How to | Replace or Remove Invalid or Missing Data

In data analysis, it is often necessary to clean a dataset before analyzing it. Data points with missing entries or that contain invalid values must be removed or replaced by some estimate.

*Mathematica* provides a rich environment for this type of preprocessing.

It is common to use a particular symbol, string, or out-of-range number to represent missing information in a dataset. The

Missing symbol is used for this purpose in

*Mathematica* built-in data functions.

The following gets the gross domestic product (GDP) for each country known by

CountryData. The large output is suppressed by a semicolon (

):

This shows the values of

for the first 10 countries ordered alphabetically:

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You may want to do some analysis on this data, such as finding the maximum GDP, but values are not available for some countries.

The maximum cannot be completely determined from the raw data:

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Data can be missing for many reasons, but

CountryData will represent them all by expressions starting with

Missing. The number of missing values in the dataset can be determined by counting the expressions with the head

Missing.

Use

Count with the pattern

to count expressions with head

Missing:

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In comparison with the number of data points in the entire set, the number of missing results is relatively small:

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It would be quite reasonable to remove the missing points and perform any analysis on the remaining points. A simple way to remove these data points is to use

DeleteCases with the same pattern used for

Missing expressions in

Count:

From the length and maximum value of the new dataset, you can see that missing values have been removed:

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Any calculations that can be carried out on a list of numbers can now be performed on the new filtered dataset.

In the previous example a very specific form for invalid data points was known: they all had head

Missing. In practice, notation used for missing information varies. A person entering data in a spreadsheet might type NA for a value that is not available or does not apply, while some data acquisition software may represent missing measurements with a specific out-of-range number. As a result, it can be important to know what the data represents to determine which values are not valid.

Here is a dataset where the first elements represent a group number for one of two groups and the other two entries are numeric measurements for an individual in that group:

Displaying the data in a grid can make it easier to see where there are problems with the data:

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In this case, there are two problems. While there are only two groups, 1 and 2, there is an entry with a group number of 4. There is also a non-numeric entry in the third column. Something will need to be done to remove or modify the problematic points.

Valid data entries for this dataset will either have 1 or 2 as the first element and numbers for the second and third elements. A pattern for data of this type is

. The

symbol indicates an alternative: the first element must be 1 or 2.

NumberQ tests whether its argument is a number, and

is a pattern for numbers. You can use this pattern along with

MatchQ and

Not to write a function that identifies bad data points.

This function identifies such points by checking if its input matches the pattern. It returns

True if the input does not match the pattern:

To remove invalid data points, you can use

as a pattern in

DeleteCases:

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It is often desirable to replace invalid values by estimates based on other data points, rather than remove them entirely. However, for the current dataset it is not clear how to replace the group 4 entry other than to randomly choose 1 or 2 to replace it. You may still want to remove the data point in such cases.

Define a function

, which determines if a group entry is invalid by checking if the entry's first element is 1 or 2:

Use the pattern

with

DeleteCases to remove data points with invalid group values:

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Since

is the only entry without data in the third column and belongs to group 1, you could replace it with the mean or median of all the other third column data points from group 1.

Replacing by the group mean could be done directly by selecting the data points for group 1, computing the mean for the third elements that are numbers, and replacing

by that mean value.

Use

Select to pick out data points from group 1:

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Use

Select again to pick out the numbers from the last elements (third column) of the

data and then compute their mean:

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Now use a replacement rule to substitute this mean for

:

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Use

Grid to display the filtered data:

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The previous steps still require a fair amount of manual effort. You can instead write a function to perform the replacement on all invalid entries in a given column based on the value from another column. The function can then be used to process each column in a dataset.

The following function takes a dataset, the number of the column to process, the number of the column to group by, and a function as its arguments. It replaces any non-numeric values in the column being processed with the result of the function applied to the elements in that column, grouped by the value in the grouping column:

First define the original data, omitting the group 4 entry:

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Replace the non-numeric entries in column 3 by their associated group mean:

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Alternatively, replace the non-numeric entries in column 3 by their associated group medians:

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You can then use

Table to process the dataset column by column. Use

Transpose with

Grid to display the newly cleaned result in tabular form. This example uses group means as the replacement values:

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Because the function is set up to operate on a single column, you can use different estimates for each column.

Here, invalid elements in the second column are replaced by the respective group mean for that column, and invalid elements in the third column are replaced by the respective group median for that column:

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