5,929 −313.94 −121.57
5,950 −314.53 −120.79
5,982 −313.24 −118.74
5,984 −312.67 −118.55
6,048 −302.36 −114.17
6,074 −296.87 −112.03
6,142 −281.77 −105.92
6,174 −276.18 −102.61
6,201 −268.80 −100.60
6,202 −268.86 −100.58
6,212 −266.46 −99.64
6,214 −266.00 −99.53
6,220 −264.90 −99.12
6,244 −259.67 −97.04
6,251 −258.19 −96.49
6,255 −257.54 −96.15
6,294 −248.52 −93.21
6,317 −241.94 −91.54
6,438 −216.46 −83.03
6,482 −208.89 −79.90
6,526 −200.45 −76.55
6,558 −195.10 −73.66
6,565 −193.28 −73.07
6,566 −193.28 −73.00
6,603 −184.06 −70.01
6,626 −179.15 −68.14
6,653 −173.41 −65.68
6,664 −170.97 −64.69
6,671 −169.10 −64.26
6,726 −153.76 −59.65
6,742 −149.29 −58.15
6,795 −134.86 −54.22
6,879 −110.27 −49.36
6,916 −100.02 −47.59
6,920 −98.94 −47.28

5D: CORRELATION, LINKS 1 and 2

ρ12 Count
−1.0 0
−0.5 0
−0.5 1
−0.4 4
−0.4 23
−0.3 47
−0.3 122
−0.2 112
−0.2 113
−0.1 39
−0.1 31
−0.6 6
0.0 0
0.5 0
1.0 0
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