Cusum Basics

Mel Alexander, ASQ Fellow, CQE

Tutorial: ASQ - Baltimore Section

January 13, 2004Phone: 410-712-7426/work

E-mail: alexanderm@dfmc.org or vze3gmwz@verizon.net

Purpose

Review how Cumulative Sum (Cusum) charts can detect small process shifts sooner than standard control charting schemes

Agenda

•Cusum Background• Why CUSUM are useful• Examples where Cusum are used• Ways of constructing Cusums• Future Trends and Developments• Q & A

Cusum Background• Roots began with Abraham Wald’s

Sequential Probability Ratio Test (SPRT) in the late 1940s

• Cusums were introduced by E.S. Page in 1954

• Popularized by Jim Lucas (et al.) at DuPont in the 1970s and 1980s

Cusum Background cont.

Wald’s SPRT used a 3-way sequential sampling where samples of n 1 were taken at stages to:

(1) declare that a process is in control;

(2) find the process out-of-control; (3) take additional observations

Cusum Background cont.

The advantage SPRT had over fixed-sized sampling was decisions regarding the two risks associated with shift detection and the size of the shift were determined in advance.

The two risks regarding shift detection are:

- finding false alarms (finding process shifts that did not occur)

- missing shifts that did occur (fail to detect process shifts that occurred)

Cusum Background cont.

With fixed-sized sampling, is usually specified first, while is computed for different process level values

Wald and G.A. Barnard (in 1945, 1946) showed that the acceptance and reject limit numbers (Ca and Cr, respectively) must satisfy the relationships: and

so that a decision interval boundary could be formed Ca < SPRT < Cr

aC

1

1

rC

Why Cusums are Useful

Cusums are more capable of detecting:

• small changes in process levels• the start when processes drifts

out-of-control

Each Cusum point gives the cumulative history of processes

• small systematic shifts easily detected

• but large abrupt shifts detected faster with Shewhart charts

Why Cusums are Useful

Cusums (Si) plot the cumulative sums of deviations of sample values (Xis) from a target value or aim (T) over time

Si = where

Xi = process output value of the i-th item or sample (sometimes = i-th mean may be used),

T = Target value or aim, T may be estimated with the in- control mean ,

n= number of samples collected, tested, or baselined

)(1

TXn

ii

iX

X

Why Cusums are Useful

Where Cusums are used• Gil Culfari presented Bioprocess Protein%

tutorial at the Sept. 2003 ASQ-Baltimore Section meeting (see http://www.asqbaltimore.org/gilcsep16.htm)

• Chemical Process industries (DuPont applied more than 10,000 cusums between 1980s-1990s)

• In Healthcare, cusums helped assess physicians’ clinical competence performing surgeries and managing hospital length of stay (los) by patients

• In business & finance, change-point analysis use cusums to monitor the impact of trade deficits on stock market portfolios and in handling product/service complaints by consumers

Cusum Construction Approaches

(1) Tabular Approach (preferred method to easily implement with spreadsheet software)

• ARL - average number of samples/items/subgroups tested before an out-of-control signal is sent or shift is detected

(2) V-mask proposed by G.A. Barnard in 1959

• Error probability considerations ( , , delta shift)

(3) Fast Initial Response (FIR) proposed by Lucas and Crosier (Technometrics, 24, 199-205,1982)

Tabular One-Sided Decision Interval Cusum

Uses deviations above (below) the target T that is calculated as:

Upper Cusum: (Shi,i) = max[0, Shi,i-1 + Xi – (T - k)]

Lower Cusum: (Slo,i) = max[0, Slo,i-1 + (T- k) - Xi]

where starting values Shi,0 = Slo,0 = 0,

Next, we find parameter values K and H

K – Reference Value

K = reference value (a.k.a. allowance or slack) equal to some constant (multiple, coefficient) times - sigma,

i.e, standard deviation estimated from values, subgroup ranges, or average moving range.

Usually, K= 0.5 x Delta = where

Delta =the amount of shift from the target (T) we seek to detect. Usually, Delta equals sigma (= )

Xout = out-of-control value of the mean (= T + K )

TX out 5.0

H Decision Interval

The parameter H serves as a decision point (like a control limit) that works as follows:

H=4 or 5 indicates an out-of-control signal

wheneverShi,i > H or Slo,i > H for sample item (or

subgroup) i

Parameters H and K are designed to yield large Average Run Lengths (number of samples before an signaling an out-of-control condition) when process is on target, denoted as ARL(0).

As the process shifts by the size of Delta, the Average Run Lengths should be small, denoted as ARL(Delta).

Tables exist that show the relationship of ARLs to H, K, and Delta

Average Run Lengths for One-Side V-Mask CUSUMs Based H, K, DeltaParameters (Delta shift in mean)

H K 0 0.25 0.5 0.75 1 1.5 2 2.5 3 4 52.5 0.25 27.27 13.43 7.96 5.42 4.06 2.71 2.06 1.68 1.42 1.11 1.01

4 0.25 77.08 26.68 13.29 8.38 6.06 3.91 2.93 2.38 2.05 1.61 1.236 0.25 350.8 51.34 20.9 12.37 8.73 5.51 4.07 3.26 2.74 2.13 1.98 0.25 736.78 84 28.76 16.37 11.39 7.11 5.21 4.15 3.48 2.67 2.14

10 0.25 2071.51 124.66 36.71 20.37 14.06 8.71 6.36 5.04 4.2 3.2 2.652 0.5 38.55 18.19 10 6.32 4.45 2.74 1.99 1.58 1.32 1.07 1.013 0.5 117.6 39.47 17.35 9.68 6.4 3.75 2.68 2.12 1.77 1.31 1.074 0.5 335.37 77.08 26.68 13.29 8.38 4.75 3.34 2.62 2.19 1.71 1.315 0.5 930.89 141.69 38.01 17.05 10.38 5.75 4.01 3.11 2.57 2.01 1.696 0.5 2553.11 250.8 51.34 20.9 12.37 6.75 4.68 3.62 2.98 2.24 1.95

1.5 0.75 42.57 21.09 11.59 7.09 4.78 2.73 1.9 1.48 1.24 1.04 12.25 0.75 139.71 51.46 22.38 11.66 7.13 3.73 2.51 1.91 1.56 1.16 1.02

3 0.75 442.8 117.6 39.47 17.35 9.68 4.73 3.12 2.36 1.93 1.41 1.113.75 0.75 1375.71 258.96 65.65 24.16 12.37 5.73 3.71 2.79 2.27 1.72 1.314.5 0.75 4251.69 559.95 105.12 32.09 15.15 6.73 4.31 3.21 2.59 1.97 1.6

1 1 35.29 19.22 11.21 7.03 4.75 2.63 1.78 1.38 1.17 1.02 11.5 1 93.85 42.57 21.09 11.59 7.09 3.5 2.24 1.66 1.34 1.07 1.01

Tabular Cusum Example

T Xout Delta= Sigma K=0.5 x Delta H= 5 x Delta1000 1080 128 64 640

i XiShi,i = Xi - T - K + Shi, i-1 max(0, Shi,i ) Shi,i Status Slo,i = T - K - Xi + Slo,i -1 max(0,Slo,i) Slo,i

Status0 0 0 0

1 1580 516 516 OK -644 0 OK2 1020 472 472 OK -728 0 OK3 550 -42 0 OK -342 0 OK4 890 -216 0 OK -296 0 OK5 1770 490 490 OK -1130 0 OK6 1610 1036 1036 Off-Aim -1804 0 OK

V-mask (Error Probability) Approach

The V-mask is the classical cusum two-sided scheme

• Estimates H and K from the error probabilities

- finding false shifts - missing real shifts• Interprets the cusum as a reverse SPRT

(working backwards through past data)• A sideways-shaped V placed a fixed

distance after the last data point

Cumulative Sum (St )

Constructing a Cusum V-mask

Subgroup Index (t )

- semi angle

Formulas for Constructing a Cusum V-mask

Lead distance d =

=

H = d tan( )

1

ln2

2Delta

K

Delta

2tan 1

Hd

• Data collected on 20 Sample means of size 4

• Each sample mean input as single data point

• Target of 325, • Delta = = 0.6325,• = 0.0027 (equivalent

to Shewhart’s 3 )• = 0.01

Cusum Example with V-mask(Source

http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc323.htm)

i data1 324.9252 324.6753 324.7254 324.355 325.356 325.2257 324.1258 324.5259 325.225

10 324.611 324.62512 325.1513 328.32514 327.2515 327.82516 328.517 326.67518 327.77519 326.87520 328.35

Cusum Example with V-mask cont.

• V-mask over last data point has cusum point below lower arm, indicating upward drift

• In-control ARL(0) 63 or 64

• Out-of-control ARL(Delta) 7

-5

0

5

10

15

20

25

Cum

ulat

ive

Sum

of C

olum

n 1

2 4 6 8 10 12 14 16 18 20 22

Sample

CUSUM of Column 1

Target

Delta

Shift

Sigma

Head Start

325.0000

1.0000

0.6350

0.6350

0.0000

Parameters

ARL (Delta)

ARL (0)

6.82922

63.26935

ARL

Control Chart

Cusum Example with V-mask cont.

• Moving V-mask backwards through past data helped find where shift-signal first occurred.

• First signal of upward shift took place at sample 14 since lower V-mask arm did not cross data at sample 13 (i.e., process was in control)

-5

0

5

10

15

20

25

Cum

ula

tive S

um

of C

olu

mn 1

2 4 6 8 10 12 14 16 18 20 22

Sample

CUSUM of Column 1

Control Chart

-5

0

5

10

15

20

25

Cum

ula

tive S

um

of C

olu

mn 1

2 4 6 8 10 12 14 16 18 20 22

Sample

CUSUM of Column 1

Control Chart

Fast Initial Response (FIR) at Headstart

• Introduced to increase cusum sensitivity upon startup

• Sets starting values of Shi,0 and Slo,0 to some nonzero value, say H/2 (a.k.a. 50 percent headstart)

• FIR detects out-of-control situations 40% faster than standard cusums

• Drifts to zero quickly for in-controlled processes

Cusums can be used to monitor process variabilty

• For Xis N(0, ), the standardized Xi is zi = (Xi - 0)/

• A new standardized quantity by Hawkins (JQT, 13, 228-231, 1981; JQT, 25, 248-261, 1993) is defined by:

vi =

• Hawkins suggested that the vi s were more sensitive to variance changes than mean changes.

349.0

822.0iy

•So vi N(0,1), and the two-sided Scale Cusum is defined as:

Shi,i = max(0, vi – k + Shi,i-1)

Slo,i = max(0, vi – k + Slo,i-1) where

Shi,i = Slo,i = 0.

If either Shi,i > H or Slo,i > H, then the process is declared out-of-control

Cusums for monitoring process variabilty cont.

Other Types of Cusums • Cusums have been been studied on

binomial and Poisson (attributes) and non-normal data.

• See Lucas(Technometrics, 27, 129-144, 1985); Ewan & Kemp (Biometrika, 47, 363-380, 1960); Wadsworth et al., Modern Methods for Quality Control and Improvement, (Wiley, 1986); Ryan, Statistical Methods for Quality Improvement, (Wiley, 1989); Bourke(1999, http://www.stat.fi/isi99/proceedings/arkisto/bork0597.pdf); and British Standards Institution (BS5703-4, 1997 or ISO/TR 7871:1997) for more information.

Cusum Limitations • Short term drifts or erratic behavior in the

process mean may not be detected.• Not as effective in detecting large process

shifts as Shewhart charts, but is corrected with a combined Cusum-Shewhart scheme, See Lucas(JQT, 14, 51-59, 1982) for more information.

• Since parameters that construct cusums depend much on the ratio of vertical and horizontal axes, this may require scales to be redrawn and redefined as more data are collected.

Cusum Future Trends and Developments

Manhattan Control • Dr. Juergen Ude of Australia modified

Woodward and Goldsmith’s (Cumulative Sum Techniques, Oliver and Boyd for ICI, 1964) approaches to detect onset and duration of changes in manufacturing processes. Visit http://www.qtechinternational.com for more information

What are Manhattan Charts?What are Manhattan Charts? Manhattan Control charts test for relative

changes.

This meanis significantrelative to thismean.

Statistical significance tests for relative changes are performed on adjacent local means that help identify new problems

Cusum Future Trends and Developments cont.

Change-Point Analysis• Wayne Taylor combined Cusum charting

scheme with bootstraping (resampling) to detect changes on various kinds of data (time-ordered, non-normal, customer complaints, and data with outliers).

• Adaptive CUSUM that adjusts to signal one-step ahead forecasts of varying location shifts in deviations from target. See Sparks (JQT, 32, 157-171, 2000) for details.

Change-Point Analysis Example: Plot of US Trade Deficit Data Showing Changes in

BackgroundFor more information, visit Wayne Taylor’s web site: http://www.variation.com/cpa/tech/changepoint.html