Classification Using Hidden Markov Model

This is a demonstration using the implemented forward algorithm in the context of a hidden Markov model to classify multiple targets.

We will attempt to classify 3 targets in an undefined region.

All Stone Soup imports will be given in order of usage.

from datetime import datetime, timedelta

import numpy as np

from stonesoup.models.transition.categorical import MarkovianTransitionModel
from stonesoup.types.groundtruth import CategoricalGroundTruthState
from stonesoup.types.groundtruth import GroundTruthPath

Ground Truth

The targets may take one of two discrete hidden classes: ‘bike’, and ‘car’. A target may be able to transition from one class to another (this could be considered as a person switching from riding a bike to driving a car and vice versa). This behaviour will be modelled in the transition matrix of the MarkovianTransitionModel. This transition matrix is a Markov process matrix, whereby it is assumed that the state of a target is wholly dependent on its previous state, and nothing else.

A CategoricalState class is used to store information on the classification/category of the targets. The state vector will define a categorical distribution over the 2 possible classes, whereby each component defines the probability that a target is of the corresponding class. For example, the state vector (0.2, 0.8), with category names (‘bike’, ‘car’) indicates that a target has a 20% probability of being class ‘bike’ and an 80% probability of being class ‘car’ etc. It does not make sense to have a true target being a distribution over the possible classes, and therefore the true categorical states will have binary state vectors indicating a specific class (i.e. a ‘1’ at one state vector index, and ‘0’s elsewhere). This can be considered as stating there is a 100% probability that the target is of a particular class. We specify that there should be noise when functioning our transition model in order to sample the resultant distribution and receive this binary vector. The CategoricalGroundTruthState class inherits directly from the base CategoricalState class.

The category and timings for one of the ground truth paths will be printed.

transition_matrix = np.array([[0.8, 0.2],  # P(bike | bike), P(bike | car)
                              [0.4, 0.6]])  # P(car | bike), P(car | car)
category_transition = MarkovianTransitionModel(transition_matrix=transition_matrix)

start = datetime.now()

hidden_classes = ['bike', 'car']

# Generating ground truth
ground_truths = list()
for i in range(1, 4):  # 4 targets
    state_vector = np.zeros(2)  # create a vector with 2 zeroes
    state_vector[np.random.choice(2, 1, p=[1 / 2, 1 / 2])] = 1  # pick a random class out of the 2
    ground_truth_state = CategoricalGroundTruthState(state_vector,
                                                     timestamp=start,
                                                     categories=hidden_classes)

    ground_truth = GroundTruthPath([ground_truth_state], id=f"GT{i}")

    for _ in range(10):
        new_vector = category_transition.function(ground_truth[-1],
                                                  noise=True,
                                                  time_interval=timedelta(seconds=1))
        new_state = CategoricalGroundTruthState(
            new_vector,
            timestamp=ground_truth[-1].timestamp + timedelta(seconds=1),
            categories=hidden_classes
        )

        ground_truth.append(new_state)
    ground_truths.append(ground_truth)

for states in np.vstack(ground_truths).T:
    print(f"{states[0].timestamp:%H:%M:%S}", end="")
    for state in states:
        print(f" -- {state.category}", end="")
    print()

11:40:27 -- car -- bike -- car
11:40:28 -- car -- bike -- car
11:40:29 -- car -- bike -- car
11:40:30 -- car -- bike -- car
11:40:31 -- car -- bike -- car
11:40:32 -- car -- car -- bike
11:40:33 -- bike -- bike -- bike
11:40:34 -- bike -- bike -- bike
11:40:35 -- bike -- bike -- car
11:40:36 -- bike -- bike -- bike
11:40:37 -- bike -- car -- bike

Measurement

Using a Hidden Markov model, it is assumed the true class of a target cannot be directly observed (hence ‘hidden’), and instead observations that are dependent on this class are taken. In this instance, observations of the targets’ sizes are taken (‘small’, ‘medium’ or ‘large’). The relationship between true class and observed size is modelled by the emission matrix of the MarkovianMeasurementModel, which is used by the HMMSensor to provide CategoricalDetection types. We will model this such that a ‘bike’ has a very small chance of being observed as a ‘big’ target etc.

from stonesoup.models.measurement.categorical import MarkovianMeasurementModel
from stonesoup.sensor.categorical import HMMSensor

E = np.array([[0.8, 0.1],  # P(small | bike), P(small | car)
              [0.19, 0.3],  # P(medium | bike), P(medium | car)
              [0.01, 0.6]])  # P(large | bike), P(large | car)

model = MarkovianMeasurementModel(emission_matrix=E,
                                  measurement_categories=['small', 'medium', 'large'])

eo = HMMSensor(measurement_model=model)

# Generating measurements
measurements = list()
for index, states in enumerate(np.vstack(ground_truths).T):
    if index == 5:
        measurements_at_time = set()  # Give tracker chance to use prediction instead
    else:
        measurements_at_time = eo.measure(states)
    timestamp = next(iter(states)).timestamp
    measurements.append((timestamp, measurements_at_time))

    print(f"{timestamp:%H:%M:%S} -- {[meas.category for meas in measurements_at_time]}")

11:40:27 -- ['medium', 'small', 'medium']
11:40:28 -- ['large', 'large', 'small']
11:40:29 -- ['medium', 'small', 'large']
11:40:30 -- ['large', 'medium', 'medium']
11:40:31 -- ['small', 'large', 'medium']
11:40:32 -- []
11:40:33 -- ['small', 'small', 'small']
11:40:34 -- ['small', 'medium', 'small']
11:40:35 -- ['large', 'medium', 'small']
11:40:36 -- ['small', 'small', 'medium']
11:40:37 -- ['large', 'medium', 'small']

Tracking Components

Predictor

A HMMPredictor specifically uses MarkovianTransitionModel types to predict.

from stonesoup.predictor.categorical import HMMPredictor

# It would be cheating to use the same transition model as in ground truth generation!
transition_matrix = np.array([[0.81, 0.19],  # P(bike | bike), P(bike | car)
                              [0.39, 0.61]])  # P(car | bike), P(car | car)
category_transition = MarkovianTransitionModel(transition_matrix=transition_matrix)

predictor = HMMPredictor(category_transition)

Updater

from stonesoup.updater.categorical import HMMUpdater

updater = HMMUpdater()

Hypothesiser

A HMMHypothesiser is used for calculating categorical hypotheses. It utilises the ObservationAccuracy measure: a multi-dimensional extension of an ‘accuracy’ score, essentially providing a measure of the similarity between two categorical distributions.

from stonesoup.hypothesiser.categorical import HMMHypothesiser

hypothesiser = HMMHypothesiser(predictor=predictor, updater=updater)

Data Associator

We will use a standard GNNWith2DAssignment data associator.

from stonesoup.dataassociator.neighbour import GNNWith2DAssignment

data_associator = GNNWith2DAssignment(hypothesiser)

Prior

As we are tracking in a categorical state space, we should initiate with a categorical state for the prior. Equal probability is given to all 3 of the possible hidden classes that a target might take (the category names are also provided here).

from stonesoup.types.state import CategoricalState

prior = CategoricalState([1 / 2, 1 / 2], categories=hidden_classes)

Initiator

For each unassociated detection, a new track will be initiated. In this instance we use a SimpleCategoricalMeasurementInitiator, which specifically handles categorical state priors.

from stonesoup.initiator.categorical import SimpleCategoricalMeasurementInitiator

initiator = SimpleCategoricalMeasurementInitiator(prior_state=prior, updater=updater)

Deleter

We can use a standard UpdateTimeStepsDeleter.

from stonesoup.deleter.time import UpdateTimeStepsDeleter

deleter = UpdateTimeStepsDeleter(2)

Tracker

We can use a standard MultiTargetTracker.

from stonesoup.tracker.simple import MultiTargetTracker

tracker = MultiTargetTracker(initiator, deleter, measurements, data_associator, updater)

Tracking

tracks = set()
for time, ctracks in tracker:
    tracks.update(ctracks)

print(f"Number of tracks: {len(tracks)}")
for track in tracks:
    certainty = track.state_vector[np.argmax(track.state_vector)][0] * 100
    print(f"id: {track.id} -- category: {track.category} -- certainty: {certainty}%")
    for state in track:
        _time = state.timestamp.strftime('%H:%M')
        _type = str(type(state)).replace("class 'stonesoup.types.", "").strip("<>'. ")
        state_string = f"{_time} -- {_type} -- {state.category}"
        try:
            meas_string = f"associated measurement: {state.hypothesis.measurement.category}"
        except AttributeError:
            pass
        else:
            state_string += f" -- {meas_string}"
        print(state_string)
    print()

Number of tracks: 3
id: f3786fca-f8e5-498a-8eb8-9ee09dddf966 -- category: bike -- certainty: 50.92087944676565%
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: medium
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: large
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: medium
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: medium
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: medium
11:40 -- prediction.CategoricalStatePrediction -- car
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: medium
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: medium

id: b5ecfe6b-a579-4aef-911b-57d7da027526 -- category: car -- certainty: 99.25153034099766%
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: medium
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: large
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: large
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: large
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: large
11:40 -- prediction.CategoricalStatePrediction -- car
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: medium
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: large
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: medium
11:40 -- update.CategoricalStateUpdate -- car -- associated measurement: large

id: e0af699c-2506-40d6-ad0a-2d28cf163cfc -- category: bike -- certainty: 93.61671325402695%
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: medium
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- prediction.CategoricalStatePrediction -- bike
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small
11:40 -- update.CategoricalStateUpdate -- bike -- associated measurement: small

Metric

Determining tracking accuracy. In calculating how many targets were classified correctly, only tracks with the highest classification certainty are considered. In the situation where probabilities are equal, a random classification is chosen.

excess_tracks = len(tracks) - len(ground_truths)  # target value = 0
sorted_tracks = sorted(tracks,
                       key=lambda track: track.state_vector[np.argmax(track.state_vector)][0],
                       reverse=True)
best_tracks = sorted_tracks[:3]
true_classifications = [ground_truth.category for ground_truth in ground_truths]
track_classifications = [track.category for track in best_tracks]

num_correct_classifications = 0  # target value = num ground truths
for true_classification in true_classifications:
    for i in range(len(track_classifications)):
        if track_classifications[i] == true_classification:
            num_correct_classifications += 1
            del track_classifications[i]
            break

print(f"Excess tracks: {excess_tracks}")
print(f"No. correct classifications: {num_correct_classifications}")

Excess tracks: 0
No. correct classifications: 3

Plotting

Plotting the probability that each one of our targets and tracks is a ‘bike’ will help to visualise this 2-hidden class problem.

Dotted lines indicate ground truth probabilities, and solid lines for tracks.

import matplotlib.pyplot as plt


def plot(path, style):
    times = list()
    probs = list()
    for state in path:
        times.append(state.timestamp)
        probs.append(state.state_vector[0])
    plt.plot(times, probs, linestyle=style)


for truth in ground_truths:
    plot(truth, '--')
for track in tracks:
    plot(track, '-')