Naive Bayes Classifiers
Naive Bayes Classifiers
A naive bayes classifier uses Bayes rule to combine information about a set of predictors. Although many Bayesian approaches can be quite complex and computationally-intensive, Naive Bayes classifiers are simple such that they can often be implemented without any special library. They are also easy to use even when you have hundreds or thousands of features, such as when trying to classify text (e.g., each word is a feature/predictor), for classification problems like sentiment analysis or spam detection. Let’s start with an example application for the iPhone data set:
dat <- read.csv("data_study1.csv")
dat$Smartphone <- factor(dat$Smartphone)
library(klaR)
nb <- NaiveBayes(Smartphone ~ ., data = dat)
library(DAAG)
summary(nb) Length Class Mode
apriori 2 table numeric
tables 12 -none- list
levels 2 -none- character
call 3 -none- call
x 12 data.frame list
usekernel 1 -none- logical
varnames 12 -none- characternb$apriorigrouping
Android iPhone
0.4139887 0.5860113 nb$tables$Gender
var
grouping female male
Android 0.5753425 0.4246575
iPhone 0.7516129 0.2483871
$Age
[,1] [,2]
Android 31.42466 13.52190
iPhone 26.84839 12.19792
$Honesty.Humility
[,1] [,2]
Android 3.598174 0.5971617
iPhone 3.351935 0.6269406
$Emotionality
[,1] [,2]
Android 3.231963 0.7187408
iPhone 3.455806 0.6648467
$Extraversion
[,1] [,2]
Android 3.196347 0.6842832
iPhone 3.288710 0.6360540
$Agreeableness
[,1] [,2]
Android 3.175799 0.5900619
iPhone 3.123226 0.6390040
$Conscientiousness
[,1] [,2]
Android 3.544292 0.5910664
iPhone 3.573871 0.6173289
$Openness
[,1] [,2]
Android 3.542922 0.6218371
iPhone 3.415806 0.6224821
$Avoidance.Similarity
[,1] [,2]
Android 2.545205 0.8413687
iPhone 2.306452 0.8150825
$Phone.as.status.object
[,1] [,2]
Android 2.086758 0.5411150
iPhone 2.375484 0.6006299
$Social.Economic.Status
[,1] [,2]
Android 5.949772 1.586132
iPhone 6.012903 1.485582
$Time.owned.current.phone
[,1] [,2]
Android 12.72146 11.497509
iPhone 11.68065 8.748811To understand how it works, let’s look at the output. First, we have the apriori values–this determines the overall base rate of the outcomes. Then, we have the mean and standard deviation of normal distribution describing of each feature for each group. To confirm, look at the values for age:
means <- tapply(as.numeric(dat$Age), list(dat$Smartphone), mean)
sds <- tapply(as.numeric(dat$Age), list(dat$Smartphone), sd)
means Android iPhone
31.42466 26.84839 sds Android iPhone
13.52190 12.19792 Naive Bayes works very similarly to qda. QDA attempts to find a decision point consistent with the most likely outcome that combines all features optimally. In contrast, NB combines all features equally–for each feature computing the likelihood of each option, and combining those one at a time. For the Age predictor, the two densities look like this:
library(ggplot2)
phone <- rep(c("iphone", "android"), each = 100)
density <- c(dnorm(1:100, mean = means[2], sd = sds[2]), dnorm(1:100, mean = means[1],
sd = sds[1]))
df <- data.frame(age = c(1:100, 1:100), phone, density)
ggplot(df, aes(x = age, y = density, colour = phone)) + geom_line()
So, in a way this is very similar to LDA/QDA, because it uses a model-based approach, assuming a normal distribution, and permits different means and standard deviation for each group. Notice how for different ages, there is a higher likelihood for one group or the other. We can compute a ratio easily:
df <- data.frame(age = c(1:100), lr = (dnorm(1:100, mean = means[2], sd = sds[2])/dnorm(1:100,
mean = means[1], sd = sds[1])))
ggplot(df, aes(x = age, y = lr)) + geom_line() + geom_hline(yintercept = 1)
This shows the likelihood of ratio in favor of being an iPhone user across age. Notice that this is highly model-based–if your assumed model is wrong, like it if it is not normally-distributed, the ratio can be wrong. This is especially true out in the tails of the distribution, where we probably have the least amount of data, but where the observations can have the greatest impact. If this were a decision tree, we’d use the point the likelihood ratio crosses 1.0 as the place to put the cut-off for that dimension. If we were doing LDA or QDA, we’d combine all axes and find a cutoff along a single vector. NB combines the likelihood ratio curves for each dimension using Bayes Rule, which also permits combining prior probability (base rate). So, like LDA, it combines all the dimensions, but like a decision tree, it computes a fit on dimension.
In contrast to QDA, the weighing of features technically differs in Naive Bayes, but they differ in details a little. One important aspect is that QDA and LDA essentially use a regression equation that incorporates base rate in an intercept value. Here, each features is ignorant of the bias. If you have a missing value, you can ignore the contribution of that feature, as the best guess is 1.0–which has no impact on the classification. For LDA and QDA, it is not as easy to deal with missing values. This is one benefit of Bayes Factor–it is useful for data that are often likely to be missing. This is one of the reasons why it is frequently used for classifying text data, which may have thousands of features (i.e., words) but most of them are missing for any given document.
We can use this for prediction:
Overall accuracy = 0.667
Confusion matrix
Predicted (cv)
Actual Android iPhone
Android 0.543 0.457
iPhone 0.245 0.755This performs about as well as any of the other models. There is no simple way to do cross-validation with this implementation–you would have to do this by hand if you wanted to implement cross-validation. But also note that if a feature is non-predictive, it will essentially have very little impact on the outcome.
Using alternative kernel
The model assumes a normal/gaussian distribution, but Naive Bayes will permit us to estimate the of each feature empirically, which happens when we specify usekernel=T.
nb2 <- NaiveBayes(Smartphone ~ ., usekernel = T, data = dat)
nb2$tables$Gender
var
grouping female male
Android 0.5753425 0.4246575
iPhone 0.7516129 0.2483871
$Age
$Age$Android
Call:
density.default(x = xx)
Data: xx (219 obs.); Bandwidth 'bw' = 4.142
x y
Min. : 3.575 Min. :5.350e-06
1st Qu.:23.537 1st Qu.:2.524e-03
Median :43.500 Median :8.884e-03
Mean :43.500 Mean :1.251e-02
3rd Qu.:63.463 3rd Qu.:1.954e-02
Max. :83.425 Max. :4.128e-02
$Age$iPhone
Call:
density.default(x = xx)
Data: xx (310 obs.); Bandwidth 'bw' = 2.292
x y
Min. : 9.123 Min. :1.259e-05
1st Qu.:27.062 1st Qu.:8.960e-04
Median :45.000 Median :7.547e-03
Mean :45.000 Mean :1.392e-02
3rd Qu.:62.938 3rd Qu.:1.111e-02
Max. :80.877 Max. :9.380e-02
$Honesty.Humility
$Honesty.Humility$Android
Call:
density.default(x = xx)
Data: xx (219 obs.); Bandwidth 'bw' = 0.1829
x y
Min. :1.151 Min. :0.0001146
1st Qu.:2.226 1st Qu.:0.0197670
Median :3.300 Median :0.1441117
Mean :3.300 Mean :0.2324787
3rd Qu.:4.374 3rd Qu.:0.4797145
Max. :5.449 Max. :0.5965253
$Honesty.Humility$iPhone
Call:
density.default(x = xx)
Data: xx (310 obs.); Bandwidth 'bw' = 0.1791
x y
Min. :1.163 Min. :0.0000963
1st Qu.:2.206 1st Qu.:0.0382611
Median :3.250 Median :0.1789519
Mean :3.250 Mean :0.2392920
3rd Qu.:4.294 3rd Qu.:0.4296896
Max. :5.337 Max. :0.6257473
$Emotionality
$Emotionality$Android
Call:
density.default(x = xx)
Data: xx (219 obs.); Bandwidth 'bw' = 0.2202
x y
Min. :0.4395 Min. :0.0000934
1st Qu.:1.7448 1st Qu.:0.0166069
Median :3.0500 Median :0.1157574
Mean :3.0500 Mean :0.1913477
3rd Qu.:4.3552 3rd Qu.:0.3759779
Max. :5.6605 Max. :0.5323054
$Emotionality$iPhone
Call:
density.default(x = xx)
Data: xx (310 obs.); Bandwidth 'bw' = 0.19
x y
Min. :0.9301 Min. :0.0000773
1st Qu.:2.0900 1st Qu.:0.0219588
Median :3.2500 Median :0.1508402
Mean :3.2500 Mean :0.2153124
3rd Qu.:4.4100 3rd Qu.:0.3922926
Max. :5.5699 Max. :0.6291699
$Extraversion
$Extraversion$Android
Call:
density.default(x = xx)
Data: xx (219 obs.); Bandwidth 'bw' = 0.2057
x y
Min. :0.7828 Min. :0.0001224
1st Qu.:1.9164 1st Qu.:0.0407787
Median :3.0500 Median :0.1578239
Mean :3.0500 Mean :0.2203167
3rd Qu.:4.1836 3rd Qu.:0.3968226
Max. :5.3172 Max. :0.5870843
$Extraversion$iPhone
Call:
density.default(x = xx)
Data: xx (310 obs.); Bandwidth 'bw' = 0.1817
x y
Min. :1.055 Min. :0.0000833
1st Qu.:2.077 1st Qu.:0.0339189
Median :3.100 Median :0.2100553
Mean :3.100 Mean :0.2442280
3rd Qu.:4.123 3rd Qu.:0.4375608
Max. :5.145 Max. :0.5799863
$Agreeableness
$Agreeableness$Android
Call:
density.default(x = xx)
Data: xx (219 obs.); Bandwidth 'bw' = 0.1807
x y
Min. :1.058 Min. :0.0001347
1st Qu.:2.104 1st Qu.:0.0270296
Median :3.150 Median :0.1628894
Mean :3.150 Mean :0.2387443
3rd Qu.:4.196 3rd Qu.:0.4663836
Max. :5.242 Max. :0.6169762
$Agreeableness$iPhone
Call:
density.default(x = xx)
Data: xx (310 obs.); Bandwidth 'bw' = 0.1826
x y
Min. :0.8522 Min. :0.000095
1st Qu.:1.9261 1st Qu.:0.027074
Median :3.0000 Median :0.181926
Mean :3.0000 Mean :0.232570
3rd Qu.:4.0739 3rd Qu.:0.445709
Max. :5.1478 Max. :0.568577
$Conscientiousness
$Conscientiousness$Android
Call:
density.default(x = xx)
Data: xx (219 obs.); Bandwidth 'bw' = 0.181
x y
Min. :0.9569 Min. :0.000114
1st Qu.:2.1034 1st Qu.:0.021708
Median :3.2500 Median :0.101329
Mean :3.2500 Mean :0.217826
3rd Qu.:4.3966 3rd Qu.:0.402436
Max. :5.5431 Max. :0.706375
$Conscientiousness$iPhone
Call:
density.default(x = xx)
Data: xx (310 obs.); Bandwidth 'bw' = 0.1706
x y
Min. :1.188 Min. :0.0001964
1st Qu.:2.244 1st Qu.:0.0359617
Median :3.300 Median :0.1567514
Mean :3.300 Mean :0.2365316
3rd Qu.:4.356 3rd Qu.:0.4422605
Max. :5.412 Max. :0.6719030
$Openness
$Openness$Android
Call:
density.default(x = xx)
Data: xx (219 obs.); Bandwidth 'bw' = 0.1905
x y
Min. :1.029 Min. :0.0001105
1st Qu.:2.139 1st Qu.:0.0234947
Median :3.250 Median :0.1359343
Mean :3.250 Mean :0.2248574
3rd Qu.:4.361 3rd Qu.:0.4378584
Max. :5.471 Max. :0.6162173
$Openness$iPhone
Call:
density.default(x = xx)
Data: xx (310 obs.); Bandwidth 'bw' = 0.1779
x y
Min. :1.166 Min. :0.0000961
1st Qu.:2.233 1st Qu.:0.0252414
Median :3.300 Median :0.1804943
Mean :3.300 Mean :0.2341143
3rd Qu.:4.367 3rd Qu.:0.4645641
Max. :5.434 Max. :0.5693733
$Avoidance.Similarity
$Avoidance.Similarity$Android
Call:
density.default(x = xx)
Data: xx (219 obs.); Bandwidth 'bw' = 0.2286
x y
Min. :0.3142 Min. :0.00027
1st Qu.:1.6071 1st Qu.:0.03275
Median :2.9000 Median :0.15129
Mean :2.9000 Mean :0.19315
3rd Qu.:4.1929 3rd Qu.:0.37572
Max. :5.4858 Max. :0.46712
$Avoidance.Similarity$iPhone
Call:
density.default(x = xx)
Data: xx (310 obs.); Bandwidth 'bw' = 0.1706
x y
Min. :0.4882 Min. :0.0004298
1st Qu.:1.7441 1st Qu.:0.0316214
Median :3.0000 Median :0.1668941
Mean :3.0000 Mean :0.1988441
3rd Qu.:4.2559 3rd Qu.:0.2669040
Max. :5.5118 Max. :0.8710047
$Phone.as.status.object
$Phone.as.status.object$Android
Call:
density.default(x = xx)
Data: xx (219 obs.); Bandwidth 'bw' = 0.1657
x y
Min. :0.5028 Min. :0.0001238
1st Qu.:1.4514 1st Qu.:0.0213441
Median :2.4000 Median :0.1708817
Mean :2.4000 Mean :0.2632744
3rd Qu.:3.3486 3rd Qu.:0.4800005
Max. :4.2972 Max. :0.6997651
$Phone.as.status.object$iPhone
Call:
density.default(x = xx)
Data: xx (310 obs.); Bandwidth 'bw' = 0.1706
x y
Min. :0.4882 Min. :0.0001104
1st Qu.:1.4441 1st Qu.:0.0385170
Median :2.4000 Median :0.2001546
Mean :2.4000 Mean :0.2612763
3rd Qu.:3.3559 3rd Qu.:0.5033981
Max. :4.3118 Max. :0.6255264
$Social.Economic.Status
$Social.Economic.Status$Android
Call:
density.default(x = xx)
Data: xx (219 obs.); Bandwidth 'bw' = 0.4572
x y
Min. :-0.3715 Min. :4.495e-05
1st Qu.: 2.5642 1st Qu.:7.736e-03
Median : 5.5000 Median :5.593e-02
Mean : 5.5000 Mean :8.507e-02
3rd Qu.: 8.4358 3rd Qu.:1.503e-01
Max. :11.3715 Max. :2.785e-01
$Social.Economic.Status$iPhone
Call:
density.default(x = xx)
Data: xx (310 obs.); Bandwidth 'bw' = 0.4245
x y
Min. :-0.2735 Min. :6.835e-05
1st Qu.: 2.6133 1st Qu.:7.145e-03
Median : 5.5000 Median :3.632e-02
Mean : 5.5000 Mean :8.652e-02
3rd Qu.: 8.3867 3rd Qu.:1.511e-01
Max. :11.2735 Max. :2.871e-01
$Time.owned.current.phone
$Time.owned.current.phone$Android
Call:
density.default(x = xx)
Data: xx (219 obs.); Bandwidth 'bw' = 2.972
x y
Min. : -8.915 Min. :0.000e+00
1st Qu.: 20.293 1st Qu.:8.400e-07
Median : 49.500 Median :5.839e-04
Mean : 49.500 Mean :8.550e-03
3rd Qu.: 78.707 3rd Qu.:1.160e-02
Max. :107.915 Max. :4.714e-02
$Time.owned.current.phone$iPhone
Call:
density.default(x = xx)
Data: xx (310 obs.); Bandwidth 'bw' = 2.5
x y
Min. :-7.50 Min. :5.810e-06
1st Qu.: 8.25 1st Qu.:9.996e-04
Median :24.00 Median :5.201e-03
Mean :24.00 Mean :1.586e-02
3rd Qu.:39.75 3rd Qu.:3.214e-02
Max. :55.50 Max. :5.031e-02 plot(nb2)
Now, instead of just the mean and standard deviation, we have estimated quantiles of the distribution. This might help us when we have skewed distributions, but we need a lot of observations in order to get reliable estimates. 200-300 observations in each group might not be enough to do well. If we look at the predictions:
pred2 <- predict(nb2)
confusion(dat$Smartphone, pred2$class)Overall accuracy = 0.699
Confusion matrix
Predicted (cv)
Actual Android iPhone
Android 0.557 0.443
iPhone 0.200 0.800Here, we do a bit better, and in contrast to some of the other classification methods, we have a 55% chance of getting Android users right (the lda model sometimes had less than a 50% chance of getting them correct.)
Again, it would be good to implement a cross-validation here.
Example: Predicting Dengue Fever
The NaiveBayes function is sensitive to missing data and zero-variances. If you have a variable with no variance, any new value will have likelihood of 0, and we have a chance of getting a likelihood ratio that is infinite. Similarly, an NA in the values can cause trouble, dependent on how the model handles it. It might be useful to impute NA data, or add small amounts of noise to the training set to smooth out the values.
library(e1071)
data(dengue)
summary(dengue) humid humid90 temp temp90
Min. : 0.6714 Min. : 1.066 Min. :-18.68 Min. :-10.07
1st Qu.:10.0088 1st Qu.:10.307 1st Qu.: 11.10 1st Qu.: 12.76
Median :16.1433 Median :16.870 Median : 20.99 Median : 22.03
Mean :16.7013 Mean :17.244 Mean : 18.41 Mean : 19.41
3rd Qu.:23.6184 3rd Qu.:24.131 3rd Qu.: 25.47 3rd Qu.: 25.98
h10pix h10pix90 trees trees90
Min. : 4.317 Min. : 5.848 Min. : 0.0 Min. : 0.00
1st Qu.:14.584 1st Qu.:14.918 1st Qu.: 1.0 1st Qu.: 6.00
Median :23.115 Median :24.130 Median :15.0 Median :30.60
Mean :21.199 Mean :21.557 Mean :22.7 Mean :35.21
3rd Qu.:28.509 3rd Qu.:28.627 3rd Qu.:37.0 3rd Qu.:63.62
NoYes Xmin Xmax Ymin
Min. :0.0000 Min. :-179.50 Min. :-172.00 Min. :-54.50
1st Qu.:0.0000 1st Qu.: -12.00 1st Qu.: -10.00 1st Qu.: 6.00
Median :0.0000 Median : 16.00 Median : 17.75 Median : 18.00
Mean :0.4155 Mean : 13.31 Mean : 15.63 Mean : 19.78
3rd Qu.:1.0000 3rd Qu.: 42.62 3rd Qu.: 44.50 3rd Qu.: 39.00
Ymax
Min. :-55.50
1st Qu.: 5.00
Median : 17.00
Mean : 18.16
3rd Qu.: 37.00
[ reached getOption("max.print") -- omitted 2 rows ]dengue$NoYes <- as.factor(dengue$NoYes)Notice that there areabout a dozen or so values with NA data.
# This doesn't work nb.dengue <- NaiveBayes(NoYes~.,data=dengue)
nb.dengue <- NaiveBayes(NoYes ~ ., data = dengue, na.action = "na.omit") ##this works
# This one works with the klaR NaiveBayes:
nb.dengue2 <- NaiveBayes(NoYes ~ h10pix + Xmin + Ymin, data = dengue)
# this one works--from e1071 package
nb.dengue3 <- naiveBayes(NoYes ~ ., data = dengue)
# when we remove the na, we need to remove it from the ground truth too:
confusion(dengue$NoYes[!is.na(rowMeans(dengue[, -9]))], predict(nb.dengue)$class)Overall accuracy = 0.882
Confusion matrix
Predicted (cv)
Actual 0 1
0 0.845 0.155
1 0.066 0.934confusion(dengue$NoYes, predict(nb.dengue2)$class)Overall accuracy = 0.883
Confusion matrix
Predicted (cv)
Actual 0 1
0 0.843 0.157
1 0.059 0.941confusion(dengue$NoYes, predict(nb.dengue3, newdata = dengue))Overall accuracy = 0.881
Confusion matrix
Predicted (cv)
Actual 0 1
0 0.844 0.156
1 0.066 0.934Naive Bayes with the mnist (handwriting) set
##this code will dowload a create a 500-letter two-class training set from
##mnist via tensorflow
library(tensorflow)
datasets <- tf$contrib$learn$datasets
mnist <- datasets$mnist$read_data_sets("MNIST-data", one_hot = TRUE)
##extract just two labels and sample images
mnist.1 <- mnist$train$labels[,1]
mnist.2 <- mnist$train$labels[,2]
mnist.img1 <- mnist$train$images[mnist.1==1,]
mnist.img2 <- mnist$train$images[mnist.2==1,]
##plot prototypes
par(mfrow=c(1,2))
image((matrix(colMeans(mnist.img1),28,28)))
image((matrix(colMeans(mnist.img2),28,28)))
##these are too many. Sample 250 from each
traintest1 <- sample(1:nrow(mnist.img1),size=500)
traintest2 <- sample(1:nrow(mnist.img2),size=500)
train <- rbind(mnist.img1[traintest1[1:250],],
mnist.img2[traintest2[1:250],])
test <- rbind(mnist.img1[traintest1[251:500],],
mnist.img2[traintest2[251:500],])
train<-as.data.frame(train)
test<- as.data.frame(test)
train$labels <- rep(0:1,each=250)
write.csv(train,"trainmnist.csv",row.names=F)
write.csv(test,"testmnist.csv",row.names=F)library(klaR)
library(DAAG)
train <- read.csv("trainmnist.csv")
test <- read.csv("testmnist.csv")
## smooth out the training set a bit by adding some noise, so no pixel has a sd of
## 0.
for (i in 1:ncol(train)) {
train[, i] <- rnorm(500, as.numeric(train[, i]), 0.1)
}
par(mfrow = c(2, 4))
image(matrix(unlist(train[1, ]), nrow = 28))
image(matrix(unlist(train[2, ]), nrow = 28))
image(matrix(unlist(train[3, ]), nrow = 28))
image(matrix(colMeans(train[1:250, ]), nrow = 28))
image(matrix(unlist(train[251, ]), nrow = 28))
image(matrix(unlist(train[252, ]), nrow = 28))
image(matrix(unlist(train[253, ]), nrow = 28))
image(matrix(colMeans(train[251:500, ]), nrow = 28))
Build Naive Bayes model
train$labels <- as.factor(rep(0:1, each = 250))
nb3 <- NaiveBayes(labels ~ ., usekernel = T, data = train)
p3a <- predict(nb3) ##this takes a while
confusion(train$labels, p3a$class) ##almost perfectOverall accuracy = 0.998
Confusion matrix
Predicted (cv)
Actual 0 1
0 0.996 0.004
1 0.000 1.000p3b <- predict(nb3, test) ##equally good
confusion(rep(0:1, each = 250), p3b$class)Overall accuracy = 0.998
Confusion matrix
Predicted (cv)
Actual [,1] [,2]
[1,] 1.000 0.000
[2,] 0.004 0.996What if we add noise to the test?
for (i in 1:ncol(test)) {
test[, i] <- rnorm(500, as.numeric(test[, i]), 0.02)
}
par(mfrow = c(2, 4))
image(matrix(unlist(test[1, ]), nrow = 28))
image(matrix(unlist(test[2, ]), nrow = 28))
image(matrix(unlist(test[3, ]), nrow = 28))
image(matrix(colMeans(test[1:250, ]), nrow = 28))
image(matrix(unlist(test[251, ]), nrow = 28))
image(matrix(unlist(test[252, ]), nrow = 28))
image(matrix(unlist(test[253, ]), nrow = 28))
image(matrix(colMeans(test[251:500, ]), nrow = 28))
p3c <- predict(nb3, test) ##This one is still pretty good
confusion(rep(0:1, each = 250), p3c$class)Overall accuracy = 0.998
Confusion matrix
Predicted (cv)
Actual [,1] [,2]
[1,] 1.000 0.000
[2,] 0.004 0.996##add larger amount of noise
for (i in 1:ncol(test)) {
test[, i] <- rnorm(500, as.numeric(test[, i]), 0.35)
}
par(mfrow = c(2, 4))
image(matrix(unlist(test[1, ]), nrow = 28))
image(matrix(unlist(test[2, ]), nrow = 28))
image(matrix(unlist(test[3, ]), nrow = 28))
image(matrix(colMeans(test[1:250, ]), nrow = 28))
image(matrix(unlist(test[251, ]), nrow = 28))
image(matrix(unlist(test[252, ]), nrow = 28))
image(matrix(unlist(test[253, ]), nrow = 28))
image(matrix(colMeans(test[251:500, ]), nrow = 28))
p3c <- predict(nb3, test) ##This one is pretty bad.
confusion(rep(0:1, each = 250), p3c$class)Overall accuracy = 0.508
Confusion matrix
Predicted (cv)
Actual [,1] [,2]
[1,] 1.000 0.000
[2,] 0.984 0.016