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# Artificial Intelligence Engineer Nanodegree - Probabilistic Models
## Project: Sign Language Recognition System
- [Introduction](#intro)
- [Part 1 Feature Selection](#part1_tutorial)
- [Tutorial](#part1_tutorial)
- [Features Submission](#part1_submission)
- [Features Unittest](#part1_test)
- [Part 2 Train the models](#part2_tutorial)
- [Tutorial](#part2_tutorial)
- [Model Selection Score Submission](#part2_submission)
- [Model Score Unittest](#part2_test)
- [Part 3 Build a Recognizer](#part3_tutorial)
- [Tutorial](#part3_tutorial)
- [Recognizer Submission](#part3_submission)
- [Recognizer Unittest](#part3_test)
- [Part 4 (OPTIONAL) Improve the WER with Language Models](#part4_info)
The overall goal of this project is to build a word recognizer for American Sign Language video sequences, demonstrating the power of probabalistic models. In particular, this project employs hidden Markov models (HMM's) to analyze a series of measurements taken from videos of American Sign Language (ASL) collected for research (see the RWTH-BOSTON-104 Database). In this video, the right-hand x and y locations are plotted as the speaker signs the sentence.
The raw data, train, and test sets are pre-defined. You will derive a variety of feature sets (explored in Part 1), as well as implement three different model selection criterion to determine the optimal number of hidden states for each word model (explored in Part 2). Finally, in Part 3 you will implement the recognizer and compare the effects the different combinations of feature sets and model selection criteria.
At the end of each Part, complete the submission cells with implementations, answer all questions, and pass the unit tests. Then submit the completed notebook for review!
A data handler designed for this database is provided in the student codebase as the AslDb class in the asl_data module. This handler creates the initial pandas dataframe from the corpus of data included in the data directory as well as dictionaries suitable for extracting data in a format friendly to the hmmlearn library. We'll use those to create models in Part 2.
To start, let's set up the initial database and select an example set of features for the training set. At the end of Part 1, you will create additional feature sets for experimentation.
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import math
import numpy as np
import pandas as pd
from asl_data import AslDb
asl = AslDb() # initializes the database
asl.df.head() # displays the first five rows of the asl database, indexed by video and frame
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asl.df.ix[98,1] # look at the data available for an individual frame
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The frame represented by video 98, frame 1 is shown here:
The objective of feature selection when training a model is to choose the most relevant variables while keeping the model as simple as possible, thus reducing training time. We can use the raw features already provided or derive our own and add columns to the pandas dataframe asl.df for selection. As an example, in the next cell a feature named 'grnd-ry' is added. This feature is the difference between the right-hand y value and the nose y value, which serves as the "ground" right y value.
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asl.df['grnd-ry'] = asl.df['right-y'] - asl.df['nose-y']
asl.df.head() # the new feature 'grnd-ry' is now in the frames dictionary
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from asl_utils import test_features_tryit
# TODO add df columns for 'grnd-rx', 'grnd-ly', 'grnd-lx' representing differences between hand and nose locations
asl.df['grnd-lx'] = asl.df['left-x'] - asl.df['nose-x']
asl.df['grnd-ly'] = asl.df['left-y'] - asl.df['nose-y']
asl.df['grnd-rx'] = asl.df['right-x'] - asl.df['nose-x']
asl.df['grnd-ry'] = asl.df['right-y'] - asl.df['nose-y']
asl.df.head()
# test the code
test_features_tryit(asl)
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# collect the features into a list
features_ground = ['grnd-rx','grnd-ry','grnd-lx','grnd-ly']
#show a single set of features for a given (video, frame) tuple
[asl.df.ix[98,1][v] for v in features_ground]
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Now that we have a feature list defined, we can pass that list to the build_training method to collect the features for all the words in the training set. Each word in the training set has multiple examples from various videos. Below we can see the unique words that have been loaded into the training set:
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training = asl.build_training(features_ground)
print("Training words: {}".format(training.words))
The training data in training is an object of class WordsData defined in the asl_data module. in addition to the words list, data can be accessed with the get_all_sequences, get_all_Xlengths, get_word_sequences, and get_word_Xlengths methods. We need the get_word_Xlengths method to train multiple sequences with the hmmlearn library. In the following example, notice that there are two lists; the first is a concatenation of all the sequences(the X portion) and the second is a list of the sequence lengths(the Lengths portion).
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training.get_word_Xlengths('CHOCOLATE')
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So far we have a simple feature set that is enough to get started modeling. However, we might get better results if we manipulate the raw values a bit more, so we will go ahead and set up some other options now for experimentation later. For example, we could normalize each speaker's range of motion with grouped statistics using Pandas stats functions and pandas groupby. Below is an example for finding the means of all speaker subgroups.
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df_means = asl.df.groupby('speaker').mean()
df_means
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To select a mean that matches by speaker, use the pandas map method:
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asl.df['left-x-mean']= asl.df['speaker'].map(df_means['left-x'])
asl.df.head()
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from asl_utils import test_std_tryit
# TODO Create a dataframe named `df_std` with standard deviations grouped by speaker
df_std = asl.df.groupby('speaker').std()
# test the code
test_std_tryit(df_std)
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Implement four feature sets and answer the question that follows.
normalized Cartesian coordinates
polar coordinates
delta difference
custom features
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# TODO add features for normalized by speaker values of left, right, x, y
# Name these 'norm-rx', 'norm-ry', 'norm-lx', and 'norm-ly'
# using Z-score scaling (X-Xmean)/Xstd
# Left mean values
asl.df['mean-lx'] = asl.df['speaker'].map(df_means['left-x'])
asl.df['mean-ly'] = asl.df['speaker'].map(df_means['left-y'])
# Left standard deviation values
asl.df['std-lx'] = asl.df['speaker'].map(df_std['left-x'])
asl.df['std-ly'] = asl.df['speaker'].map(df_std['left-y'])
# Right mean values
asl.df['mean-rx'] = asl.df['speaker'].map(df_means['right-x'])
asl.df['mean-ry'] = asl.df['speaker'].map(df_means['right-y'])
# Right standard deviation values
asl.df['std-rx'] = asl.df['speaker'].map(df_std['right-x'])
asl.df['std-ry'] = asl.df['speaker'].map(df_std['right-y'])
# Calculating normalized values using the standard score equation
asl.df['norm-lx'] = (asl.df['left-x'] - asl.df['mean-lx']) / asl.df['std-lx']
asl.df['norm-ly'] = (asl.df['left-y'] - asl.df['mean-ly']) / asl.df['std-ly']
asl.df['norm-rx'] = (asl.df['right-x'] - asl.df['mean-rx']) / asl.df['std-rx']
asl.df['norm-ry'] = (asl.df['right-y'] - asl.df['mean-ry']) / asl.df['std-ry']
features_norm = ['norm-rx', 'norm-ry', 'norm-lx', 'norm-ly']
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import math
# TODO add features for polar coordinate values where the nose is the origin
# Name these 'polar-rr', 'polar-rtheta', 'polar-lr', and 'polar-ltheta'
# Note that 'polar-rr' and 'polar-rtheta' refer to the radius and angle
# Left polar coordinate values
asl.df['polar-lr'] = ((asl.df['grnd-lx'] ** 2) + (asl.df['grnd-ly'] ** 2)) ** (0.5)
asl.df['polar-ltheta'] = np.arctan2(asl.df['grnd-lx'], asl.df['grnd-ly'])
# Right polar coordinate values
asl.df['polar-rr'] = ((asl.df['grnd-rx'] ** 2) + (asl.df['grnd-ry'] ** 2)) ** (0.5)
asl.df['polar-rtheta'] = np.arctan2(asl.df['grnd-rx'], asl.df['grnd-ry'])
features_polar = ['polar-rr', 'polar-rtheta', 'polar-lr', 'polar-ltheta']
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# TODO add features for left, right, x, y differences by one time step, i.e. the "delta" values discussed in the lecture
# Name these 'delta-rx', 'delta-ry', 'delta-lx', and 'delta-ly'
# Left delta values
asl.df['delta-lx'] = asl.df['left-x'].diff().fillna(0.0)
asl.df['delta-ly'] = asl.df['left-y'].diff().fillna(0.0)
# Right delta values
asl.df['delta-rx'] = asl.df['right-x'].diff().fillna(0.0)
asl.df['delta-ry'] = asl.df['right-y'].diff().fillna(0.0)
features_delta = ['delta-rx', 'delta-ry', 'delta-lx', 'delta-ly']
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# TODO add features of your own design, which may be a combination of the above or something else
# Name these whatever you would like
# Normalized values using the feature scaling equation
# Speaker min and max values
df_min = asl.df.groupby('speaker').min()
df_max= asl.df.groupby('speaker').max()
# Features min and max values
asl.df['left-x-min'] = asl.df['speaker'].map(df_min['left-x'])
asl.df['left-y-min'] = asl.df['speaker'].map(df_min['left-y'])
asl.df['right-x-min'] = asl.df['speaker'].map(df_min['right-x'])
asl.df['right-y-min'] = asl.df['speaker'].map(df_min['right-y'])
asl.df['left-x-max'] = asl.df['speaker'].map(df_max['left-x'])
asl.df['left-y-max'] = asl.df['speaker'].map(df_max['left-y'])
asl.df['right-x-max'] = asl.df['speaker'].map(df_max['right-x'])
asl.df['right-y-max'] = asl.df['speaker'].map(df_max['right-y'])
# Feature scaling using the rescaling method: x' = x - min(x) / max(x) - min(x)
asl.df['resc-lx'] = (asl.df['left-x'] - asl.df['left-x-min']) / (asl.df['left-x-max'] - asl.df['left-x-min'])
asl.df['resc-ly'] = (asl.df['left-y'] - asl.df['left-y-min']) / (asl.df['left-y-max'] - asl.df['left-y-min'])
asl.df['resc-rx'] = (asl.df['right-x'] - asl.df['right-x-min']) / (asl.df['right-x-max'] - asl.df['right-x-min'])
asl.df['resc-ry'] = (asl.df['right-y'] - asl.df['right-y-min']) / (asl.df['right-y-max'] - asl.df['right-y-min'])
# TODO define a list named 'features_custom' for building the training set
features_custom = ['resc-lx', 'resc-ly', 'resc-rx', 'resc-ry']
Question 1: What custom features did you choose for the features_custom set and why?
Answer 1: I choose to use feature rescaling because is widely used, it's simple and it helps to reduce noisy features in support vector machines, which are models that I've used before. Also, being simple means computationally simple.
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import unittest
# import numpy as np
class TestFeatures(unittest.TestCase):
def test_features_ground(self):
sample = (asl.df.ix[98, 1][features_ground]).tolist()
self.assertEqual(sample, [9, 113, -12, 119])
def test_features_norm(self):
sample = (asl.df.ix[98, 1][features_norm]).tolist()
np.testing.assert_almost_equal(sample, [ 1.153, 1.663, -0.891, 0.742], 3)
def test_features_polar(self):
sample = (asl.df.ix[98,1][features_polar]).tolist()
np.testing.assert_almost_equal(sample, [113.3578, 0.0794, 119.603, -0.1005], 3)
def test_features_delta(self):
sample = (asl.df.ix[98, 0][features_delta]).tolist()
self.assertEqual(sample, [0, 0, 0, 0])
sample = (asl.df.ix[98, 18][features_delta]).tolist()
self.assertTrue(sample in [[-16, -5, -2, 4], [-14, -9, 0, 0]], "Sample value found was {}".format(sample))
suite = unittest.TestLoader().loadTestsFromModule(TestFeatures())
unittest.TextTestRunner().run(suite)
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The objective of Model Selection is to tune the number of states for each word HMM prior to testing on unseen data. In this section you will explore three methods:
Now that we have built a training set with sequence data, we can "train" models for each word. As a simple starting example, we train a single word using Gaussian hidden Markov models (HMM). By using the fit method during training, the Baum-Welch Expectation-Maximization (EM) algorithm is invoked iteratively to find the best estimate for the model for the number of hidden states specified from a group of sample seequences. For this example, we assume the correct number of hidden states is 3, but that is just a guess. How do we know what the "best" number of states for training is? We will need to find some model selection technique to choose the best parameter.
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import warnings
from hmmlearn.hmm import GaussianHMM
def train_a_word(word, num_hidden_states, features):
warnings.filterwarnings("ignore", category=DeprecationWarning)
training = asl.build_training(features)
X, lengths = training.get_word_Xlengths(word)
model = GaussianHMM(n_components=num_hidden_states, n_iter=1000).fit(X, lengths)
logL = model.score(X, lengths)
return model, logL
demoword = 'BOOK'
model, logL = train_a_word(demoword, 3, features_ground)
print("Number of states trained in model for {} is {}".format(demoword, model.n_components))
print("logL = {}".format(logL))
The HMM model has been trained and information can be pulled from the model, including means and variances for each feature and hidden state. The log likelihood for any individual sample or group of samples can also be calculated with the score method.
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def show_model_stats(word, model):
print("Number of states trained in model for {} is {}".format(word, model.n_components))
variance=np.array([np.diag(model.covars_[i]) for i in range(model.n_components)])
for i in range(model.n_components): # for each hidden state
print("hidden state #{}".format(i))
print("mean = ", model.means_[i])
print("variance = ", variance[i])
print()
show_model_stats(demoword, model)
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my_testword = 'CHOCOLATE'
model, logL = train_a_word(my_testword, 3, features_delta) # Experiment here with different parameters
show_model_stats(my_testword, model)
print("logL = {}".format(logL))
We can plot the means and variances for each state and feature. Try varying the number of states trained for the HMM model and examine the variances. Are there some models that are "better" than others? How can you tell? We would like to hear what you think in the classroom online.
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%matplotlib inline
%load_ext autoreload
%autoreload 2
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import math
from matplotlib import (cm, pyplot as plt, mlab)
def visualize(word, model):
""" visualize the input model for a particular word """
variance=np.array([np.diag(model.covars_[i]) for i in range(model.n_components)])
figures = []
for parm_idx in range(len(model.means_[0])):
xmin = int(min(model.means_[:,parm_idx]) - max(variance[:,parm_idx]))
xmax = int(max(model.means_[:,parm_idx]) + max(variance[:,parm_idx]))
fig, axs = plt.subplots(model.n_components, sharex=True, sharey=False)
colours = cm.rainbow(np.linspace(0, 1, model.n_components))
for i, (ax, colour) in enumerate(zip(axs, colours)):
x = np.linspace(xmin, xmax, 100)
mu = model.means_[i,parm_idx]
sigma = math.sqrt(np.diag(model.covars_[i])[parm_idx])
ax.plot(x, mlab.normpdf(x, mu, sigma), c=colour)
ax.set_title("{} feature {} hidden state #{}".format(word, parm_idx, i))
ax.grid(True)
figures.append(plt)
for p in figures:
p.show()
visualize(my_testword, model)
Review the ModelSelector class from the codebase found in the my_model_selectors.py module. It is designed to be a strategy pattern for choosing different model selectors. For the project submission in this section, subclass SelectorModel to implement the following model selectors. In other words, you will write your own classes/functions in the my_model_selectors.py module and run them from this notebook:
SelectorCV: Log likelihood with CVSelectorBIC: BIC SelectorDIC: DICYou will train each word in the training set with a range of values for the number of hidden states, and then score these alternatives with the model selector, choosing the "best" according to each strategy. The simple case of training with a constant value for n_components can be called using the provided SelectorConstant subclass as follow:
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from my_model_selectors import SelectorConstant
training = asl.build_training(features_delta) # Experiment here with different feature sets defined in part 1
word = 'CHOCOLATE' # Experiment here with different words
model = SelectorConstant(training.get_all_sequences(), training.get_all_Xlengths(), word, n_constant=3).select()
print("Number of states trained in model for {} is {}".format(word, model.n_components))
If we simply score the model with the Log Likelihood calculated from the feature sequences it has been trained on, we should expect that more complex models will have higher likelihoods. However, that doesn't tell us which would have a better likelihood score on unseen data. The model will likely be overfit as complexity is added. To estimate which topology model is better using only the training data, we can compare scores using cross-validation. One technique for cross-validation is to break the training set into "folds" and rotate which fold is left out of training. The "left out" fold scored. This gives us a proxy method of finding the best model to use on "unseen data". In the following example, a set of word sequences is broken into three folds using the scikit-learn Kfold class object. When you implement SelectorCV, you will use this technique.
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from sklearn.model_selection import KFold
training = asl.build_training(features_custom) # Experiment here with different feature sets
word = 'VEGETABLE' # Experiment here with different words
word_sequences = training.get_word_sequences(word)
split_method = KFold()
for cv_train_idx, cv_test_idx in split_method.split(word_sequences):
print("Train fold indices:{} Test fold indices:{}".format(cv_train_idx, cv_test_idx)) # view indices of the folds
Tip: In order to run hmmlearn training using the X,lengths tuples on the new folds, subsets must be combined based on the indices given for the folds. A helper utility has been provided in the asl_utils module named combine_sequences for this purpose.
Scoring model topologies with BIC balances fit and complexity within the training set for each word. In the BIC equation, a penalty term penalizes complexity to avoid overfitting, so that it is not necessary to also use cross-validation in the selection process. There are a number of references on the internet for this criterion. These slides include a formula you may find helpful for your implementation.
The advantages of scoring model topologies with DIC over BIC are presented by Alain Biem in this reference (also found here). DIC scores the discriminant ability of a training set for one word against competing words. Instead of a penalty term for complexity, it provides a penalty if model liklihoods for non-matching words are too similar to model likelihoods for the correct word in the word set.
Implement SelectorCV, SelectorBIC, and SelectorDIC classes in the my_model_selectors.py module. Run the selectors on the following five words. Then answer the questions about your results.
Tip: The hmmlearn library may not be able to train or score all models. Implement try/except contructs as necessary to eliminate non-viable models from consideration.
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words_to_train = ['FISH', 'BOOK', 'VEGETABLE', 'FUTURE', 'JOHN']
import timeit
%load_ext autoreload
%autoreload 2
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# TODO: Implement SelectorCV in my_model_selector.py
from my_model_selectors import SelectorCV
training = asl.build_training(features_ground) # Experiment here with different feature sets defined in part 1
sequences = training.get_all_sequences()
Xlengths = training.get_all_Xlengths()
for word in words_to_train:
start = timeit.default_timer()
model = SelectorCV(sequences, Xlengths, word,
min_n_components=2, max_n_components=15, random_state = 14).select()
end = timeit.default_timer()-start
if model is not None:
print("Training complete for {} with {} states with time {} seconds".format(word, model.n_components, end))
else:
print("Training failed for {}".format(word))
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# TODO: Implement SelectorBIC in module my_model_selectors.py
from my_model_selectors import SelectorBIC
training = asl.build_training(features_ground) # Experiment here with different feature sets defined in part 1
sequences = training.get_all_sequences()
Xlengths = training.get_all_Xlengths()
for word in words_to_train:
start = timeit.default_timer()
model = SelectorBIC(sequences, Xlengths, word,
min_n_components=2, max_n_components=15, random_state = 14).select()
end = timeit.default_timer()-start
if model is not None:
print("Training complete for {} with {} states with time {} seconds".format(word, model.n_components, end))
else:
print("Training failed for {}".format(word))
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# TODO: Implement SelectorDIC in module my_model_selectors.py
from my_model_selectors import SelectorDIC
training = asl.build_training(features_ground) # Experiment here with different feature sets defined in part 1
sequences = training.get_all_sequences()
Xlengths = training.get_all_Xlengths()
for word in words_to_train:
start = timeit.default_timer()
model = SelectorDIC(sequences, Xlengths, word,
min_n_components=2, max_n_components=15, random_state = 14).select()
end = timeit.default_timer()-start
if model is not None:
print("Training complete for {} with {} states with time {} seconds".format(word, model.n_components, end))
else:
print("Training failed for {}".format(word))
Question 2: Compare and contrast the possible advantages and disadvantages of the various model selectors implemented.
Answer 2:
Overall SelectorCV and SelectorBIC offered much better results than SelectorDIC. Although Running time and states was very similar between SelectorCV and Selector BIC, I'd choose SeletorCV because some test performed in other computers showed less states than BIC and also in some talks in the forums many other students agreed that SelectorCV performs better than the other selectors.
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from asl_test_model_selectors import TestSelectors
suite = unittest.TestLoader().loadTestsFromModule(TestSelectors())
unittest.TextTestRunner().run(suite)
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The objective of this section is to "put it all together". Using the four feature sets created and the three model selectors, you will experiment with the models and present your results. Instead of training only five specific words as in the previous section, train the entire set with a feature set and model selector strategy.
The following example trains the entire set with the example features_ground and SelectorConstant features and model selector. Use this pattern for you experimentation and final submission cells.
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# autoreload for automatically reloading changes made in my_model_selectors and my_recognizer
from my_model_selectors import SelectorConstant
def train_all_words(features, model_selector):
training = asl.build_training(features) # Experiment here with different feature sets defined in part 1
sequences = training.get_all_sequences()
Xlengths = training.get_all_Xlengths()
model_dict = {}
for word in training.words:
model = model_selector(sequences, Xlengths, word,
n_constant=3).select()
model_dict[word]=model
return model_dict
models = train_all_words(features_ground, SelectorConstant)
print("Number of word models returned = {}".format(len(models)))
The build_test method in ASLdb is similar to the build_training method already presented, but there are a few differences:
SinglesData get_all_sequences, get_all_Xlengths, get_item_sequences and get_item_Xlengths
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test_set = asl.build_test(features_ground)
print("Number of test set items: {}".format(test_set.num_items))
print("Number of test set sentences: {}".format(len(test_set.sentences_index)))
For the final project submission, students must implement a recognizer following guidance in the my_recognizer.py module. Experiment with the four feature sets and the three model selection methods (that's 12 possible combinations). You can add and remove cells for experimentation or run the recognizers locally in some other way during your experiments, but retain the results for your discussion. For submission, you will provide code cells of only three interesting combinations for your discussion (see questions below). At least one of these should produce a word error rate of less than 60%, i.e. WER < 0.60 .
Tip: The hmmlearn library may not be able to train or score all models. Implement try/except contructs as necessary to eliminate non-viable models from consideration.
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# TODO implement the recognize method in my_recognizer
from my_recognizer import recognize
from asl_utils import show_errors
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# TODO Choose a feature set and model selector
features = features_norm # change as needed
model_selector = SelectorCV # change as needed
# TODO Recognize the test set and display the result with the show_errors method
models = train_all_words(features, model_selector)
test_set = asl.build_test(features)
probabilities, guesses = recognize(models, test_set)
show_errors(guesses, test_set)
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# TODO Choose a feature set and model selector
features = features_norm # change as needed
model_selector = SelectorBIC # change as needed
# TODO Recognize the test set and display the result with the show_errors method
models = train_all_words(features, model_selector)
test_set = asl.build_test(features)
probabilities, guesses = recognize(models, test_set)
show_errors(guesses, test_set)
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# TODO Choose a feature set and model selector
features = features_norm # change as needed
model_selector = SelectorDIC # change as needed
# TODO Recognize the test set and display the result with the show_errors method
models = train_all_words(features, model_selector)
test_set = asl.build_test(features)
probabilities, guesses = recognize(models, test_set)
show_errors(guesses, test_set)
Question 3: Summarize the error results from three combinations of features and model selectors. What was the "best" combination and why? What additional information might we use to improve our WER? For more insight on improving WER, take a look at the introduction to Part 4.
Answer 3:
After testing all combinations I noticed that using Features Polar gives better results. The Best combination I think would be Features Polar with Selector CV. Features Polar with Selector BIC could also be a good choice.
This challenge reminds me of the short story "The Gold-Bug" by Edgar Allan Poe. In this history the protagonists find a simple substitution cipher that can be solved using letter frequencies. So having this in mind I would suggest to do something similar computig the probability of some words to appear in phrase or sequence (I'm still not sure the current method used to archieve this in AI).
Features Custom. SelectorCV WER = 0.6741573033707865 Features Custom. SelectorBIC WER = 0.6741573033707865 Features Custom. SelectorDIC WER = 0.6741573033707865
Features Ground. SelectorCV WER = 0.6685393258426966 Features Ground. SelectorBIC WER = 0.6685393258426966 Features Ground. SelectorDIC WER = 0.6685393258426966
Features Polar. SelectorCV WER = 0.6179775280898876 Features Polar. SelectorBIC WER = 0.6179775280898876 Features Polar. SelectorDIC WER = 0.6179775280898876
Features Delta. SelectorCV WER = 0.6404494382022472 Features Delta. SelectorBIC WER = 0.6404494382022472 Features Delta. SelectorDIC WER = 0.6404494382022472
Features Norm. SelectorCV WER = 0.6235955056179775 Features Norm. SelectorBIC WER = 0.6235955056179775 Features Norm. SelectorDIC WER = 0.6235955056179775
In [120]:
from asl_test_recognizer import TestRecognize
suite = unittest.TestLoader().loadTestsFromModule(TestRecognize())
unittest.TextTestRunner().run(suite)
Out[120]:
We've squeezed just about as much as we can out of the model and still only get about 50% of the words right! Surely we can do better than that. Probability to the rescue again in the form of statistical language models (SLM). The basic idea is that each word has some probability of occurrence within the set, and some probability that it is adjacent to specific other words. We can use that additional information to make better choices.
The recognizer you implemented in Part 3 is equivalent to a "0-gram" SLM. Improve the WER with the SLM data provided with the data set in the link above using "1-gram", "2-gram", and/or "3-gram" statistics. The probabilities data you've already calculated will be useful and can be turned into a pandas DataFrame if desired (see next cell).
Good luck! Share your results with the class!
In [ ]:
# create a DataFrame of log likelihoods for the test word items
df_probs = pd.DataFrame(data=probabilities)
df_probs.head()