Handling Missing Data and Outliers

Handling Missing Data

Data acquisition failures sometimes result in missing measurements both in the input and the output signals. When you import data that contains missing values using the MATLAB® Import Wizard, these values are automatically set to NaN. NaN serves as a flag for nonexistent or undefined data. When you plot data on a time-plot that contains missing values, gaps appear on the plot where missing data exists.

You can use misdata to estimate missing values. This command linearly interpolates missing values to estimate the first model. Then, it uses this model to estimate the missing data as parameters by minimizing the output prediction errors obtained from the reconstructed data. You can specify the model structure you want to use in the misdata argument or estimate a default-order model using the n4sid method. For more information, see the misdata reference page.

For example, suppose y and u are output and input signals that contain NaNs. This data is sampled at 0.2 s. The following syntax creates a new iddata object with these input and output signals.

dat = iddata(y,u,0.2) % y and u contain NaNs 
                      % representing missing data

Apply the misdata command to the new data object. For example:

dat1 = misdata(dat);
plot(dat,dat1)        % Check how the missing data
                      % was estimated on a time plot

Handling Outliers

Malfunctions can produce errors in measured values, called outliers. Such outliers might be caused by signal spikes or by measurement malfunctions. If you do not remove outliers from your data, this can adversely affect the estimated models.

To identify the presence of outliers, perform one of the following tasks:

  • Before estimating a model, plot the data on a time plot and identify values that appear out of range.

  • After estimating a model, plot the residuals and identify unusually large values. For more information about plotting residuals, see Residual Analysis. Evaluate the original data that is responsible for large residuals. For example, for the model Model and validation data Data, you can use the following commands to plot the residuals:

% Compute the residuals
  E = resid(Model,Data)
% Plot the residuals
  plot(E)

Next, try these techniques for removing or minimizing the effects of outliers:

  • Extract the informative data portions into segments and merge them into one multiexperiment data set (see Extract and Model Specific Data Segments). For more information about selecting and extracting data segments, see Selecting Subsets of Data.

      Tip   The inputs in each of the data segments must be consistently exciting the system. Splitting data into meaningful segments for steady-state data results in minimum information loss. Avoid making data segments too small.

  • Manually replace outliers with NaNs and then use the misdata command to reconstruct flagged data. This approach treats outliers as missing data and is described in Handling Missing Data. Use this method when your data contains several inputs and outputs, and when you have difficulty finding reliable data segments in all variables.

  • Remove outliers by prefiltering the data for high-frequency content because outliers often result from abrupt changes. For more information about filtering, see Filtering Data.

    Note:   The estimation algorithm can handle outliers by assigning a smaller weight to outlier data. A robust error criterion applies an error penalty that is quadratic for small and moderate prediction errors, and is linear for large prediction errors. Because outliers produce large prediction errors, this approach gives a smaller weight to the corresponding data points during model estimation. Set the ErrorThreshold estimation option (see Advanced.ErrorThreshold in, for example, polyestOptions) to a nonzero value to activate the correction for outliers in the estimation algorithm.

Extract and Model Specific Data Segments

This example shows how to create a multi-experiment, time-domain data set by merging only the accurate data segments and ignoring the rest.

Assume that the data has poor or no measurements for some sample ranges (for example 341–499). You cannot simply concatenate the good data segments because the transients at the connection points compromise the model. Instead, you must create a multiexperiment iddata object, where each experiment corresponds to a good segment of data, as follows:

% Plot the data in a MATLAB Figure window
plot(data)

% Create multiexperiment data set
% by merging data segments
  datam = merge(data(1:340),...
                data(500:897),...
                data(1001:1200),...
                data(1550:2000));

% Model the multiexperiment data set
% using "experiments" 1, 2, and 4
m = n4sid(getexp(datam,[1,2,4]))

% Validate the model by comparing its output to
% the output data of experiment 3
compare(getexp(datam,3),m)

See Also

To learn more about the theory of handling missing data and outliers, see the chapter on preprocessing data in System Identification: Theory for the User, Second Edition, by Lennart Ljung, Prentice Hall PTR, 1999.

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