Обработка данных. Выбор экземпляров
1. Выполните классификацию k-ближайших соседей с использованием
функции knn() из пакета class на наборе данных iris. Проведите
нормализацию данных, разделите выборку на обучающую и тестовую. Оцените
построенную модель с использованием функции CrossTable() из пакета
gmodels. Постройте матрицу ошибок и диагональную оценку качества
прогноза (diagonal mark quality prediction).
str(iris)
## 'data.frame': 150 obs. of 5 variables:
## $ Sepal.Length: num 5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 ...
## $ Sepal.Width : num 3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 ...
## $ Petal.Length: num 1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 ...
## $ Petal.Width : num 0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 ...
## $ Species : Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 1 1 1 ...
table(iris$Species)
##
## setosa versicolor virginica
## 50 50 50
round(prop.table(table(iris$Species)) * 100, digits = 1)
##
## setosa versicolor virginica
## 33.3 33.3 33.3
plot(iris$Sepal.Length, iris$Sepal.Width, col=iris$Species, pch=19)
legend("topright", legend=levels(iris$Species), bty="n", pch=19, col=palette())
normMinMax = function(x) {
return((x-min(x))/max(x)-min(x))
}
iris_norm <- as.data.frame(lapply(iris[1:4], normMinMax))
set.seed(1234)
indexes = sample(2, nrow(iris), replace=TRUE, prob=c(0.7, 0.3))
iris_train = iris[indexes==1, 1:4]
iris_test = iris[indexes==2, 1:4]
iris_train_labels = iris[indexes==1, 5]
iris_test_labels = iris[indexes==2, 5]
#install.packages("class")
library("class")
iris_mdl = knn(train=iris_train, test=iris_test, cl=iris_train_labels, k=3)
#install.packages("gmodels")
library("gmodels")
## Warning: пакет 'gmodels' был собран под R версии 4.2.2
CrossTable(x=iris_test_labels, y=iris_mdl, prop.chisq = FALSE)
##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 38
##
##
## | iris_mdl
## iris_test_labels | setosa | versicolor | virginica | Row Total |
## -----------------|------------|------------|------------|------------|
## setosa | 10 | 0 | 0 | 10 |
## | 1.000 | 0.000 | 0.000 | 0.263 |
## | 1.000 | 0.000 | 0.000 | |
## | 0.263 | 0.000 | 0.000 | |
## -----------------|------------|------------|------------|------------|
## versicolor | 0 | 12 | 0 | 12 |
## | 0.000 | 1.000 | 0.000 | 0.316 |
## | 0.000 | 0.923 | 0.000 | |
## | 0.000 | 0.316 | 0.000 | |
## -----------------|------------|------------|------------|------------|
## virginica | 0 | 1 | 15 | 16 |
## | 0.000 | 0.062 | 0.938 | 0.421 |
## | 0.000 | 0.077 | 1.000 | |
## | 0.000 | 0.026 | 0.395 | |
## -----------------|------------|------------|------------|------------|
## Column Total | 10 | 13 | 15 | 38 |
## | 0.263 | 0.342 | 0.395 | |
## -----------------|------------|------------|------------|------------|
##
##
CM = table(iris_test_labels, iris_mdl)
CM
## iris_mdl
## iris_test_labels setosa versicolor virginica
## setosa 10 0 0
## versicolor 0 12 0
## virginica 0 1 15
accuracy = (sum(diag(CM)))/sum(CM)
accuracy
## [1] 0.9736842
2. Рассмотрите пример реализации метода опорных векторов с
использованием функции svm() из пакета e1071. Постройте линейный
классификатор для прогнозирования. Для подбора параметров модели
выполните перекрестную проверку с делением исходной выборки на 10 равных
частей (cross=10).
#install.packages("e1071")
library("e1071")
## Warning: пакет 'e1071' был собран под R версии 4.2.2
DGlass <- read.table(file="Glass.txt", sep=",", header=TRUE, row.names=1)
DGlass$F <- as.factor(ifelse(DGlass$Class==2,2,1))
svm.all <- svm(formula = F ~ ., data=DGlass[,-10], cross=10, kernel = "linear")
table(Факт=DGlass$F, Прогноз=predict(svm.all))
## Прогноз
## Факт 1 2
## 1 72 15
## 2 27 49
Acc = mean(predict(svm.all) == DGlass$F)
paste("Точность=", round(100*Acc, 2), "%", sep="")
## [1] "Точность=74.23%"
library(e1071)
CVsvm <- function(x, y) {
n <- nrow(x) ; Err_S <- 0
for(i in 1:n) {
svm.temp <- svm(x=x[-i,], y = y[-i], kernel = "linear")
if (predict(svm.temp, newdata=x[i,]) != y[i])
Err_S <- Err_S +1 }
Err_S/n }
Acc <- 1-CVsvm(DGlass[,1:9],DGlass$F)
paste("Точность=", round(100*Acc, 2), "%", sep="")
## [1] "Точность=71.17%"
3. Выполните расчет главных компонент с использованием пакета
vegan() и его функции rda(). Постройте ординационную диаграмму методом
PCA [3, c. 49] и сделайте выводы.
#install.packages("vegan")
library(vegan)
## Warning: пакет 'vegan' был собран под R версии 4.2.2
## Загрузка требуемого пакета: permute
## Warning: пакет 'permute' был собран под R версии 4.2.2
## Загрузка требуемого пакета: lattice
## This is vegan 2.6-4
Y <- as.data.frame(DGlass[,1:9])
mod.pca <- rda(Y ~ 1)
summary(mod.pca)
##
## Call:
## rda(formula = Y ~ 1)
##
## Partitioning of variance:
## Inertia Proportion
## Total 3.656 1
## Unconstrained 3.656 1
##
## Eigenvalues, and their contribution to the variance
##
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Eigenvalue 2.6056 0.5828 0.21129 0.18895 0.04308 0.014033 0.008907
## Proportion Explained 0.7126 0.1594 0.05779 0.05168 0.01178 0.003838 0.002436
## Cumulative Proportion 0.7126 0.8721 0.92984 0.98152 0.99330 0.997137 0.999573
## PC8 PC9
## Eigenvalue 0.0015623 6.952e-07
## Proportion Explained 0.0004273 1.901e-07
## Cumulative Proportion 0.9999998 1.000e+00
##
## Scaling 2 for species and site scores
## * Species are scaled proportional to eigenvalues
## * Sites are unscaled: weighted dispersion equal on all dimensions
## * General scaling constant of scores: 4.93332
##
##
## Species scores
##
## PC1 PC2 PC3 PC4 PC5 PC6
## RI -0.006663 0.003040 -0.001587 -0.0007297 0.0005398 9.075e-05
## Na 0.461583 1.180084 0.833131 0.0499341 0.0109644 2.782e-03
## Mg 2.037591 0.659746 -0.616211 -0.5223737 -0.0598691 4.080e-02
## Al 0.240546 -0.337483 -0.107770 0.6588300 -0.2282742 1.616e-01
## Si 0.691702 -1.339222 0.458279 -0.4986449 0.0521649 4.062e-02
## K 0.265988 -0.302647 -0.111021 0.2512089 -0.1346813 -2.370e-01
## Ca -3.512043 0.228783 -0.158561 -0.3491594 -0.0734353 2.515e-02
## Ba -0.190199 -0.009844 -0.267021 0.3372595 0.4523324 1.708e-02
## Fe -0.039086 -0.017137 -0.046526 0.0299297 0.0074709 -8.279e-02
##
##
## Site scores (weighted sums of species scores)
##
## PC1 PC2 PC3 PC4 PC5 PC6
## 1 0.1468069 0.653541 -0.452476 -0.2986962 -0.0429202 0.936689
## 2 0.2831309 0.140664 0.445786 0.1797183 0.1566509 0.104893
## 3 0.2892866 -0.078679 0.338106 0.1733055 0.1124002 0.711760
## 4 0.1922588 0.012620 -0.081285 0.0359569 0.0035938 -0.172746
## 5 0.2337499 -0.146211 0.160402 -0.1074701 0.1936305 -0.073273
## 6 0.2214789 -0.297171 -0.200230 0.1444547 -0.1806841 0.058841
## 7 0.2114214 -0.131683 0.184216 -0.1760887 0.2406837 -0.298989
## 8 0.1990103 -0.214651 0.139648 -0.3142412 0.3185712 -0.345347
## 9 0.1650442 0.427873 0.271529 0.3424207 -0.1176032 -0.241228
## 10 0.1555260 -0.193697 -0.071725 -0.0963416 -0.0154747 0.019020
## 11 0.2067898 -0.419220 -0.100392 0.0813228 -0.0821635 -0.065561
## 12 0.1251108 -0.235822 -0.219846 -0.2253187 -0.0182958 -0.043307
## 13 0.2169013 -0.393270 0.048795 -0.0002951 0.0730060 -0.377453
## 14 0.1580811 -0.310301 -0.055911 -0.2186649 0.1199783 -0.045091
## 15 0.1353949 -0.407632 -0.205212 -0.2811839 0.0329116 0.146036
## 16 0.1540141 -0.337567 -0.062588 -0.2440226 0.1337667 -0.059614
## 17 0.0973589 -0.288098 -0.270874 -0.3683717 0.0402427 -0.177028
## 18 -0.0084986 0.943531 0.081491 -0.0408565 0.0643035 0.011934
## 19 0.0507622 0.487314 0.123418 -0.1044787 0.0880175 0.936470
## 20 0.1347476 -0.131896 -0.144416 0.1872670 -0.3094050 0.573642
## 21 0.1121537 -0.176349 -0.258422 0.0431257 -0.1736372 0.149900
## 22 0.0371994 0.876116 0.655614 -0.5413318 0.7933986 -0.506454
## 23 0.0828807 -0.165469 -0.302383 -0.1526636 -0.0978158 -0.059376
## 24 0.1105440 -0.258288 -0.182322 -0.1539343 -0.0780710 0.017767
## 25 0.1384949 -0.020805 0.173453 -0.1191470 0.1839849 -0.154338
## 26 0.1214141 -0.198476 -0.059864 -0.1774827 0.0437900 -0.333474
## 27 0.1282319 -0.033062 -0.013949 0.1198376 -0.1058145 -0.038016
## 28 0.1335665 -0.265261 -0.076520 -0.1001442 0.0317380 0.063529
## 29 0.1136685 -0.393717 -0.262697 -0.1488337 -0.0809386 0.296854
## 30 0.1209217 -0.132781 -0.021245 -0.0594555 0.0105849 -0.175989
## 31 0.0860924 -0.318960 -0.264456 -0.2325393 -0.0486448 -0.123568
## 32 0.1173561 -0.326776 -0.027078 -0.3336688 0.1887918 -0.125028
## 33 0.1018155 -0.235916 -0.146484 -0.1223026 0.1887853 -0.410948
## 34 0.1152386 -0.479691 -0.158309 -0.2443691 -0.0017352 0.166560
## 35 0.0677300 -0.261521 -0.276400 -0.1781288 -0.0986615 0.123308
## 36 0.0992427 -0.020357 0.081085 -0.0537938 0.0591605 -0.237457
## 37 0.0445475 0.503223 0.049375 0.3243428 -0.0481309 -0.195238
## 38 0.0777972 -0.270426 -0.215633 -0.1165364 -0.1053155 -0.076241
## 39 -0.0911489 0.815696 0.127367 -0.5414945 0.3850218 -0.339415
## 40 -0.0911489 0.815696 0.127367 -0.5414945 0.3850218 -0.339415
## 41 0.0628550 -0.250665 -0.164489 -0.2960750 0.0652168 -0.409279
## 42 0.0847742 -0.358011 -0.108959 -0.2654146 0.1028795 -0.142373
## 43 0.0889362 -0.073396 0.052679 0.0233180 -0.0424783 -0.120474
## 44 -0.1320087 0.660142 -0.212061 -0.4691470 0.0899143 -0.023284
## 45 0.0518571 -0.259647 -0.208012 -0.1950786 0.0270265 -0.573156
## 46 -0.0087009 0.332246 -0.129429 0.2068569 -0.3000674 -0.178591
## 47 0.0331130 -0.041018 0.017281 -0.0986988 0.0251734 -0.433167
## 48 -0.1679795 0.797858 -0.058337 -0.3803800 0.1550659 0.136950
## 49 -0.2012391 0.424215 -0.448261 -0.6009582 0.0178371 0.328678
## 50 0.0007798 0.294650 0.039392 0.1829980 -0.1460324 -0.509506
## 51 -0.2160692 0.682819 -0.187630 -0.6267501 0.2397490 -0.296316
## 52 -0.0464905 0.086955 -0.119905 0.0321635 -0.2059905 -0.382238
## 53 -0.0547600 -0.081275 0.401190 0.0128336 0.1066234 -0.328492
## 54 -0.0724550 -0.183510 0.234393 0.0458549 0.0217208 -0.179401
## 55 -0.0594847 -0.213291 0.336418 0.0228252 0.0802899 -0.108306
## 56 -0.0704153 -0.712215 0.151214 -0.2488176 0.1732766 -0.110566
## 57 0.1467028 -0.202253 -0.010477 -0.1358693 0.1950615 -0.775254
## 58 0.1300355 -0.233834 -0.105940 -0.0774267 0.0284228 -0.146323
## 59 0.2575892 -0.030250 0.209615 -0.1308647 0.2074165 -0.287136
## 60 0.1889440 0.011296 0.095530 -0.0794457 0.1191626 -0.359259
## 61 0.0771736 0.190036 0.177048 -0.2614640 0.2455654 0.746507
## 62 0.0546356 0.540926 -0.116183 0.4442499 1.0483509 0.851199
## 63 -0.0909277 0.555389 -0.339461 -0.3441461 -0.0186931 -0.034835
## 64 -0.1318250 0.935502 -0.057407 -0.2653125 0.0833017 0.320174
## 65 -0.1116236 0.456380 -0.237121 -0.4026143 0.0338339 0.223323
## 66 -0.0812986 0.487817 -0.047051 -0.1582556 -0.0168286 0.756854
## 67 -0.1743517 0.253329 -0.412842 -0.5435280 0.0347130 0.161232
## 68 -0.1703635 0.218810 -0.380257 -0.5831745 0.0529063 0.204720
## 69 -0.1743832 0.262167 -0.348965 -0.4719521 0.0131173 0.075231
## 70 -0.2510780 0.429206 -0.325188 -0.5452860 0.0045123 0.339750
## 71 0.3781360 0.708146 0.754661 0.7985880 -0.0060587 1.026982
## 72 0.1772181 0.406967 -0.135191 0.1993558 -0.1230611 -0.579174
## 73 0.2805550 -0.260841 0.069565 0.1271804 -0.0211697 0.032264
## 74 0.2633097 -0.104933 0.145619 0.2341497 -0.0762680 0.185875
## 75 0.2624362 -0.292199 0.031129 0.1339269 -0.0730998 -0.052011
## 76 0.2521590 -0.283762 0.023527 0.0832923 -0.0528479 0.001452
## 77 0.2235291 0.104945 0.013489 0.3672157 -0.2035813 -0.143518
## 78 0.2230509 -0.181389 -0.087822 0.1750000 -0.1131681 0.151199
## 79 0.2547114 0.108423 0.547569 0.0478701 0.3165085 0.130437
## 80 0.2328024 -0.298385 -0.184396 0.4048680 -0.4083247 0.533670
## 81 0.2289419 -0.236317 -0.242730 0.6009904 -0.6183316 0.828517
## 82 0.2629144 -0.245865 0.256749 0.0964700 0.1190682 0.010953
## 83 0.2145421 -0.031571 0.171617 0.0570823 0.0877684 -0.521617
## 84 0.2189771 -0.184163 -0.005580 0.2062197 -0.1309796 -0.092345
## 85 0.3862146 0.163549 0.716313 1.2934362 -0.4773278 -0.832405
## 86 0.1906738 -0.007059 0.086890 0.1275616 -0.0479922 0.479376
## 87 0.2331419 -0.245233 0.255292 -0.0244364 0.1575381 0.761558
## 88 0.2082620 -0.013912 0.118891 0.2924066 -0.1317270 -0.219202
## 89 0.1991266 -0.201947 -0.030927 0.1292000 -0.0531897 0.085757
## 90 0.2118602 -0.501657 -0.214594 0.2079756 -0.3167420 0.719267
## 91 0.0257571 0.102732 -0.459035 -0.1455973 -0.1509585 -0.787595
## 92 0.1626865 -0.280356 -0.016387 -0.0069597 0.0466821 0.524716
## 93 0.1443948 -0.263786 0.191686 -0.0996536 0.1422599 1.632912
## 94 0.1712175 -0.208537 0.249926 0.0465840 0.1082097 0.656590
## 95 0.1636377 -0.455989 -0.026896 0.0191808 -0.0297872 0.044148
## 96 0.0758216 0.149762 -0.041392 0.2445195 -0.1806563 0.062746
## 97 -0.0185118 0.103673 -0.346609 -0.1910232 -0.1171016 -0.789928
## 98 0.0119047 -0.647457 -0.259098 -0.4352474 0.1399667 -0.367660
## 99 0.0501388 -0.526643 0.068348 0.2761857 -0.2302582 0.125117
## 100 -0.0023463 -0.275835 0.087541 0.1649489 0.1521819 -0.019062
## 101 -0.0073871 -0.478662 0.125407 0.0599733 0.1953284 -0.028196
## 102 -0.1320233 -0.483833 -0.231198 0.2252823 -0.3698801 0.021329
## 103 -0.1380376 -0.612170 0.278419 -0.6902410 0.5609710 -0.166406
## 104 -0.6456153 1.099167 -0.458552 -0.2848663 -0.3764022 -0.203957
## 105 -0.4718632 0.771074 -0.085493 0.0933298 -0.4058715 0.593518
## 106 -1.3102496 -0.780759 -0.285356 0.6366394 -1.2289235 -0.557524
## 107 -1.4713648 -0.183581 -2.088964 2.4475244 3.1958580 -0.004478
## 108 -1.9885037 0.483326 -0.650701 0.0550906 -1.2991646 -0.303698
## 109 -0.8860189 -0.008354 1.962377 0.4331774 0.4401990 -0.213398
## 110 -0.7340592 -0.823594 2.225817 -0.4059408 1.2678094 -0.039731
## 111 -1.5940657 -0.953610 -0.085414 -0.9275683 -0.1652429 0.703606
## 112 -1.6618145 -0.954359 -0.282213 -0.9725733 -0.2527127 0.681596
## 113 -1.5477689 -0.126432 0.396581 -0.3624646 -0.2046791 -0.172564
## 114 0.2142594 0.134586 -0.015287 0.0019161 0.0031011 -0.308931
## 115 0.1719956 0.104202 -0.342907 -0.1219932 -0.0782661 -0.320273
## 116 0.1993896 0.191601 -0.130478 0.0443299 -0.0913907 0.058157
## 117 0.1894690 0.150099 -0.267837 0.0971972 -0.1966340 -0.007936
## 118 0.2598551 0.285161 0.038955 0.6591716 -0.4346719 0.229392
## 119 0.2279653 0.019504 -0.045009 0.2933579 -0.1695131 -0.322663
## 120 0.2373132 0.117454 0.136488 0.3435757 -0.0966723 -0.220522
## 121 0.1532426 0.117587 -0.182327 0.0539188 -0.1275014 -0.151936
## 122 0.2247944 -0.265176 -0.083721 0.1929649 -0.1535907 0.061639
## 123 0.2111228 -0.109025 0.070992 0.1387069 -0.0383132 0.184159
## 124 0.2243494 0.036104 0.131145 0.4634214 -0.2575650 0.242295
## 125 0.1332503 -0.008237 -0.057939 -0.1784048 0.0794937 -0.201848
## 126 0.1140993 -0.048467 -0.329250 0.1316202 -0.3200861 0.186569
## 127 0.1037421 -0.105641 -0.200311 -0.1202447 -0.0333008 -0.079069
## 128 -0.3126771 0.249680 0.478328 0.4989058 -0.2799120 -0.307813
## 129 -0.2758630 0.040628 0.445996 0.7771977 0.0733266 -0.027311
## 130 -0.5683353 0.263987 0.841529 0.8722168 -0.4137612 -0.025969
## 131 -0.7189561 0.050941 0.960357 0.5151030 -0.1834270 0.006499
## 132 -1.3449351 0.338028 0.812137 0.6451526 -0.6679765 0.036613
## 133 0.2401993 0.174939 -0.101594 -0.0643698 0.0304291 -0.326262
## 134 0.2065498 0.461018 -0.171555 0.4046095 -0.3514169 -0.027916
## 135 0.2399565 0.019130 -0.006621 -0.0963094 0.0866466 0.005394
## 136 0.1598833 0.151330 -0.309638 0.0057856 -0.1395767 -0.324282
## 137 0.1822030 -0.163387 -0.109118 -0.3540032 0.1903190 -0.332152
## 138 0.2162485 -0.251403 -0.138341 0.0987608 -0.1438860 0.292678
## 139 0.2606658 -0.450579 -0.001395 0.0303083 0.0001248 0.184652
## 140 0.2417294 -0.346111 -0.060238 0.1294935 -0.1431601 0.331452
## 141 0.2034381 0.004872 0.011912 0.3509053 -0.2514941 -0.020241
## 142 0.1508226 -0.047205 -0.016507 -0.1665515 0.3286651 -0.567277
## 143 0.1780054 -0.287868 -0.126302 0.0576450 0.0277000 -0.315677
## 144 0.1816259 -0.179515 -0.114358 0.3663742 -0.3685873 0.398663
## 145 0.0201565 -0.227126 0.058505 -0.0911967 0.0653796 -0.428547
## 146 0.0848711 -0.060447 -0.362373 -0.0897477 -0.1056451 -0.609365
## 147 0.1203272 0.160076 0.251911 -0.2892059 0.3178440 0.855055
## 148 0.1573150 0.004117 0.064015 0.0770438 -0.0107786 -0.071461
## 149 0.1388705 -0.024247 -0.015826 0.0377586 -0.0745257 -0.033439
## 150 0.0935700 -0.419268 -0.577115 -0.1007629 -0.0772871 0.030768
## 151 0.1312131 -0.039152 -0.122003 0.3962154 -0.3974487 0.293251
## 152 -0.0700853 0.928509 0.059614 -0.2744927 0.1089498 0.461842
## 153 0.0540899 0.139315 0.327306 -0.7191940 0.6665850 0.370274
## 154 0.1460560 0.009990 0.188761 0.0725898 0.1074519 -0.403299
## 155 0.0554919 -0.083795 -0.309378 -0.0720243 -0.1675207 -0.144663
## 156 0.0955381 -0.198745 0.041074 -0.1345727 0.0855845 0.041386
## 157 0.0755571 0.042210 0.134649 0.0220057 0.0006557 -0.059419
## 158 -0.1168639 0.738986 0.036095 -0.4962379 0.2862393 -0.150431
## 159 0.0330710 0.271928 -0.037453 0.3551775 -0.3632008 -0.007375
## 160 0.0195299 0.280813 -0.078775 0.4666121 -0.4577559 0.067714
## 161 -0.0170152 0.176284 -0.115178 0.2876232 -0.3957898 0.138785
## 162 0.0360425 0.221129 0.213986 -0.4053523 0.7524278 -0.106796
## 163 -0.0185512 0.869712 -0.007506 0.0195031 0.0311207 -0.522748
library(ggplot2)
F <- as.factor(ifelse(DGlass$Class == 2, 2, 1))
pca.scores <- as.data.frame(summary(mod.pca)$sites[,1:2])
pca.scores <- cbind(pca.scores, F)
l <- lapply(unique(pca.scores$F), function(c)
{ f <- subset(pca.scores, F == c); f[chull(f), ]})
hull <- do.call(rbind, l)
axX <- paste("PC1 (",
as.integer(100*mod.pca$CA$eig[1]/sum(mod.pca$CA$eig)),"%)")
axY <- paste("PC2 (",
as.integer(100*mod.pca$CA$eig[2]/sum(mod.pca$CA$eig)),"%)")
ggplot() +
geom_polygon(data=hull,aes(x=PC1,y=PC2, fill=F),
alpha=0.4, linetype=0) +
geom_point(data=pca.scores,aes(x=PC1,y=PC2,shape=F,
colour=F),size=3) +
scale_colour_manual( values = c('red', 'orange'))+
xlab(axX) + ylab(axY) + coord_equal() + theme_bw()
