Cusum

Types of cusum

Cusum is a method of monitoring the process, which is used in quality control to detect small changes in the average value of a process over time. This method is based on the idea of cumulative sums, where the sum of deviations from a target value is calculated cumulatively. Here are some types of cusums:

• Sequential Cumulative Sum

  This is a CUSUM analysis in which a CUSUM statistic is defined as the cumulative sum of the deviations from the target value. When the measurements are taken to be independent and identically distributed (IID) random variables from a known distribution, the expected value of the cumulative sum (CUSUM) statistic can be expressed in terms of the mean of the distribution.

• Two-Sided CUSUM

  A two-sided CUSUM chart can be used to detect shifts in both directions. It has two cumulative sums, one for each direction:

  • Positive Cumulative Sum (CUSUM+)

  • Negative Cumulative Sum (CUSUM-)

  Each time a value is observed, it is added to the appropriate CUSUM. If the observed value is greater than the target, it adds to CUSUM+ and resets CUSUM-. If it's less than the target, it adds to CUSUM- and resets CUSUM+. If either CUSUM exceeds a predetermined threshold, it indicates a potential shift in the process mean.

• One-Sided CUSUM

  One-sided CUSUM is used when shifts are only anticipated in one direction. For example, if a process is only expected to increase, then only the positive CUSUM would be used:

  • Positive Cumulative Sum (CUSUM+)

  In this case, deviations from the target are accumulated only in one direction. If a process is only expected to decrease, then only the negative CUSUM would be calculated.

• Exponential CUSUM

  Exponential CUSUM (also known as CUSUM with forgetting factor) is used when recent observations are considered more informative than older ones. This is achieved by applying a discount factor to previous cumulative sums:

  Mathematically, if the current CUSUM is represented by C, the discount factor is λ (0 < λ < 1), and the last CUSUM value was C', then:

  C = λC' + (1 - λ) \times (X - \mu)

  Here, X is the current observation, and μ is the target mean. The discount factor λ gives more weight to recent observations, allowing the CUSUM to adapt more quickly to changes.

• Multivariate CUSUM

  Multivariate CUSUM is used when monitoring several related processes simultaneously. It takes into account the correlation between different variables:

  It uses a vector of cumulative sums, one for each variable, and a multivariate normal distribution to model the joint behavior. This allows it to detect shifts in any of the variables while considering their interdependencies.

• Adaptive CUSUM

  Adaptive CUSUM adjusts its parameters dynamically based on the observed process variability. It might change the target or the threshold for the cumulative sum based on recent performance. This allows it to maintain sensitivity even when the process behavior changes over time.

Design of cusum

Here are some of the following designs of Cumulative Sum:

• Strip CUSUM: In the design of strip cusum, it is the most common and simplest CUSUM method. It uses two boundaries, one upper and one lower, to detect shifts in the mean of a process. The boundaries are set based on the acceptable level of risk for false positives and false negatives. The CUSUM chart is updated with each new observation. If the cumulative sum exceeds the upper boundary, it signals a potential shift in the process mean in the positive direction. Conversely, if it falls below the lower boundary, it indicates a shift in the negative direction. This design is straightforward and effective for detecting mean shifts.
• Multivariate CUSUM: Processes often have multiple correlated quality characteristics. This design takes into account the multivariate nature of the data. It uses a joint CUSUM statistic that combines information from all quality characteristics. The CUSUM statistic is based on the Mahalanobis distance, which measures how far the process vector is from its target vector in multivariate space. Control limits are set based on the distribution of this statistic. If the statistic exceeds the control limits, it signals a shift in one or more quality characteristics. Multivariate CUSUM is more sensitive to shifts because it considers the correlation between variables.
• Adaptive CUSUM: This design adjusts its parameters dynamically based on the process behavior. It uses a feedback loop to update the CUSUM parameters in real-time. For example, the target value and control limits may be adjusted based on recent process performance. This allows the CUSUM to be more sensitive during periods of high variability and less sensitive during stable periods. The adaptive parameters are typically estimated using statistical methods such as maximum likelihood estimation. This design is useful for processes with varying levels of noise.
• Sequential CUSUM: This design processes data sequentially, updating the CUSUM statistic with each new observation without waiting for a fixed sample size. It uses a stopping rule based on the CUSUM statistic to detect shifts as soon as they occur. The stopping rule specifies a threshold for the CUSUM statistic. If the threshold is exceeded, it indicates a shift, and the process is stopped for investigation. The design is efficient for real-time monitoring and allows for quick detection of shifts without waiting for a predefined sample size.
• Robust CUSUM: This design is less sensitive to outliers and non-normality in the data. It uses robust statistical methods to calculate the CUSUM statistic. For example, instead of the sample mean, it may use the median to estimate the process center. Control limits are set based on the distribution of the robust estimator. This design is suitable for processes known to have outliers or non-normal distributions.

Wearing/Matching suggestions of cusum

Cusum can be worn and matched in a variety of ways depending on the occasion. Here are some suggestions:

• Formal Occasions

  Cusum shirts can be dressed up for formal events by pairing them with dress pants or a skirt. Opt for a solid-colored cusum shirt in a neutral tone like white, black, or navy for a classic and sophisticated look. Match it with tailored trousers or a pencil skirt in a complementary color. Add a blazer for an extra touch of professionalism and elegance. Complete the outfit with polished dress shoes and minimalistic accessories such as a watch or simple jewelry.

• Business Casual

  For a business casual setting, cusum shirts offer versatility. Pair a striped or patterned cusum shirt with chinos or khaki pants for a balanced look. Choose colors that are not too bold or bright to maintain a professional appearance. Roll up the sleeves slightly for a more relaxed vibe. Opt for loafers or smart casual shoes to complete the outfit. Accessories like a leather belt and a classic watch can add a touch of sophistication.

• Casual Outings

  For casual outings, cusum shirts can be styled in various ways. A short-sleeved cusum shirt can be paired with denim jeans or shorts for a laid-back and comfortable look. Choose a cusum shirt with a fun print or bright colors to add a playful element to the outfit. Sneakers or casual slip-on shoes work well for this style. Consider adding a baseball cap or sunglasses for a trendy touch.

• Layering

  Layering is another way to wear and match cusum shirts. During colder months, layer a long-sleeved cusum shirt under a sweater or cardigan. Choose a sweater with a V-neck so that the collar of the cusum shirt is visible, adding a touch of class to the ensemble. Pair this layered look with jeans or chinos and finish off with boots or closed-toe sandals. A scarf can be added for extra warmth and style.

• Accessories

  Accessories play a crucial role in enhancing the look of cusum shirts. Add a tie to a long-sleeved cusum shirt for a more polished appearance, suitable for formal events or business meetings. Choose a tie that complements the color or pattern of the cusum shirt. Pocket squares, cufflinks, and belts can also elevate the style of cusum shirts for formal and business casual occasions. For casual looks, consider adding a watch, bracelets, or a casual belt to complete the outfit.

Q&A

Q1: What are the steps to create a cusum chart?

A1: Typically, the first step is to determine the average value of the variable being monitored. Next, this average is used as a baseline to calculate deviations for each subsequent data point. These deviations are then cumulated, hence the name CUSUM. The cumulative values are plotted on a chart to create the CUSUM control chart.

Q2: What is the purpose of CUSUM?

A2: The main purpose of CUSUM is to detect shifts or trends in a process or system over time. By accumulating the deviations from a target or average, it can highlight small shifts that might indicate a problem or change in the process.

Q3: What does CUSUM stand for in statistics?

A3: CUSUM stands for "Cumulative Sum." It refers to a technique used in statistical quality control and process monitoring.

Q4: What are the advantages of CUSUM?

A4: Some advantages of CUSUM include its sensitivity to small shifts, its ability to detect changes quickly, and its cumulative nature, which allows it to highlight trends over time. It's also versatile and can be used in various fields, including manufacturing, healthcare, and finance.

Q5: Are there any limitations to CUSUM?

A5: While CUSUM is a powerful tool, it also has limitations. For example, it can be affected by variability in the process, leading to false positives. It also requires a clear understanding of the process and the factors being monitored to set up correctly and interpret the results.