Main Content

detectdrift

Detect drifts between baseline and target data using permutation testing

    Description

    example

    DDiagnostics = detectdrift(Baseline,Target) performs Permutation Testing to detect drift for each variable in the Baseline and Target data sets, and returns the results in DDiagnostics.

    DDiagnostics is a DriftDiagnostics object.

    example

    DDiagnostics = detectdrift(Baseline,Target,Name=Value) specifies additional options using one or more of the name-value arguments. For example, you can specify the metrics to use for the variables or the maximum number of permutations.

    Examples

    collapse all

    Generate baseline and target data with two variables, where the distribution parameters of the second variable change for the target data.

    rng('default') % For reproducibility
    baseline = [normrnd(0,1,100,1),wblrnd(1.1,1,100,1)];
    target = [normrnd(0,1,100,1),wblrnd(1.2,2,100,1)];

    Compare the two data sets for any drift.

    DDiagnostics = detectdrift(baseline,target)
    DDiagnostics = 
      DriftDiagnostics
    
                  VariableNames: ["x1"    "x2"]
           CategoricalVariables: []
                    DriftStatus: ["Stable"    "Drift"]
                        PValues: [0.2850 0.0030]
            ConfidenceIntervals: [2×2 double]
        MultipleTestDriftStatus: "Drift"
                 DriftThreshold: 0.0500
               WarningThreshold: 0.1000
    
    
      Properties, Methods
    
    

    DDiagnostics is a DriftDiagnostics object. detectdrift displays some of the object properties.

    Display the confidence intervals for the estimated p-values.

    DDiagnostics.ConfidenceIntervals
    ans = 2×2
    
        0.2572    0.0006
        0.3141    0.0087
    
    

    For the first variable, the lower bound of the confidence interval for the estimated p-value is greater than the warning threshold value of 0.1. Therefore, detectdrift determines that the target data for the first variable is stable compared to the baseline data. For the second variable, the upper bound of the confidence interval for the estimated p-value is smaller than the drift threshold of 0.05. Therefore, the drift status for this variable is Drift, which indicates that detectdrift detects the shift in the distribution parameters.

    detectdrift uses the default Bonferroni method for testing multiple hypotheses. The function first divides the warning and drift thresholds by the number of p-values, which in this case is two. Then the function determines if any p-value is still lower than either threshold. Here, the second p-value is still lower than the modified drift threshold, so the function sets the MultipleTestDriftStatus to Drift for the overall data.

    Visualize the permutation results for both variables.

    tiledlayout(1,2);
    ax1 = nexttile;
    plotPermutationResults(DDiagnostics,ax1,Variable="x1")
    ax2 = nexttile;
    plotPermutationResults(DDiagnostics,ax2,Variable="x2")

    Bars to the right of the dashed line show the metric values that are greater than the threshold, which is the initial metric value detectdrift computes using the baseline and target data for each variable. The amount of the bars greater than the threshold is much more for variable x1, which indicates that there is not a significant drift between the baseline and target data for this variable.

    Load the sample data.

    load humanactivity

    For details on the data set, enter Description at the command line.

    Assign the first 250 observations as baseline data and next 250 as target data.

    baseline = feat(1:250,:);
    target = feat(251:500,:);

    Test for drift on variables 5 to 10 using a warning threshold of 0.05 and a drift threshold of 0.01. All variables are continuous, so use the Kolmogorov-Smirnov metric for all variables. Specify the False Discovery Rate method as the multiple test correction.

    DDiagnostics = detectdrift(baseline(:,5:10),target(:,5:10),WarningThreshold=0.05, ...
            DriftThreshold=0.01,ContinuousMetric="ks",MultipleTestCorrection="fdr")
    DDiagnostics = 
      DriftDiagnostics
    
                  VariableNames: ["x1"    "x2"    "x3"    "x4"    "x5"    "x6"]
           CategoricalVariables: []
                    DriftStatus: ["Drift"    "Drift"    "Drift"    "Stable"    "Warning"    "Drift"]
                        PValues: [1.0000e-03 1.0000e-03 1.0000e-03 0.8810 0.0110 1.0000e-03]
            ConfidenceIntervals: [2×6 double]
        MultipleTestDriftStatus: "Drift"
                 DriftThreshold: 0.0100
               WarningThreshold: 0.0500
    
    
      Properties, Methods
    
    

    Display the confidence intervals for the estimated p-values.

    DDiagnostics.ConfidenceIntervals
    ans = 2×6
    
        0.0000    0.0000    0.0000    0.8593    0.0055    0.0000
        0.0056    0.0056    0.0056    0.9004    0.0196    0.0056
    
    

    The lower confidence bound of the p-value for the 8th variable (variable name x4) is greater than the warning threshold. Therefore, detectdrift determines that the drift status for this variable is "Stable". The upper confidence bound of the p-value for the 9th variable (variable name x5) is greater than the drift threshold, but lower than the warning threshold. Therefore, detectdrift determines that the drift status for this variable is "Warning". Confidence intervals of all other variables are smaller than the drift threshold, so they have a drift status of "Drift". Based on the False Discovery Rate method for multiple test correction, the function determines that the drift status for the overall data is "Drift".

    Visualize the p-values with the confidence intervals and corresponding drift status.

    plotDriftStatus(DDiagnostics)

    The plot shows the estimated p-values with the confidence intervals against the warning and drift thresholds. The estimated p-value for variable x4 and its confidence intervals are higher than the warning threshold. Therefore, the drift status for this variable is "Stable". The upper confidence bound of the p-value for x5 is greater than the drift threshold, but lower than the warning threshold. Therefore, the drift status for this variable is "Warning". Confidence intervals of all other variables are smaller than the drift threshold, so they have a drift status of "Drift".

    Load the data set NYCHousing2015.

    load NYCHousing2015

    The data set includes 10 variables with information on the sales of properties in New York City in 2015.

    Remove outliers and convert the datetime array (SALEDATE) to the month numbers.

    idx = isoutlier(NYCHousing2015.SALEPRICE);
    NYCHousing2015(idx,:) = [];
    NYCHousing2015.SALEDATE = month(NYCHousing2015.SALEDATE);

    Define the baseline and target data as information on the sales made in January and July, respectively.

    tbl = NYCHousing2015;
    baseline = tbl(tbl.SALEDATE==1,:);
    target = tbl(tbl.SALEDATE==7,:);

    Shuffle the data.

    n = numel(baseline(:,1));
    rng(1); % For reproducibility
    idx = randsample(n,n);
    baseline = baseline(idx,:);
    n = numel(target(:,1));
    idx = randsample(n,n);
    target = target(idx,:);

    Test for potential drift between the baseline and target data. Specify the categorical variables and the metrics to use with each variable.

    DDiagnostics = detectdrift(baseline(1:1500,:),target(1:1500,:), ...
        VariableNames=["BOROUGH","BUILDINGCLASSCATEGORY","LANDSQUAREFEET","GROSSSQUAREFEET","SALEPRICE"], ...
        CategoricalVariables=["BOROUGH","BUILDINGCLASSCATEGORY"], ...
        Metrics=["Hellinger","Hellinger","ad","ks","energy"])
    DDiagnostics = 
      DriftDiagnostics
    
                  VariableNames: ["BOROUGH"    "BUILDINGCLASSCATEGORY"    "LANDSQUAREFEET"    "GROSSSQUAREFEET"    "SALEPRICE"]
           CategoricalVariables: [1 2]
                    DriftStatus: ["Drift"    "Stable"    "Drift"    "Drift"    "Drift"]
                        PValues: [0.0260 0.1440 0.0070 0.0230 0.0110]
            ConfidenceIntervals: [2×5 double]
        MultipleTestDriftStatus: "Drift"
                 DriftThreshold: 0.0500
               WarningThreshold: 0.1000
    
    
      Properties, Methods
    
    

    detectdrift identifies drift between the baseline and target data for all variables except BUILDINGCLASSCATEGORY.

    Display the confidence intervals for the estimated p-values.

    DDiagnostics.ConfidenceIntervals
    ans = 2×5
    
        0.0171    0.1228    0.0028    0.0146    0.0055
        0.0379    0.1673    0.0144    0.0343    0.0196
    
    

    Plot a histogram for SALEPRICE.

    plotHistogram(DDiagnostics,Variable="SALEPRICE")

    The histogram shows the shift in the sale prices for the month of July compared to January.

    Plot the empirical cumulative distribution function for the baseline and target data of SALEPRICE.

    plotEmpiricalCDF(DDiagnostics,Variable="SALEPRICE")

    Plot the permutation results for SALEPRICE.

    plotPermutationResults(DDiagnostics,Variable="SALEPRICE")

    Generate baseline and target data with three variables, where the distribution parameters of the second and third variables change for the target data.

    rng('default') % For reproducibility
    baseline = [normrnd(0,1,100,1),wblrnd(1.1,1,100,1),betarnd(1,2,100,1)];
    target = [normrnd(0,1,100,1),wblrnd(1.2,2,100,1),betarnd(1.7,2.8,100,1)];

    Compute the initial metrics for all variables between the baseline and target data without estimating the p-values.

    DDiagnostics = detectdrift(baseline,target,EstimatePValues=false)
    DDiagnostics = 
      DriftDiagnostics
    
               VariableNames: ["x1"    "x2"    "x3"]
        CategoricalVariables: []
                     Metrics: ["Wasserstein"    "Wasserstein"    "Wasserstein"]
                MetricValues: [0.2022 0.3468 0.0559]
    
    
      Properties, Methods
    
    

    detectdrift computes only the initial metric value for each variable using the baseline and target data. The properties associated with permutation testing and p-value estimation are either empty or contain NaNs.

    summary(DDiagnostics)
              MetricValue       Metric    
              ___________    _____________
    
        x1      0.20215      "Wasserstein"
        x2      0.34676      "Wasserstein"
        x3     0.055922      "Wasserstein"
    

    summary function displays only the initial metric value and the metric used for each specified variable.

    plotDriftStatus and plotPermutationResults do not produce plots and return warning messages when you compute metrics without estimating p-values. plotEmpiricalCDF and plotHistogram plot the ecdf and the histogram, respectively, for the first variable by default. They both return NaN for the p-value and drift status associated with the variable.

    plotEmpiricalCDF(DDiagnostics)

    plotHistogram(DDiagnostics)

    Input Arguments

    collapse all

    Baseline data, specified as a numeric array, categorical array, or table. Baseline and Target data must have the same data type. When the input data is a categorical array, detectdrift treats each column as an independent categorical variable.

    Data Types: single | double | categorical | table

    Target data, specified as a numeric array, categorical array, or table. Baseline and Target data must have the same data type. When the input data is a categorical array, detectdrift treats each column as an independent categorical variable.

    Data Types: single | double | categorical | table

    Name-Value Arguments

    Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

    Example: detectdrift(Baseline,Target,WarningThreshold=0.05,DriftThreshold=0.01,VariableNames=["Weight","MPG"],ContinuousMetrics="ad") sets the warning threshold to 0.05 and drift threshold to 0.01, specifies Weight and MPG as the variables to test for drift detection, and Anderson-Darling as the metric to use in testing all continuous variables.

    Variables to analyze for drift, specified as a string, array of unique strings, character vector, or cell array of character vectors.

    Example: VariableNames=["x1","x3"]

    Data Types: string | char | cell

    List of categorical variables, specified as "all", a string, array of unique strings, character vector, cell array of unique character vectors, vector of integer indices, or vector of logical indices.

    detectdrift treats the following as categorical variables: ordinal or nominal data types, or the categorical data type with the ordinal indicator set to true as categorical variables.

    Example: CategoricalVariables="Zone"

    Data Types: single | double | logical | string | cell

    Threshold for detecting drift, specified as a scalar value from 0 to 1.

    detectdrift uses the drift threshold together with warning threshold to determine the drift status. The DriftThreshold value must be strictly lower than WarningThreshold value.

    If the confidence interval for an estimated p-value is (Lower,Upper), then detectdrift determines the drift status as follows.

    Drift StatusCondition
    DriftUpper < DriftThreshold
    WarningDriftThreshold < Lower < WarningThreshold or DriftThreshold < Upper < WarningThreshold
    StableLower > WarningThreshold

    Example: DriftThreshold=0.01

    Data Types: single | double

    Threshold for potential drift warning, specified as a scalar value between 0 and 1.

    detectdrift uses the warning threshold together with drift threshold to determine the drift status. The WarningThreshold value must be strictly greater than the DriftThreshold value.

    If the confidence interval for an estimated p-value is (Lower,Upper), then detectdrift determines the drift status as follows.

    Drift StatusCondition
    DriftUpper < DriftThreshold
    WarningDriftThreshold < Lower < WarningThreshold or DriftThreshold < Upper < WarningThreshold
    StableLower > WarningThreshold

    Example: WarningThreshold=0.05

    Data Types: single | double

    Maximum number of permutations, specified as a positive integer value. detectdrift increases the number of trials for permutation logarithmically, according to a heuristic algorithm, until it determines the drift status or reaches MaxNumPermutations. If detectdrift cannot determine the drift status by the end of the maximum number of permutations, then it sets the drift status to "Warning".

    Example: MaxNumPermutations=1500

    Data Types: single | double

    Metrics used to detect drift for each variable, specified as one of the following:

    • String, string vector, character vector, or cell array of character vectors representing one or more of the built-in metrics.

      Built-in metrics for continuous variables

      ValueDefinition
      "wasserstein"Wasserstein
      "energy"Energy
      "ks"Kolmogorov-Smirnov
      "ad"Anderson-Darling

      Built-in metrics for categorical variables

      ValueDefinition
      "tv"Total Variation
      "psi"Population Stability Index
      "hellinger"Hellinger
      "chi2"Chi-Square
      "bhattacharyya"Bhattacharyya
    • Function handle or a cell array of function handles. If you provide a function handle FUN as a metric, detectdrift calls it as:

      FUN(BaselineVariable,TargetVariable),

      where BaselineVariable is the variable in Baseline and TargetVariable is the variable in Target. The output of FUN must be a scalar representing the metric value.

    • Structure or a cell array of structures, where each structure contains a single field whose value is a function handle. If you pass a structure, detectdrift uses the field name as the metric name. If the function handle is anonymous, detectdrift names it 'CustomMetric_i', where i is the position of the variable in Metrics.

    Metrics must contain one value for each variable in VariableNames and its size must be equal to the size of VariableNames.

    If you specify metrics using Metrics, you cannot specify them using ContinuousMetric or CategoricalMetric.

    Example: Metrics=["wasserstein","psi","hellinger"]

    Data Types: string | cell | function_handle | struct

    Metric for drift detection in continuous variables, specified as one of the following:

    • String or a character vector representing one or more of the built-in metrics.

      Built-in metrics for continuous variables

      ValueDefinition
      "wasserstein"Wasserstein
      "energy"Energy
      "ks"Kolmogorov-Smirnov
      "ad"Anderson-Darling
    • Function handle called as:

      FUN(BaselineVariable,TargetVariable),

      where BaselineVariable is the variable in Baseline and TargetVariable is the variable in Target. The output of FUN must be a scalar representing the metric value.

      If the function handle is not anonymous, detectdrift extracts the metric name from the provided function handle. If the function handle is not anonymous, then it names the metric 'CustomContinuousMetric'.

    • Structure with a single field whose value is a function handle. In this case, detectdrift uses the field name as the metric name.

    If you specify ContinuousMetric, then you cannot specify other metrics using Metrics.

    Example: ContinuousMetric="ks"

    Data Types: string | char | function_handle | struct

    Metric for drift detection in categorical variables, specified as one of the following:

    • String or a character vector representing one or more of the built-in metrics.

      Built-in metrics for categorical variables

      ValueDefinition
      "tv"Total Variation
      "psi"Population Stability Index
      "hellinger"Hellinger
      "chi2"Chi-Square
      "bhattacharyya"Bhattacharyya
    • Function handle called as follows:

      FUN(BaselineVariable,TargetVariable),

      where BaselineVariable is the variable in Baseline and TargetVariable is the variable in Target. The output of FUN must be a scalar representing the metric value.

      If the function handle is not anonymous, detectdrift extracts the metric name from the provided function handle. If The function handle is anonymous, then it names the metric 'CustomCategoricalMetric'.

    • Structure with a single field whose value is a function handle. In this case, detectdrift uses the field name as the metric name.

    If you specify CategoricalMetric, then you cannot specify other metrics using Metrics.

    Example: CategoricalMetric="chi2"

    Data Types: string | char | function_handle | struct

    Correction method for multiple hypothesis tests, specified as one of the following.

    • "bonferroni" – Bonferroni correction. If k variables are specified for drift detection, detectdrift modifies the warning threshold and drift threshold by dividing each by k. Then, the function checks if any p-values are smaller than the modified threshold values to determine the drift status.

    • "fdr" – False discovery rate (FDR) method. detectdrift uses the Benjamini-Hochberg procedure to compute the false discovery rate. If k variables are specified for drift detection, the FDR method takes these steps:

      1. Rank the p-values corresponding to the specified variables.

      2. Divide the ranks 1 to k by the number of variables k to obtain Q = [1/k, 2/k, 3/k ,…, k/k].

      3. Modify the warning and drift thresholds for each sorted p-value by multiplying the initial warning and drift threshold values by the corresponding q value. For example, the modified warning threshold for rank 3 is (WarningThreshold)*3/k.

      4. Check if any sorted p-values are smaller than the corresponding modified warning or drift thresholds to determine the drift status.

    The multiple test correction methods provide a conservative estimate of the multivariable drift.

    Example: MultipleTestCorrection="fdr"

    Flag to run in parallel, specified as true or false. If you specify UseParallel=true, the detectdrift function executes for-loop iterations in parallel by using parfor. This option requires Parallel Computing Toolbox™.

    Example: UseParallel=true

    Indicator to estimate the p-values during permutation testing, specified as true or false. If you specify EstimatePValues=false, then detectdrift computes the metrics only.

    Example: EstimatePValues=false

    Output Arguments

    collapse all

    Results of permutation testing for drift detection, returned as a DriftDiagnostics object. detectdrift displays the following properties.

    Property NameDescription
    VariableNamesVariables analyzed for drift detection
    CategoricalVariablesIndices of categorical variables in the data
    DriftStatusDrift status for each variable
    PValuesEstimated p-value for each variable
    ConfidenceIntervals95% confidence interval bounds for the estimated p-values
    MultipleTestDriftStatusDrift status for the overall data
    DriftThresholdThreshold to determine the drift status
    WarningThresholdThreshold to determine the warning status

    For a full list of the properties and their descriptions, see the DriftDiagnostics reference page.

    Algorithms

    collapse all

    Permutation Testing

    detectdrift uses permutation testing to determine the drift status for each variable in the Baseline data and its counterpart in the Target data. A permutation test is a nonparametric statistical significance test in which the function obtains distribution of a metric (test statistic) under the null hypothesis by computing the values of that metric under all possible rearrangements of a variable in Baseline and Target. Depending on the number of variables and observations, trying all possible permutations of a variable might be infeasible. Therefore detectdrift performs a sufficient number of permutations to obtain a good estimate of the metric for the variable.

    Under null hypothesis (no drift), many values of the metric recorded during permutation testing can be as extreme as the initial test statistic. This suggests sufficiently high confidence that the observations of the specified variable in the baseline and target data come from the same distribution. Therefore, no evidence of drift is found, and detectdrift fails to reject the null hypothesis.

    If the initial test statistic is identified as an outlier, then detectdrift rejects the null hypothesis. This suggest sufficiently high confidence that the observations of the specified variable in the baseline and target data come from different distributions. Therefore, drift is detected.

    detectdrift takes these steps in permutation testing:

    • For a given variable with m observations in the baseline data and n observations in the target data, detectdrift computes an initial value of the metric from the original data.

    • The function then permutes the observations of the variable in the baseline and target data and separates them into two vectors with sizes m and n, respectively. Next, the function computes the same metric value. detectdrift repeats this step for MaxNumPermutations times to obtain a distribution of the specified metric.

    • An estimate of the p-value is p = x/perm, where x is the number of times a metric value obtained from a permutation is greater than the value of the initial metric value, and perm is the number of permutations. With the binomial distribution assumption for x, detectdrift estimates the 95% confidence interval for the p-value by using [~,CI] = binofit(x,perm,0.05).

    Given the confidence intervals (Lower, Upper) of the p-values, detectdrift determines the drift status based on the following conditions.

    Drift StatusCondition
    DriftUpper < DriftThreshold
    WarningDriftThreshold < Lower < WarningThreshold or DriftThreshold < Upper < WarningThreshold
    StableLower > WarningThreshold

    Metrics

    detectdrift uses the following metrics as test statistics in permutation testing for detecting drift between the baseline and target data.

    Metrics for Continuous Variables

    The detectdrift function first defines the following:

    • Eb(x) as the empirical cumulative distribution function (ecdf) of the baseline data over the common domain

    • Et(x) as the ecdf of the target data over the common domain

    • D(x) as the joint ecdf of all data, and w as the difference between the edges of the bins

    Next, detectdrift computes the metrics for continuous variables as follows:

    • Wasserstein

      W=xw*|Eb(x)Et(x)|

    • Energy

      En=2*xw*|Eb(x)Et(x)|2

    • Kolmogorov-Smirnov

      KS=max|Eb(x)Et(x)|

    • Anderson-Darling

      AD=x(|Eb(x)Et(x)|(m+n)D(x)*(1D(x)))2

      m and n are the number of observations in the baseline data and target data, respectively.

    Metrics for Categorical Variables

    The detectdrift function defines the following:

    • Hb(x) as the percentage of the baseline data in the bins determined by combining the baseline and target data (jointly considering them across the same domain)

    • Ht(x) as the percentage of the baseline data in the bins determined by combining the baseline and target data

    Next, detectdrift computes the metrics for categorical variables as follows:

    • Total Variation

      TV=0.5*x|Hb(x)Ht(x)|

    • Population Stability Index

      PSI=max(0,xlog(Ht(x)Hb(x))(Ht(x)Hb(x)))

    • Chi-Square

      χ2=x(Ht(x)Hb(x))2Hb(x)

    • Bhattacharyya

      B=max(0,log(min(1,xHb(x)*Ht(x))))

    • Hellinger

      H=max(0,1(min(1,xHb(x)*Ht(x))))

    To handle empty bins (categories), detectdrift adds a 0.5 correction factor to histogram bin counts for each bin. This is equivalent to the assumption that the parameter p, which is the probability that the value of the variable is in that category, has the prior distribution Beta(0.5,0.5) (Jeffreys prior assumption for the distribution parameter).

    References

    [1] Benjamini, Yoav, and Yosef Hochberg. "Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing." Journal of the Royal Statistical Society, Series B (Methodological). Vol. 57, No. 1, pp. 289-300, 1995.

    [2] Villani, Cédric. Topics in Optimal Transportation. Graduate Studies in Mathematics. Vol. 58, American Mathematical Society, 2000.

    [3] Deza, Elena, and Michel Marie Deza. Encyclopedia of Distances, Springer Berlin Heidelberg, 2009.

    Extended Capabilities

    Version History

    Introduced in R2022a