Step Detection Algorithm (SDA): A toolbox for detecting steps of unknown size and location in a noisy signal.

How to use

Here is a sample code to get the estimate of step signals from noisy measurements.

# Noise standard deviation
sigma = 0.3162 #0.3162 Corresponds to an SNR of 5 for signal of size 1 and a duty ratio of 0.5 


# =============================================================================
# Generating noisy stepping data 
# This section is needed only for sample signal generation. Replace this section 
# with your data. That is, instead of generating the measured signal 'y', 
# assign 'y' to your signal of interest 
# =============================================================================
d = 10 # No. of steps
dt = 0.1 # Sampling interval in s
N = 1000 # total number of samples 
L = 1 # Step size
iterations = 50 # No. of SDA iterations
resolution = 0.05 # Resolution of the SDA
seed = 5 # For reproducibility (change seed for a new signal)
include_dynamics = False # Choose to simulate sensor dynamics 

# Simulating sensor dynamics
SDynObj = SDADynamics()
num = [1] # Specify numerator of sensor transfer function
den = [0.1, 0.8, 1, 1] # Specify denominator of sensor transfer function
sysd = SDynObj.get_DT_sys(num, den, dt)


# Generating a sample stepping signal
SIGObj = SampleSignals()
# Stepping signal
x = SIGObj.get_step_sample(step_size = L, no_steps = d, no_samples = N, bidirectional = 0, seed = seed)
# Noise
noise = SIGObj.get_gaussian_noise(std = sigma, no_samples = N, seed = seed)

if(include_dynamics == True):
    [T, x_sensor] = SIGObj.sensor_dynamics(num, den, dt, x)
    x_sensor = np.expand_dims(x_sensor, axis = 0)
else:
    x_sensor = x
    sysd = None

y = x_sensor + noise  # Noisy measurements

# =============================================================================

# To use the SDA to estimate the stepping signal from data, 
# you would need the standard deviation of noise in the signal.
# Here the standard deviation of noise is denoted by 'sigma'.  
SDAObj = SDA()
[step_estimate, est_array, hist_array]= SDAObj.SDA_dynamics(sigma, y)
SDAObj.plot_SDA_stages(x.T, y, est_array, hist_array)

Example

Here is an example of the Step Detection Algorithm in action. The SDA works in stages, refining & learning about the underlying signal in each stage. Shown below is the signal estimate and step size histogram after the first stage.

After two stages, the step size histogram has been refined, with some of the erroneous sizes eleminated.

The refinement process continues in stage 3...

and in stage 4.

At stage 5, no further refinements are achieved in this example and the algorithm stops. The result of this final stage is the output of the SDA.

Citation

To cite SDA, please use the Automatica publication:

• Rajaganapathy, Sivaraman, James Melbourne, and Murti V. Salapaka. "Change detection using an iterative algorithm with guarantees." Automatica 136 (2022): 110075.

Additional References

• Rajaganapathy, Sivaraman, James Melbourne, Tanuj Aggarwal, Rachit Shrivastava, and Murti V. Salapaka. "Learning and Estimation of Single Molecule Behavior." In 2018 Annual American Control Conference (ACC), pp. 5125-5130. IEEE, 2018.

• Aggarwal, Tanuj, Donatello Materassi, Robert Davison, Thomas Hays, and Murti Salapaka. "Detection of steps in single molecule data." Cellular and molecular bioengineering 5, no. 1 (2012): 14-31.