HW3_njs48_report
Nicole Salomons
February 4, 2017
First we will load in all the data into R:
train = read.table("digits_train.csv", header = TRUE, sep = ',')
test = read.table("digits_test.csv", header = TRUE, sep = ',')
validation = read.table("digits_valid.csv", header = TRUE, sep = ',')plotDigit <- function(k, dat)
{
p <- matrix(as.numeric(dat[k,1:256]),16,16)
image(x=1:16, y=1:16, p[,16:1], xlab="", ylab="",
main=paste("Row: ", k, " | Digit: ", dat[k,257]))
}Part 1 - Which digits will be the hardest to classify?
I think one of the digits that will be the hardest to classify is the number 1. For two different reasons:
- The first being that it is only composed of one line, and algorithms usually look for things that share similar things. In this case it is comparing pixels, but these pixels have to be in exactly the same place. So even if two ones are identical but in one the pixels are all in column 6 and in the other all the pixels are in column 7, then most algorithms will consider that they are completely different from one another. Consider the images of 1 bellow:
fig.width = 50
fig.height = 100
plotDigit(348, train)
plotDigit(156, train)
2)The second reason why 1 might be hard to classify is that it might often get confused with the number 7. Both the 1 and 7 can be drawn as having a vertical line with a nearly horizontal line at the top.
plotDigit(156, train)
plotDigit(621, train)
Another digit that might often be misclassified is 8, because it is similar to 6 and 9. Some examples of eights that might be mistaken for a 6 or a 9:
plotDigit(84, train)
plotDigit(129, train)
plotDigit(137, train)
plotDigit(139, train)
Part 2:
Train a knn nearest neighbor to find the best k using the validation set:
library(class)
library(gmodels)
# y will be all the y values for the training set
y = train[257]
#tr is the x values or the training set
tr = train[c(0:256)]
# vy is the y value for the validation set
vy = validation[257]
# v are the x values for the validation set
v = validation[c(0:256)]
yy = y[,1]
knn_predict1 <- knn(tr, v, yy, k = 1)
knn_predict2 <- knn(tr, v, yy, k = 2)
knn_predict3 <- knn(tr, v, yy, k = 3)
knn_predict4 <- knn(tr, v, yy, k = 4)
knn_predict5 <- knn(tr, v, yy, k = 5)
knn_predict6 <- knn(tr, v, yy, k = 6)
knn_predict7 <- knn(tr, v, yy, k = 7)
knn_predict8 <- knn(tr, v, yy, k = 8)
knn_predict9 <- knn(tr, v, yy, k = 9)
knn_predict10 <- knn(tr, v, yy, k = 10)
knn_predict11 <- knn(tr, v, yy, k = 11)
knn_predict12 <- knn(tr, v, yy, k = 12)
mean(knn_predict1[1:318] != t(vy[1]))## [1] 0.1194969mean(knn_predict2[1:318] != t(vy[1]))## [1] 0.1540881mean(knn_predict3[1:318] != t(vy[1]))## [1] 0.1226415mean(knn_predict4[1:318] != t(vy[1]))## [1] 0.1257862mean(knn_predict5[1:318] != t(vy[1]))## [1] 0.1194969mean(knn_predict6[1:318] != t(vy[1]))## [1] 0.1194969mean(knn_predict7[1:318] != t(vy[1]))## [1] 0.1163522mean(knn_predict8[1:318] != t(vy[1]))## [1] 0.1194969mean(knn_predict9[1:318] != t(vy[1]))## [1] 0.1257862mean(knn_predict10[1:318] != t(vy[1]))## [1] 0.1352201mean(knn_predict11[1:318] != t(vy[1]))## [1] 0.1320755mean(knn_predict12[1:318] != t(vy[1]))## [1] 0.1352201It appears that the best value found for k on the validation set is k=5. Which has around a 10% error rate.
Create models using knn and LDA and run it on the validation data to see which one works best:
library("MASS")
test_y_knn_val = knn(tr, v, yy, k = 5)
lda_model = lda(tr, yy)
test_y_lda_val = predict(lda_model,v)$class
print ("error percentage on knn and its confusion matrix")## [1] "error percentage on knn and its confusion matrix"mean(test_y_knn_val[1:318] != t(vy[1]))## [1] 0.1226415table(predicted = test_y_knn_val, actual = t(vy[1]))## actual
## predicted 0 1 2 3 4 5 6 7 8 9
## 0 40 0 0 0 0 0 3 0 0 2
## 1 0 26 1 0 6 0 0 4 1 0
## 2 0 0 30 0 0 0 0 0 3 0
## 3 0 0 0 24 0 0 0 0 1 4
## 4 0 0 0 0 26 0 0 0 0 0
## 5 0 0 0 0 1 37 0 0 0 3
## 6 0 0 0 0 0 1 33 0 2 0
## 7 0 0 0 2 0 0 0 26 0 0
## 8 0 0 1 0 0 0 0 0 17 0
## 9 0 0 0 1 0 0 0 0 3 20print ("error percentage on lda and its confusion matrix:")## [1] "error percentage on lda and its confusion matrix:"mean(test_y_lda_val != t(vy[1]))## [1] 0.1981132table(predicted = test_y_lda_val, actual = t(vy[1]))## actual
## predicted 0 1 2 3 4 5 6 7 8 9
## 0 37 0 0 0 0 0 1 0 1 1
## 1 0 21 1 0 7 0 1 2 2 0
## 2 0 2 25 0 0 0 0 0 1 2
## 3 0 0 0 24 0 6 0 0 0 4
## 4 0 0 0 0 24 0 0 0 0 1
## 5 0 1 0 0 0 28 1 1 0 0
## 6 1 0 2 0 1 1 31 0 0 0
## 7 0 1 0 3 0 2 0 27 2 0
## 8 1 0 3 0 1 1 2 0 18 1
## 9 1 1 1 0 0 0 0 0 3 20Run KNN and LDA on the test data and save it to csv files:
#create test predition using knn and the best k=6.
test_y_knn = knn(tr, test, yy, k = 6)
#create test prediction using LDA
l = lda(tr, yy)
test_y_lda = predict(l,test)$class
cols = c(test_y_knn , test_y_lda)
knn_pred = cols[1:638]
lda_pred = cols[639:1276]
df = data.frame(knn_pred,lda_pred)
#write the results to a csv file
write.csv(df, file = "HW3_njs48.csv", row.names = FALSE)- Summarize the performance of the 2 approaches. What error rates do you achieve from each? I tested both approaches using the training data to train the data, and then testing it on the validation set. The percentage of error of each was: knn error rate: 0.1257862 lda error rate: 0.1981132 So it appears that KNN is doing a better job at categorizing this type of data.
2)Examine a confusion matrix for your best model. Which digit(s) is(are) relatively difficult to classify? How does this compare to your initial guesses from Part 1?
KNN actual predicted 0 1 2 3 4 5 6 7 8 9 0 40 0 0 0 0 0 2 0 0 2 1 0 25 1 0 7 0 0 4 1 0 2 0 0 29 0 0 0 0 0 3 0 3 0 0 0 24 0 0 0 0 0 4 4 0 0 1 0 26 0 0 0 0 0 5 0 0 0 0 0 37 0 0 0 4 6 0 0 1 0 0 1 34 0 1 0 7 0 1 0 2 0 0 0 26 0 0 8 0 0 0 0 0 0 0 0 19 0 9 0 0 0 1 0 0 0 0 3 19
LDA actual predicted 0 1 2 3 4 5 6 7 8 9 0 37 0 0 0 0 0 1 0 1 1 1 0 21 1 0 7 0 1 2 2 0 2 0 2 25 0 0 0 0 0 1 2 3 0 0 0 24 0 6 0 0 0 4 4 0 0 0 0 24 0 0 0 0 1 5 0 1 0 0 0 28 1 1 0 0 6 1 0 2 0 1 1 31 0 0 0 7 0 1 0 3 0 2 0 27 2 0 8 1 0 3 0 1 1 2 0 18 1 9 1 1 1 0 0 0 0 0 3 20
It appears that both models have difficulty in classifying the number 1. As I guessed, it did often confuse it with the number 7. But in addition to 7, it also often confused it to the number 4. A number that both models had difficulty in predicting was the number 9. But suprisingly it didnt confuse it with the number 8. It often confused it with the number 3, which makes sense because the only difference is the top left line.
- Why it would be difficult (or impossible) to apply a multinomial logistic regression to this problem? First there are better algorithms than multinomial logistic regression for multiple class problems. And second, MLR works well when there are clearly defined separable classes, but when it comes to pixel points for numbers, often the pixels will be overlapping in different classes and thus not clearly separable.
Part 3:
Run LDA on the training set. And use predict on training data and on validation data.
library("MASS")
lda_train <- lda(formula = digit ~ ., data = train)
lda_train## Call:
## lda(digit ~ ., data = train)
##
## Prior probabilities of groups:
## 0 1 2 3 4 5
## 0.08320251 0.11930926 0.11302983 0.10047096 0.10518053 0.08791209
## 6 7 8 9
## 0.09890110 0.09890110 0.08634223 0.10675039
##
## Group means:
## pix1 pix2 pix3 pix4 pix5 pix6
## 0 0.00000000 0.03773585 0.09433962 0.15094340 0.37735849 0.56603774
## 1 0.00000000 0.00000000 0.00000000 0.00000000 0.03947368 0.07894737
## 2 0.08333333 0.23611111 0.37500000 0.51388889 0.58333333 0.68055556
## 3 0.35937500 0.54687500 0.68750000 0.78125000 0.82812500 0.85937500
## 4 0.00000000 0.07462687 0.16417910 0.23880597 0.16417910 0.10447761
## 5 0.07142857 0.14285714 0.28571429 0.37500000 0.50000000 0.60714286
## 6 0.00000000 0.00000000 0.00000000 0.03174603 0.06349206 0.15873016
## 7 0.46031746 0.46031746 0.53968254 0.53968254 0.52380952 0.57142857
## 8 0.03636364 0.05454545 0.10909091 0.27272727 0.36363636 0.54545455
## 9 0.04411765 0.08823529 0.19117647 0.33823529 0.48529412 0.64705882
## pix7 pix8 pix9 pix10 pix11 pix12 pix13
## 0 0.75471698 0.83018868 0.9056604 0.8867925 0.7547170 0.6415094 0.6415094
## 1 0.06578947 0.07894737 0.1052632 0.1578947 0.1710526 0.1842105 0.2631579
## 2 0.75000000 0.86111111 0.8333333 0.7500000 0.5972222 0.4722222 0.3611111
## 3 0.90625000 0.93750000 0.9218750 0.7968750 0.7031250 0.5312500 0.3437500
## 4 0.16417910 0.16417910 0.2089552 0.2835821 0.2985075 0.3283582 0.1940299
## 5 0.64285714 0.71428571 0.8214286 0.8571429 0.9107143 0.8750000 0.9107143
## 6 0.22222222 0.28571429 0.3492063 0.4285714 0.5873016 0.7142857 0.7142857
## 7 0.61904762 0.58730159 0.5396825 0.5396825 0.6507937 0.6031746 0.5714286
## 8 0.60000000 0.74545455 0.9272727 0.9454545 0.9818182 0.9636364 0.8363636
## 9 0.75000000 0.86764706 0.9264706 0.9558824 0.9558824 0.8529412 0.7941176
## pix14 pix15 pix16 pix17 pix18 pix19
## 0 0.45283019 0.18867925 0.01886792 0.03773585 0.1509434 0.24528302
## 1 0.57894737 0.67105263 0.61842105 0.00000000 0.0000000 0.00000000
## 2 0.26388889 0.20833333 0.04166667 0.19444444 0.3194444 0.38888889
## 3 0.25000000 0.10937500 0.03125000 0.50000000 0.5468750 0.48437500
## 4 0.07462687 0.07462687 0.04477612 0.00000000 0.1044776 0.22388060
## 5 0.82142857 0.76785714 0.64285714 0.10714286 0.2678571 0.42857143
## 6 0.71428571 0.66666667 0.52380952 0.00000000 0.0000000 0.04761905
## 7 0.50793651 0.42857143 0.33333333 0.30158730 0.3968254 0.42857143
## 8 0.74545455 0.50909091 0.20000000 0.09090909 0.1818182 0.30909091
## 9 0.63235294 0.44117647 0.19117647 0.13235294 0.3088235 0.50000000
## pix20 pix21 pix22 pix23 pix24 pix25
## 0 0.49056604 0.64150943 0.77358491 0.66037736 0.52830189 0.4150943
## 1 0.01315789 0.06578947 0.07894737 0.06578947 0.09210526 0.1315789
## 2 0.34722222 0.44444444 0.40277778 0.45833333 0.52777778 0.5972222
## 3 0.37500000 0.35937500 0.28125000 0.35937500 0.39062500 0.5000000
## 4 0.19402985 0.19402985 0.16417910 0.23880597 0.26865672 0.3731343
## 5 0.53571429 0.58928571 0.60714286 0.73214286 0.69642857 0.6428571
## 6 0.09523810 0.22222222 0.23809524 0.34920635 0.41269841 0.5079365
## 7 0.44444444 0.39682540 0.34920635 0.41269841 0.31746032 0.4285714
## 8 0.41818182 0.47272727 0.41818182 0.47272727 0.43636364 0.4000000
## 9 0.58823529 0.61764706 0.63235294 0.55882353 0.44117647 0.3088235
## pix26 pix27 pix28 pix29 pix30 pix31 pix32
## 0 0.3773585 0.3207547 0.4528302 0.5660377 0.60377358 0.43396226 0.11320755
## 1 0.1447368 0.1973684 0.2368421 0.4473684 0.67105263 0.69736842 0.47368421
## 2 0.5694444 0.5138889 0.4027778 0.3611111 0.33333333 0.27777778 0.13888889
## 3 0.6093750 0.6718750 0.5625000 0.5468750 0.35937500 0.20312500 0.06250000
## 4 0.3432836 0.2835821 0.2089552 0.1044776 0.07462687 0.05970149 0.04477612
## 5 0.5714286 0.4642857 0.4285714 0.2857143 0.23214286 0.12500000 0.05357143
## 6 0.5238095 0.5396825 0.4920635 0.3809524 0.30158730 0.26984127 0.15873016
## 7 0.4761905 0.5079365 0.6190476 0.5396825 0.44444444 0.41269841 0.25396825
## 8 0.2363636 0.2363636 0.2909091 0.4363636 0.60000000 0.74545455 0.61818182
## 9 0.3382353 0.4264706 0.4705882 0.5294118 0.70588235 0.70588235 0.44117647
## pix33 pix34 pix35 pix36 pix37 pix38
## 0 0.05660377 0.20754717 0.5094340 0.67924528 0.66037736 0.54716981
## 1 0.00000000 0.00000000 0.0000000 0.01315789 0.07894737 0.07894737
## 2 0.26388889 0.37500000 0.2500000 0.26388889 0.25000000 0.27777778
## 3 0.20312500 0.18750000 0.0781250 0.04687500 0.04687500 0.03125000
## 4 0.02985075 0.23880597 0.2835821 0.25373134 0.22388060 0.28358209
## 5 0.26785714 0.48214286 0.5000000 0.44642857 0.39285714 0.35714286
## 6 0.00000000 0.06349206 0.1269841 0.22222222 0.36507937 0.39682540
## 7 0.22222222 0.30158730 0.3174603 0.33333333 0.33333333 0.28571429
## 8 0.14545455 0.25454545 0.4363636 0.38181818 0.32727273 0.32727273
## 9 0.33823529 0.58823529 0.7058824 0.54411765 0.47058824 0.36764706
## pix39 pix40 pix41 pix42 pix43 pix44
## 0 0.3584906 0.1886792 0.1132075 0.18867925 0.24528302 0.28301887
## 1 0.1052632 0.1710526 0.1842105 0.21052632 0.30263158 0.35526316
## 2 0.2777778 0.2361111 0.3194444 0.40277778 0.45833333 0.41666667
## 3 0.0312500 0.0781250 0.1562500 0.34375000 0.46875000 0.59375000
## 4 0.3880597 0.3731343 0.4626866 0.31343284 0.25373134 0.20895522
## 5 0.3214286 0.2500000 0.1964286 0.12500000 0.08928571 0.07142857
## 6 0.4126984 0.5238095 0.4920635 0.38095238 0.30158730 0.14285714
## 7 0.2857143 0.3333333 0.3968254 0.46031746 0.53968254 0.63492063
## 8 0.3090909 0.2545455 0.1090909 0.05454545 0.03636364 0.12727273
## 9 0.2647059 0.1617647 0.1176471 0.11764706 0.20588235 0.26470588
## pix45 pix46 pix47 pix48 pix49 pix50
## 0 0.50943396 0.67924528 0.64150943 0.24528302 0.18867925 0.4905660
## 1 0.56578947 0.71052632 0.60526316 0.25000000 0.00000000 0.0000000
## 2 0.36111111 0.29166667 0.27777778 0.18055556 0.20833333 0.1805556
## 3 0.54687500 0.39062500 0.18750000 0.10937500 0.03125000 0.0468750
## 4 0.13432836 0.07462687 0.04477612 0.01492537 0.07462687 0.2835821
## 5 0.03571429 0.03571429 0.01785714 0.01785714 0.41071429 0.6428571
## 6 0.07936508 0.01587302 0.00000000 0.01587302 0.04761905 0.1269841
## 7 0.53968254 0.42857143 0.31746032 0.11111111 0.12698413 0.1587302
## 8 0.27272727 0.45454545 0.74545455 0.67272727 0.16363636 0.3454545
## 9 0.47058824 0.63235294 0.79411765 0.63235294 0.47058824 0.7500000
## pix51 pix52 pix53 pix54 pix55 pix56
## 0 0.64150943 0.64150943 0.49056604 0.33962264 0.1509434 0.01886792
## 1 0.01315789 0.01315789 0.03947368 0.09210526 0.1578947 0.15789474
## 2 0.18055556 0.16666667 0.16666667 0.25000000 0.1805556 0.12500000
## 3 0.04687500 0.01562500 0.04687500 0.04687500 0.0625000 0.09375000
## 4 0.31343284 0.22388060 0.31343284 0.31343284 0.3432836 0.35820896
## 5 0.42857143 0.35714286 0.30357143 0.21428571 0.1428571 0.08928571
## 6 0.17460317 0.31746032 0.46031746 0.46031746 0.4761905 0.42857143
## 7 0.15873016 0.19047619 0.17460317 0.15873016 0.1904762 0.25396825
## 8 0.45454545 0.41818182 0.29090909 0.29090909 0.3090909 0.18181818
## 9 0.66176471 0.55882353 0.41176471 0.33823529 0.2058824 0.17647059
## pix57 pix58 pix59 pix60 pix61 pix62
## 0 0.03773585 0.07547170 0.09433962 0.2264151 0.3018868 0.71698113
## 1 0.22368421 0.32894737 0.38157895 0.4868421 0.6184211 0.64473684
## 2 0.25000000 0.37500000 0.44444444 0.4861111 0.3055556 0.27777778
## 3 0.29687500 0.45312500 0.57812500 0.5312500 0.4375000 0.32812500
## 4 0.35820896 0.23880597 0.17910448 0.1343284 0.1194030 0.08955224
## 5 0.05357143 0.03571429 0.00000000 0.0000000 0.0000000 0.00000000
## 6 0.36507937 0.14285714 0.09523810 0.0000000 0.0000000 0.00000000
## 7 0.31746032 0.39682540 0.57142857 0.6031746 0.4920635 0.25396825
## 8 0.09090909 0.05454545 0.10909091 0.2363636 0.3454545 0.52727273
## 9 0.16176471 0.19117647 0.25000000 0.3823529 0.5294118 0.67647059
## pix63 pix64 pix65 pix66 pix67 pix68
## 0 0.69811321 0.47169811 0.39622642 0.64150943 0.64150943 0.52830189
## 1 0.42105263 0.07894737 0.00000000 0.01315789 0.01315789 0.01315789
## 2 0.20833333 0.16666667 0.09722222 0.09722222 0.11111111 0.12500000
## 3 0.17187500 0.07812500 0.01562500 0.01562500 0.01562500 0.04687500
## 4 0.02985075 0.02985075 0.20895522 0.35820896 0.34328358 0.28358209
## 5 0.00000000 0.00000000 0.53571429 0.67857143 0.51785714 0.33928571
## 6 0.00000000 0.00000000 0.11111111 0.20634921 0.39682540 0.44444444
## 7 0.17460317 0.03174603 0.04761905 0.06349206 0.04761905 0.03174603
## 8 0.78181818 0.54545455 0.16363636 0.29090909 0.32727273 0.43636364
## 9 0.75000000 0.57352941 0.61764706 0.77941176 0.64705882 0.54411765
## pix69 pix70 pix71 pix72 pix73 pix74
## 0 0.30188679 0.18867925 0.05660377 0.01886792 0.01886792 0.05660377
## 1 0.03947368 0.10526316 0.19736842 0.28947368 0.35526316 0.46052632
## 2 0.15277778 0.22222222 0.22222222 0.20833333 0.34722222 0.43055556
## 3 0.06250000 0.07812500 0.17187500 0.26562500 0.43750000 0.54687500
## 4 0.31343284 0.37313433 0.37313433 0.28358209 0.25373134 0.20895522
## 5 0.35714286 0.28571429 0.16071429 0.14285714 0.10714286 0.05357143
## 6 0.60317460 0.50793651 0.44444444 0.28571429 0.14285714 0.06349206
## 7 0.06349206 0.04761905 0.06349206 0.22222222 0.36507937 0.57142857
## 8 0.40000000 0.38181818 0.34545455 0.27272727 0.23636364 0.18181818
## 9 0.39705882 0.36764706 0.32352941 0.30882353 0.30882353 0.29411765
## pix75 pix76 pix77 pix78 pix79 pix80
## 0 0.05660377 0.03773585 0.16981132 0.49056604 0.84905660 0.58490566
## 1 0.47368421 0.52631579 0.68421053 0.57894737 0.27631579 0.05263158
## 2 0.52777778 0.48611111 0.34722222 0.29166667 0.19444444 0.05555556
## 3 0.54687500 0.51562500 0.31250000 0.23437500 0.10937500 0.00000000
## 4 0.17910448 0.11940299 0.10447761 0.10447761 0.07462687 0.02985075
## 5 0.05357143 0.03571429 0.03571429 0.01785714 0.00000000 0.00000000
## 6 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
## 7 0.58730159 0.47619048 0.34920635 0.19047619 0.07936508 0.01587302
## 8 0.29090909 0.40000000 0.56363636 0.63636364 0.58181818 0.29090909
## 9 0.41176471 0.54411765 0.73529412 0.76470588 0.70588235 0.42647059
## pix81 pix82 pix83 pix84 pix85 pix86
## 0 0.52830189 0.71698113 0.64150943 0.47169811 0.22641509 0.13207547
## 1 0.01315789 0.01315789 0.01315789 0.02631579 0.10526316 0.23684211
## 2 0.04166667 0.06944444 0.08333333 0.06944444 0.12500000 0.15277778
## 3 0.00000000 0.04687500 0.10937500 0.14062500 0.25000000 0.39062500
## 4 0.31343284 0.40298507 0.43283582 0.38805970 0.37313433 0.38805970
## 5 0.50000000 0.64285714 0.57142857 0.51785714 0.51785714 0.37500000
## 6 0.14285714 0.39682540 0.47619048 0.61904762 0.53968254 0.41269841
## 7 0.01587302 0.01587302 0.01587302 0.03174603 0.06349206 0.07936508
## 8 0.09090909 0.18181818 0.32727273 0.43636364 0.50909091 0.49090909
## 9 0.57352941 0.67647059 0.63235294 0.52941176 0.48529412 0.39705882
## pix87 pix88 pix89 pix90 pix91 pix92
## 0 0.03773585 0.03773585 0.03773585 0.03773585 0.05660377 0.03773585
## 1 0.26315789 0.38157895 0.50000000 0.47368421 0.55263158 0.64473684
## 2 0.26388889 0.34722222 0.50000000 0.55555556 0.51388889 0.37500000
## 3 0.45312500 0.54687500 0.62500000 0.60937500 0.48437500 0.34375000
## 4 0.28358209 0.14925373 0.17910448 0.17910448 0.17910448 0.11940299
## 5 0.28571429 0.26785714 0.17857143 0.17857143 0.16071429 0.08928571
## 6 0.23809524 0.07936508 0.04761905 0.00000000 0.00000000 0.00000000
## 7 0.20634921 0.30158730 0.42857143 0.58730159 0.57142857 0.44444444
## 8 0.50909091 0.49090909 0.41818182 0.41818182 0.56363636 0.58181818
## 9 0.36764706 0.36764706 0.39705882 0.47058824 0.60294118 0.72058824
## pix93 pix94 pix95 pix96 pix97 pix98
## 0 0.07547170 0.39622642 0.90566038 0.75471698 0.67924528 0.86792453
## 1 0.63157895 0.46052632 0.19736842 0.03947368 0.05263158 0.03947368
## 2 0.27777778 0.25000000 0.11111111 0.02777778 0.02777778 0.02777778
## 3 0.28125000 0.10937500 0.01562500 0.00000000 0.04687500 0.15625000
## 4 0.11940299 0.20895522 0.16417910 0.11940299 0.40298507 0.61194030
## 5 0.03571429 0.03571429 0.01785714 0.00000000 0.41071429 0.51785714
## 6 0.00000000 0.00000000 0.00000000 0.00000000 0.30158730 0.52380952
## 7 0.22222222 0.12698413 0.07936508 0.04761905 0.03174603 0.06349206
## 8 0.65454545 0.47272727 0.29090909 0.10909091 0.05454545 0.10909091
## 9 0.73529412 0.73529412 0.57352941 0.19117647 0.44117647 0.57352941
## pix99 pix100 pix101 pix102 pix103 pix104
## 0 0.62264151 0.28301887 0.09433962 0.05660377 0.01886792 0.03773585
## 1 0.06578947 0.11842105 0.18421053 0.30263158 0.40789474 0.47368421
## 2 0.05555556 0.04166667 0.11111111 0.23611111 0.36111111 0.48611111
## 3 0.26562500 0.45312500 0.62500000 0.68750000 0.73437500 0.79687500
## 4 0.47761194 0.52238806 0.37313433 0.26865672 0.14925373 0.11940299
## 5 0.55357143 0.57142857 0.55357143 0.53571429 0.46428571 0.44642857
## 6 0.69841270 0.58730159 0.44444444 0.20634921 0.12698413 0.12698413
## 7 0.06349206 0.14285714 0.19047619 0.23809524 0.31746032 0.46031746
## 8 0.20000000 0.29090909 0.52727273 0.67272727 0.70909091 0.80000000
## 9 0.57352941 0.58823529 0.63235294 0.61764706 0.55882353 0.52941176
## pix105 pix106 pix107 pix108 pix109 pix110
## 0 0.03773585 0.03773585 0.03773585 0.03773585 0.09433962 0.35849057
## 1 0.47368421 0.51315789 0.61842105 0.67105263 0.55263158 0.32894737
## 2 0.51388889 0.48611111 0.45833333 0.27777778 0.23611111 0.11111111
## 3 0.78125000 0.59375000 0.48437500 0.34375000 0.18750000 0.06250000
## 4 0.14925373 0.19402985 0.17910448 0.19402985 0.22388060 0.34328358
## 5 0.39285714 0.41071429 0.33928571 0.32142857 0.17857143 0.08928571
## 6 0.06349206 0.09523810 0.11111111 0.09523810 0.07936508 0.03174603
## 7 0.58730159 0.65079365 0.58730159 0.41269841 0.20634921 0.09523810
## 8 0.78181818 0.74545455 0.74545455 0.67272727 0.45454545 0.29090909
## 9 0.58823529 0.66176471 0.69117647 0.75000000 0.77941176 0.69117647
## pix111 pix112 pix113 pix114 pix115 pix116
## 0 0.94339623 0.86792453 0.86792453 0.84905660 0.43396226 0.20754717
## 1 0.15789474 0.01315789 0.05263158 0.09210526 0.14473684 0.17105263
## 2 0.04166667 0.00000000 0.02777778 0.02777778 0.01388889 0.04166667
## 3 0.04687500 0.00000000 0.06250000 0.21875000 0.35937500 0.43750000
## 4 0.35820896 0.25373134 0.53731343 0.67164179 0.58208955 0.35820896
## 5 0.03571429 0.01785714 0.37500000 0.37500000 0.35714286 0.37500000
## 6 0.00000000 0.00000000 0.46031746 0.66666667 0.71428571 0.49206349
## 7 0.06349206 0.06349206 0.14285714 0.22222222 0.31746032 0.34920635
## 8 0.09090909 0.03636364 0.03636364 0.14545455 0.21818182 0.38181818
## 9 0.35294118 0.13235294 0.20588235 0.35294118 0.38235294 0.42647059
## pix117 pix118 pix119 pix120 pix121 pix122
## 0 0.09433962 0.07547170 0.05660377 0.05660377 0.03773585 0.03773585
## 1 0.22368421 0.30263158 0.34210526 0.40789474 0.44736842 0.53947368
## 2 0.11111111 0.25000000 0.38888889 0.50000000 0.45833333 0.40277778
## 3 0.53125000 0.54687500 0.64062500 0.64062500 0.59375000 0.56250000
## 4 0.25373134 0.07462687 0.07462687 0.13432836 0.14925373 0.22388060
## 5 0.30357143 0.30357143 0.37500000 0.33928571 0.35714286 0.37500000
## 6 0.30158730 0.22222222 0.19047619 0.22222222 0.26984127 0.22222222
## 7 0.38095238 0.44444444 0.50793651 0.60317460 0.68253968 0.60317460
## 8 0.52727273 0.67272727 0.85454545 0.83636364 0.83636364 0.76363636
## 9 0.50000000 0.54411765 0.58823529 0.57352941 0.50000000 0.52941176
## pix123 pix124 pix125 pix126 pix127 pix128
## 0 0.01886792 0.01886792 0.09433962 0.30188679 0.92452830 0.77358491
## 1 0.64473684 0.60526316 0.52631579 0.32894737 0.13157895 0.00000000
## 2 0.34722222 0.22222222 0.11111111 0.02777778 0.01388889 0.00000000
## 3 0.51562500 0.48437500 0.31250000 0.23437500 0.07812500 0.04687500
## 4 0.22388060 0.25373134 0.34328358 0.47761194 0.41791045 0.31343284
## 5 0.48214286 0.41071429 0.30357143 0.25000000 0.14285714 0.03571429
## 6 0.22222222 0.19047619 0.15873016 0.09523810 0.07936508 0.01587302
## 7 0.49206349 0.30158730 0.20634921 0.15873016 0.14285714 0.09523810
## 8 0.56363636 0.40000000 0.30909091 0.12727273 0.03636364 0.00000000
## 9 0.61764706 0.63235294 0.63235294 0.52941176 0.35294118 0.13235294
## pix129 pix130 pix131 pix132 pix133 pix134
## 0 0.92452830 0.90566038 0.37735849 0.1698113 0.1320755 0.05660377
## 1 0.18421053 0.17105263 0.19736842 0.2500000 0.2763158 0.36842105
## 2 0.02777778 0.04166667 0.06944444 0.2083333 0.3472222 0.45833333
## 3 0.04687500 0.14062500 0.21875000 0.2500000 0.2812500 0.26562500
## 4 0.61194030 0.88059701 0.68656716 0.4477612 0.2686567 0.17910448
## 5 0.19642857 0.26785714 0.28571429 0.3392857 0.3571429 0.32142857
## 6 0.60317460 0.92063492 0.71428571 0.5079365 0.3968254 0.39682540
## 7 0.23809524 0.34920635 0.36507937 0.4603175 0.5396825 0.61904762
## 8 0.12727273 0.34545455 0.45454545 0.5272727 0.6909091 0.74545455
## 9 0.07352941 0.16176471 0.19117647 0.2647059 0.3235294 0.38235294
## pix135 pix136 pix137 pix138 pix139 pix140
## 0 0.05660377 0.03773585 0.01886792 0.0000000 0.01886792 0.09433962
## 1 0.42105263 0.53947368 0.52631579 0.6184211 0.59210526 0.52631579
## 2 0.52777778 0.54166667 0.45833333 0.3472222 0.29166667 0.15277778
## 3 0.29687500 0.42187500 0.40625000 0.5000000 0.54687500 0.57812500
## 4 0.16417910 0.14925373 0.22388060 0.2388060 0.29850746 0.40298507
## 5 0.28571429 0.21428571 0.23214286 0.2857143 0.33928571 0.39285714
## 6 0.42857143 0.42857143 0.50793651 0.5396825 0.42857143 0.34920635
## 7 0.73015873 0.74603175 0.76190476 0.6825397 0.57142857 0.47619048
## 8 0.76363636 0.76363636 0.74545455 0.6545455 0.54545455 0.41818182
## 9 0.39705882 0.35294118 0.35294118 0.3970588 0.48529412 0.51470588
## pix141 pix142 pix143 pix144 pix145 pix146
## 0 0.16981132 0.56603774 0.90566038 0.67924528 0.92452830 0.92452830
## 1 0.51315789 0.31578947 0.09210526 0.00000000 0.15789474 0.15789474
## 2 0.02777778 0.01388889 0.01388889 0.00000000 0.06944444 0.13888889
## 3 0.64062500 0.53125000 0.32812500 0.12500000 0.01562500 0.03125000
## 4 0.55223881 0.68656716 0.68656716 0.38805970 0.77611940 0.83582090
## 5 0.37500000 0.41071429 0.33928571 0.21428571 0.03571429 0.08928571
## 6 0.25396825 0.28571429 0.22222222 0.09523810 0.79365079 0.95238095
## 7 0.39682540 0.33333333 0.23809524 0.22222222 0.17460317 0.23809524
## 8 0.32727273 0.20000000 0.10909091 0.03636364 0.34545455 0.52727273
## 9 0.58823529 0.50000000 0.38235294 0.11764706 0.04411765 0.04411765
## pix147 pix148 pix149 pix150 pix151 pix152
## 0 0.35849057 0.09433962 0.1132075 0.05660377 0.03773585 0.01886792
## 1 0.13157895 0.18421053 0.3157895 0.32894737 0.46052632 0.53947368
## 2 0.20833333 0.37500000 0.4722222 0.47222222 0.38888889 0.44444444
## 3 0.04687500 0.07812500 0.0625000 0.06250000 0.09375000 0.12500000
## 4 0.67164179 0.37313433 0.2835821 0.25373134 0.28358209 0.31343284
## 5 0.07142857 0.07142857 0.1250000 0.16071429 0.14285714 0.12500000
## 6 0.66666667 0.41269841 0.4126984 0.42857143 0.53968254 0.53968254
## 7 0.33333333 0.39682540 0.4920635 0.58730159 0.65079365 0.77777778
## 8 0.60000000 0.63636364 0.6181818 0.50909091 0.32727273 0.30909091
## 9 0.08823529 0.08823529 0.1470588 0.19117647 0.17647059 0.22058824
## pix153 pix154 pix155 pix156 pix157 pix158
## 0 0.01886792 0.0000000 0.07547170 0.13207547 0.28301887 0.62264151
## 1 0.53947368 0.4078947 0.42105263 0.43421053 0.44736842 0.30263158
## 2 0.34722222 0.2361111 0.08333333 0.04166667 0.01388889 0.01388889
## 3 0.20312500 0.2812500 0.32812500 0.40625000 0.57812500 0.71875000
## 4 0.38805970 0.4179104 0.46268657 0.59701493 0.79104478 0.80597015
## 5 0.14285714 0.1250000 0.23214286 0.25000000 0.21428571 0.33928571
## 6 0.50793651 0.5396825 0.53968254 0.46031746 0.46031746 0.42857143
## 7 0.71428571 0.5873016 0.49206349 0.42857143 0.36507937 0.34920635
## 8 0.40000000 0.3818182 0.43636364 0.45454545 0.40000000 0.23636364
## 9 0.25000000 0.2941176 0.39705882 0.38235294 0.45588235 0.45588235
## pix159 pix160 pix161 pix162 pix163 pix164
## 0 0.86792453 0.45283019 0.79245283 0.90566038 0.39622642 0.16981132
## 1 0.09210526 0.00000000 0.17105263 0.14473684 0.17105263 0.23684211
## 2 0.02777778 0.02777778 0.12500000 0.29166667 0.52777778 0.55555556
## 3 0.68750000 0.34375000 0.00000000 0.00000000 0.04687500 0.06250000
## 4 0.64179104 0.32835821 0.59701493 0.70149254 0.56716418 0.47761194
## 5 0.44642857 0.25000000 0.05357143 0.01785714 0.01785714 0.03571429
## 6 0.36507937 0.19047619 0.87301587 0.95238095 0.73015873 0.49206349
## 7 0.33333333 0.25396825 0.17460317 0.17460317 0.23809524 0.28571429
## 8 0.18181818 0.05454545 0.61818182 0.78181818 0.63636364 0.45454545
## 9 0.36764706 0.17647059 0.01470588 0.01470588 0.04411765 0.04411765
## pix165 pix166 pix167 pix168 pix169 pix170
## 0 0.13207547 0.05660377 0.03773585 0.01886792 0.0000000 0.03773585
## 1 0.36842105 0.43421053 0.51315789 0.50000000 0.4078947 0.31578947
## 2 0.55555556 0.51388889 0.43055556 0.29166667 0.1944444 0.11111111
## 3 0.03125000 0.04687500 0.01562500 0.03125000 0.0312500 0.04687500
## 4 0.40298507 0.38805970 0.43283582 0.50746269 0.5671642 0.61194030
## 5 0.08928571 0.10714286 0.10714286 0.16071429 0.1428571 0.16071429
## 6 0.55555556 0.42857143 0.34920635 0.25396825 0.2539683 0.26984127
## 7 0.41269841 0.53968254 0.58730159 0.61904762 0.6031746 0.39682540
## 8 0.21818182 0.12727273 0.12727273 0.18181818 0.2909091 0.21818182
## 9 0.04411765 0.04411765 0.07352941 0.11764706 0.2205882 0.23529412
## pix171 pix172 pix173 pix174 pix175 pix176
## 0 0.11320755 0.26415094 0.54716981 0.67924528 0.67924528 0.33962264
## 1 0.38157895 0.38157895 0.39473684 0.28947368 0.10526316 0.01315789
## 2 0.04166667 0.01388889 0.01388889 0.01388889 0.05555556 0.04166667
## 3 0.14062500 0.21875000 0.29687500 0.78125000 0.92187500 0.62500000
## 4 0.61194030 0.65671642 0.76119403 0.70149254 0.49253731 0.20895522
## 5 0.21428571 0.25000000 0.19642857 0.28571429 0.42857143 0.33928571
## 6 0.31746032 0.34920635 0.38095238 0.52380952 0.50793651 0.33333333
## 7 0.31746032 0.23809524 0.20634921 0.20634921 0.17460317 0.12698413
## 8 0.32727273 0.40000000 0.43636364 0.41818182 0.30909091 0.10909091
## 9 0.38235294 0.30882353 0.39705882 0.41176471 0.39705882 0.22058824
## pix177 pix178 pix179 pix180 pix181 pix182
## 0 0.73584906 0.86792453 0.56603774 0.22641509 0.13207547 0.07547170
## 1 0.13157895 0.15789474 0.22368421 0.28947368 0.35526316 0.43421053
## 2 0.29166667 0.52777778 0.62500000 0.51388889 0.45833333 0.43055556
## 3 0.01562500 0.04687500 0.07812500 0.06250000 0.03125000 0.03125000
## 4 0.38805970 0.44776119 0.37313433 0.35820896 0.38805970 0.38805970
## 5 0.08928571 0.08928571 0.07142857 0.03571429 0.07142857 0.07142857
## 6 0.88888889 1.00000000 0.68253968 0.57142857 0.39682540 0.22222222
## 7 0.06349206 0.15873016 0.20634921 0.28571429 0.38095238 0.47619048
## 8 0.72727273 0.87272727 0.50909091 0.23636364 0.09090909 0.03636364
## 9 0.02941176 0.01470588 0.01470588 0.01470588 0.04411765 0.05882353
## pix183 pix184 pix185 pix186 pix187 pix188
## 0 0.03773585 0.0000000 0.01886792 0.11320755 0.22641509 0.39622642
## 1 0.42105263 0.3157895 0.28947368 0.27631579 0.32894737 0.38157895
## 2 0.37500000 0.1805556 0.11111111 0.02777778 0.01388889 0.02777778
## 3 0.01562500 0.0000000 0.00000000 0.03125000 0.03125000 0.06250000
## 4 0.41791045 0.4328358 0.49253731 0.44776119 0.46268657 0.58208955
## 5 0.14285714 0.1250000 0.19642857 0.12500000 0.21428571 0.28571429
## 6 0.14285714 0.1428571 0.19047619 0.25396825 0.23809524 0.33333333
## 7 0.52380952 0.4603175 0.30158730 0.26984127 0.20634921 0.11111111
## 8 0.10909091 0.1636364 0.23636364 0.18181818 0.20000000 0.25454545
## 9 0.13235294 0.1911765 0.23529412 0.27941176 0.30882353 0.27941176
## pix189 pix190 pix191 pix192 pix193 pix194
## 0 0.58490566 0.64150943 0.41509434 0.07547170 0.62264151 0.75471698
## 1 0.39473684 0.28947368 0.14473684 0.01315789 0.09210526 0.22368421
## 2 0.04166667 0.06944444 0.11111111 0.12500000 0.50000000 0.63888889
## 3 0.21875000 0.60937500 0.92187500 0.79687500 0.09375000 0.12500000
## 4 0.70149254 0.64179104 0.35820896 0.07462687 0.16417910 0.25373134
## 5 0.23214286 0.35714286 0.39285714 0.35714286 0.16071429 0.21428571
## 6 0.33333333 0.49206349 0.49206349 0.31746032 0.84126984 0.92063492
## 7 0.07936508 0.07936508 0.06349206 0.03174603 0.03174603 0.07936508
## 8 0.40000000 0.41818182 0.41818182 0.23636364 0.72727273 0.87272727
## 9 0.32352941 0.39705882 0.41176471 0.23529412 0.07352941 0.05882353
## pix195 pix196 pix197 pix198 pix199 pix200
## 0 0.71698113 0.35849057 0.28301887 0.16981132 0.1132075 0.1320755
## 1 0.28947368 0.32894737 0.35526316 0.39473684 0.3289474 0.2368421
## 2 0.58333333 0.47222222 0.34722222 0.36111111 0.3611111 0.2083333
## 3 0.10937500 0.04687500 0.03125000 0.07812500 0.0625000 0.0156250
## 4 0.23880597 0.29850746 0.29850746 0.28358209 0.2835821 0.2686567
## 5 0.10714286 0.07142857 0.08928571 0.12500000 0.1607143 0.2142857
## 6 0.77777778 0.52380952 0.33333333 0.20634921 0.1428571 0.1428571
## 7 0.19047619 0.28571429 0.39682540 0.44444444 0.4920635 0.3650794
## 8 0.47272727 0.29090909 0.16363636 0.12727273 0.1090909 0.1272727
## 9 0.05882353 0.08823529 0.02941176 0.08823529 0.1911765 0.2205882
## pix201 pix202 pix203 pix204 pix205 pix206
## 0 0.2075472 0.35849057 0.54716981 0.66037736 0.75471698 0.49056604
## 1 0.2631579 0.26315789 0.31578947 0.32894737 0.38157895 0.28947368
## 2 0.1527778 0.09722222 0.06944444 0.11111111 0.16666667 0.22222222
## 3 0.0468750 0.04687500 0.14062500 0.21875000 0.40625000 0.70312500
## 4 0.3582090 0.32835821 0.38805970 0.52238806 0.64179104 0.56716418
## 5 0.2500000 0.19642857 0.28571429 0.32142857 0.37500000 0.41071429
## 6 0.2063492 0.28571429 0.31746032 0.36507937 0.44444444 0.50793651
## 7 0.2698413 0.25396825 0.15873016 0.07936508 0.01587302 0.01587302
## 8 0.2363636 0.20000000 0.18181818 0.21818182 0.36363636 0.40000000
## 9 0.2647059 0.23529412 0.32352941 0.29411765 0.29411765 0.36764706
## pix207 pix208 pix209 pix210 pix211 pix212 pix213
## 0 0.2452830 0.00000000 0.30188679 0.67924528 0.7547170 0.7358491 0.5660377
## 1 0.1578947 0.02631579 0.17105263 0.27631579 0.3421053 0.3289474 0.2894737
## 2 0.2638889 0.19444444 0.48611111 0.58333333 0.5833333 0.5277778 0.4583333
## 3 0.8437500 0.67187500 0.23437500 0.28125000 0.2656250 0.2343750 0.2031250
## 4 0.3283582 0.07462687 0.07462687 0.08955224 0.1194030 0.1044776 0.1194030
## 5 0.3928571 0.21428571 0.25000000 0.37500000 0.3392857 0.2142857 0.2857143
## 6 0.4444444 0.25396825 0.63492063 0.79365079 0.8253968 0.6825397 0.4444444
## 7 0.0000000 0.00000000 0.06349206 0.11111111 0.1904762 0.2222222 0.3968254
## 8 0.4181818 0.21818182 0.50909091 0.70909091 0.5636364 0.4545455 0.3818182
## 9 0.3676471 0.19117647 0.11764706 0.17647059 0.1470588 0.1617647 0.2500000
## pix214 pix215 pix216 pix217 pix218 pix219 pix220
## 0 0.45283019 0.3773585 0.4528302 0.5094340 0.6037736 0.6981132 0.60377358
## 1 0.30263158 0.3026316 0.2500000 0.2236842 0.2631579 0.3421053 0.31578947
## 2 0.41666667 0.4305556 0.4861111 0.4027778 0.3611111 0.4305556 0.43055556
## 3 0.14062500 0.1093750 0.1718750 0.2031250 0.2343750 0.3125000 0.54687500
## 4 0.08955224 0.1343284 0.1641791 0.2388060 0.2537313 0.2985075 0.43283582
## 5 0.35714286 0.3214286 0.3214286 0.2857143 0.3035714 0.3571429 0.35714286
## 6 0.28571429 0.2539683 0.3015873 0.3015873 0.3809524 0.4444444 0.46031746
## 7 0.49206349 0.4285714 0.3333333 0.2857143 0.2063492 0.1111111 0.04761905
## 8 0.27272727 0.2363636 0.2727273 0.2909091 0.2545455 0.2727273 0.29090909
## 9 0.29411765 0.2941176 0.2500000 0.3088235 0.2941176 0.3529412 0.35294118
## pix221 pix222 pix223 pix224 pix225 pix226
## 0 0.50943396 0.2830189 0.05660377 0.00000000 0.01886792 0.37735849
## 1 0.35526316 0.3289474 0.30263158 0.15789474 0.23684211 0.34210526
## 2 0.38888889 0.4166667 0.38888889 0.22222222 0.40277778 0.56944444
## 3 0.71875000 0.7812500 0.62500000 0.31250000 0.40625000 0.48437500
## 4 0.55223881 0.4776119 0.29850746 0.04477612 0.01492537 0.02985075
## 5 0.39285714 0.3214286 0.19642857 0.08928571 0.28571429 0.60714286
## 6 0.44444444 0.4285714 0.30158730 0.14285714 0.26984127 0.60317460
## 7 0.01587302 0.0000000 0.00000000 0.00000000 0.07936508 0.14285714
## 8 0.38181818 0.4181818 0.32727273 0.21818182 0.30909091 0.50909091
## 9 0.38235294 0.3676471 0.30882353 0.14705882 0.25000000 0.32352941
## pix227 pix228 pix229 pix230 pix231 pix232
## 0 0.62264151 0.75471698 0.81132075 0.86792453 0.90566038 0.9056604
## 1 0.32894737 0.27631579 0.22368421 0.25000000 0.28947368 0.2368421
## 2 0.65277778 0.76388889 0.69444444 0.70833333 0.80555556 0.8472222
## 3 0.57812500 0.67187500 0.73437500 0.75000000 0.76562500 0.8281250
## 4 0.05970149 0.05970149 0.04477612 0.04477612 0.08955224 0.1194030
## 5 0.73214286 0.80357143 0.80357143 0.67857143 0.69642857 0.6250000
## 6 0.71428571 0.85714286 0.88888889 0.90476190 0.90476190 0.8730159
## 7 0.14285714 0.25396825 0.31746032 0.46031746 0.42857143 0.2857143
## 8 0.63636364 0.80000000 0.80000000 0.81818182 0.80000000 0.7636364
## 9 0.39705882 0.47058824 0.52941176 0.58823529 0.60294118 0.5882353
## pix233 pix234 pix235 pix236 pix237 pix238 pix239
## 0 0.8490566 0.6792453 0.5660377 0.41509434 0.16981132 0.01886792 0.0000000
## 1 0.2631579 0.2763158 0.3289474 0.32894737 0.36842105 0.39473684 0.3552632
## 2 0.8611111 0.9027778 0.8888889 0.79166667 0.76388889 0.68055556 0.5555556
## 3 0.8750000 0.8750000 0.8593750 0.82812500 0.76562500 0.53125000 0.2812500
## 4 0.2089552 0.2238806 0.2686567 0.38805970 0.56716418 0.46268657 0.3283582
## 5 0.6607143 0.5892857 0.5535714 0.41071429 0.23214286 0.16071429 0.1250000
## 6 0.8412698 0.7619048 0.6825397 0.55555556 0.41269841 0.25396825 0.1428571
## 7 0.2698413 0.1904762 0.1269841 0.03174603 0.01587302 0.00000000 0.0000000
## 8 0.7636364 0.6727273 0.5272727 0.50909091 0.43636364 0.36363636 0.3090909
## 9 0.6323529 0.5735294 0.5294118 0.47058824 0.38235294 0.30882353 0.2205882
## pix240 pix241 pix242 pix243 pix244 pix245
## 0 0.00000000 0.00000000 0.00000000 0.16981132 0.35849057 0.5849057
## 1 0.21052632 0.18421053 0.21052632 0.14473684 0.11842105 0.1315789
## 2 0.44444444 0.09722222 0.19444444 0.19444444 0.29166667 0.3888889
## 3 0.04687500 0.23437500 0.26562500 0.39062500 0.48437500 0.5781250
## 4 0.11940299 0.00000000 0.02985075 0.02985075 0.01492537 0.0000000
## 5 0.03571429 0.17857143 0.30357143 0.41071429 0.53571429 0.6607143
## 6 0.03174603 0.03174603 0.09523810 0.26984127 0.46031746 0.6349206
## 7 0.00000000 0.09523810 0.09523810 0.12698413 0.17460317 0.2539683
## 8 0.12727273 0.05454545 0.18181818 0.29090909 0.41818182 0.5090909
## 9 0.08823529 0.16176471 0.22058824 0.25000000 0.29411765 0.2941176
## pix246 pix247 pix248 pix249 pix250 pix251 pix252
## 0 0.64150943 0.64150943 0.5849057 0.4905660 0.3584906 0.1886792 0.03773585
## 1 0.11842105 0.13157895 0.1447368 0.2105263 0.1973684 0.1842105 0.21052632
## 2 0.37500000 0.37500000 0.3888889 0.4583333 0.4444444 0.3888889 0.37500000
## 3 0.62500000 0.60937500 0.6406250 0.6250000 0.5312500 0.4531250 0.26562500
## 4 0.01492537 0.02985075 0.1044776 0.1343284 0.1641791 0.1492537 0.34328358
## 5 0.55357143 0.53571429 0.5000000 0.4285714 0.3750000 0.2321429 0.12500000
## 6 0.73015873 0.69841270 0.6349206 0.6031746 0.4285714 0.2857143 0.19047619
## 7 0.34920635 0.38095238 0.2539683 0.1746032 0.1587302 0.0952381 0.01587302
## 8 0.58181818 0.67272727 0.5454545 0.5636364 0.5090909 0.4181818 0.34545455
## 9 0.32352941 0.41176471 0.3529412 0.3970588 0.3970588 0.3088235 0.26470588
## pix253 pix254 pix255 pix256
## 0 0.00000000 0.00000000 0.00000000 0.00000000
## 1 0.25000000 0.23684211 0.23684211 0.10526316
## 2 0.43055556 0.33333333 0.29166667 0.18055556
## 3 0.15625000 0.06250000 0.01562500 0.00000000
## 4 0.41791045 0.43283582 0.29850746 0.11940299
## 5 0.05357143 0.05357143 0.01785714 0.00000000
## 6 0.06349206 0.01587302 0.00000000 0.00000000
## 7 0.01587302 0.00000000 0.00000000 0.00000000
## 8 0.27272727 0.12727273 0.07272727 0.00000000
## 9 0.19117647 0.13235294 0.08823529 0.02941176
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3 LD4 LD5
## pix1 -0.6774323345 0.028918581 0.269821010 -0.313166721 0.511230907
## pix2 0.5248079175 -0.293828312 -0.421932623 0.487565490 0.221840800
## pix3 -0.0067647267 0.580953968 -0.075497931 0.029613048 1.075341097
## pix4 1.0028755585 -0.399856442 -0.292829306 -0.321841486 -0.015717447
## pix5 0.1691228471 -0.020414951 0.493834195 0.157323286 -0.326440207
## pix6 0.0450394565 -1.092117531 -0.412589974 0.166294811 0.382967882
## pix7 0.1973499731 0.495128187 0.425732370 -0.023907354 0.342822227
## pix8 0.2405920375 -0.339290483 -0.087481647 -0.452203393 0.374625881
## pix9 0.1008891341 -1.073254230 0.013015663 0.031743997 -0.250343095
## pix10 -0.1437228243 -0.021539411 -0.166528525 -0.322025256 -0.283122154
## pix11 0.6387454991 -0.165218438 0.058085677 0.781736962 0.033942780
## pix12 -0.0001669454 -0.080262871 0.152941198 0.203953728 0.075547883
## pix13 0.2694009251 -0.730369431 -0.282321843 0.982722415 0.035997942
## pix14 -0.4661609012 -0.275797245 -0.144874775 -0.464392036 0.414403722
## pix15 -0.1184250776 -0.832456249 -0.492436278 0.270951622 -0.040823323
## pix16 0.0726299428 -0.173431803 -0.916501533 0.989533850 -0.417440259
## pix17 0.0163458577 -0.148898743 0.065357824 -0.102169889 0.207313422
## pix18 0.1156500788 -0.622889934 0.197915837 0.304292140 -0.600915756
## pix19 -0.5263600764 0.054314883 -0.182230231 -0.565437602 0.384305160
## pix20 -0.0974543967 -0.144349869 0.220945707 0.381126206 0.105543501
## pix21 0.2418097664 0.180623530 -0.426825849 -0.005978922 0.242710160
## pix22 -0.2804036263 0.127164730 0.592700152 -0.141435245 0.105082132
## pix23 -0.0329323320 -0.164091302 -0.235205338 0.162921816 0.180295291
## pix24 0.0351286700 -0.178929852 -0.010104264 0.284631091 -0.111756852
## pix25 0.1486689958 -0.059892050 0.348450423 -0.018494328 0.275836196
## pix26 0.0777385485 -0.571628698 -0.309220318 -0.002880698 0.078489209
## pix27 -0.0446724850 0.145289989 0.071115125 -0.031006078 -0.011505633
## pix28 -0.0235831706 -0.343986981 -0.465523678 0.365851109 -0.153483635
## pix29 0.2044920731 -0.124540702 -0.389198262 -0.205551416 -0.067637048
## pix30 0.0149679173 0.221105573 -0.129453773 0.219110455 0.088440358
## pix31 0.2257018793 0.250100209 0.361722176 -0.246509288 0.289005550
## pix32 -0.1541888500 -0.145756868 0.086990824 -0.105248701 -0.117225787
## pix33 -0.2949812301 -0.028385571 -0.333234539 -0.223627737 0.178543896
## pix34 0.4077037986 0.154113183 0.534147654 0.383243152 0.047217843
## pix35 -0.3605081089 -0.148586856 -0.036578637 0.309303170 -0.256014047
## pix36 -0.0566531090 -0.628901226 -0.122037372 -0.525741610 0.429497559
## pix37 0.3931713692 -0.014952177 -0.376991699 -0.349075664 -0.005628246
## pix38 -0.0661800193 -0.171403213 0.610536294 0.033876805 0.022942317
## pix39 0.2391389000 -0.159626282 -0.083776309 0.203801375 0.058934863
## pix40 -0.0604124758 -0.235649622 0.218217953 -0.324400660 0.358506808
## pix41 -0.0776184451 0.125969942 0.223885672 -0.651363378 0.085285419
## pix42 0.2853815037 0.002046776 0.239760509 0.178874326 0.615013384
## pix43 0.1100389168 -0.354636132 0.055691683 0.181640317 -0.075779616
## pix44 -0.3925752940 -0.607903763 0.261155094 -0.076772660 0.089690224
## pix45 0.2370534957 0.318681297 0.445972520 -0.072390868 0.524247772
## pix46 -0.1095995041 0.048817359 0.145026956 -0.045300357 0.053828830
## pix47 0.4659557602 -0.100083056 0.445894755 -0.434047040 -0.033525429
## pix48 0.7069479392 0.059295617 0.060575844 0.504876656 -0.764408687
## pix49 0.3235800131 -0.180688835 -0.499685028 -0.028779767 -0.187618312
## pix50 0.5432754182 -0.511101434 0.047521713 0.025000570 0.135401937
## pix51 -0.0242035143 0.156182354 0.861050533 -0.038724999 -0.194845560
## pix52 0.3506754887 -0.182003930 -0.144429092 0.436499247 -0.082056139
## pix53 -0.2323103483 -0.135239664 -0.292631552 0.469971019 0.022633437
## pix54 0.0173101639 -0.329743580 0.229973140 -0.625587669 0.091013363
## pix55 0.3534908004 0.301565568 -0.083976910 0.251502677 -0.074018827
## pix56 0.4031067890 0.233355393 0.120410859 -0.111810013 0.151730652
## pix57 -0.1005762381 0.032799047 0.084420568 0.310761640 0.147292288
## pix58 -0.3815996041 -0.172790835 0.344871167 -0.579449825 -0.057677513
## pix59 -0.2590350200 0.652238206 0.326977610 -0.571652586 0.401031424
## pix60 0.4844095542 0.521252902 0.681114041 -0.703616381 -0.079608676
## pix61 -0.0929872666 -0.475440637 0.551717722 0.124491545 -0.098872998
## pix62 -0.3422526561 0.083415693 0.583558420 -0.246060150 0.023475987
## pix63 -0.2748427039 -0.291140769 0.713644812 -0.588242634 0.100792391
## pix64 0.3123971479 0.113099443 0.623727925 -0.263801567 -0.100237758
## pix65 -0.2359788587 0.187698190 0.027067957 0.206129152 -0.256860356
## pix66 -0.5306082007 -0.037483459 0.301667171 0.802265963 -0.492635654
## pix67 -0.4907162808 0.389182185 -0.489295344 0.374638467 -0.328931371
## pix68 -0.0681671542 -0.029778099 0.579207528 0.197804731 -0.486335463
## pix69 0.0253525234 0.088162990 -0.082256996 -0.251148789 0.085032205
## pix70 -0.4065967371 -0.061350511 -0.042988128 0.526307317 -0.121115205
## pix71 -0.2984739788 0.060770158 0.104581261 -0.173525683 -0.930816729
## pix72 0.1277685498 -0.162088485 0.297740400 0.082067277 -0.164306669
## pix73 0.5059203893 0.335199696 0.344081021 0.467813190 -0.255747664
## pix74 0.6202373406 -0.207974721 -0.305125707 -0.162261918 -0.165897740
## pix75 0.0230252737 0.182682655 0.450835953 0.188097987 0.031017638
## pix76 -0.0888259324 -0.288395923 -0.254790054 0.013230681 -0.012845748
## pix77 0.1575227448 0.034021070 0.180504093 -0.202509716 -0.344875399
## pix78 -0.0212433666 -0.315450449 0.040863486 -0.359869067 0.196920645
## pix79 0.4679874441 -0.381883683 0.618506419 0.040640423 0.350793556
## pix80 -0.0589127482 -0.413101448 0.669327045 -0.298207312 -0.504417690
## pix81 -0.8635338920 -0.349529653 0.201858224 0.226034601 -1.006818584
## pix82 -0.3767551437 -0.291064707 -0.197495789 -0.063524708 0.294839106
## pix83 0.6810768423 -0.027235992 0.524646633 -0.319409095 -0.359646278
## pix84 -0.5566487051 -0.365462449 -0.096072396 0.240859850 -0.608614759
## pix85 0.1077206525 0.306392376 0.373516225 0.403672175 -0.170321460
## pix86 -0.2189548048 0.006778994 0.096110370 0.136725152 0.051713943
## pix87 0.1335302175 -0.126666501 -0.042828425 -0.169785707 -0.338300740
## pix88 -0.0143360340 0.216069397 0.371426823 0.450614663 0.403304912
## pix89 0.1365919496 -0.090107089 -0.167647707 -0.356581078 -0.409318430
## pix90 -0.0234993072 0.290431833 0.069311192 -0.254446767 0.245736469
## pix91 -0.1304175878 -0.154808270 0.243894980 0.093440489 -0.402772367
## pix92 0.7923618639 0.152394266 0.198627593 -0.254424825 0.591635293
## pix93 -0.0582116600 0.118113478 0.679336463 -0.204721067 -0.386972150
## pix94 -0.0808849180 0.332020384 -0.204574412 0.428700906 -0.511882100
## pix95 -0.1606256992 -0.370075846 0.402512646 0.080289561 -0.105862275
## pix96 -0.8524654792 -0.353903322 0.218250939 -1.234213054 1.164325465
## pix97 -0.3061344774 -0.034192883 0.234439440 -0.329262555 0.353391866
## pix98 0.3205179440 0.008212990 0.124109229 0.309905057 -0.379390521
## pix99 -0.3451779398 -0.160964306 -0.005140511 0.290720926 0.435480095
## pix100 -0.6219153485 0.167511678 -0.395109658 0.007579569 -0.161045330
## pix101 0.4607949649 0.010587879 -0.011641177 0.271394600 0.180873614
## pix102 0.2394053441 0.178410009 0.186254098 -0.215012477 0.007842007
## pix103 -0.0709033969 -0.300422680 -0.359094352 -0.125826663 -0.503031451
## pix104 0.0004240968 0.255919439 0.359800161 0.198881681 -0.063689648
## pix105 0.2553450334 0.223094520 -0.052639261 0.502926402 0.363612347
## pix106 0.1323246044 0.041027069 -0.210787698 -0.050316222 0.157758363
## pix107 -0.0341997516 0.131883901 -0.165113978 0.115322470 -0.294891390
## pix108 -0.0769782209 -0.025170296 0.131200692 0.632866593 -0.362703828
## pix109 0.2174973632 0.086283980 -0.006818414 -0.573824215 0.063374624
## pix110 0.3798311566 -0.249904415 -0.219050872 0.262934844 -0.297266454
## pix111 -0.5283347187 0.579435196 0.278651876 1.009365155 -0.651677619
## pix112 -0.5892106763 -0.139418102 0.757434689 -0.500593807 0.680141349
## pix113 -0.5069353646 0.391476328 -0.509854928 0.523258956 0.176941188
## pix114 -0.0871935875 -0.306776898 0.064571039 0.045604495 0.057413249
## pix115 0.3581806240 0.104218404 -0.094949837 -0.052239355 0.299434962
## pix116 0.3060240648 -0.314080419 0.462653266 0.115387397 -0.316572966
## pix117 -0.4669494618 0.296439932 -0.085780655 0.494769286 -0.226214235
## pix118 -0.0285167515 -0.148475121 0.102143795 -0.167271425 0.129433161
## pix119 0.2529712415 0.150316002 1.037983519 0.520302690 -0.079947282
## pix120 -0.3190205599 0.088625255 -0.779527327 -0.117700925 -0.748166895
## pix121 0.2231181874 0.137414551 0.006031118 -0.359275346 0.003248771
## pix122 0.0698130119 -0.279818505 0.427392745 0.688095080 -0.252905941
## pix123 0.1367495900 0.114898610 -0.346736332 -0.373565158 0.082704176
## pix124 0.0448224078 0.188120797 0.024942518 -0.034532539 -0.477648925
## pix125 0.0797433503 -0.311310519 -0.034622251 -0.301029790 -0.363051159
## pix126 0.1340873894 -0.075018774 -0.343140599 0.076667783 -0.240371210
## pix127 -0.5303835990 0.110477813 0.304593903 -0.387919641 0.730369993
## pix128 -0.2936007144 0.246348297 0.675545087 0.233513539 0.137718664
## pix129 -0.1663248181 -0.091744662 -0.207461760 -0.680808632 0.282301003
## pix130 -0.7326220606 0.404024298 0.386487056 0.588181098 -0.212577558
## pix131 0.1497092455 -0.012459204 0.069033401 -0.027287817 -0.357868283
## pix132 0.3595344907 -0.017700318 -0.852435410 0.005967905 0.179037769
## pix133 -0.2848436001 -0.267787802 0.467976594 -0.023660749 -0.511314277
## pix134 0.3732048161 0.499822584 -0.676605958 0.364428649 -0.045956850
## pix135 -0.0459654338 0.096397819 0.689813606 -0.075034879 -0.207981983
## pix136 -0.0039028283 -0.192419618 -0.128468111 -0.317896195 0.111248257
## pix137 -0.1508497989 0.095127554 0.097505281 0.220411380 -0.156469472
## pix138 -0.0599164211 0.456317067 -0.140229938 0.097202850 0.147500831
## pix139 0.1488107987 -0.242487143 -0.394039419 -0.016510706 -0.024825262
## pix140 -0.2692926361 0.372400703 -0.307476792 0.139344493 0.090118838
## pix141 0.1780059789 -0.085902698 0.544577833 0.487366024 0.321202023
## pix142 0.2811098508 0.301696260 0.240759708 -0.102060792 0.065347889
## pix143 0.1025713571 0.592422395 -0.084654692 -0.148280453 -0.234857644
## pix144 -0.3756799368 0.612970350 0.383726990 0.337812724 0.022993313
## pix145 -0.3172947930 -0.035840064 -0.169716735 0.345076559 -0.055084303
## pix146 -1.0935168669 0.157471093 0.178372431 -0.577211211 -0.285743665
## pix147 -0.1507098181 0.175404227 -0.217685670 0.111332286 0.162486386
## pix148 0.4430263388 0.169679819 -0.065919902 0.387741925 0.285040989
## pix149 -0.3352573571 -0.013303060 0.112018863 -0.524465032 -0.675463237
## pix150 -0.2566838605 0.155571811 0.283947001 0.156388939 -0.141695409
## pix151 0.0689192589 -0.352050770 0.072448001 0.333334962 -0.568103928
## pix152 0.0963022513 0.185152342 -0.437953266 -0.196348440 0.325904778
## pix153 0.0407763426 0.212124285 -0.003271360 -0.333478545 0.081282338
## pix154 0.2160190955 -0.148114157 -0.457361385 -0.050890850 -0.253760690
## pix155 -0.4655916186 0.340775359 0.331226722 0.334143594 -0.166820742
## pix156 0.3118502743 -0.191912447 -0.101917167 0.131527307 0.119496944
## pix157 0.3461781334 0.444586590 -0.455890920 -0.125374903 -0.393223346
## pix158 -0.5687069052 0.381926278 0.654619852 0.469747041 0.379155735
## pix159 -0.4342374538 -0.240155055 -0.175118442 -0.264296157 0.589809085
## pix160 0.1042546008 0.042793211 0.266687674 0.028181444 0.443191404
## pix161 -0.0580573739 0.101817470 -0.010517484 -0.553208006 -0.048997949
## pix162 -0.5431454039 -0.340975617 0.300808796 -0.149870111 -0.558754132
## pix163 -0.1590090726 -0.389692606 -0.552687485 -0.591203008 -0.061286212
## pix164 -0.2268198050 0.224965538 0.457487281 0.051811804 -0.224782686
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##
## Proportion of trace:
## LD1 LD2 LD3 LD4 LD5 LD6 LD7 LD8 LD9
## 0.2666 0.1785 0.1394 0.1271 0.0949 0.0700 0.0575 0.0423 0.0238lda_train <- lda(digit ~ ., data = train)
LDs_train <- predict(lda_train)$x
LDs_validation <- predict(lda_train, newdata = validation)$x
train_LDA1 <- data.frame(y = train$digit, LD1 = LDs_train[,1])
val_LDA1 <- data.frame(y = validation$digit, LD1 = LDs_validation[,1])
lda1_train <- lda(y~., data = train_LDA1)
pred_val1 <- predict(lda1_train, newdata = val_LDA1)$class
mean(pred_val1 != validation$digit)## [1] 0.6446541train_LDA2 <- data.frame(y = train$digit, LD1 = LDs_train[,1], LD2 = LDs_train[,2])
val_LDA2 <- data.frame(y = validation$digit, LD1 = LDs_validation[,1], LD2 = LDs_validation[,2])
lda2_train <- lda(y~., data = train_LDA2)
pred_val2 <- predict(lda2_train, newdata = val_LDA2)$class
mean(pred_val2 != validation$digit)## [1] 0.4937107train_LDA3 <- data.frame(y = train$digit, LD1 = LDs_train[,1], LD2 = LDs_train[,2], LD3 = LDs_train[,3])
val_LDA3 <- data.frame(y = validation$digit, LD1 = LDs_validation[,1], LD2 = LDs_validation[,2], LD3 = LDs_validation[,3])
lda3_train <- lda(y~., data = train_LDA3)
pred_val3 <- predict(lda3_train, newdata = val_LDA3)$class
mean(pred_val3 != validation$digit)## [1] 0.3962264train_LDA4 <- data.frame(y = train$digit, LD1 = LDs_train[,1], LD2 = LDs_train[,2], LD3 = LDs_train[,3], LD4 = LDs_train[,4])
val_LDA4 <- data.frame(y = validation$digit, LD1 = LDs_validation[,1], LD2 = LDs_validation[,2], LD3 = LDs_validation[,3], LD4 = LDs_validation[,4])
lda4_train <- lda(y~., data = train_LDA4)
pred_val4 <- predict(lda4_train, newdata = val_LDA4)$class
mean(pred_val4 != validation$digit)## [1] 0.2798742train_LDA5 <- data.frame(y = train$digit, LD1 = LDs_train[,1], LD2 = LDs_train[,2], LD3 = LDs_train[,3], LD4 = LDs_train[,4],LD5 = LDs_train[,5] )
val_LDA5 <- data.frame(y = validation$digit, LD1 = LDs_validation[,1], LD2 = LDs_validation[,2], LD3 = LDs_validation[,3], LD4 = LDs_validation[,4],LD5 = LDs_validation[,5])
lda5_train <- lda(y~., data = train_LDA5)
pred_val5 <- predict(lda5_train, newdata = val_LDA5)$class
mean(pred_val5 != validation$digit)## [1] 0.2421384train_LDA6 <- data.frame(y = train$digit, LD1 = LDs_train[,1], LD2 = LDs_train[,2], LD3 = LDs_train[,3], LD4 = LDs_train[,4],LD5 = LDs_train[,5],LD6 = LDs_train[,6] )
val_LDA6 <- data.frame(y = validation$digit, LD1 = LDs_validation[,1], LD2 = LDs_validation[,2], LD3 = LDs_validation[,3], LD4 = LDs_validation[,4],LD5 = LDs_validation[,5],LD6 = LDs_validation[,6])
lda6_train <- lda(y~., data = train_LDA6)
pred_val6 <- predict(lda6_train, newdata = val_LDA6)$class
mean(pred_val6 != validation$digit)## [1] 0.1949686train_LDA7 <- data.frame(y = train$digit, LD1 = LDs_train[,1], LD2 = LDs_train[,2], LD3 = LDs_train[,3], LD4 = LDs_train[,4],LD5 = LDs_train[,5],LD6 = LDs_train[,6],LD7 = LDs_train[,7] )
val_LDA7 <- data.frame(y = validation$digit, LD1 = LDs_validation[,1], LD2 = LDs_validation[,2], LD3 = LDs_validation[,3], LD4 = LDs_validation[,4],LD5 = LDs_validation[,5],LD6 = LDs_validation[,6],LD7 = LDs_validation[,7])
lda7_train <- lda(y~., data = train_LDA7)
pred_val7 <- predict(lda7_train, newdata = val_LDA7)$class
mean(pred_val7 != validation$digit)## [1] 0.1949686train_LDA8 <- data.frame(y = train$digit, LD1 = LDs_train[,1], LD2 = LDs_train[,2], LD3 = LDs_train[,3], LD4 = LDs_train[,4],LD5 = LDs_train[,5],LD6 = LDs_train[,6],LD7 = LDs_train[,7],LD8 = LDs_train[,8] )
val_LDA8 <- data.frame(y = validation$digit, LD1 = LDs_validation[,1], LD2 = LDs_validation[,2], LD3 = LDs_validation[,3], LD4 = LDs_validation[,4],LD5 = LDs_validation[,5],LD6 = LDs_validation[,6],LD7 = LDs_validation[,7],LD8 = LDs_validation[,8])
lda8_train <- lda(y~., data = train_LDA8)
pred_val8 <- predict(lda8_train, newdata = val_LDA8)$class
mean(pred_val8 != validation$digit)## [1] 0.1949686train_LDA9 <- data.frame(y = train$digit, LD1 = LDs_train[,1], LD2 = LDs_train[,2], LD3 = LDs_train[,3], LD4 = LDs_train[,4],LD5 = LDs_train[,5],LD6 = LDs_train[,6],LD7 = LDs_train[,7],LD8 = LDs_train[,8],LD9 = LDs_train[,9] )
val_LDA9 <- data.frame(y = validation$digit, LD1 = LDs_validation[,1], LD2 = LDs_validation[,2], LD3 = LDs_validation[,3], LD4 = LDs_validation[,4],LD5 = LDs_validation[,5],LD6 = LDs_validation[,6],LD7 = LDs_validation[,7],LD8 = LDs_validation[,8],LD9 = LDs_validation[,9])
lda9_train <- lda(y~., data = train_LDA9)
pred_val9 <- predict(lda9_train, newdata = val_LDA9)$class
mean(pred_val9 != validation$digit)## [1] 0.1981132Using LDs 1-6 increasingly made the results better, and it didnt make a difference if 7 or 8 were being used or not. But LD9 actually made the result worse. So the best option is to use LD1,LD2,LD3,LD4,LD5 and LD6. From which the lowest error was found: 0.1949686