MATLAB Arrays :: Organizing Data (Statistics Toolbox™)

MATLAB Arrays

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Numerical Data Heterogeneous Data Statistical Functions

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Numerical Data

MATLAB two-dimensional numerical arrays (matrices) containing statistical data use rows to represent observations and columns to represent measured variables. For example,

load fisheriris % Fisher's iris data (1936)

loads the variables meas and species into the MATLAB workspace. The meas variable is a 150-by-4 numerical matrix, representing 150 observations of 4 different measured variables (by column: sepal length, sepal width, petal length, and petal width, respectively).

The observations in meas are of three different species of iris (setosa, versicolor, and virginica), which can be separated from one another using the 150-by-1 cell array of strings species:

setosa_indices = strcmp('setosa',species);
setosa = meas(setosa_indices,:);

The resulting setosa variable is 50-by-4, representing 50 observations of the 4 measured variables for iris setosa.

To access and display the first five observations in the setosa data, use row, column parenthesis indexing:

SetosaObs = setosa(1:5,:)
SetosaObs =
    5.1000    3.5000    1.4000    0.2000
    4.9000    3.0000    1.4000    0.2000
    4.7000    3.2000    1.3000    0.2000
    4.6000    3.1000    1.5000    0.2000
    5.0000    3.6000    1.4000    0.2000

The data are organized into a table with implicit column headers "Sepal Length," "Sepal Width," "Petal Length," and "Petal Width." Implicit row headers are "Observation 1," "Observation 2," "Observation 3," etc.

Similarly, 50 observations for iris versicolor and iris virginica can be extracted from the meas container variable:

versicolor_indices = strcmp('versicolor',species);
versicolor = meas(versicolor_indices,:);

virginica_indices = strcmp('virginica',species);
virginica = meas(virginica_indices,:);

Because the data sets for the three species happen to be of the same size, they can be reorganized into a single 50-by-4-by-3 multidimensional array:

iris = cat(3,setosa,versicolor,virginica);

The iris array is a three-layer table with the same implicit row and column headers as the setosa, versicolor, and virginica arrays. The implicit layer names, along the third dimension, are "Setosa," "Versicolor," and "Virginica." The utility of such a multidimensional organization depends on assigning meaningful properties of the data to each dimension.

To access and display data in a multidimensional array, use parenthesis indexing, as for 2-D arrays. The following gives the first five observations of sepal lengths in the setosa data:

SetosaSL = iris(1:5,1,1)
SetosaSL =
    5.1000
    4.9000
    4.7000
    4.6000
    5.0000

Multidimensional arrays provide a natural way to organize numerical data for which the observations, or experimental designs, have many dimensions. If, for example, data with the structure of iris are collected by multiple observers, in multiple locations, over multiple dates, the entirety of the data can be organized into a single higher dimensional array with dimensions for "Observer," "Location," and "Date." Likewise, an experimental design calling for m observations of n p-dimensional variables could be stored in an m-by-n-by-p array.

Numerical arrays have limitations when organizing more general statistical data. One limitation is the implicit nature of the metadata. Another is the requirement that multidimensional data be of commensurate size across all dimensions. If variables have different lengths, or the number of variables differs by layer, then multidimensional arrays must be artificially padded with NaNs to indicate "missing values." These limitations are addressed by dataset arrays (see Dataset Arrays), which are specifically designed for statistical data.

Heterogeneous Data

MATLAB data types include two container variables—cell arrays and structure arrays—that allow you to combine metadata with variables of different types and sizes.

The data in the variables setosa, versicolor, and virginica created in Numerical Data can be organized in a cell array, as follows:

iris1 = cell(51,5,3); % Container variable

obsnames = strcat({'Obs'},num2str((1:50)','%-d'));
iris1(2:end,1,:) = repmat(obsnames,[1 1 3]);

varnames = {'SepalLength','SepalWidth',...
            'PetalLength','PetalWidth'};
iris1(1,2:end,:) = repmat(varnames,[1 1 3]);

iris1(2:end,2:end,1) = num2cell(setosa);
iris1(2:end,2:end,2) = num2cell(versicolor);
iris1(2:end,2:end,3) = num2cell(virginica);

iris1{1,1,1} = 'Setosa';
iris1{1,1,2} = 'Versicolor';
iris1{1,1,3} = 'Virginica';

To access and display the cells, use parenthesis indexing. The following displays the first five observations in the setosa sepal data:

SetosaSLSW = iris1(1:6,1:3,1)
SetosaSLSW = 
    'Setosa'    'SepalLength'    'SepalWidth'
    'Obs1'      [     5.1000]    [    3.5000]
    'Obs2'      [     4.9000]    [         3]
    'Obs3'      [     4.7000]    [    3.2000]
    'Obs4'      [     4.6000]    [    3.1000]
    'Obs5'      [          5]    [    3.6000]

Here, the row and column headers have been explicitly labeled with metadata.

To extract the data subset, use row, column curly brace indexing:

subset = reshape([iris1{2:6,2:3,1}],5,2)
subset =
    5.1000    3.5000
    4.9000    3.0000
    4.7000    3.2000
    4.6000    3.1000
    5.0000    3.6000

While cell arrays are useful for organizing heterogeneous data, they may be cumbersome when it comes to manipulating and analyzing the data. MATLAB and Statistics Toolbox statistical functions do not accept data in the form of cell arrays. For processing, data must be extracted from the cell array to a numerical container variable, as in the preceding example. The indexing can become complicated for large, heterogeneous data sets. This limitation of cell arrays is addressed by dataset arrays (see Dataset Arrays), which are designed to store general statistical data and provide easy access.

The data in the preceding example can also be organized in a structure array, as follows:

iris2.data = cat(3,setosa,versicolor,virginica);
iris2.varnames = {'SepalLength','SepalWidth',...
                  'PetalLength','PetalWidth'};
iris2.obsnames = strcat({'Obs'},num2str((1:50)','%-d'));
iris2.species = {'setosa','versicolor','virginica'};

The data subset is then returned using a combination of dot and parenthesis indexing:

subset = iris2.data(1:5,1:2,1)
subset =
    5.1000    3.5000
    4.9000    3.0000
    4.7000    3.2000
    4.6000    3.1000
    5.0000    3.6000

For statistical data, structure arrays have many of the same limitations as cell arrays. Once again, dataset arrays (see Dataset Arrays), designed specifically for general statistical data, address these limitations.

Statistical Functions

One of the advantages of working in the MATLAB language is that functions operate on entire arrays of data, not just on single scalar values. The functions are said to be vectorized. Vectorization allows for both efficient problem formulation, using array-based data, and efficient computation, using vectorized statistical functions.

When MATLAB and Statistics Toolbox statistical functions operate on a vector of numerical data (either a row vector or a column vector), they return a single computed statistic:

% Fisher's setosa data:
load fisheriris
setosa_indices = strcmp('setosa',species);
setosa = meas(setosa_indices,:);

% Single variable from the data:
setosa_sepal_length = setosa(:,1);

% Standard deviation of the variable:
std(setosa_sepal_length)
ans =
    0.3525

When statistical functions operate on a matrix of numerical data, they treat the columns independently, as separate measured variables, and return a vector of statistics—one for each variable:

std(setosa)
ans =
    0.3525    0.3791    0.1737    0.1054

The four standard deviations are for measurements of sepal length, sepal width, petal length, and petal width, respectively.

Compare this to

std(setosa(:))
ans =
    1.8483

which gives the standard deviation across the entire array (all measurements).

Compare the preceding statistical calculations to the more generic mathematical operation

sin(setosa)

This operation returns a 50-by-4 array the same size as setosa. The sin function is vectorized in a different way than the std function, computing one scalar value for each element in the array.

MATLAB and Statistics Toolbox statistical functions, like std, must be distinguished from general mathematical functions like sin. Both are vectorized, and both are useful for working with array-based data, but only statistical functions summarize data across observations (rows) while preserving variables (columns). This property of statistical functions may be explicit, as with std, or implicit, as with regress. To see how a particular function handles array-based data, consult its reference page.

MATLAB statistical functions expect data input arguments to be in the form of numerical arrays. If data is stored in a cell or structure array, it must be extracted to a numerical array, via indexing, for processing. Statistics Toolbox functions are more flexible. Many toolbox functions accept data input arguments in the form of both numerical arrays and dataset arrays (see Dataset Arrays), which are specifically designed for storing general statistical data.

Introduction

Statistical Arrays

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