Control Charts for Six Sigma and AI with Python

1. Introduction to Control Charts

Control charts are statistical tools used to monitor process stability and control over time. They help distinguish between common cause variation (inherent to the process) and special cause variation (indicating potential issues).

2. Importing Required Libraries

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import stats
    

3. Generating Sample Data

np.random.seed(42)

# Generate sample data
n_points = 100
data = np.random.normal(loc=10, scale=1, size=n_points)

# Add some out-of-control points
data[80] += 4  # Point above upper control limit
data[90] -= 4  # Point below lower control limit

df = pd.DataFrame({'Measurement': data})
print(df.head())
    

4. Implementing X-bar and R Charts

def create_control_charts(data, subgroup_size=5):
    subgroups = [data[i:i+subgroup_size] for i in range(0, len(data), subgroup_size)]
    
    x_bar = [np.mean(subgroup) for subgroup in subgroups]
    ranges = [np.ptp(subgroup) for subgroup in subgroups]
    
    grand_mean = np.mean(x_bar)
    mean_range = np.mean(ranges)
    
    # Constants for control limits (assuming subgroup size of 5)
    A2, D3, D4 = 0.577, 0, 2.114
    
    # X-bar chart limits
    x_ucl = grand_mean + A2 * mean_range
    x_lcl = grand_mean - A2 * mean_range
    
    # R chart limits
    r_ucl = D4 * mean_range
    r_lcl = D3 * mean_range
    
    return x_bar, ranges, grand_mean, mean_range, x_ucl, x_lcl, r_ucl, r_lcl

x_bar, ranges, grand_mean, mean_range, x_ucl, x_lcl, r_ucl, r_lcl = create_control_charts(df['Measurement'])

# Plotting X-bar chart
plt.figure(figsize=(12, 6))
plt.plot(x_bar, marker='o')
plt.axhline(grand_mean, color='g', linestyle='--')
plt.axhline(x_ucl, color='r', linestyle='--')
plt.axhline(x_lcl, color='r', linestyle='--')
plt.title('X-bar Chart')
plt.xlabel('Subgroup')
plt.ylabel('Subgroup Mean')
plt.show()

# Plotting R chart
plt.figure(figsize=(12, 6))
plt.plot(ranges, marker='o')
plt.axhline(mean_range, color='g', linestyle='--')
plt.axhline(r_ucl, color='r', linestyle='--')
plt.axhline(r_lcl, color='r', linestyle='--')
plt.title('R Chart')
plt.xlabel('Subgroup')
plt.ylabel('Subgroup Range')
plt.show()
    

5. Implementing Individual and Moving Range (I-MR) Charts

def create_imr_charts(data):
    individual = data
    moving_range = np.abs(np.diff(individual))
    
    mean_i = np.mean(individual)
    mean_mr = np.mean(moving_range)
    
    # Constants for control limits
    E2 = 2.66  # For moving range of 2
    
    # Individual chart limits
    i_ucl = mean_i + 3 * mean_mr / 1.128
    i_lcl = mean_i - 3 * mean_mr / 1.128
    
    # Moving Range chart limits
    mr_ucl = E2 * mean_mr
    mr_lcl = 0  # Lower control limit for MR chart is always 0
    
    return individual, moving_range, mean_i, mean_mr, i_ucl, i_lcl, mr_ucl, mr_lcl

individual, moving_range, mean_i, mean_mr, i_ucl, i_lcl, mr_ucl, mr_lcl = create_imr_charts(df['Measurement'])

# Plotting Individual chart
plt.figure(figsize=(12, 6))
plt.plot(individual, marker='o')
plt.axhline(mean_i, color='g', linestyle='--')
plt.axhline(i_ucl, color='r', linestyle='--')
plt.axhline(i_lcl, color='r', linestyle='--')
plt.title('Individual Chart')
plt.xlabel('Observation')
plt.ylabel('Individual Value')
plt.show()

# Plotting Moving Range chart
plt.figure(figsize=(12, 6))
plt.plot(moving_range, marker='o')
plt.axhline(mean_mr, color='g', linestyle='--')
plt.axhline(mr_ucl, color='r', linestyle='--')
plt.axhline(mr_lcl, color='r', linestyle='--')
plt.title('Moving Range Chart')
plt.xlabel('Observation')
plt.ylabel('Moving Range')
plt.show()
    

6. Interpreting Control Charts

When interpreting control charts, look for these patterns that may indicate special cause variation:

• Points outside the control limits
• Runs of 7 or more points on one side of the centerline
• Trends (6 or more points consistently increasing or decreasing)
• Cyclic patterns
• Unnatural lack of variation

7. AI Applications in Control Charts

AI can enhance the use of control charts in several ways:

• Automated Pattern Recognition: Use machine learning algorithms to automatically detect and classify patterns in control charts.
• Predictive Maintenance: Implement AI models to predict when a process might go out of control based on historical data and current trends.
• Multi-variate Control Charts: Use AI techniques to monitor and analyze multiple variables simultaneously, detecting complex relationships and interactions.
• Real-time Monitoring and Alerting: Develop AI systems for continuous monitoring of processes and instant alerting when potential issues are detected.
• Adaptive Control Limits: Implement machine learning algorithms to dynamically adjust control limits based on changing process conditions and historical performance.

8. Monitoring AI Model Performance with Control Charts

Control charts can also be used to monitor the performance of AI models over time:

• Track key performance metrics (e.g., accuracy, F1 score) of your AI model using I-MR charts.
• Monitor prediction errors or residuals to detect shifts in model performance.
• Use EWMA (Exponentially Weighted Moving Average) charts for early detection of small shifts in model accuracy.
• Implement multivariate control charts to simultaneously monitor multiple model performance indicators.

Conclusion

Control charts are powerful tools for monitoring process stability and detecting special cause variation. By integrating AI techniques with traditional control chart methods, we can enhance our ability to monitor, analyze, and control processes in real-time. This integration is particularly valuable in the context of Industry 4.0 and IoT, where large amounts of data are generated continuously. Furthermore, applying control chart principles to AI model performance monitoring ensures that our AI systems remain reliable and accurate over time.