Introduction
This vignette visualizes classification results of neural nets using tools from the classmap package. We will fit neural nets by the nnet package, which allows the training of simple neural networks with a single hidden layer.
library(nnet)
library(classmap)
Iris data
Training data
As a first small example, we consider the Iris data. We start by training a neural network with one hidden layer containing 3 neurons on the full iris data.
set.seed(123)
nn.iris <- nnet(Species ~ ., data = iris, size = 3)
## # weights: 27
## initial value 207.903843
## iter 10 value 69.852587
## iter 20 value 8.126663
## iter 30 value 5.901328
## iter 40 value 5.299180
## iter 50 value 4.942638
## iter 60 value 4.747845
## iter 70 value 4.379577
## iter 80 value 4.221971
## iter 90 value 3.738670
## iter 100 value 3.572490
## final value 3.572490
## stopped after 100 iterations
names(nn.iris)
## [1] "n" "nunits" "nconn" "conn"
## [5] "nsunits" "decay" "entropy" "softmax"
## [9] "censored" "value" "wts" "convergence"
## [13] "fitted.values" "residuals" "lev" "call"
## [17] "terms" "coefnames" "xlevels"
head(nn.iris$fitted.values) # matrix of posterior probabilities.
## setosa versicolor virginica
## 1 1 6.016990e-17 1.968299e-78
## 2 1 1.091170e-16 1.603084e-78
## 3 1 6.068970e-17 1.967292e-78
## 4 1 1.211741e-16 1.701564e-78
## 5 1 6.005968e-17 1.968701e-78
## 6 1 6.186116e-17 2.008031e-78
nn.iris$wts # the "weights" (coefficients, some are negative).
## [1] 8.461321 1.361977 -12.051401 16.284091 -36.812648 -659.229209
## [7] -55.372159 -69.510090 197.535633 143.586681 2.651707 1.568175
## [13] 6.055571 -13.504329 -9.530820 23.554323 -28.893963 -6.098134
## [19] 49.173270 75.460324 120.079027 -189.946588 -40.085310 -97.126609
## [25] -90.777198 195.965473 -9.070470
summary(nn.iris)
## a 4-3-3 network with 27 weights
## options were - softmax modelling
## b->h1 i1->h1 i2->h1 i3->h1 i4->h1
## 8.46 1.36 -12.05 16.28 -36.81
## b->h2 i1->h2 i2->h2 i3->h2 i4->h2
## -659.23 -55.37 -69.51 197.54 143.59
## b->h3 i1->h3 i2->h3 i3->h3 i4->h3
## 2.65 1.57 6.06 -13.50 -9.53
## b->o1 h1->o1 h2->o1 h3->o1
## 23.55 -28.89 -6.10 49.17
## b->o2 h1->o2 h2->o2 h3->o2
## 75.46 120.08 -189.95 -40.09
## b->o3 h1->o3 h2->o3 h3->o3
## -97.13 -90.78 195.97 -9.07
In the above output, the coefficients are listed with their arrows: i1,…,i4 are the 4 _i_nput variables. h1, h2, h3 are the 3 _h_idden nodes. o1, o2, o3 are the _o_utput nodes, one for each class (before the softmax function is applied). b->h1, b->h2, b->h3 are the intercepts to the hidden layer. b->o1, b->o2, b->o3 are the intercepts to the output layer.
We can use this to reconstruct the model from the coefficients. We first extract the coefficients from each layer:
train_x <- as.matrix(iris[, -5]) # original input: 4 variables.
# extract "weights" (coefficients) from each layer:
bias1 <- nn.iris$wts[c(1, 6, 11)] # intercepts to hidden layer
betas1 <- cbind(nn.iris$wts[2:5], nn.iris$wts[7:10],
nn.iris$wts[12:15]) # slopes to hidden layer
bias2 <- nn.iris$wts[c(16, 20, 24)] # intercepts to output layer
betas2 <- cbind(nn.iris$wts[17:19], nn.iris$wts[21:23],
nn.iris$wts[25:27]) # slopes to output layer.
Now we use the weights and the sigmoid function (for activation in the hidden layer) to reconstruct the representations of the data in the hidden layer (H), as well as in the final layer before softmax (X). X is the set of G-dimensional points v_i given by (6) in the paper. This X will be used for the farness in the class maps.
sigmoid <- function(x) 1 / (1 + exp(-x))
H <- t(t(train_x %*% betas1) + bias1) # = hidden layer
X <- t(t(sigmoid(H) %*% betas2) + bias2) # = layer before softmax.
pairs(X, col = unclass(iris$Species)) # G=3 classes ==> 3 dimensions.
To obtain the posterior probabilities, we can apply the softmax function to the last layer. We verify that this yields the same result as the probabilities returned by nnet().
softmax <- function(x) exp(x) / sum(exp(x))
outprobs <- t(apply(X, 1, softmax))
trainprobs <- nn.iris$fitted.values # this is after softmax
range(trainprobs - outprobs) # near 0, so we have a match.
## [1] -3.418358e-06 3.418358e-06
Now we prepare the output for visualizations.
y <- iris[, 5]
vcrtrain <- vcr.neural.train(X, y, trainprobs)
##
## There is at least one class with (near-) singular covariance matrix,
## so farness will be computed from PCA.
names(vcrtrain)
## [1] "X" "yint" "y" "levels" "predint" "pred"
## [7] "altint" "altlab" "PAC" "figparams" "fig" "farness"
## [13] "ofarness"
vcrtrain$predint[c(1:10, 51:60, 101:110)] # prediction as integer
## 1 2 3 4 5 6 7 8 9 10 51 52 53 54 55 56 57 58 59 60
## 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
## 101 102 103 104 105 106 107 108 109 110
## 3 3 3 3 3 3 3 3 3 3
vcrtrain$pred[c(1:10, 51:60, 101:110)] # prediction as label
## [1] "setosa" "setosa" "setosa" "setosa" "setosa"
## [6] "setosa" "setosa" "setosa" "setosa" "setosa"
## [11] "versicolor" "versicolor" "versicolor" "versicolor" "versicolor"
## [16] "versicolor" "versicolor" "versicolor" "versicolor" "versicolor"
## [21] "virginica" "virginica" "virginica" "virginica" "virginica"
## [26] "virginica" "virginica" "virginica" "virginica" "virginica"
vcrtrain$altint[c(1:10, 51:60, 101:110)] # alternative label as integer
## [1] 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 2 2 1
vcrtrain$altlab[c(1:10, 51:60, 101:110)] # alternative label
## [1] "versicolor" "versicolor" "versicolor" "versicolor" "versicolor"
## [6] "versicolor" "versicolor" "versicolor" "versicolor" "versicolor"
## [11] "setosa" "setosa" "setosa" "setosa" "setosa"
## [16] "setosa" "setosa" "setosa" "setosa" "setosa"
## [21] "setosa" "setosa" "setosa" "versicolor" "setosa"
## [26] "versicolor" "setosa" "versicolor" "versicolor" "setosa"
# Probability of Alternative Class (PAC) of each object:
vcrtrain$PAC[1:3]
## [1] 6.01699e-17 1.09117e-16 6.06897e-17
#
summary(vcrtrain$PAC)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.0000 0.0000 0.0124 0.0000 0.9258
# f(i, g) is the distance from case i to class g:
vcrtrain$fig[1:5, ] # for the first 5 objects:
## [,1] [,2] [,3]
## [1,] 0.20737485 0.9395651 0.9876833
## [2,] 0.85045110 0.9395364 0.9876831
## [3,] 0.08450745 0.9395643 0.9876833
## [4,] 0.86189533 0.9395163 0.9876827
## [5,] 0.23301865 0.9395652 0.9876833
# The farness of an object i is the f(i, g) to its own class:
vcrtrain$farness[1:5]
## [1] 0.20737485 0.85045110 0.08450745 0.86189533 0.23301865
#
summary(vcrtrain$farness)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0368 0.2432 0.2599 0.4864 0.8557 0.9680
# The "overall farness" of an object is defined as the
# lowest f(i, g) it has to any class g (including its own):
summary(vcrtrain$ofarness)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0368 0.2432 0.2599 0.4848 0.8557 0.9389
confmat.vcr(vcrtrain)
##
## Confusion matrix:
## predicted
## given setosa versicolor virginica
## setosa 50 0 0
## versicolor 0 49 1
## virginica 0 0 50
##
## The accuracy is 99.33%.
The vcrtrain object can now be used to create the visualizations.
cols <- c("red", "darkgreen", "blue")
# stacked mosaic plot:
stplot <- stackedplot(vcrtrain, classCols = cols,
main = "Stacked plot of nnet on iris data")
stplot
# silhouette plot:
silplot(vcrtrain, classCols = cols, main =
"Silhouette plot of nnet on iris data")
## classNumber classLabel classSize classAveSi
## 1 setosa 50 1.00
## 2 versicolor 50 0.96
## 3 virginica 50 0.96
# class maps:
classmap(vcrtrain, "setosa", classCols = cols)
# Very tight class (low PAC, no high farness).
classmap(vcrtrain, "versicolor", classCols = cols)
# Not so tight, one point is predicted as virginica.
classmap(vcrtrain, "virginica", classCols = cols)
# Also tight.
New data
To illustrate the use of new data we create a “fake” dataset which is a subset of the training data, where not all classes occur, and ynew has NA’s.
test_x <- train_x[1:100, ]
ynew <- y[1:100]
ynew[c(1:10, 51:60)] <- NA
pairs(train_x, col = as.numeric(y) + 1, pch = 19) # 3 colors
pairs(test_x, col = as.numeric(ynew) + 1, pch = 19) # only red and green
We now calculate the predictions of this test data, and reconstruct the representation of the test data in the hidden and final layer.
predprobs <- predict(nn.iris, test_x, type = "raw")
range(predprobs - trainprobs[1:100, ]) # perfect match
## [1] 0 0
# Reconstruct this prediction:
Hnew <- t(t(test_x %*% betas1) + bias1) # hidden layer
Xnew <- t(t(sigmoid(Hnew) %*% betas2) + bias2) # layer before softmax
probsnew <- t(apply(Xnew, 1, softmax))
range(probsnew - predprobs) # ~0, so we have a match
## [1] -1.110223e-15 9.714451e-16
Now prepare for visualization:
vcrtest <- vcr.neural.newdata(Xnew, ynew, probsnew, vcrtrain)
plot(vcrtest$predint, vcrtrain$predint[1:100]); abline(0, 1) # identical, OK
plot(vcrtest$altint, vcrtrain$altint[1:100]); abline(0, 1) # identical when not NA, OK
plot(vcrtest$PAC, vcrtrain$PAC[1:100]); abline(0, 1) # OK
vcrtest$farness # length 100, with NA's where ynew is NA
## [1] NA NA NA NA NA NA
## [7] NA NA NA NA 0.22279874 0.72023725
## [13] 0.96321287 0.53001183 0.25870493 0.25729970 0.25553649 0.18077135
## [19] 0.26115206 0.20655762 0.89671342 0.20830291 0.25856230 0.95779043
## [25] 0.96190111 0.96171838 0.86977892 0.05423371 0.13796909 0.89553878
## [31] 0.93788017 0.61105777 0.25501513 0.25848322 0.88010290 0.22171244
## [37] 0.24876810 0.16330724 0.52374495 0.22309847 0.23270883 0.95775904
## [43] 0.09137918 0.92827252 0.93314093 0.78883032 0.16967184 0.41060696
## [49] 0.22014483 0.03679866 NA NA NA NA
## [55] NA NA NA NA NA NA
## [61] 0.25978151 0.88984549 0.25993220 0.24565279 0.88991058 0.82758963
## [67] 0.88294537 0.25991897 0.25718649 0.25830565 0.88104884 0.82941910
## [73] 0.92115278 0.25992966 0.09838746 0.73693813 0.25961446 0.83092147
## [79] 0.85324862 0.17433168 0.25708124 0.25956760 0.27683565 0.92785948
## [85] 0.88458022 0.88992431 0.71927806 0.25990656 0.87408023 0.30086342
## [91] 0.25986061 0.05544259 0.22655780 0.10915362 0.06261839 0.08496130
## [97] 0.67823299 0.06594331 0.88983510 0.73933757
plot(vcrtest$farness, vcrtrain$farness[1:100]); abline(0, 1) # identical where not NA
plot(vcrtest$fig, vcrtrain$fig[1:100, ]); abline(0, 1) # same, OK
vcrtest$ofarness # 100, withOUT NA's: ofarness even exists for cases with missing ynew, for which farness cannot exist.
## [1] 0.20737485 0.85045110 0.08450745 0.86189533 0.23301865 0.52757739
## [7] 0.29437232 0.21839328 0.88896177 0.93850072 0.22279874 0.72023725
## [13] 0.93891234 0.53001183 0.25870493 0.25729970 0.25553649 0.18077135
## [19] 0.26115206 0.20655762 0.89671342 0.20830291 0.25856230 0.93859265
## [25] 0.93834530 0.93888981 0.86977892 0.05423371 0.13796909 0.89553878
## [31] 0.93788017 0.61105777 0.25501513 0.25848322 0.88010290 0.22171244
## [37] 0.24876810 0.16330724 0.52374495 0.22309847 0.23270883 0.93857185
## [43] 0.09137918 0.92827252 0.93314093 0.78883032 0.16967184 0.41060696
## [49] 0.22014483 0.03679866 0.10278070 0.88802994 0.17433818 0.20547488
## [55] 0.22641177 0.25516339 0.88960053 0.28551006 0.25902041 0.88957763
## [61] 0.25978151 0.88984549 0.25993220 0.24565279 0.88991058 0.82758963
## [67] 0.88294537 0.25991897 0.25718649 0.25830565 0.88104884 0.82941910
## [73] 0.92115278 0.25992966 0.09838746 0.73693813 0.25961446 0.83092147
## [79] 0.85324862 0.17433168 0.25708124 0.25956760 0.27683565 0.85571604
## [85] 0.88458022 0.88992431 0.71927806 0.25990656 0.87408023 0.30086342
## [91] 0.25986061 0.05544259 0.22655780 0.10915362 0.06261839 0.08496130
## [97] 0.67823299 0.06594331 0.88983510 0.73933757
plot(vcrtest$ofarness, vcrtrain$ofarness[1:100]); abline(0, 1) # same, OK
confmat.vcr(vcrtest) # as expected:
##
## Confusion matrix:
## predicted
## given setosa versicolor virginica
## setosa 40 0 0
## versicolor 0 39 1
##
## The accuracy is 98.75%.
Now we can visualize the classification on the test data, and compare the resulting plots with their counterparts on the training data.
stplot # to compare with:
stackedplot(vcrtest, classCols = cols, separSize = 1.5,
minSize = 1, main = "Stacked plot of nnet on subset of iris data")
##
## Not all classes occur in these data. The classes to plot are:
## [1] 1 2
silplot(vcrtest, classCols = cols, main = "Silhouette plot of nnet on subset of iris data")
## classNumber classLabel classSize classAveSi
## 1 setosa 40 1.00
## 2 versicolor 40 0.95
classmap(vcrtrain, 1, classCols = cols)
classmap(vcrtest, 1, classCols = cols) # same, but fewer points
#
classmap(vcrtrain, 2, classCols = cols)
classmap(vcrtest, 2, classCols = cols) # same, but fewer points
#
# classmap(vcrtrain, 3, classCols = cols)
# classmap(vcrtest, 3, classCols = cols)
# # Class number 3 with label virginica has no objects to visualize.
floral buds data:
As a second example, we consider the floral buds data.
data("data_floralbuds")
We start by training a neural network with one hidden layer consisting of 2 neurons.
set.seed(123)
nn.buds <- nnet(y ~ ., data = data_floralbuds, size = 2)
## # weights: 26
## initial value 1041.109691
## iter 10 value 536.818801
## iter 20 value 459.928280
## iter 30 value 277.176449
## iter 40 value 245.861545
## iter 50 value 142.979121
## iter 60 value 90.662364
## iter 70 value 79.756728
## iter 80 value 75.207876
## iter 90 value 74.102871
## iter 100 value 73.179742
## final value 73.179742
## stopped after 100 iterations
names(nn.buds)
## [1] "n" "nunits" "nconn" "conn"
## [5] "nsunits" "decay" "entropy" "softmax"
## [9] "censored" "value" "wts" "convergence"
## [13] "fitted.values" "residuals" "lev" "call"
## [17] "terms" "coefnames" "xlevels"
head(nn.buds$fitted.values) # matrix of posterior probabilities of each class
## branch bud scales support
## 1 1.058835e-07 0.9999986 7.774815e-26 1.276449e-06
## 2 4.339590e-07 0.9999954 6.576856e-24 4.199437e-06
## 3 6.458451e-07 0.9999923 2.847148e-23 7.039262e-06
## 4 7.031223e-07 0.9999923 3.364849e-23 6.949520e-06
## 5 3.429871e-07 0.9999959 3.466130e-24 3.744776e-06
## 6 1.847493e-07 0.9999976 5.016766e-25 2.246819e-06
nn.buds$wts # the "weights" (coefficients, some negative)
## [1] -0.2624278 -35.7878683 -21.6885293 51.5692666 -0.8608722 20.4026110
## [7] -26.4567237 2.5410791 -8.3654734 18.6048565 11.7291332 -13.5435723
## [13] 36.6413025 -27.1683942 22.3796811 6.0538747 -22.1938387 -16.6302472
## [19] 22.5568173 16.5677260 33.2761417 -33.1952834 -35.9826056 -38.9852649
## [25] 5.3913482 42.3781984
summary(nn.buds) # A hidden layer with 2 neurons:
## a 6-2-4 network with 26 weights
## options were - softmax modelling
## b->h1 i1->h1 i2->h1 i3->h1 i4->h1 i5->h1 i6->h1
## -0.26 -35.79 -21.69 51.57 -0.86 20.40 -26.46
## b->h2 i1->h2 i2->h2 i3->h2 i4->h2 i5->h2 i6->h2
## 2.54 -8.37 18.60 11.73 -13.54 36.64 -27.17
## b->o1 h1->o1 h2->o1
## 22.38 6.05 -22.19
## b->o2 h1->o2 h2->o2
## -16.63 22.56 16.57
## b->o3 h1->o3 h2->o3
## 33.28 -33.20 -35.98
## b->o4 h1->o4 h2->o4
## -38.99 5.39 42.38
In the output above, the coefficients are listed with their arrows: i1, …, i6 are the 6 input variables h1, h2 are the 2 hidden nodes o1, .., o4 are the output nodes, one for each class (before softmax) b->h1 and b->h2 are both intercepts to the hidden layer b_o1, …, b->o4 are the intercepts to the output layer
We first extract these weights from the output of nnet():
train_x <- as.matrix(data_floralbuds[, -7])
# extract weights for each layer:
bias1 <- nn.buds$wts[c(1, 8)]
betas1 <- cbind(nn.buds$wts[2:7], nn.buds$wts[9:14])
bias2 <- nn.buds$wts[c(15, 18, 21, 24)]
betas2 <- cbind(nn.buds$wts[16:17], nn.buds$wts[19:20],
nn.buds$wts[22:23], nn.buds$wts[25:26])
We now use these weights and the sigmoid activation function to reconstruct the data in the hidden and final layer (before applying the softmax function). We verify that, after applying softmax, we obtain the same posterior probabilities as those in the output of nnet().
H <- t(t(train_x %*% betas1) + bias1) # hidden layer
X <- t(t(sigmoid(H) %*% betas2) + bias2) # layer before softmax
outprobs <- t(apply(X, 1, softmax))
trainprobs <- nn.buds$fitted.values # posterior probabilities
range(trainprobs - outprobs) # ~0
## [1] -2.126469e-10 2.126470e-10
# OK, we have reconstructed the posterior probabilities
pairs(X, col = data_floralbuds$y)
Now prepare for visualization:
y <- data_floralbuds$y
vcrtrain <- vcr.neural.train(X, y, trainprobs)
##
## There is at least one class with (near-) singular covariance matrix,
## so farness will be computed from PCA.
cols <- c("saddlebrown", "orange", "olivedrab4", "royalblue3")
stackedplot(vcrtrain, classCols = cols, main =
"Stacked plot of nnet on floral buds data")
# Silhouette plot:
silplot(vcrtrain, classCols = cols,
main = "Silhouette plot of nnet on floral buds data")
## classNumber classLabel classSize classAveSi
## 1 branch 49 0.45
## 2 bud 363 0.96
## 3 scales 94 0.86
## 4 support 44 0.57
# Quasi residual plot:
PAC <- vcrtrain$PAC
feat <- rowSums(train_x); xlab <- "rowSums(X)"
# pdf("Floralbuds_quasi_residual_plot.pdf", width = 5, height = 4.8)
qresplot(PAC, feat, xlab = xlab, plotErrorBars = TRUE, fac = 2, main = "Floral buds: quasi residual plot")
# images with higher sum are easier to classify
# dev.off()
cor.test(feat, PAC, method = "spearman")
##
## Spearman's rank correlation rho
##
## data: feat and PAC
## S = 34339314, p-value = 1.68e-08
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
## rho
## -0.2383866
# rho = -0.238, p-value < 2e-8
# ==> decreasing trend is significant
# Class maps:
classmap(vcrtrain, "branch", classCols = cols)
classmap(vcrtrain, "bud", classCols = cols)
classmap(vcrtrain, "scales", classCols = cols)
classmap(vcrtrain, "support", classCols = cols)
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