Monitoring statistics

AEWMA(λ, k, value)

Adaptive exponentially weighted moving average with design parameters λ, k, and initial value value.

The update mechanism for the statistic $C_t$ based on a new observation x is given by

$C_t = (1-\phi(e))\cdot C_{t-1} + \phi(e) * x$,

where \phi(e) is a forecast error function based on the Huber score function.

Arguments

• λ::Float64: The smoothing constant. Default is 0.1.
• k::Float64: The threshold value in the Huber score. Default is 3.0`.
• value::Float64: The initial value of the statistic. Default is 0.0.

References

Capizzi, G. & Masarotto, G. (2003). An Adaptive Exponentially Weighted Moving Average Control Chart. Technometrics, 45(3), 199-207.

CUSUM(k, value, upw::Bool)

CUSUM statistic with design parameter k and initial value value.

The update mechanism based on a new observation x is given by:

• if upw == true, then $C_t = \max\{0, C_{t-1} + x - k\}$;
• if upw == false, then $C_t = \min\{0, C_{t-1} + x + k\}$.

Arguments

• k::Float64: The allowance constant of the CUSUM statistic. Defaults to 1.0.
• value::Float64: The current value of the cumulative sum. Defaults to 0.0.
• upw::Bool: A boolean indicating whether the CUSUM statistic is increasing or decreasing. Defaults to true.

References

Page, E. S. (1954). Continuous Inspection Schemes. Biometrika, 41(1/2), 100. https://doi.org/10.2307/2333009

EWMA(λ, value)

Exponentially weighted moving average with design parameter λ and initial value value.

The update mechanism for the statistic $C_t$ based on a new observation x is given by

$C_t = (1-λ)\cdot C_{t-1} + λ \cdot x$.

Arguments

• λ::Float64: The smoothing constant. Defaults to 0.1.
• value::Float64: The initial value for the EWMA statistic. Defaults to 0.0.

References

Roberts, S. W. (1959). Control Chart Tests Based on Geometric Moving Averages. Technometrics, 1(3), 239-250. https://doi.org/10.1080/00401706.1959.10489860

OneSidedEWMA(λ, value, upw::Bool)

OneSidedEWMA statistic with design parameter λ and initial value value.

The update mechanism based on a new observation x is given by:

• if upw == true, then $C_t = \max\{0, (1-λ)\cdot C_{t-1} + λ\cdot x\}$;
• if upw == true, then $C_t = \min\{0, (1-λ)\cdot C_{t-1} + λ\cdot x\}$;

Arguments

• λ::Float64: The smoothing constant. Default is 0.1.
• value::Float64: The current value of the statistic. Default is 0.0.
• upw::Bool: Whether the statistic should monitor increases (true) or decrease (falses) in the process mean. Default is true.

Shewhart(value)

Shewhart control chart.

The update mechanism of $C_t$ based on a new observation x is given by

$C_t = x$.

References

Shewhart, W. A. (1931). Economic Control of Quality Of Manufactured Product. D. Van Nostrand Company.

WCUSUM <: AbstractStatistic

A weighted cumulative sum statistic.

Arguments

• k::Float64: The allowance constant of the CUSUM used for calculating the WCUSUM statistic. Default is 1.0.
• λ::Float64: The smoothing constant for updating the estimate of the fault signature. Must be a value between 0 and 1. Default is 0.2.
• value::Float64: The initial value of the weighted cumulative sum statistic. Default is 0.0.
• Q::Float64: The residual value of the weighted cumulative sum. Default is 0.0.
• upw::Bool: Whether to monitor increases in the mean (true) or decreases (false). Default is true.

References

Shu, L., Jiang, W., & Tsui, K.-L. (2008). A Weighted CUSUM Chart for Detecting Patterned Mean Shifts. Journal of Quality Technology, 40(2), 194-213. https://doi.org/10.1080/00224065.2008.11917725

Examples

stat = WCUSUM()
get_design(stat)            # returns [1.0, 0.2]
update_statistic(stat, 3.0) # returns 1.2
stat.Q                      # returns 0.6

ALT{M}

A Generalized Likelihood Ratio statistic for monitoring changes in the variance-covariance matrix.

Fields

• value::Float64: The value of the statistic, initialized to 0.0.
• Ω::M: The precision matrix of the in-control process.
• detΣ::Float64: The determinant of the in-control process variance.

References

Alt, F. A. (1984). Multivariate quality control. In N. L. Johnson, S. Kotz, & C. R. Read (Eds.), The encyclopedia of statistical sciences (Vol. 6, pp. 110-122). Wiley.

AMCUSUM(λ, p; minshift = 0.1, shift = 0.0, Et = zeros(p), t = 0, stat = MCUSUM(k=0.1, p=p))

An Adaptive Multivariate Cumulative Sum (MCUSUM) statistic.

Arguments

• λ::Float64: The smoothing constant, such that 0.0 <= λ <= 1.0.
• p::Int: The number of quality variables to monitor.
• minshift::Float64: The minimum shift value to be detected. Default is 0.1.
• shift::Float64: The current shift value. Default is 0.0.
• Et::Vector{Float64}: The vector Et of smoothed deviations from the zero mean. Has to be exactly equal to zeros(p)
• t::Int: The current value of time. Default is 0.
• stat::C: The underlying classical MCUSUM statistic. Default is MCUSUM(k=0.1, p=p).

References

Dai, Y., Luo, Y., Li, Z., & Wang, Z. (2011). A new adaptive CUSUM control chart for detecting the multivariate process mean. Quality and Reliability Engineering International, 27(7), 877-884. https://doi.org/10.1002/qre.1177

DiagMEWMA(Λ, value)

Exponentially weighted moving average with diagonal smoothing matrix Λ and initial value value.

The update mechanism based on a new observation x is given by

$Z_t = (I-Λ)Z_{t-1} + Λ x_t$,

and the chart value is defined as

$C_t = Z_t' Λ^{-1} Z_t$.

Arguments

• Λ::Vector{Float64}: The vector of smoothing constants.
• value::Float64: Current value of the statistic (default = 0.0).
• z::Vector{Float64}: Vector of smoothed observations (Default: zeros(length(Λ))).
• inv_Σz::Matrix{Float64}: Inverse of the covariance matrix of the control variates.

References

Lowry, C. A., Woodall, W. H., Champ, C. W., & Rigdon, S. E. (1992). A Multivariate Exponentially Weighted Moving Average Control Chart. Technometrics, 34(1), 46-53. https://doi.org/10.1080/00401706.1992.10485232

MAEWMA(λ, k, value, z::Vector{Float64})

A Multivariate Adaptive Exponentially Weighted Moving Average.

The update mechanism based on a new observation x is given by

$Z_t = (I-Ω)*Z_{t-1} + Ω * x_t$,

where Ω = ω(e)*I is an adaptive generalization of the classical MEWMA smoothing matrix. The chart value is defined as

$value_t = Z_t' Z_t$.

Arguments

• λ::Float64: The value of the EWMA smoothing parameter.
• k::Float64: The value of the parameter of the Huber score.
• value::Float64: The value of the statistic. (default: 0.0)
• z::Vector{Float64}: The vector of smoothed observations.

References

Mahmoud, M. A., & Zahran, A. R. (2010). A Multivariate Adaptive Exponentially Weighted Moving Average Control Chart. Communications in Statistics - Theory and Methods, 39(4), 606-625. https://doi.org/10.1080/03610920902755813

MCUSUM(k, p, value = 0.0, St = zeros(p))

A Multivariate Cumulative Sum (MCUSUM) statistic.

Arguments

• k::Float64: The value of the allowance parameter.
• p::Int: The number of variables to be monitored.
• value::Float64: The initial value of the statistic. Default is 0.0.
• St::Vector{Float64}: A vector representing the multivariate cumulative sum at the current time t.

Examples

stat = MCUSUM(0.25, 2, 0.0, [0.0, 0.0])

References

Crosier, R. B. (1988). Multivariate Generalizations of Cumulative Sum Quality-Control Schemes. Technometrics, 30(3), 291-303. https://doi.org/10.2307/1270083

MEWMC(λ, value)

An Exponentially Weighted Moving Covariance Matrix statistic with smoothing constant λ and initial value value.

The update mechanism for $C_t$ based on a new observation x \in \mathbb{R}^p is given by

$Z_t = (1 - λ)Z_{t-1} + λ \cdot xx'$,

and the chart value is defined as

$C_t = \text{tr}(Z_t) - \log|Z_t| - p$.

References

Hawkins, D. M., & Maboudou-Tchao, E. M. (2008). Multivariate Exponentially Weighted Moving Covariance Matrix. Technometrics, 50(2), 155-166.

MEWMS(λ, value)

Exponentially weighted moving covariance matrix with smoothing constant λ.

The update mechanism based on a new observation x \in \mathbb{R}^p is given by

$Z_t = (1 - λ)*Z_{t-1} + λ \cdot xx'$,

and the chart value is defined as

``valuet = \text{tr}(Zt).

References

Huwang, L., Yeh, A. B., & Wu, C.-W. (2007). Monitoring Multivariate Process Variability for Individual Observations. Journal of Quality Technology, 39(3), 258-278. https://doi.org/10.1080/00224065.2007.11917692

MShewhart(value)

Shewhart control chart for monitoring multivariate observations with initial value value.

The update mechanism of $C_t$ based on a new observation x is given by

$C_t = x'x$.

References

Shewhart, W. A. (1931). Economic Control of Quality Of Manufactured Product. D. Van Nostrand Company.

RiskAdjustedCUSUM{G} <: AbstractStatistic

Risk-adjusted CUSUM monitoring statistic.

Arguments

• Δ::Float64: Shift in the linear predictor of the logistic regression model to be detected.
• model::G: Logistic regression model used for prediction. Must have a predict(model, x) function.
• response::Symbol: Name of the response variable in the DataFrame.
• value::Float64: Initial value of the statistic. Defaults to 0.0.

References

Steiner, S. H., Cook, R. J., Farewell, V. T., Treasure, T. (2000). Monitoring surgical performance using risk-adjusted cumulative sum charts. Biostatistics, 1(4), 441-452. https://doi.org/10.1093/biostatistics/1.4.441 ```

LocationScaleStatistic{S, M, P}

A mutable struct representing a statistic applied to a location-scale family.

Fields

• stat::S: The statistic.
• μ::M: The location parameter.
• Ω::P: The inverse square root of the variance.

Examples

STAT = EWMA(λ = 0.2)
RSTAT = LocationScaleStatistic(STAT, 1.0, 2.5)

categorize_data(x::AbstractVector, qtls::Vector{Vector{Float64}})
categorize_data(x::AbstractMatrix, qtls::Vector{Vector{Float64}})

Categorize continuous data using the medians of the continuous variables.

compose(lhs::Symbol, rhs::Tuple)
compose(lhs::Symbol, rhs::Term)
compose(lhs::Symbol, rhs::ConstantTerm)

Compose a response variable lhs with a tuple or vector of predictors rhs terms.

Arguments

• lhs::Symbol: The left-hand side of the equation.
• rhs::Tuple: The right-hand side of the equation.

Returns

• The composed equation.

If rhs is empty, the function returns lhs composed with an intercept. Otherwise, the function returns lhs composed with the terms in rhs.

get_removable_terms(rhs)

Given an iterable collection of terms rhs, this function returns the indices of the terms in rhs that can be removed in a backward elimination step.

Arguments

• rhs: An iterable collection of terms in the equation.

Returns

• An array of indices of the terms in rhs that have the maximum order.

kronecker_matrix(qtls::AbstractVector)

Compute the matrix given by the Kronecker vectors according to Equation (6) of Wang et Al. (2017). This matrix is used to compute the approximate GLRT statistic to test the null hypothesis that the main effects and the second order interactions of a log-linear model are stable.

References

Wang, J., Li, J., & Su, Q. (2017). Multivariate Ordinal Categorical Process Control Based on Log-Linear Modeling. Journal of Quality Technology, 49(2), 108-122. https://doi.org/10.1080/00224065.2017.11917983

quantile_range(lower, upper, m)

Get the set of quantiles needed to divide the data in m classes.

LLCUSUM(x::AbstractMatrix, k::Real; ncuts::AbstractVector = [2 for _ in eachcol(x)])

A distibution-free Log-Linear CUSUM monitoring statistic based on data categorization.

Fields

• k::Float64: The allowance constant of the CUSUM statistic.
• value::Float64: The initial value of the statistic (default: 0.0).
• Sobs::Vector{Float64}: The vector of cumulative observed categories.
• Sexp::Vector{Float64}: The vector of expected observed categories.
• qtls::Vector{Vector{Float64}}: A vector of quantiles used for data categorization.
• f0::Vector{Float64}: The vector of IC cell probabilities, estimated using a loglinear model.
• table::Matrix{Float64}: The matrix of cell combinations.

References

Qiu, P. (2008). Distribution-free multivariate process control based on log-linear modeling. IIE Transactions, 40(7), 664-677. https://doi.org/10.1080/07408170701744843

categorical_to_index(g::AbstractVector, table::AbstractMatrix)

Converts a categorized variable g to its corresponding index in a table.

Arguments

• g::AbstractVector: The categorical variable to convert.
• table::AbstractMatrix: The table containing all the possible categories.

Returns

• Int: The index of the category in the table.

Raises

• ErrorException if the index is not found

Example

x = randn(500, 3)
df, table, qtls = create_table(x, [2 for _ in eachcol(x)])
xnew = randn(3)
g = categorize_data(xnew, qtls)
idx = categorical_to_index(g, table)

estimate_ordinal_model_probabilities(df, table)

Estimates the probabilities of an ordinal loglinear model based on the observed cell counts.

Arguments

• df::DataFrame: The data frame containing the predictor variables.
• table::AbstractMatrix: The matrix of predictor values for which probabilities are to be estimated.

Returns

• prob::Vector{Float64}: A vector of estimated probabilities.

LLD(x::AbstractMatrix, l::Real; ncuts::AbstractVector = [3 for _ in eachcol(x)], N = 1)

A Log-Linear Directional monitoring statistic based on data categorization.

Fields

• l::Float64: The exponentially weighted smoothing constant of the statistic.
• value::Float64: The initial value of the statistic (default: 0.0).
• qtls::Vector{Vector{Float64}}: A vector of quantiles used for data categorization.
• f0::Vector{Float64}: The vector of IC cell probabilities, estimated using a loglinear model.
• table::Matrix{Float64}: The matrix of cell combinations.
• Sigma::Matrix{Float64}: The covariance matrix of the cell counts.
• directions::Matrix{Float64}: The matrix of directions to be used in the hypohesis test (see Equation (6) of [Li, 2012]). This is calculated according to the procedure described in Equations (5) and (6) of [Wang, 2017].
• N::Int: The number of observations at each time point (default: 1)
• z_k::Vector{Float64}: The current vector of smoothed values.

References

Li, J., Tsung, F., & Zou, C. (2012). Directional Control Schemes for Multivariate Categorical Processes. Journal of Quality Technology, 44(2), 136–154. https://doi.org/10.1080/00224065.2012.11917889

Wang, J., Li, J., & Su, Q. (2017). Multivariate Ordinal Categorical Process Control Based on Log-Linear Modeling. Journal of Quality Technology, 49(2), 108-122. https://doi.org/10.1080/00224065.2017.11917983

MOC(x::AbstractMatrix, l::Real; ncuts::AbstractVector = [3 for _ in eachcol(x)], N = 1)

A Multivariate Ordinal Categorical monitoring statistic based on data categorization.

Fields

• l::Float64: The exponentially weighted smoothing constant of the statistic.
• value::Float64: The initial value of the statistic (default: 0.0).
• qtls::Vector{Vector{Float64}}: A vector of quantiles used for data categorization.
• f0::Vector{Float64}: The vector of IC cell probabilities, estimated using a loglinear model.
• table::Matrix{Float64}: The matrix of cell combinations.
• inv_VCOV::Matrix{Float64}: The matrix containing the inverse covariance to be used in the running statistic.
• N::Int: The number of observations at each time point (default: 1)
• z_k::Vector{Float64}: The current vector of smoothed values.

References

Wang, J., Li, J., & Su, Q. (2017). Multivariate Ordinal Categorical Process Control Based on Log-Linear Modeling. Journal of Quality Technology, 49(2), 108-122. https://doi.org/10.1080/00224065.2017.11917983

estimate_ordinal_model_probabilities(df, table)

Estimates the probabilities of an ordinal loglinear model based on the observed cell counts.

Arguments

• df::DataFrame: The data frame containing the predictor variables.
• table::AbstractMatrix: The matrix of predictor values for which probabilities are to be estimated.

Returns

• prob::Vector{Float64}: A vector of estimated probabilities.

NEWMA

A Nonparametric Exponentially Weighted Moving Average statistic for monitoring profiles.

Fields

• λ::Float64: The EWMA smoothing constant. Must be between 0 and 1.
• value::Float64: The current value of the NEWMA statistic.
• σ::Float64: The standard deviation of the error term.
• cdf_σ::E: A function that computes the (estimated) cdf of the error term σ with signature cdf_σ(x::Real).
• g::F: The estimated regression function object. Must have a method of signature predict(g::F, x::AbstractVector).
• Ej::Vector{Float64}: The current smoothed observations.

References

Zou, C., Tsung, F., & Wang, Z. (2008). Monitoring Profiles Based on Nonparametric Regression Methods. Technometrics, 50(4), 512-526. https://doi.org/10.1198/004017008000000433

NEWMA(λ::Float64, g::F, dat::FunctionalData)

Construct a NEWMA control chart by calculating the standard errors for a functional regression model.

Arguments

• λ::Float64: The EWMA smoothing constant. Must be between 0 and 1.
• g::F: The estimated regression function object. Must have a method of signature predict(g::F, x::AbstractVector).
• dat::FunctionalData: A FunctionalData object containing observations of the regression curves.
• Ej::Vector{Float64}: The current smoothed observations.

Returns

The constructed NEWMA control chart.

Examples

using Loess
xs = 10 .* rand(100)
ys = sin.(xs) .+ rand(100)
g = loess(xs, ys)
dat = FunctionalData(xs, ys)
NEWMA(0.2, g, dat)

AbstractSampling

Abstract type representing a generic sampling method for partially-observed data.

ThompsonSampling <: AbstractSampling

Type representing the Thompson Sampling method.

Fields

• β::Float64: A parameter regulating the concentration of the Dirichlet distribution for Thompson sampling.

Examples

ts = ThompsonSampling(2.0)

TopQ <: AbstractSampling

Type representing the TopQ sampling method. New data is generating by taking the q local statistics with highest value.

References

Mei, Y. (2011). Quickest detection in censoring sensor networks. 2011 IEEE International Symposium on Information Theory Proceedings, 2148-2152. https://doi.org/10.1109/ISIT.2011.6034390

new_layout(sampling::AbstractSampling, local_statistics) -> layout

Abstract function that creates a new layout based on the given sampling method and local statistics.

Arguments

• sampler::AbstractSampling: An instance of an object subtype of AbstractSampling.
• local_statistics: Array of local statistics used for generating the layout.
• q::Int: The number of items to include in the layout.

Returns

• layout: The generated layout to observe at the next iteration.

Examples

# Create an instance of ThompsonSampling
ts = ThompsonSampling(2.0)

# Generate a layout using ThompsonSampling
stats = [1, 2, 3, 4]
layout = new_layout(ts, stats, 2)

RSADA{D,S}

A R-SADA monitoring statistic, whcih is used for monitoring the mean of partially-observed independent data streams. The monitoring statistic iteratively samples arms (data streams) and updates a set of local statistics based on the observed values of the data streams. The RSADA algorithm uses the local statistics to make decisions on which arms to sample at each iteration.

Fields

• k: The allowance constant of the CUSUM Control chart. Default value is 0.1.
• mu_min: The minimum value of the mean shift to detect. Default value is 0.2.
• p::Int: The number of independent data streams. Default value is 1.
• q::Int: The number of data streams that are observable at each iteration. Default value is 1.
• dist::Distribution: The distribution of the data streams. Default value is Normal(0,1).
• value: The current value of the monitoring statistic. Default value is 0.0.
• eta: The vector of augmented variables. Initialized to be a vector of zeros for each data stream.
• obs::Vector{Int}: The indices of the data streams to sample. Default value is an array of q random integers between 1 and p.
• S1::Vector{F}: The sum of the rewards for each arm.
• S2::Vector{F}: The sum of the squared rewards for each arm.

References

Xian, X., Zhang, C., Bonk, S., & Liu, K. (2019). Online monitoring of big data streams: A rank-based sampling algorithm by data augmentation. Journal of Quality Technology, 53(2), 135-153. https://doi.org/10.1080/00224065.2019.1681924

augmented_vector(y, obs, mu, dist)

Compute the augmented vector eta (Equation 10 of [Xian et Al., 2019]) based on the given parameters.

Arguments

• y: The input vector.
• obs: The indices of the observed elements in y.
• mu: The mean parameter.
• dist: The IC distribution of the individual data streams.

Returns

• eta: The computed augmented vector.

References

Xian, X., Zhang, C., Bonk, S., & Liu, K. (2019). Online monitoring of big data streams: A rank-based sampling algorithm by data augmentation. Journal of Quality Technology, 53(2), 135-153. https://doi.org/10.1080/00224065.2019.1681924

update_sampling!(STAT::RSADA)

Update the monitored data streams of the RSADA monitoring statistic STAT by selecting the top-q values of the local monitoring statistics in STAT.S1.

Arguments

• STAT: The RSADA object to update.

Returns

• STAT.obs: The selected indices of the top STAT.q values of STAT.S1.

TRAS{F,D}

Fields

A Top-r-based Adaptive Sampling monitoring statistic for monitoring multiple data streams with partial observations.

Fields

• k::Float64: The allowance constant for the CUSUM statistic. Defaults to 0.1.
• mu_min::Float64: The minimum shift to be detected. Defaults to 1.0.
• value::Float64: The current value of the monitoring statistic. Defaults to 0.0.
• Δ::Float64: The compensation coefficient for non-observed variables.
• r::Int: The number of largest variables to sum.
• q::Int: The total number of observable variables.
• p::Int: The total number of variables.
• W::Vector{Float64}: The vector of local monitoring statistics. Defaults to a vector of zeros.
• W1::Vector{Float64}: The vector used to store the local monitoring statistics to detect increases in the mean. Defaults to a vector of zeros.
• W2::Vector{Float64}: The vector used to store the local monitoring statistics to detect decreases in the mean. Defaults to a vector of zeros.
• obs::Vector{Int}: The selected indices of the top q values of local monitoring statistics. Defaults to a random sample of size q from the range 1:p.
• sampler::S: The sampling strategy used to select the next set of obs indices. Defaults to ThompsonSampling().

References

Liu, K., Mei, Y., & Shi, J. (2015). An Adaptive Sampling Strategy for Online High-Dimensional Process Monitoring. Technometrics, 57(3), 305-319. https://doi.org/10.1080/00401706.2014.947005

update_sampling!(STAT::TRAS)

Update the monitored data streams of the TRAS monitoring statistic STAT using its associated AbstractSampler object.

Arguments

• STAT: The TRAS object to update.

Returns

• STAT.obs: The selected indices of the top STAT.q values of STAT.S1.