Question set

📊 Industrial Problems Using Decaying Windows

In industrial analytics, streaming data must be processed in real time. One powerful tool is the decaying window model, which gives more weight to recent data while fading older values. Below are four problems designed for students to solve with paper and pen.

Problem 1: Machine Vibration Monitoring

A smart factory monitors machine vibration every second. Engineers want to detect anomalies using a decaying window average.

Formula: W_t = ( Σ x_i · e^-λ(t-i) ) / ( Σ e^-λ(t-i) )

• Sensor readings: [5, 7, 6, 20, 8, 7, 6]
• Decay rate: λ = 0.1
• Threshold: θ = 10

Tasks

1. Compute W_t step by step.
2. Identify anomaly points (W_t > θ).
3. Discuss effect of changing λ.

Problem 2: Energy Consumption in Smart Grid

A smart grid records household energy usage every minute. Recent spikes should be detected quickly, but older data should fade.

• Energy readings (kWh): [2, 3, 2, 10, 4, 3, 2]
• Decay rate: λ = 0.2
• Threshold: θ = 5

Tasks

1. Calculate decaying averages for each time step.
2. Mark when consumption exceeds threshold.
3. Explain why decaying windows are better than simple averages here.

Problem 3: Network Traffic Monitoring

A company tracks packets per second on its server. Sudden surges may indicate a cyber attack. Use decaying windows to detect anomalies.

• Packet counts: [100, 120, 110, 500, 130, 125, 115]
• Decay rate: λ = 0.15
• Threshold: θ = 250

Tasks

1. Compute weighted averages.
2. Identify suspicious surges.
3. Discuss how λ affects detection speed.

Problem 4: Financial Transaction Fraud Detection

A bank monitors transaction amounts in real time. Recent large transactions should trigger alerts more strongly than older ones.

• Transaction amounts ($): [50, 60, 55, 1000, 70, 65, 60]
• Decay rate: λ = 0.25
• Threshold: θ = 300

Tasks

1. Calculate decaying averages for each transaction.
2. Flag anomalies where W_t > θ.
3. Explain how decay helps reduce false alarms from old transactions.

Conclusion

These problems show how decaying windows apply across industries: predictive maintenance, energy management, cybersecurity, and fraud detection. By solving them, students gain practical insight into real-time data analysis.