NelsonRules.jl Documentation

Nelson rules are a method in process control of determining whether some measured variable is out of control (unpredictable versus consistent). Rules for detecting "out-of-control" or non-random conditions were first postulated by Walter A. Shewhart in the 1920s. The Nelson rules were first published in the October 1984 issue of the Journal of Quality Technology in an article by Lloyd S Nelson.

The rules are applied to a control chart on which the magnitude of some variable is plotted against time. The rules are based on the mean value and the standard deviation of the samples.

More information on the Nelson Rules are available at https://en.wikipedia.org/wiki/Nelson_rules

All rule methods in this module accept the same type of argument and return the same type of return value:

Arguments

series::Real[] : A timeseries vector to check for rule violation. Each entry is for a single unit of time.

Returns

Zip{Tuple{Integer[],Integer[]}}: An iterator of indices and sequence length of sequences that violate the particular Nelson Rule.

See the individual method documentation for rule description, examples and nuances.

Methods are named ruleN where N goes from 1-8. To make it easier to call multiple rules programmatically, methods may also be called via rule(::Val{N}, series) where N goes from 1-8.

Detects violation of nelson rule 1: One point is more than 3 standard deviations from the mean.

Problem Indicated: One or more samples are grossly out of control.

The sequence length in the return value will always be 1.

Example

julia> NelsonRules.rule1([1, 2, 4, 5, 6, 7, -205, 9, -10, 12, 13, 200, 10, -5, 8, 3, -5, 5, 3, 9, -12, 17])
zip([7, 12], [1, 1])

Detects violation of nelson rule 2: Nine (or more) points in a row are on the same side of the mean.

Problem Indicated: Some prolonged bias exists.

Example

julia> NelsonRules.rule2([39, 398, 4, 76, 435, 188, 236, 283, 481, 271, 270, 274, 270, 272, 273, 273, 271, 271, 384, 194, 57, 232, 494, 468, 417, 104, 323, 469, 136, 214, 393, 267, 160, 385, 253, 155, 289, 455, 104, 289, 138, 184, 356, 186, 146, 268, 76, 258])
zip([8], [12])

julia> NelsonRules.rule2([26, 31, 46, 47, 81, 6, 88, 23, 73, 1, 66, 73, 6, 84, 70, 36, 80, 94, 63, 37, 62, 84, 53, 54, 80, 75, 26, 56, 48, 3, 6, 56, 21, 43, 87, 28, 47, 73, 63, 48, 68, 60, 63, 70, 60, 67, 61, 61, 66])
zip([41], [9])

Detects violation of nelson rule 3: Six (or more) points in a row are continually increasing (or decreasing).

Problem Indicated: A trend exists.

Example

julia> NelsonRules.rule3([62, 79, 70, 81, 82, 83, 84, 87, 13, 83, 32, 5, 13, 36, 93, 74, 34, 20, 69, 96, 98, 101, 104, 107, 110])
zip([3, 18], [6, 8])

Detects violation of nelson rule 4: Fourteen (or more) points in a row alternate in direction, increasing then decreasing.

Problem Indicated: This much oscillation is beyond noise.

Example

julia> NelsonRules.rule4([1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 2])
zip([1], [14])

Detects violation of nelson rule 5: Two (or three) out of three points in a row are more than 2 standard deviations from the mean in the same direction.

Problem Indicated: There is a medium tendency for samples to be mediumly out of control.

Note that the same sequence might show up for consecutive points as 2 out of 3 points can occur in two separate groups of 3 points.

Example

julia> NelsonRules.rule5([1524, 1583, 2284, -882, 2184, -485, 57, -13, -3494, -3150, 1148, 2182, -953, 863, -31, -621, 947, -65, 323, -237])
zip([8, 9], [2, 2])

Detects violation of nelson rule 6: Four (or five) out of five points in a row are more than 1 standard deviation from the mean in the same direction.

Problem Indicated: There is a strong tendency for samples to be slightly out of control.

Example

julia> NelsonRules.rule6([816, 555, 712, 883, 397, 717, 165, 135, 261, 751, 1765, 1858, 1395, 1263, 1969, 253, 783, 631, 145, 924, -914, -701, -361, -590, 252, 848, 371, 546, 113, 984])
zip([10, 11, 12, 20, 21], [4, 5, 4, 4, 4])

Detects violation of nelson rule 7: Fifteen points in a row are all within 1 standard deviation of the mean on either side of the mean.

Problem Indicated: With 1 standard deviation, greater variation would be expected.

The sequence length in the return value will always be 15. Note that a sequence of more than 15 points will show as multiple consecutive sequences of 15 points each.

Example

julia> NelsonRules.rule7([13, 81, 96, 40, 24, 66, 24, 34, 27, 72, 32, 73, 74, 22, 59, 39, 69, 62, 60, 2, 52, 51, 48, 25, 40, 60, 23, 109, -15, 57])
zip([4, 5], [15, 15])

Detects violation of nelson rule 8: Eight points in a row exist, but none within 1 standard deviation of the mean, and the points are in both directions from the mean.

Problem Indicated: Jumping from above to below while missing the first standard deviation band is rarely random.

The sequence length in the return value will always be 8. Note that a sequence of more than 8 points will show as multiple consecutive sequences of 8 points each.

Example

julia> NelsonRules.rule8([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1])
zip([19, 20], [8, 8])