ML Case-study Interview Question: Fourier Transform-Based Defrost Cycle Prediction for Refrigeration System Anomaly Detection.

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Case-Study question

A major retail organization operates thousands of supermarket refrigeration systems that maintain specific temperatures for perishable items. The organization wants to automate anomaly detection in these refrigeration units. Some temperature spikes are legitimate defrost events, while others may indicate mechanical or operational issues. The team has access to univariate time-series sensor data from each refrigeration case. The data includes temperature readings captured at regular intervals, along with partial defrost commands for a subset of refrigeration cases. They observe that defrost cycles shift gradually because of external factors such as weather, store traffic, and repairs. Formulate a comprehensive approach to detect and predict these defrost cycles so that non-defrost anomalies can be flagged and handled proactively. Propose a detailed solution outline that covers the data preprocessing steps, the method to extract periodic components, the technique to isolate defrost signals, and a plan to handle irregular defrost patterns or noisy sensor data. Provide your recommended architecture, details about each algorithmic component, the final outputs, and a short proposal for how to validate and benchmark the system’s results.

Detailed solution

High-level idea

Use a Fourier Transform-based approach to decompose the temperature time-series signal into its major frequency components. Identify which frequency component corresponds to periodic defrost cycles, extract that component via filtering, and then derive a binary defrost indicator from the reconstructed signal. Extend this indicator into future time windows to predict probable defrost times. Compare predictions against real defrost commands (where available) to refine or validate the approach.

Fourier transform

Any time-domain signal can be expressed as a weighted sum of sinusoids of varying frequencies, amplitudes, and phases. The mathematical core is shown below.

Here F(omega) is the frequency-domain representation of the time-domain signal f(t). The parameter t is time, and omega is the angular frequency. The exponential term encodes the sinusoidal basis functions used to decompose f(t). In practice, we use the Discrete Fourier Transform or its optimized version, Fast Fourier Transform (FFT), for digital signals.

Steps to isolate defrost signals

Convert temperature data to frequency domain using FFT. Identify the highest amplitude (zero frequency) representing the average or steady-state temperature. Locate the second-highest amplitude corresponding to the strongest periodic cycle, which is often the defrost frequency. Zero out all other frequency components. Perform the inverse Fourier transform to get the defrost signal in time domain. Set a threshold to convert that sinusoidal curve to a binary indicator for defrost vs. non-defrost. Forecast future defrost periods by extrapolating the derived sinusoid.

Python code snippet

import numpy as np
import matplotlib.pyplot as plt

def get_defrost(x):
    fhat = np.fft.fft(x, len(x))
    A = fhat * np.conj(fhat) / len(x)
    idx = A >= sorted(A)[-2]  # second-highest amplitude
    fhat = idx * fhat
    ffilt = np.fft.ifft(fhat)
    return abs(ffilt)

Explain in simple terms. The function above computes the FFT of the temperature array x. It calculates the magnitude (A) of each frequency component. It finds the frequency component with the second-highest amplitude, keeps only that frequency, and removes others by zeroing them out. It then applies the inverse FFT to retrieve the defrost-specific periodic signal.

Convert the resulting array to a binary defrost indicator by picking a threshold. Larger values indicate a defrost period. This threshold can be tuned via experimentation or domain knowledge.

Handling irregular defrost cycles

When defrost cycles are irregular, a pure single-frequency approach may fail. Incorporate additional domain knowledge:

• Check other frequencies beyond the second-highest amplitude to see if some cases have composite defrost patterns.

• Segment data into shorter windows and apply the same FFT-based approach to handle gradually shifting defrost cycles.

• Use historical data of each case’s temperature profile to dynamically adjust thresholds.

• Combine Fourier-based features with a supervised learning model if you have enough labeled defrost data.

Validating and benchmarking

Compare predicted defrost intervals against any known defrost commands in sensor data. Compute metrics such as precision and recall to measure how closely the predicted intervals align with actual defrost times. Identify systematic mismatches to refine thresholds or incorporate secondary periodicities. Where no defrost command is available, perform manual checks on temperature patterns or consult domain experts to gauge correctness.

How would you handle noisy temperature signals?

Design a denoising step before extracting defrost cycles. In the frequency domain, identify and remove very low amplitude frequencies that represent random noise. Use domain knowledge about typical defrost periods to guide an upper bound on permissible frequencies. If signals remain unpredictable, combine the Fourier-based approach with moving averages in time domain or wavelet transforms, adjusting for noise in multiple scales.

How would you address data quality problems?

Investigate discrepancies where the predicted defrost intervals do not match the sensor-commanded intervals. This may expose synchronization errors or missing data. For serious data gaps, backfill or interpolate if possible. If the data is entirely corrupt, flag it for hardware inspection. Incorporate data validation checks to ensure correct time-stamping and sensor calibration. Build robust error-handling routines so the model can operate even if partial data is unavailable.

How can you predict future defrost periods once you isolate the defrost signal?

Focus on the isolated sinusoidal component. It will have a phase, amplitude, and frequency that can be extended forward in time. Use that periodic pattern to create a schedule of future high points, each high point indicating a probable defrost window. If minor drifts exist, build a short sliding window to recalculate frequency changes. Maintain a rolling forecast that updates when new data arrives.

What if the periodic pattern is not strictly sinusoidal?

Consider a more generalized basis (for example, multiple harmonics) so that abrupt defrost starts and stops can be captured. Another option is to fit a piecewise function. Some cases may display irregularities because of store conditions or machine configurations. In those cases, watch the spectral components for multiple peaks or broadband signals, and refine the filter logic to accommodate multiple defrost-related frequencies.

Would you recommend adding extra sensor data to improve results?

Yes. Pull in additional streams such as door sensor data, outside temperature, or power consumption. If a door stays open or outside conditions are extreme, the measured temperature could distort the model. Adding these variables helps separate genuine anomalies from normal events. Use the sensor fusion approach as an additional signal or as part of a multi-variate time-series model.

Could a supervised learning approach replace the Fourier-based method?

A purely supervised approach can work if labeled data is extensive and reliable. However, the Fourier-based approach is quick for capturing cyclical patterns with less labeled data. A hybrid approach can be valuable. Use the Fourier transform to isolate cyclical behavior, then feed that into a machine learning model that handles outliers, transitional states, and unusual conditions.

How would you scale this solution to thousands of refrigeration cases?

Implement a pipeline that processes each time-series in small batches or microservices. Each pipeline step can run independently on separate nodes. For large-scale data ingestion, store sensor data in a distributed system. The FFT step is computationally efficient, but be mindful of how often you run it. Parallelize or distribute the computations if the load is huge. Monitor system performance with real-time dashboards to spot any backlog.

How do you ensure minimal false alarms?

Analyze the precision/recall tradeoff. Adjust the binary threshold for defrost detection to maximize F1 score. Add rules-based checks (for instance, an improbable defrost cycle lasting too long or too short). Revisit domain knowledge to confirm typical defrost durations. Inspect misclassified intervals and refine parameters. Iterate until the solution reaches an acceptable false alarm rate for field deployment.

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