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Related Concept Videos

Mean Absolute Deviation01:13

Mean Absolute Deviation

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The mean absolute deviation is also a measure of the variability of data in a sample. It is the absolute value of the average difference between the data values and the mean.
Let us consider a dataset containing the number of unsold cupcakes in five shops: 10, 15, 8, 7, and 10. Initially, calculate the sample mean. Then calculate the deviation, or the difference, between each data value and the mean. Next, the absolute values of these deviations are added and divided by the sample size to...
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Estimating Population Mean with Known Standard Deviation01:16

Estimating Population Mean with Known Standard Deviation

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To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need sample mean as an estimate for μ and we need the margin of error. Here, the margin of error (EBM) is called the error bound for a population mean (abbreviated EBM). The sample mean is the point estimate of the unknown population mean μ.
The confidence interval estimate will have the form as follows:
(point estimate - error bound, point estimate +...
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Estimating Population Mean with Unknown Standard Deviation01:22

Estimating Population Mean with Unknown Standard Deviation

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In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Gosset (1876–1937) of the...
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Empirical Method to Interpret Standard Deviation01:09

Empirical Method to Interpret Standard Deviation

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The empirical rule, also known as the three-sigma rule, allows a statistician to interpret the standard deviation in a normally distributed dataset. The rule states that 68% of the data lies within one standard deviation from the mean, 95% lies within two standard deviations from the mean, and 99.7% lies within three standard deviations from the mean. Additionally, this rule is also called the 68-95-99.7 rule.
This rule is used widely in statistics to calculate the proportion of data values...
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Coefficient of Correlation01:12

Coefficient of Correlation

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The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y.
If you suspect a linear relationship between x and y, then r can measure how strong the linear relationship is.
What the VALUE of r tells us:
The value of r is always between –1 and +1: –1 ≤ r ≤ 1.
The size of the correlation r indicates the...
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Sampling Distribution01:12

Sampling Distribution

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Given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution. How much the statistic varies from one sample to another is known as the sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example...
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How to Get MAD: Generating Uniformly Sampled Correlation Matrices with a Fixed Mean Absolute Discrepancy.

Niels G Waller1

  • 1Department of Psychology, University of Minnesota, Minneapolis, MN, USA.

Multivariate Behavioral Research
|August 4, 2025
PubMed
Summary

This study introduces a fast algorithm for generating correlation matrices with controlled error. This method is useful for model robustness, portfolio stress testing, and analyzing model-approximation error.

Keywords:
Correlation matricesMonte Carlo studieselliptope geometrymean absolute discrepancy

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Area of Science:

  • Statistics
  • Computational Mathematics
  • Financial Modeling

Background:

  • Generating correlation matrices with specific properties is crucial for various statistical and financial applications.
  • Existing methods may lack efficiency or control over the degree of deviation from a target matrix.
  • Understanding model-approximation error is key in robust analysis and stress testing.

Purpose of the Study:

  • To present a simple and fast algorithm for generating uniformly sampled correlation matrices (R) with a fixed mean absolute discrepancy (MAD) relative to a target population matrix (R_pop).
  • To demonstrate the utility of this algorithm in applications such as model robustness studies, investment portfolio stress testing, and dynamic model-fit analyses.
  • To provide a theoretical underpinning using higher-dimensional geometry to characterize the properties of matrices generated by the algorithm.

Main Methods:

  • Development of a novel algorithm for sampling correlation matrices with a controlled MAD.
  • Application of higher-dimensional geometry to define the mathematical space where matrices with a fixed MAD reside.
  • Implementation of the algorithm using R code, with accompanying materials for reproducibility.

Main Results:

  • The algorithm efficiently generates correlation matrices with a specified MAD.
  • Matrices with a fixed MAD are shown to lie in the intersection of an elliptope and the surface of a cross-polytope.
  • For n=3, these geometric sets can be visualized as an elliptical tetrahedron and an octahedron surface.

Conclusions:

  • The developed algorithm provides a practical tool for generating correlation matrices with precise control over approximation error.
  • The theoretical framework offers insights into the geometric structure of correlation matrices with bounded discrepancies.
  • The algorithm and its implementation are valuable for researchers and practitioners in statistics, finance, and computational modeling.