Data, Covariance, and Correlation Matrix
Data, Covariance, and Correlation Matrix
Nathaniel E. Helwig Assistant Professor of Psychology and Statistics
University of Minnesota (Twin Cities)
Updated 16-Jan-2017
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 1
Copyright
Copyright c 2017 by Nathaniel E. Helwig
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 2
Outline of Notes
1) The Data Matrix Definition Properties R code
2) The Covariance Matrix Definition Properties R code
3) The Correlation Matrix Definition Properties R code
4) Miscellaneous Topics Crossproduct calculations Vec and Kronecker Visualizing data
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 3
The Data Matrix
The Data Matrix
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 4
The Data Matrix Definition
The Organization of Data
The data matrix refers to the array of numbers
x11 x12 ? ? ? x1p
x21 x22 ? ? ? x2p
X = x31
x32
???
x3p
...
...
...
...
xn1 xn2 ? ? ? xnp
where xij is the j-th variable collected from the i-th item (e.g., subject). items/subjects are rows variables are columns
X is a data matrix of order n ? p (# items by # variables).
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 5
The Data Matrix Definition
Collection of Column Vectors
We can view a data matrix as a collection of column vectors:
X = x1
x2
???
xp
where xj is the j-th column of X for j {1, . . . , p}.
The n ? 1 vector xj gives the j-th variable's scores for the n items.
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 6
The Data Matrix Definition
Collection of Row Vectors
We can view a data matrix as a collection of row vectors:
x1
X=
x2 ...
xn
where xi is the i-th row of X for i {1, . . . , n}.
The 1 ? p vector xi gives the i-th item's scores for the p variables.
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 7
The Data Matrix Properties
Calculating Variable (Column) Means
The sample mean of the j-th variable is given by
x?j
=
1 n
n
xij
i =1
= n-11nxj
where 1n denotes an n ? 1 vector of ones xj denotes the j-th column of X
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 8
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