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Uncertainty: Confidence Intervals

Uncertainty: Confidence Intervals

The confidence interval is the range of values around the mean that contains the true mean. It is expressed as a probability percentage. The interpretation of a 95% confidence interval, for instance, is that the statistician is 95% confident that the true mean falls within the interval. The upper and lower limits of this range are known as confidence limits. The confidence limits for the true mean are estimated from the sample's mean, the standard deviation, and the statistical factor...

Propagation of Uncertainty from Random Error

Propagation of Uncertainty from Random Error

An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...

Uncertainty: Overview

Uncertainty: Overview

In analytical chemistry, we often perform repetitive measurements to detect and minimize inaccuracies caused by both determinate and indeterminate errors. Despite the cares we take, the presence of random errors means that repeated measurements almost never have exactly the same magnitude. The collective difference between these measurements - observed values - and the estimated or expected value is called uncertainty. Uncertainty is conventionally written after the estimated or expected value.

Propagation of Uncertainty from Systematic Error

Propagation of Uncertainty from Systematic Error

The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this...

Interpretation of Confidence Intervals

Interpretation of Confidence Intervals

A confidence interval is a better estimate of the population than a point estimate, as it uses a range of values from a sample instead of a single value.
Confidence intervals have confidence coefficients that are crucial for their interpretation. The most common confidence coefficients are 0.90, 0.95, and 0.99, which can be written as percentages–90%, 95%, and 99%, respectively.
Suppose a person calculates a confidence interval with a confidence coefficient of 0.95. In that case, they can...

Confidence Intervals

Confidence Intervals

An unbiased point estimate is often insufficient to predict a population estimate, such as population mean or population proportion. In this scenario, a confidence interval is used. A confidence interval is an estimate similar to a sample proportion. However, unlike the point estimate which is a single value, the confidence interval contains a range of values. These values have lower and upper limits, known as confidence limits, and can be designated as L1 and L2, respectively.
A...

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Related Experiment Video

Updated: Apr 4, 2026

An R-Based Landscape Validation of a Competing Risk Model

An R-Based Landscape Validation of a Competing Risk Model

Published on: September 16, 2022

Approximate Uncertainty Modeling in Risk Analysis with Vine Copulas.

Tim Bedford¹, Alireza Daneshkhah², Kevin J Wilson¹

¹Department of Management Science, University of Strathclyde, Glasgow, UK.

Risk Analysis : an Official Publication of the Society for Risk Analysis

|September 3, 2015

Summary

This summary is machine-generated.

This study introduces novel vine-based copula methods for joint uncertainty modeling in risk analysis. These methods offer superior approximation capabilities for complex, high-dimensional probability distributions, especially with non-constant dependencies.

Keywords:

Copula entropy information risk modeling vine

Related Experiment Videos

Last Updated: Apr 4, 2026

An R-Based Landscape Validation of a Competing Risk Model

An R-Based Landscape Validation of a Competing Risk Model

Published on: September 16, 2022

Area of Science:

Probability theory
Statistical modeling
Risk analysis

Background:

Jointly modeling multiple uncertain quantities is crucial for risk analysis.
Bayesian networks and copulas are standard methods for this, but have limitations.

Purpose of the Study:

To develop new methodologies for copulas using vine structures.
To address limitations of existing approaches like the multivariate Gaussian copula.

Main Methods:

Utilizing vine structures for constructing higher-dimensional probability distributions.
Applying minimum information copulas for parametric approximation.
Extending vine methods to include non-constant conditional dependencies.

Main Results:

Demonstrated a fundamental approximation result: any density can be closely approximated using vines.
Developed parametric copula classes with strong approximation properties.
Showcased applicability to financial risk modeling with non-constant dependencies.

Conclusions:

Vine-based copulas provide a flexible and powerful framework for joint uncertainty modeling.
The proposed methods enhance the approximation of complex probability distributions.
These techniques are valuable for financial risk analysis and can be quantified via expert judgment or data fitting.