文章目录

写在前面独立性检验χ2\chi^2χ2独立性检验Fisher独立性检验Cochran-Mantel-Haenszel χ2\chi^2χ2独立性检验相关性分析相关性检验相关性检验偏相关检验分组数据的相关性检验两组数据多组数据图形表示轮廓图星图(雷达图、蜘蛛图)调和曲线图

写在前面

总结学习R语言的笔记，这次的主要学习内容是：多元数据的数据特征、独立性检验、相关分析、相关性检验与图形表示。由于使用R Markdown编写，所以在导出为.md文件时直接将运行结果也导出了，显得有点不美观，见谅。

独立性检验

原假设：未发生；

备择假设：发生了。

p-value大于0.050.050.05则不拒绝原假设，认为两变量不具有相关关系；

p-value小于0.050.050.05则拒绝原假设，认为两变量具有相关关系。

数值设置不绝对，也可选取KaTeX parse error: Undefined control sequence: \mbox at position 5: 0.01\̲m̲b̲o̲x̲{或者}0.1，要根据具体数据的精度来进行调整。

χ2\chi^2χ2独立性检验

使用chisq.test()函数进行χ2\chi^2χ2独立性检验：

(交换两变量的顺序，结果不变)

# 加载数据包library(vcd)

## Loading required package: grid

mytable <- table(Arthritis$Treatment, Arthritis$Improved)# 进行卡方独立性检验(p-value小于0.05，具有相关关系)chisq.test(mytable)

## ## Pearson's Chi-squared test## ## data: mytable## X-squared = 13.055, df = 2, p-value = 0.001463

mytable <- table(Arthritis$Improved, Arthritis$Sex)# 进行卡方独立性检验(p-value大于0.05，不具有相关关系)chisq.test(mytable)

## ## Pearson's Chi-squared test## ## data: mytable## X-squared = 4.8407, df = 2, p-value = 0.08889

Fisher独立性检验

使用fisherq.test()函数进行FisherFisherFisher独立性检验：

(交换两变量的顺序，结果不变)

# 加载数据包library(vcd)mytable <- table(Arthritis$Treatment, Arthritis$Improved)# 进行Fisher独立性检验fisher.test(mytable)

## ## Fisher's Exact Test for Count Data## ## data: mytable## p-value = 0.001393## alternative hypothesis: two.sided

mytable <- table(Arthritis$Sex, Arthritis$Improved)# 进行Fisher独立性检验fisher.test(mytable)

## ## Fisher's Exact Test for Count Data## ## data: mytable## p-value = 0.1094## alternative hypothesis: two.sided

Cochran-Mantel-Haenszel χ2\chi^2χ2独立性检验

使用mantelhaen.test()函数进行Cochran-Mantel-Haenszel χ2\chi^2χ2独立性检验：

(变量的顺序会影响p-value的值)

# 加载数据包library(vcd)# 三个变量列联表mytable <- xtabs(~Treatment+Improved+Sex, data = Arthritis)# 三个变量进行Cochran-Mantel-Haenszel 卡方独立性检验mantelhaen.test(mytable)

## ## Cochran-Mantel-Haenszel test## ## data: mytable## Cochran-Mantel-Haenszel M^2 = 14.632, df = 2, p-value = 0.0006647

mytable <- xtabs(~Treatment+Sex+Improved, data = Arthritis)# 三个变量进行Cochran-Mantel-Haenszel 卡方独立性检验mantelhaen.test(mytable)

## ## Mantel-Haenszel chi-squared test with continuity correction## ## data: mytable## Mantel-Haenszel X-squared = 2.0863, df = 1, p-value = 0.1486## alternative hypothesis: true common odds ratio is not equal to 1## 95 percent confidence interval:## 0.8566711 8.0070521## sample estimates:## common odds ratio ##2.619048

相关性分析

指对两个或多个具备相关性的变量元素进行分析，从而衡量两个变量因素的相关密切程度。

包括：Pearson相关系数、Spearman相关系数、Kendall相关系数等。

# 使用state.x77数据集，推测谋杀率与哪些因素有关state.x77

##Population Income Illiteracy Life Exp Murder HS Grad Frost## Alabama 3615 3624 2.1 69.05 15.1 41.3 20## Alaska365 6315 1.5 69.31 11.3 66.7 152## Arizona 2212 4530 1.8 70.55 7.8 58.1 15## Arkansas 2110 3378 1.9 70.66 10.1 39.9 65## California21198 5114 1.1 71.71 10.3 62.6 20## Colorado 2541 4884 0.7 72.06 6.8 63.9 166## Connecticut3100 5348 1.1 72.48 3.1 56.0 139## Delaware 579 4809 0.9 70.06 6.2 54.6 103## Florida 8277 4815 1.3 70.66 10.7 52.6 11## Georgia 4931 4091 2.0 68.54 13.9 40.6 60## Hawaii868 4963 1.9 73.60 6.2 61.90## Idaho 813 4119 0.6 71.87 5.3 59.5 126## Illinois 11197 5107 0.9 70.14 10.3 52.6 127## Indiana 5313 4458 0.7 70.88 7.1 52.9 122## Iowa 2861 4628 0.5 72.56 2.3 59.0 140## Kansas2280 4669 0.6 72.58 4.5 59.9 114## Kentucky 3387 3712 1.6 70.10 10.6 38.5 95## Louisiana 3806 3545 2.8 68.76 13.2 42.2 12## Maine1058 3694 0.7 70.39 2.7 54.7 161## Maryland 4122 5299 0.9 70.22 8.5 52.3 101## Massachusetts 5814 4755 1.1 71.83 3.3 58.5 103## Michigan 9111 4751 0.9 70.63 11.1 52.8 125## Minnesota 3921 4675 0.6 72.96 2.3 57.6 160## Mississippi2341 3098 2.4 68.09 12.5 41.0 50## Missouri 4767 4254 0.8 70.69 9.3 48.8 108## Montana746 4347 0.6 70.56 5.0 59.2 155## Nebraska 1544 4508 0.6 72.60 2.9 59.3 139## Nevada590 5149 0.5 69.03 11.5 65.2 188## New Hampshire 812 4281 0.7 71.23 3.3 57.6 174## New Jersey 7333 5237 1.1 70.93 5.2 52.5 115## New Mexico 1144 3601 2.2 70.32 9.7 55.2 120## New York 18076 4903 1.4 70.55 10.9 52.7 82## North Carolina 5441 3875 1.8 69.21 11.1 38.5 80## North Dakota637 5087 0.8 72.78 1.4 50.3 186## Ohio10735 4561 0.8 70.82 7.4 53.2 124## Oklahoma 2715 3983 1.1 71.42 6.4 51.6 82## Oregon2284 4660 0.6 72.13 4.2 60.0 44## Pennsylvania 11860 4449 1.0 70.43 6.1 50.2 126## Rhode Island931 4558 1.3 71.90 2.4 46.4 127## South Carolina 2816 3635 2.3 67.96 11.6 37.8 65## South Dakota681 4167 0.5 72.08 1.7 53.3 172## Tennessee 4173 3821 1.7 70.11 11.0 41.8 70## Texas12237 4188 2.2 70.90 12.2 47.4 35## Utah 1203 4022 0.6 72.90 4.5 67.3 137## Vermont472 3907 0.6 71.64 5.5 57.1 168## Virginia 4981 4701 1.4 70.08 9.5 47.8 85## Washington 3559 4864 0.6 71.72 4.3 63.5 32## West Virginia 1799 3617 1.4 69.48 6.7 41.6 100## Wisconsin 4589 4468 0.7 72.48 3.0 54.5 149## Wyoming376 4566 0.6 70.29 6.9 62.9 173## Area## Alabama 50708## Alaska 566432## Arizona 113417## Arkansas 51945## California156361## Colorado 103766## Connecticut4862## Delaware 1982## Florida 54090## Georgia 58073## Hawaii 6425## Idaho 82677## Illinois 55748## Indiana 36097## Iowa 55941## Kansas81787## Kentucky 39650## Louisiana 44930## Maine 30920## Maryland 9891## Massachusetts 7826## Michigan 56817## Minnesota 79289## Mississippi47296## Missouri 68995## Montana 145587## Nebraska 76483## Nevada 109889## New Hampshire 9027## New Jersey 7521## New Mexico121412## New York 47831## North Carolina 48798## North Dakota 69273## Ohio 40975## Oklahoma 68782## Oregon96184## Pennsylvania 44966## Rhode Island1049## South Carolina 30225## South Dakota 75955## Tennessee 41328## Texas262134## Utah 82096## Vermont9267## Virginia 39780## Washington66570## West Virginia 24070## Wisconsin 54464## Wyoming 97203

# 计算相关矩阵（对角矩阵）cor(state.x77)

## PopulationIncome Illiteracy Life ExpMurder## Population 1.00000000 0.2082276 0.10762237 -0.06805195 0.3436428## Income0.20822756 1.0000000 -0.43707519 0.34025534 -0.2300776## Illiteracy 0.10762237 -0.4370752 1.00000000 -0.58847793 0.7029752## Life Exp -0.06805195 0.3402553 -0.58847793 1.00000000 -0.7808458## Murder0.34364275 -0.2300776 0.70297520 -0.78084575 1.0000000## HS Grad -0.09848975 0.6199323 -0.65718861 0.58221620 -0.4879710## Frost-0.33215245 0.2262822 -0.67194697 0.26206801 -0.5388834## Area 0.02254384 0.3633154 0.07726113 -0.10733194 0.2283902##HS GradFrost Area## Population -0.09848975 -0.3321525 0.02254384## Income0.61993232 0.2262822 0.36331544## Illiteracy -0.65718861 -0.6719470 0.07726113## Life Exp 0.58221620 0.2620680 -0.10733194## Murder-0.48797102 -0.5388834 0.22839021## HS Grad1.00000000 0.3667797 0.33354187## Frost 0.36677970 1.0000000 0.05922910## Area 0.33354187 0.0592291 1.00000000

# 计算协方差矩阵cov(state.x77)

##Population Income IlliteracyLife Exp Murder## Population 19931683.7588 571229.7796 292.8679592 -4.078425e+02 5663.523714## Income 571229.7796 377573.3061 -163.7020408 2.806632e+02 -521.894286## Illiteracy292.8680-163.7020 0.3715306 -4.815122e-011.581776## Life Exp -407.8425280.6632 -0.4815122 1.80e+00 -3.869480## Murder 5663.5237-521.8943 1.5817755 -3.869480e+00 13.627465## HS Grad -3551.50963076.7690 -3.2354694 6.312685e+00 -14.549616## Frost -77081.97277227.6041 -21.2900000 1.828678e+01 -103.406000## Area 8587916.9494 19049013.7510 4018.3371429 -1.229410e+04 71940.429959## HS Grad FrostArea## Population -3551.509551 -77081.97265 8.587917e+06## Income 3076.768980 7227.60408 1.904901e+07## Illiteracy-3.235469 -21.29000 4.018337e+03## Life Exp 6.31268518.28678 -1.229410e+04## Murder -14.549616 -103.40600 7.194043e+04## HS Grad 65.237894 153.99216 2.298732e+05## Frost 153.992163 2702.00857 2.627039e+05## Area 229873.192816 262703.89306 7.280748e+09

colnames(state.x77)

## [1] "Population" "Income""Illiteracy" "Life Exp" "Murder" ## [6] "HS Grad" "Frost""Area"

x <- state.x77[, c(1, 2, 3, 6)]y <- state.x77[, c(4, 5)]head(x)

## Population Income Illiteracy HS Grad## Alabama3615 3624 2.1 41.3## Alaska 365 6315 1.5 66.7## Arizona2212 4530 1.8 58.1## Arkansas 2110 3378 1.9 39.9## California21198 5114 1.1 62.6## Colorado 2541 4884 0.7 63.9

head(y)

## Life Exp Murder## Alabama 69.05 15.1## Alaska 69.31 11.3## Arizona 70.55 7.8## Arkansas70.66 10.1## California 71.71 10.3## Colorado72.06 6.8

# 计算变量x与y之间的相关性cor(x, y)

##Life ExpMurder## Population -0.06805195 0.3436428## Income0.34025534 -0.2300776## Illiteracy -0.58847793 0.7029752## HS Grad0.58221620 -0.4879710

# 使用ggm外部包中的pcor()函数计算偏相关系数library(ggm)colnames(state.x77)

## [1] "Population" "Income""Illiteracy" "Life Exp" "Murder" ## [6] "HS Grad" "Frost""Area"

pcor(c(1, 5, 2, 3), cov(state.x77))

## [1] 0.3621683

相关性检验

相关性检验

使用cor.test()函数进行相关性的检验

cor.test(state.x77[, 3], state.x77[, 5])

## ## Pearson's product-moment correlation## ## data: state.x77[, 3] and state.x77[, 5]## t = 6.8479, df = 48, p-value = 1.258e-08## alternative hypothesis: true correlation is not equal to 0## 95 percent confidence interval:## 0.5279280 0.8207295## sample estimates:## cor ## 0.7029752

# 给出p值、相关系数与置信区间（概率可能发生的范围）

使用psych外部包的corr.test()函数进行多个变量的相关性检验

library(psych)print(corr.test(state.x77), short = F)

## Call:corr.test(x = state.x77)## Correlation matrix ## Population Income Illiteracy Life Exp Murder HS Grad Frost Area## Population 1.00 0.21 0.11 -0.07 0.34 -0.10 -0.33 0.02## Income 0.21 1.00-0.440.34 -0.23 0.62 0.23 0.36## Illiteracy 0.11 -0.44 1.00 -0.59 0.70 -0.66 -0.67 0.08## Life Exp -0.07 0.34-0.591.00 -0.78 0.58 0.26 -0.11## Murder 0.34 -0.23 0.70 -0.78 1.00 -0.49 -0.54 0.23## HS Grad -0.10 0.62-0.660.58 -0.49 1.00 0.37 0.33## Frost -0.33 0.23-0.670.26 -0.54 0.37 1.00 0.06## Area 0.02 0.36 0.08 -0.11 0.23 0.33 0.06 1.00## Sample Size ## [1] 50## Probability values (Entries above the diagonal are adjusted for multiple tests.) ## Population Income Illiteracy Life Exp Murder HS Grad Frost Area## Population 0.00 1.00 1.001.00 0.23 1.00 0.25 1.00## Income 0.15 0.00 0.030.23 1.00 0.00 1.00 0.16## Illiteracy 0.46 0.00 0.000.00 0.00 0.00 0.00 1.00## Life Exp 0.64 0.02 0.000.00 0.00 0.00 0.79 1.00## Murder 0.01 0.11 0.000.00 0.00 0.01 0.00 1.00## HS Grad0.50 0.00 0.000.00 0.00 0.00 0.16 0.25## Frost 0.02 0.11 0.000.07 0.00 0.01 0.00 1.00## Area 0.88 0.01 0.590.46 0.11 0.02 0.68 0.00## ## Confidence intervals based upon normal theory. To get bootstrapped values, try cor.ci## raw.lower raw.r raw.upper raw.p lower.adj upper.adj## Ppltn-Incom-0.07 0.210.46 0.15-0.190.54## Ppltn-Illtr-0.18 0.110.37 0.46-0.280.46## Ppltn-LfExp-0.34 -0.070.21 0.64-0.390.27## Ppltn-Murdr0.07 0.340.57 0.01-0.070.66## Ppltn-HSGrd-0.37 -0.100.18 0.50-0.440.27## Ppltn-Frost-0.56 -0.33-0.06 0.02-0.650.08## Ppltn-Area-0.26 0.020.30 0.88-0.260.30## Incom-Illtr-0.64 -0.44-0.18 0.00-0.72-0.03## Incom-LfExp0.07 0.340.57 0.02-0.070.65## Incom-Murdr-0.48 -0.230.05 0.11-0.570.18## Incom-HSGrd0.41 0.620.77 0.000.270.83## Incom-Frost-0.06 0.230.47 0.11-0.170.56## Incom-Area 0.09 0.360.58 0.01-0.050.67## Illtr-LfExp-0.74 -0.59-0.37 0.00-0.81-0.22## Illtr-Murdr0.53 0.700.82 0.000.400.87## Illtr-HSGrd-0.79 -0.66-0.46 0.00-0.85-0.32## Illtr-Frost-0.80 -0.67-0.48 0.00-0.85-0.35## Illtr-Area-0.21 0.080.35 0.59-0.280.42## LfExp-Murdr-0.87 -0.78-0.64 0.00-0.91-0.53## LfExp-HSGrd0.36 0.580.74 0.000.220.80## LfExp-Frost-0.02 0.260.50 0.07-0.150.60## LfExp-Area-0.37 -0.110.18 0.46-0.460.27## Murdr-HSGrd-0.67 -0.49-0.24 0.00-0.75-0.09## Murdr-Frost-0.71 -0.54-0.31 0.00-0.78-0.16## Murdr-Area-0.05 0.230.48 0.11-0.180.57## HSGrd-Frost0.10 0.370.59 0.01-0.050.68## HSGrd-Area 0.06 0.330.56 0.02-0.080.65## Frost-Area-0.22 0.060.33 0.68-0.260.37

偏相关检验

使用ggm外部包中的pcor.test()函数进行偏相关的检验(与pcor()函数对应)

library(ggm)x <- pcor(c(1, 5, 2, 3, 6), cov(state.x77))# 需要三个值：`pcor()`函数计算出的偏相关系数，变量个数与样本数pcor.test(x, 3, n=50) # 输出t检验、自由度和p值

## $tval## [1] 2.476049## ## $df## [1] 45## ## $pvalue## [1] 0.01711252

分组数据的相关性检验

两组数据

student-t检验

主要用于样本数较小(<30)、总体标准差未知的数据

# 导入数据集library(MASS)UScrime

##M So Ed Po1 Po2 LF M.F Pop NW U1 U2 GDP IneqProb Time y## 1 151 1 91 58 56 510 950 33 301 108 41 394 261 0.084602 26. 791## 2 143 0 113 103 95 583 1012 13 102 96 36 557 194 0.029599 25.2999 1635## 3 142 1 89 45 44 533 969 18 219 94 33 318 250 0.083401 24.3006 578## 4 136 0 121 149 141 577 994 157 80 102 39 673 167 0.015801 29.9012 1969## 5 141 0 121 109 101 591 985 18 30 91 20 578 174 0.041399 21.2998 1234## 6 121 0 110 118 115 547 964 25 44 84 29 689 126 0.034201 20.9995 682## 7 127 1 111 82 79 519 982 4 139 97 38 620 168 0.042100 20.6993 963## 8 131 1 109 115 109 542 969 50 179 79 35 472 206 0.040099 24.5988 1555## 9 157 1 90 65 62 553 955 39 286 81 28 421 239 0.071697 29.4001 856## 10 140 0 118 71 68 632 1029 7 15 100 24 526 174 0.044498 19.5994 705## 11 124 0 105 121 116 580 966 101 106 77 35 657 170 0.016201 41.6000 1674## 12 134 0 108 75 71 595 972 47 59 83 31 580 172 0.031201 34.2984 849## 13 128 0 113 67 60 624 972 28 10 77 25 507 206 0.045302 36.2993 511## 14 135 0 117 62 61 595 986 22 46 77 27 529 190 0.053200 21.5010 664## 15 152 1 87 57 53 530 986 30 72 92 43 405 264 0.069100 22.7008 798## 16 142 1 88 81 77 497 956 33 321 116 47 427 247 0.052099 26.0991 946## 17 143 0 110 66 63 537 977 10 6 114 35 487 166 0.076299 19.1002 539## 18 135 1 104 123 115 537 978 31 170 89 34 631 165 0.119804 18.1996 929## 19 130 0 116 128 128 536 934 51 24 78 34 627 135 0.019099 24.9008 750## 20 125 0 108 113 105 567 985 78 94 130 58 626 166 0.034801 26.4010 1225## 21 126 0 108 74 67 602 984 34 12 102 33 557 195 0.022800 37.5998 742## 22 157 1 89 47 44 512 962 22 423 97 34 288 276 0.089502 37.0994 439## 23 132 0 96 87 83 564 953 43 92 83 32 513 227 0.030700 25.1989 1216## 24 131 0 116 78 73 574 1038 7 36 142 42 540 176 0.041598 17.6000 968## 25 130 0 116 63 57 641 984 14 26 70 21 486 196 0.069197 21.9003 523## 26 131 0 121 160 143 631 1071 3 77 102 41 674 152 0.041698 22.1005 1993## 27 135 0 109 69 71 540 965 6 4 80 22 564 139 0.036099 28.4999 342## 28 152 0 112 82 76 571 1018 10 79 103 28 537 215 0.038201 25.8006 1216## 29 119 0 107 166 157 521 938 168 89 92 36 637 154 0.023400 36.7009 1043## 30 166 1 89 58 54 521 973 46 254 72 26 396 237 0.075298 28.3011 696## 31 140 0 93 55 54 535 1045 6 20 135 40 453 200 0.041999 21.7998 373## 32 125 0 109 90 81 586 964 97 82 105 43 617 163 0.042698 30.9014 754## 33 147 1 104 63 64 560 972 23 95 76 24 462 233 0.049499 25.5005 1072## 34 126 0 118 97 97 542 990 18 21 102 35 589 166 0.040799 21.6997 923## 35 123 0 102 97 87 526 948 113 76 124 50 572 158 0.020700 37.4011 653## 36 150 0 100 109 98 531 964 9 24 87 38 559 153 0.006900 44.0004 1272## 37 177 1 87 58 56 638 974 24 349 76 28 382 254 0.045198 31.6995 831## 38 133 0 104 51 47 599 1024 7 40 99 27 425 225 0.053998 16.6999 566## 39 149 1 88 61 54 515 953 36 165 86 35 395 251 0.047099 27.3004 826## 40 145 1 104 82 74 560 981 96 126 88 31 488 228 0.038801 29.3004 1151## 41 148 0 122 72 66 601 998 9 19 84 20 590 144 0.025100 30.0001 880## 42 141 0 109 56 54 523 968 4 2 107 37 489 170 0.088904 12.1996 542## 43 162 1 99 75 70 522 996 40 208 73 27 496 224 0.054902 31.9989 823## 44 136 0 121 95 96 574 1012 29 36 111 37 622 162 0.028100 30.0001 1030## 45 139 1 88 46 41 480 968 19 49 135 53 457 249 0.056202 32.5996 455## 46 126 0 104 106 97 599 989 40 24 78 25 593 171 0.046598 16.6999 508## 47 130 0 121 90 91 623 1049 3 22 113 40 588 160 0.052802 16.0997 849

# 独立样本的t检验# 参数格式为y ~ x；y为数值型变量，x为二分型变量t.test(Prob ~ So, data = UScrime)

## ## Welch Two Sample t-test## ## data: Prob by So## t = -3.8954, df = 24.925, p-value = 0.0006506## alternative hypothesis: true difference in means is not equal to 0## 95 percent confidence interval:## -0.03852569 -0.01187439## sample estimates:## mean in group 0 mean in group 1 ##0.038512650.06371269

多组数据

已知样本分布：方差分析

未知样本分布：非参数检验

⋯\cdots⋯

图形表示

轮廓图

需要编写函数

# 编写轮廓绘制函数outline <- function(x, txt = TRUE) {# 数据转换为矩阵形式if (is.data.frame(x) == TRUE)x <- as.matrix(x)# 读取矩阵的行数、列数m <- nrow(x); n <- ncol(x)# 绘制图形的轮廓plot(c(1, n), c(min(x), max(x)), type = "n", main = "The outline graph of Data",xlab = "Number", ylab = "Value")# 遍历样本，依次绘制轮廓线for (i in 1:m) {lines(x[i, ], col = i)if (txt == TRUE) {k <- dimnames(x)[[1]][i]text(1 + (i - 1) %% n, x[i, 1 + (i-1) %% n], k)}}}# 读入数据(数据框格式)X <- read.table("test.data")# 轮廓图绘制outline(X)

星图(雷达图、蜘蛛图)

stars(mtcars, full = F, cex=0.6, col.segments = 2:6, draw.segments = T, xpd = T, key.loc = c(13, .5), mar = c(1, 0, 0, 0))

调和曲线图

# 构造绘制调和曲线图的函数unison <- function(x){if (is.data.frame(x) == TRUE)x <- as.matrix(x)t <- seq(-pi, pi, pi/30)# m为观测数据总数# n为数据的维数m <- nrow(x); n<-ncol(x)f <- array(0, c(m, length(t)))for (i in 1:m){# 设定循环初始值f[i,] <- x[i,1] / sqrt(2)for (j in 2:n) {if (j %% 2 == 0)f[i, ] <- f[i, ] + x[i, j] * sin(j / 2 * t)elsef[i, ] <- f[i, ] + x[i, j] * cos(j %/% 2 * t)}}# 绘制图形的轮廓以及图形参数的调整plot(c(-pi,pi), c(min(f), max(f)), type = "n",main = "The Unison graph of Data",xlab = "t", ylab = "f(t)")# 低水平绘图，添加曲线for(i in 1:m) lines(t, f[i,] , col = i)}# 读入数据X <- read.table("course.data")# 调和曲线图绘制unison(X)