5.2.8 CuSum Chart

CuSum Chart Introduction The CuSum chart (Cumulative Sum Control Chart) is a statistical process control tool designed to detect small, sustained shifts in process performance more quickly than traditional Shewhart control charts (like X̄ or Individuals charts). This article explains: - What a CuSum chart is - How it is constructed - How to choose key parameters - How to interpret and act on signals - How it compares to Shewhart charts The focus is on practical, analysis-ready understanding aligned with IASSC Black Belt expectations for CuSum charts. --- Concept of Cumulative Sum Basic Idea A CuSum chart monitors the cumulative sum of deviations of individual data points from a target or reference value. Instead of looking at each point alone, it accumulates the history of small departures from the target. - Standard control charts: Compare each point to control limits. - CuSum: Adds up small deviations over time; many small deviations in the same direction generate a strong signal. This makes CuSum charts effective for detecting: - Small mean shifts (often around 1–2 standard deviations) - Gradual drifts in the process average Reference Value and Target CuSum requires a target (or reference mean) and an estimate of process standard deviation. - Target (μ₀): The desired process mean or historical in-control mean. - σ (sigma): The process standard deviation when the process is in control. - Shift of interest (δ): The size of mean shift (in multiples of σ) that you want rapid detection for. The chart is then tuned to be sensitive to detecting this δ-sigma shift. --- CuSum Formulations There are two commonly used CuSum formulations: - Tabular (V-mask equivalent) CuSum - Standardized CuSum Both achieve similar objectives but use slightly different calculations and visualizations. Tabular (One-Sided) CuSum The tabular CuSum uses recursive formulas for upper and lower cumulative sums. Let: - xᵢ = i-th observation - μ₀ = target mean - σ = in-control standard deviation - k = reference value (also called allowance) - H = decision interval (control limit for the CuSum) The one-sided CuSum statistics are: - Upper CuSum (C⁺): - C⁺₀ = 0 - C⁺ᵢ = max[0, C⁺ᵢ₋₁ + (xᵢ − μ₀ − k)] - Lower CuSum (C⁻): - C⁻₀ = 0 - C⁻ᵢ = min[0, C⁻ᵢ₋₁ + (xᵢ − μ₀ + k)] Interpretation: - C⁺ accumulates evidence that the mean has shifted upward. - C⁻ accumulates evidence that the mean has shifted downward. - Each is reset toward zero by the max/min functions when the data move in the opposite direction. A signal occurs when: - C⁺ᵢ > H (process mean likely increased), or - C⁻ᵢ < −H (process mean likely decreased) Standardized CuSum Sometimes it is convenient to work with standardized data: - zᵢ = (xᵢ − μ₀) / σ Then use: - C⁺₀ = 0 - C⁺ᵢ = max[0, C⁺ᵢ₋₁ + (zᵢ − k′)] - C⁻₀ = 0 - C⁻ᵢ = min[0, C⁻ᵢ₋₁ + (zᵢ + k′)] where k′ is defined in terms of the standardized shift you want to detect. The logic is identical; only the units change (now in standard deviations rather than original units). --- Choosing CuSum Parameters Reference Value k The reference value k determines how much of a deviation from target is “counted” at each point. It is usually set to half the shift (in process units) that you want to detect. Let: - δ = shift of interest in standard deviation units (e.g., δ = 1 means a 1σ shift) Then: - In original units: k = δσ / 2 - In standardized form: k′ = δ / 2 Common choices: - Detect 1σ shift: k′ = 0.5 - Detect 1.5σ shift: k′ = 0.75 - Detect 2σ shift: k′ = 1.0 Larger k makes the chart less sensitive to small shifts and more focused on larger ones. Decision Interval H The decision interval H is the control limit applied to the CuSum statistics (C⁺ and C⁻). When the CuSum crosses H (or −H), an out-of-control signal is triggered. H is chosen to balance: - Sensitivity (detect shifts quickly) - False alarm rate (keep in-control signals rare) Typical guidelines in standardized units: - For detecting a 1σ shift: - H′ ≈ 4 to 5 - For detecting around 1.5σ: - H′ ≈ 4 to 5 - For detecting a 2σ shift: - H′ ≈ 4 to 5 More precisely, H and k can be chosen based on desired Average Run Length (ARL): - In-control ARL (ARL₀): Average number of samples until a false alarm when the process is stable. - Out-of-control ARL (ARL₁): Average number of samples until detection of a specified shift. Increasing H raises ARL₀ (fewer false alarms) but also raises ARL₁ (slower detection). --- Interpreting CuSum Charts Visual Structure A typical tabular CuSum chart displays: - Time or sample number on the horizontal axis. - CuSum values (C⁺ and/or C⁻) on the vertical axis. - Horizontal decision limits (H and −H) or an equivalent plotted tolerance band. Key elements: - C⁺ line: Trends upward when data tend to be above the target by more than k. - C⁻ line: Trends downward when data tend to be below the target by more than k. - Reset to zero region: When the process moves back toward target, the CuSum is pushed back toward zero. Out-of-Control Signals Out-of-control conditions include: - Threshold crossing: - C⁺ > H → suggests a sustained upward shift in the mean. - C⁻ < −H → suggests a sustained downward shift in the mean. - Persistent drift toward H: - Even before crossing H, a strong and consistent trend toward one limit indicates buildup of evidence of a shift. When a signal occurs: - Investigate potential special causes that could have shifted the mean. - Consider when the run began to consistently deviate; often, the change point precedes the crossing of H by several points. Comparing to Shewhart Charts CuSum is not a replacement for Shewhart charts; they are complementary. - CuSum strengths: - Very sensitive to small, persistent shifts in the mean. - Uses cumulative information from all past points. - Good when detecting drifts is more critical than detecting single extreme points. - Shewhart strengths: - Fast detection of large, sudden shifts or single-point extremes. - Simpler visual interpretation of individual outliers. Typical usage: - Shewhart chart for quick detection of large or sudden changes. - CuSum chart layered on the same data (or run in parallel) to detect smaller, systematic drifts. --- Types of CuSum Applications One-Sided vs Two-Sided CuSum - One-sided CuSum: - Monitors only upward or only downward shifts. - Useful when only one direction of change is critical (e.g., too high impurity). - Two-sided CuSum: - Monitors both C⁺ and C⁻ simultaneously. - Standard when deviations in either direction are important. Choice depends on: - Process requirements - Specification risks (upper, lower, or both sides critical) Individual Measurements vs Averages CuSum can be applied to: - Individual measurements: - Use the raw data xᵢ. - Estimate σ from an appropriate individuals-based method (e.g., moving ranges). - Sample means: - Use subgroup averages (e.g., X̄). - σ is then often σₓ̄ = σ / √n, where n is subgroup size. Main effect: - Using averages typically reduces σ, making the chart more sensitive, but requires formed subgroups and associated sampling costs. --- Constructing a CuSum Step by Step Data and Parameters To build a tabular CuSum chart: - Collect sequential data: x₁, x₂, …, xₙ. - Estimate: - Target mean μ₀ (from specs or stable historical data). - Standard deviation σ (from historical in-control data). - Choose: - Shift of interest δ (e.g., detect 1σ shift). - Compute k = δσ / 2. - Choose decision interval H to achieve desired ARL behavior. Recursive Calculation For each data point xᵢ: - Compute the increment for C⁺: d⁺ᵢ = xᵢ − μ₀ − k - Compute the increment for C⁻: d⁻ᵢ = xᵢ − μ₀ + k Then: - C⁺ᵢ = max[0, C⁺ᵢ₋₁ + d⁺ᵢ] - C⁻ᵢ = min[0, C⁻ᵢ₋₁ + d⁻ᵢ] Start with C⁺₀ = 0 and C⁻₀ = 0. At each step: - Update C⁺ and C⁻. - Check whether: - C⁺ᵢ > H, or - C⁻ᵢ < −H. If a limit is crossed, interpret it as signal of a shift and investigate. --- Practical Considerations Assumptions CuSum performance relies on certain data conditions: - Data are in time order (sequence matters). - Observations are approximately independent. - The in-control distribution of the statistic (individual or average) is approximately normal or at least symmetric around μ₀. - The estimate of σ remains valid and reasonably stable while the chart is used. When these are violated: - CuSum parameters and ARL properties can be distorted. - Pre-analysis of data structure (e.g., autocorrelation, non-normality) may be necessary before relying on CuSum performance metrics. Starting Point and Warm-Up Often, CuSum charts are started with: - C⁺₀ = 0 - C⁻₀ = 0 Alternative approaches: - Start with a short historical baseline and initialize near empirical values. - Allow an initial “warm-up” period where the chart is not used for formal decision-making, until behavior stabilizes. Resetting After an Alarm After an out-of-control signal: - Investigate and address the special cause. - Once the process is believed to be back in control and centered on μ₀: - Re-estimate μ₀ and σ if necessary. - Reset C⁺ and C⁻ to zero and resume monitoring. This prevents past out-of-control history from biasing future monitoring. --- Summary CuSum charts monitor cumulative deviations from a target to provide rapid detection of small, sustained shifts in a process mean. They rely on: - A defined target mean and standard deviation - A reference value k that sets the shift size of interest - A decision interval H that controls sensitivity and false alarms Tabular CuSum uses recursive formulas for upper (C⁺) and lower (C⁻) cumulative sums, signaling out-of-control conditions when these exceed defined limits. CuSum charts excel at identifying subtle drifts that may be missed or detected late by traditional Shewhart charts, making them a powerful complement in process monitoring and improvement focused specifically on mean shifts.