Cumulative Sum Control Chart

Understanding CUSUM charts

CUSUM charts work by calculating and plotting the cumulative sum of deviations from a target value. The basic CUSUM statistic $C_i$ at time $i$ is calculated as:

$C_i = C_{i-1} + (x_i - \mu_0)$

Where:

• $x_i$ is the current observation
• $\mu_0$ is the target mean
• $C_{i-1}$ is the previous CUSUM value

One-sided and two-sided CUSUM

CUSUM charts can be implemented as one-sided or two-sided monitoring schemes:

One-sided CUSUM

Tracks either positive or negative shifts:

$S_i^+ = \max[0, S_{i-1}^+ + (x_i - \mu_0 - K)]$

Where:

• $K$ is the reference value (usually set to $\frac{\delta}{2}$)
• $\delta$ is the shift size to detect

Two-sided CUSUM

Monitors both positive and negative shifts simultaneously:

$S_i^+ = \max[0, S_{i-1}^+ + (x_i - \mu_0 - K)]$ $S_i^- = \max[0, S_{i-1}^- - (x_i - \mu_0 + K)]$

Applications in financial markets

CUSUM charts are particularly valuable in:

1. Anomaly Detection in Time Series Data for identifying market regime changes
2. Risk Management in Swaps Trading for monitoring collateral values
3. Real-time Trade Surveillance for detecting unusual trading patterns

Decision intervals and implementation

The decision interval $H$ determines when the CUSUM signals an out-of-control condition:

1. If $S_i^+$ or $S_i^-$ exceeds $H$, the process is considered out of control
2. The choice of $H$ affects the Average Run Length (ARL) properties
3. Typical values for $H$ are between 4 and 5 standard deviations

Performance considerations

When implementing CUSUM charts in real-time systems:

1. Memory efficiency: Only need to store previous CUSUM value
2. Computational simplicity: Basic arithmetic operations
3. Sensitivity adjustment through parameters $K$ and $H$

Extensions and variations

Fast Initial Response (FIR)

Modifies the standard CUSUM to provide faster detection during startup:

$S_i^+ = \max[0, S_{i-1}^+ + (x_i - \mu_0 - K)] + \text{FIR}$

Adaptive CUSUM

Dynamically adjusts parameters based on observed data:

1. Variable reference value $K$
2. Adaptive decision interval $H$
3. Time-varying target mean $\mu_0$

Best practices

1. Choose appropriate values for $K$ and $H$ based on:

   • Expected shift magnitude
   • Desired ARL properties
   • False alarm tolerance

2. Regular parameter review:

   • Validate target mean $\mu_0$
   • Assess detection performance
   • Adjust for changing market conditions

3. Integration with other monitoring tools:

   • Anomaly Score systems
   • Alert Thresholding mechanisms
   • Real-time Analytics platforms