by Nikolai V. Shokhirev
ABC TutorialsIn science and technology systems (objects) are characterized by a finite set of parameters: x_{i} , i = 1, ... , N. Consequently, the measurements for these parameters can be arranged into a rectangular matrix:
(1) |
Here M is the number of experiments. Each experiment corresponds to the measurement of a system , sample, individual, etc. All such terms are used interchangeably.
Examples
Systems | Parameters |
Human individuals | Age, sex, education, income, weight, height, etc. |
Chemical solutions | Spectral intensities at selected wavelength |
Microchips in a control sample | Voltage and current at certain pins |
Clinical test participants | Lab test results |
The sample (population) mean vector of parameters is defined as:
(2) |
For each measurement the vector of deviations can be defined as:
(3) |
In the case of clinical research, one of the components of μ is an average patient temperature in a hospital. Obviously, more interesting is a deviation from this average.
The vectors of deviations form the matrix D similar to the initial matrix X:
(4) |
The sample covariance matrix is defined as averaged products of the deviation vector components:
(5) |
Here d_{i}_{,m} is the i-th parameter of the m-th system.
Eq (5) can be rewritten in the following matrix form:
(6) |
The superscript "T" denotes the matrix transposition.
The maximum likelihood covariance matrix C^{ML} differs by the factor M /(M-1) from the above definition:
. | (7) |
The advantage of this definition is that the i-th diagonal element is the estimation for the variances of the i-th parameter:
(8) |
Regardless of the covariance definition, the correlation coefficients are:
(9 |
or in a matrix form:
(10) |
Here is a diagonal matrix with the following matrix elements:
(1) |
The correlation coefficient is a measure of the quality of a linear least squares fit for the original data. A higher σ value means a better linear fit.
This approach is implemented in a program called "Correlations". This program is available in the Download section. You can also use more general "Stat Analysis" program.
Remark: In "Correlations" the meaning of the columns and rows is opposite to that of the tutorial.
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ABC Tutorials
Up: ABC Stat
- Definitions
- Distributions
- Multivariate correlations
- Principal Component Analysis
- Panel Data Analysis
©Nikolai V. Shokhirev, 2001-2008