A bit greater than a yr in the past, in his stunning visitor submit, Nick Strayer confirmed classify a set of on a regular basis actions utilizing smartphone-recorded gyroscope and accelerometer knowledge. Accuracy was excellent, however Nick went on to examine classification outcomes extra carefully. Have been there actions extra vulnerable to misclassification than others? And the way about these inaccurate outcomes: Did the community report them with equal, or much less confidence than people who had been right?
Technically, once we communicate of confidence in that method, we’re referring to the rating obtained for the “profitable” class after softmax activation. If that profitable rating is 0.9, we would say “the community is certain that’s a gentoo penguin”; if it’s 0.2, we’d as an alternative conclude “to the community, neither possibility appeared becoming, however cheetah regarded greatest.”
This use of “confidence” is convincing, but it surely has nothing to do with confidence – or credibility, or prediction, what have you ever – intervals. What we’d actually like to have the ability to do is put distributions over the community’s weights and make it Bayesian. Utilizing tfprobability’s variational Keras-compatible layers, that is one thing we really can do.
Including uncertainty estimates to Keras fashions with tfprobability reveals use a variational dense layer to acquire estimates of epistemic uncertainty. On this submit, we modify the convnet utilized in Nick’s submit to be variational all through. Earlier than we begin, let’s rapidly summarize the duty.
The duty
To create the Smartphone-Based mostly Recognition of Human Actions and Postural Transitions Information Set (Reyes-Ortiz et al. 2016), the researchers had topics stroll, sit, stand, and transition from a type of actions to a different. In the meantime, two kinds of smartphone sensors had been used to report movement knowledge: Accelerometers measure linear acceleration in three dimensions, whereas gyroscopes are used to trace angular velocity across the coordinate axes. Listed below are the respective uncooked sensor knowledge for six kinds of actions from Nick’s authentic submit:
Identical to Nick, we’re going to zoom in on these six kinds of exercise, and attempt to infer them from the sensor knowledge. Some knowledge wrangling is required to get the dataset right into a type we are able to work with; right here we’ll construct on Nick’s submit, and successfully begin from the information properly pre-processed and cut up up into coaching and check units:
Observations: 289
Variables: 6
$ experiment 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 17, 18, 19, 2…
$ userId 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 9, 10, 10, 11…
$ exercise 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7…
$ knowledge [, , STAND_TO_SIT, STAND_TO_SIT, STAND_TO_SIT, STAND_TO_S…
$ observationId 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 17, 18, 19, 2…
Observations: 69
Variables: 6
$ experiment 11, 12, 15, 16, 32, 33, 42, 43, 52, 53, 56, 57, 11, …
$ userId 6, 6, 8, 8, 16, 16, 21, 21, 26, 26, 28, 28, 6, 6, 8,…
$ activity 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8…
$ data [, , STAND_TO_SIT, STAND_TO_SIT, STAND_TO_SIT, STAND_TO_S…
$ observationId 11, 12, 15, 16, 31, 32, 41, 42, 51, 52, 55, 56, 71, …
The code required to arrive at this stage (copied from Nick’s post) may be found in the appendix at the bottom of this page.
Training pipeline
The dataset in question is small enough to fit in memory – but yours might not be, so it can’t hurt to see some streaming in action. Besides, it’s probably safe to say that with TensorFlow 2.0, tfdatasets pipelines are the way to feed data to a model.
Once the code listed in the appendix has run, the sensor data is to be found in trainData$data, a list column containing data.frames where each row corresponds to a point in time and each column holds one of the measurements. However, not all time series (recordings) are of the same length; we thus follow the original post to pad all series to length pad_size (= 338). The expected shape of training batches will then be (batch_size, pad_size, 6).
We initially create our training dataset:
train_x <- train_data$data %>%
map(as.matrix) %>%
pad_sequences(maxlen = pad_size, dtype = "float32") %>%
tensor_slices_dataset()
train_y <- train_data$activity %>%
one_hot_classes() %>%
tensor_slices_dataset()
train_dataset <- zip_datasets(train_x, train_y)
train_datasetThen shuffle and batch it:
n_train <- nrow(train_data)
# the highest possible batch size for this dataset
# chosen because it yielded the best performance
# alternatively, experiment with e.g. different learning rates, ...
batch_size <- n_train
train_dataset <- train_dataset %>%
dataset_shuffle(n_train) %>%
dataset_batch(batch_size)
train_datasetSame for the test data.
test_x <- test_data$data %>%
map(as.matrix) %>%
pad_sequences(maxlen = pad_size, dtype = "float32") %>%
tensor_slices_dataset()
test_y <- test_data$activity %>%
one_hot_classes() %>%
tensor_slices_dataset()
n_test <- nrow(test_data)
test_dataset <- zip_datasets(test_x, test_y) %>%
dataset_batch(n_test)Using tfdatasets does not mean we cannot run a quick sanity check on our data:
first <- test_dataset %>%
reticulate::as_iterator() %>%
# get first batch (= whole test set, in our case)
reticulate::iter_next() %>%
# predictors only
.[[1]] %>%
# first merchandise in batch
.[1,,]
firsttf.Tensor(
[[ 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. ]
...
[ 1.00416672 0.2375 0.12916666 -0.40225476 -0.20463985 -0.14782938]
[ 1.04166663 0.26944447 0.12777779 -0.26755899 -0.02779437 -0.1441642 ]
[ 1.0250001 0.27083334 0.15277778 -0.19639318 0.35094208 -0.16249016]],
form=(338, 6), dtype=float64)Now let’s construct the community.
A variational convnet
We construct on the easy convolutional structure from Nick’s submit, simply making minor modifications to kernel sizes and numbers of filters. We additionally throw out all dropout layers; no extra regularization is required on prime of the priors utilized to the weights.
Word the next concerning the “Bayesified” community.
-
Every layer is variational in nature, the convolutional ones (layer_conv_1d_flipout) in addition to the dense layers (layer_dense_flipout).
-
With variational layers, we are able to specify the prior weight distribution in addition to the type of the posterior; right here the defaults are used, leading to a regular regular prior and a default mean-field posterior.
-
Likewise, the person might affect the divergence operate used to evaluate the mismatch between prior and posterior; on this case, we really take some motion: We scale the (default) KL divergence by the variety of samples within the coaching set.
-
One last item to notice is the output layer. It’s a distribution layer, that’s, a layer wrapping a distribution – the place wrapping means: Coaching the community is enterprise as typical, however predictions are distributions, one for every knowledge level.
library(tfprobability)
num_classes <- 6
# scale the KL divergence by variety of coaching examples
n <- n_train %>% tf$forged(tf$float32)
kl_div <- operate(q, p, unused)
tfd_kl_divergence(q, p) / n
mannequin <- keras_model_sequential()
mannequin %>%
layer_conv_1d_flipout(
filters = 12,
kernel_size = 3,
activation = "relu",
kernel_divergence_fn = kl_div
) %>%
layer_conv_1d_flipout(
filters = 24,
kernel_size = 5,
activation = "relu",
kernel_divergence_fn = kl_div
) %>%
layer_conv_1d_flipout(
filters = 48,
kernel_size = 7,
activation = "relu",
kernel_divergence_fn = kl_div
) %>%
layer_global_average_pooling_1d() %>%
layer_dense_flipout(
models = 48,
activation = "relu",
kernel_divergence_fn = kl_div
) %>%
layer_dense_flipout(
num_classes,
kernel_divergence_fn = kl_div,
title = "dense_output"
) %>%
layer_one_hot_categorical(event_size = num_classes)We inform the community to reduce the unfavorable log chance.
nll <- operate(y, mannequin) - (mannequin %>% tfd_log_prob(y))This can turn into a part of the loss. The way in which we arrange this instance, this isn’t its most substantial half although. Right here, what dominates the loss is the sum of the KL divergences, added (mechanically) to mannequin$losses.
In a setup like this, it’s attention-grabbing to observe each components of the loss individually. We are able to do that by way of two metrics:
# the KL a part of the loss
kl_part <- operate(y_true, y_pred) {
kl <- tf$reduce_sum(mannequin$losses)
kl
}
# the NLL half
nll_part <- operate(y_true, y_pred) {
cat_dist <- tfd_one_hot_categorical(logits = y_pred)
nll <- - (cat_dist %>% tfd_log_prob(y_true) %>% tf$reduce_mean())
nll
}We prepare considerably longer than Nick did within the authentic submit, permitting for early stopping although.
mannequin %>% compile(
optimizer = "rmsprop",
loss = nll,
metrics = c("accuracy",
custom_metric("kl_part", kl_part),
custom_metric("nll_part", nll_part)),
experimental_run_tf_function = FALSE
)
train_history <- mannequin %>% match(
train_dataset,
epochs = 1000,
validation_data = test_dataset,
callbacks = listing(
callback_early_stopping(endurance = 10)
)
)Whereas the general loss declines linearly (and possibly would for a lot of extra epochs), this isn’t the case for classification accuracy or the NLL a part of the loss:
Closing accuracy just isn’t as excessive as within the non-variational setup, although nonetheless not unhealthy for a six-class downside. We see that with none extra regularization, there’s little or no overfitting to the coaching knowledge.
Now how will we receive predictions from this mannequin?
Probabilistic predictions
Although we gained’t go into this right here, it’s good to know that we entry extra than simply the output distributions; by their kernel_posterior attribute, we are able to entry the hidden layers’ posterior weight distributions as effectively.
Given the small dimension of the check set, we compute all predictions without delay. The predictions at the moment are categorical distributions, one for every pattern within the batch:
test_data_all <- dataset_collect(test_dataset) %>% { .[[1]][[1]]}
one_shot_preds <- mannequin(test_data_all)
one_shot_predstfp.distributions.OneHotCategorical(
"sequential_one_hot_categorical_OneHotCategorical_OneHotCategorical",
batch_shape=[69], event_shape=[6], dtype=float32)We prefixed these predictions with one_shot to point their noisy nature: These are predictions obtained on a single cross by the community, all layer weights being sampled from their respective posteriors.
From the anticipated distributions, we calculate imply and normal deviation per (check) pattern.
The usual deviations thus obtained could possibly be stated to replicate the general predictive uncertainty. We are able to estimate one other sort of uncertainty, known as epistemic, by making a lot of passes by the community after which, calculating – once more, per check pattern – the usual deviations of the anticipated means.
Placing all of it collectively, we’ve
# A tibble: 414 x 6
obs class imply sd mc_sd label
1 1 V1 0.945 0.227 0.0743 STAND_TO_SIT
2 1 V2 0.0534 0.225 0.0675 SIT_TO_STAND
3 1 V3 0.00114 0.0338 0.0346 SIT_TO_LIE
4 1 V4 0.00000238 0.00154 0.000336 LIE_TO_SIT
5 1 V5 0.0000132 0.00363 0.00164 STAND_TO_LIE
6 1 V6 0.0000305 0.00553 0.00398 LIE_TO_STAND
7 2 V1 0.993 0.0813 0.149 STAND_TO_SIT
8 2 V2 0.00153 0.0390 0.102 SIT_TO_STAND
9 2 V3 0.00476 0.0688 0.108 SIT_TO_LIE
10 2 V4 0.00000172 0.00131 0.000613 LIE_TO_SIT
# … with 404 extra rows Evaluating predictions to the bottom reality:
# A tibble: 69 x 7
obs maxprob maxprob_sd maxprob_mc_sd predicted reality right
1 1 0.945 0.227 0.0743 STAND_TO_SIT STAND_TO_SIT TRUE
2 2 0.993 0.0813 0.149 STAND_TO_SIT STAND_TO_SIT TRUE
3 3 0.733 0.443 0.131 STAND_TO_SIT STAND_TO_SIT TRUE
4 4 0.796 0.403 0.138 STAND_TO_SIT STAND_TO_SIT TRUE
5 5 0.843 0.364 0.358 SIT_TO_STAND STAND_TO_SIT FALSE
6 6 0.816 0.387 0.176 SIT_TO_STAND STAND_TO_SIT FALSE
7 7 0.600 0.490 0.370 STAND_TO_SIT STAND_TO_SIT TRUE
8 8 0.941 0.236 0.0851 STAND_TO_SIT STAND_TO_SIT TRUE
9 9 0.853 0.355 0.274 SIT_TO_STAND STAND_TO_SIT FALSE
10 10 0.961 0.195 0.195 STAND_TO_SIT STAND_TO_SIT TRUE
11 11 0.918 0.275 0.168 STAND_TO_SIT STAND_TO_SIT TRUE
12 12 0.957 0.203 0.150 STAND_TO_SIT STAND_TO_SIT TRUE
13 13 0.987 0.114 0.188 SIT_TO_STAND SIT_TO_STAND TRUE
14 14 0.974 0.160 0.248 SIT_TO_STAND SIT_TO_STAND TRUE
15 15 0.996 0.0657 0.0534 SIT_TO_STAND SIT_TO_STAND TRUE
16 16 0.886 0.318 0.0868 SIT_TO_STAND SIT_TO_STAND TRUE
17 17 0.773 0.419 0.173 SIT_TO_STAND SIT_TO_STAND TRUE
18 18 0.998 0.0444 0.222 SIT_TO_STAND SIT_TO_STAND TRUE
19 19 0.885 0.319 0.161 SIT_TO_STAND SIT_TO_STAND TRUE
20 20 0.930 0.255 0.271 SIT_TO_STAND SIT_TO_STAND TRUE
# … with 49 extra rows Are normal deviations larger for misclassifications?
# A tibble: 2 x 5
right rely avg_mean avg_sd avg_mc_sd
1 FALSE 19 0.775 0.380 0.237
2 TRUE 50 0.879 0.264 0.183 They’re; although maybe to not the extent we would want.
With simply six lessons, we are able to additionally examine normal deviations on the person prediction-target pairings degree.
# A tibble: 14 x 7
# Teams: reality [6]
reality predicted cnt avg_mean avg_sd avg_mc_sd right
1 SIT_TO_STAND SIT_TO_STAND 12 0.935 0.205 0.184 TRUE
2 STAND_TO_SIT STAND_TO_SIT 9 0.871 0.284 0.162 TRUE
3 LIE_TO_SIT LIE_TO_SIT 9 0.765 0.377 0.216 TRUE
4 SIT_TO_LIE SIT_TO_LIE 8 0.908 0.254 0.187 TRUE
5 STAND_TO_LIE STAND_TO_LIE 7 0.956 0.144 0.132 TRUE
6 LIE_TO_STAND LIE_TO_STAND 5 0.809 0.353 0.227 TRUE
7 SIT_TO_LIE STAND_TO_LIE 4 0.685 0.436 0.233 FALSE
8 LIE_TO_STAND SIT_TO_STAND 4 0.909 0.271 0.282 FALSE
9 STAND_TO_LIE SIT_TO_LIE 3 0.852 0.337 0.238 FALSE
10 STAND_TO_SIT SIT_TO_STAND 3 0.837 0.368 0.269 FALSE
11 LIE_TO_STAND LIE_TO_SIT 2 0.689 0.454 0.233 FALSE
12 LIE_TO_SIT STAND_TO_SIT 1 0.548 0.498 0.0805 FALSE
13 SIT_TO_STAND LIE_TO_STAND 1 0.530 0.499 0.134 FALSE
14 LIE_TO_SIT LIE_TO_STAND 1 0.824 0.381 0.231 FALSE Once more, we see larger normal deviations for unsuitable predictions, however to not a excessive diploma.
Conclusion
We’ve proven construct, prepare, and acquire predictions from a totally variational convnet. Evidently, there’s room for experimentation: Various layer implementations exist; a unique prior could possibly be specified; the divergence could possibly be calculated otherwise; and the standard neural community hyperparameter tuning choices apply.
Then, there’s the query of penalties (or: resolution making). What will occur in high-uncertainty circumstances, what even is a high-uncertainty case? Naturally, questions like these are out-of-scope for this submit, but of important significance in real-world functions.
Thanks for studying!
Appendix
To be executed earlier than operating this submit’s code. Copied from Classifying bodily exercise from smartphone knowledge.
library(keras)
library(tidyverse)
activity_labels <- learn.desk("knowledge/activity_labels.txt",
col.names = c("quantity", "label"))
one_hot_to_label <- activity_labels %>%
mutate(quantity = quantity - 7) %>%
filter(quantity >= 0) %>%
mutate(class = paste0("V",quantity + 1)) %>%
choose(-quantity)
labels <- learn.desk(
"knowledge/RawData/labels.txt",
col.names = c("experiment", "userId", "exercise", "startPos", "endPos")
)
dataFiles <- listing.information("knowledge/RawData")
dataFiles %>% head()
fileInfo <- data_frame(
filePath = dataFiles
) %>%
filter(filePath != "labels.txt") %>%
separate(filePath, sep = '_',
into = c("sort", "experiment", "userId"),
take away = FALSE) %>%
mutate(
experiment = str_remove(experiment, "exp"),
userId = str_remove_all(userId, "person|.txt")
) %>%
unfold(sort, filePath)
# Learn contents of single file to a dataframe with accelerometer and gyro knowledge.
readInData <- operate(experiment, userId){
genFilePath = operate(sort) {
paste0("knowledge/RawData/", sort, "_exp",experiment, "_user", userId, ".txt")
}
bind_cols(
learn.desk(genFilePath("acc"), col.names = c("a_x", "a_y", "a_z")),
learn.desk(genFilePath("gyro"), col.names = c("g_x", "g_y", "g_z"))
)
}
# Operate to learn a given file and get the observations contained alongside
# with their lessons.
loadFileData <- operate(curExperiment, curUserId) {
# load sensor knowledge from file into dataframe
allData <- readInData(curExperiment, curUserId)
extractObservation <- operate(startPos, endPos){
allData[startPos:endPos,]
}
# get remark places on this file from labels dataframe
dataLabels <- labels %>%
filter(userId == as.integer(curUserId),
experiment == as.integer(curExperiment))
# extract observations as dataframes and save as a column in dataframe.
dataLabels %>%
mutate(
knowledge = map2(startPos, endPos, extractObservation)
) %>%
choose(-startPos, -endPos)
}
# scan by all experiment and userId combos and collect knowledge right into a dataframe.
allObservations <- map2_df(fileInfo$experiment, fileInfo$userId, loadFileData) %>%
right_join(activityLabels, by = c("exercise" = "quantity")) %>%
rename(activityName = label)
write_rds(allObservations, "allObservations.rds")
allObservations <- readRDS("allObservations.rds")
desiredActivities <- c(
"STAND_TO_SIT", "SIT_TO_STAND", "SIT_TO_LIE",
"LIE_TO_SIT", "STAND_TO_LIE", "LIE_TO_STAND"
)
filteredObservations <- allObservations %>%
filter(activityName %in% desiredActivities) %>%
mutate(observationId = 1:n())
# get all customers
userIds <- allObservations$userId %>% distinctive()
# randomly select 24 (80% of 30 people) for coaching
set.seed(42) # seed for reproducibility
trainIds <- pattern(userIds, dimension = 24)
# set the remainder of the customers to the testing set
testIds <- setdiff(userIds,trainIds)
# filter knowledge.
# observe S.Okay.: renamed to train_data for consistency with
# variable naming used on this submit
train_data <- filteredObservations %>%
filter(userId %in% trainIds)
# observe S.Okay.: renamed to test_data for consistency with
# variable naming used on this submit
test_data <- filteredObservations %>%
filter(userId %in% testIds)
# observe S.Okay.: renamed to pad_size for consistency with
# variable naming used on this submit
pad_size <- trainData$knowledge %>%
map_int(nrow) %>%
quantile(p = 0.98) %>%
ceiling()
# observe S.Okay.: renamed to one_hot_classes for consistency with
# variable naming used on this submit
one_hot_classes <- . %>%
{. - 7} %>% # deliver integers right down to 0-6 from 7-12
to_categorical() # One-hot encode
