What is Circular Error Probable (CEP)?

Circular Error Probable (CEP) is a measure used in military science, specifically in ballistics, to quantify the precision of a weapon system. It is defined as the radius of a circle, centered on the aimpoint, within which 50% of the rounds fired are expected to land. Essentially, it represents the median error radius, serving as a 50% confidence interval for the impact points of munitions.

How is CEP used to understand weapon system precision?

CEP provides a standardized way to express the precision of a weapon system. For example, if a munition has a CEP of 10 meters, it implies that when 100 projectiles are targeted at the same point, an average of 50 will fall within a 10-meter radius of that point. A smaller CEP value indicates a more precise weapon system.

What are alternative names for Circular Error Probable?

Alternative names for Circular Error Probable include 'circular error probability' and 'circle of equal probability'. These terms all refer to the same statistical measure used to define the precision of a weapon system's impact points.

Besides ballistics, what other field utilizes the concept of CEP?

The concept of CEP is also applied when measuring the accuracy of positions obtained from navigation systems. Examples include GPS (Global Positioning System) and older systems like LORAN and Loran-C, where CEP helps quantify the reliability and precision of the location data.

What is the statistical distribution underlying the original concept of CEP?

The original concept of CEP was based on a circular bivariate normal distribution (CBN). This statistical model assumes that the errors in two dimensions (like range and azimuth) are normally distributed, uncorrelated, and have equal variances, causing impacts to cluster symmetrically around the mean impact point.

What are the expected proportions of rounds landing within multiples of the CEP radius under a circular bivariate normal distribution?

Under a circular bivariate normal distribution, if the CEP is 'n' meters: approximately 50% of shots land within 'n' meters of the mean impact point; about 43.7% land between 'n' and '2n' meters; roughly 6.1% land between '2n' and '3n' meters; and only about 0.2% of shots land farther than three times the CEP radius from the mean.

When might CEP not be an ideal measure of accuracy?

CEP may not be a suitable measure of accuracy when the distribution of impact points deviates from the circular bivariate normal model. This can occur if the standard deviations of range errors differ significantly from azimuth errors, resulting in an elliptical confidence region, or if there is a substantial bias, meaning the mean impact point is not centered on the aimpoint.

How can CEP be defined to account for bias and non-circular error distributions?

To incorporate factors like bias and differing error variances, CEP can be defined as the square root of the Mean Square Error (MSE). The MSE aggregates the variance of range error, variance of azimuth error, covariance between them, and the square of the bias, providing a single value that geometrically corresponds to the radius within which 50% of rounds will land.

What are some methods mentioned for estimating CEP from shot data?

Several methods exist for estimating CEP from observed impact data. These include the plug-in approach by Blischke and Halpin (1966), the Bayesian approach by Spall and Maryak (1992), and the maximum likelihood approach by Winkler and Bickert (2012). The Spall and Maryak method is particularly useful for data from mixed sources.

What is DRMS and how does it relate to standard deviation?

DRMS stands for Distance Root Mean Square. It is calculated as the square root of the average squared distance error. In the context of a circular bivariate normal distribution, DRMS is equal to the standard deviation (σ) multiplied by the square root of 2 (σd = √2 * σ). It represents the radius encompassing approximately 63.2% of the impacts.

What is R95, and how does it differ from CEP?

R95 is a measure representing the radius of a circle within which 95% of the impact points are expected to fall. Unlike CEP, which defines a 50% probability radius, R95 provides a much higher confidence interval, indicating the outer bounds for the vast majority of a weapon system's shots.

What does the formula Q(F, σ) = σ * sqrt(-2 * ln(1 - F/100%)) calculate?

This formula calculates the radius (Q) that encompasses a specific percentile probability (F) of impacts within a circular bivariate normal distribution, based on the standard deviation (σ) of the error components. It allows for the determination of radii for any desired probability, such as R95 or R99.7, not just the 50% for CEP.

According to the conversion table, what is the factor to convert DRMS to R95?

According to the conversion table, to convert a DRMS value to an R95 value, you multiply the DRMS by approximately 1.73. This means the radius containing 95% of impacts is about 1.73 times larger than the DRMS value.

What is the statistical basis for the conversion table relating different accuracy measures?

The conversion table is based on the properties of the Rayleigh distribution, which models the magnitude of distance error when the horizontal position errors are independent Gaussian variables with equal standard deviations. The table provides coefficients derived from this distribution to convert measures like DRMS or CEP to other percentile-based radii like R95 or R99.7.

What does the image caption 'CEP concept and hit probability. 0.2% outside the outmost circle' illustrate?

The caption describes a visual representation demonstrating the concept of Circular Error Probable (CEP) and its relation to hit probability. It highlights that, under the assumed distribution, a very small fraction (0.2%) of rounds are expected to land beyond the outermost depicted boundary, implying high overall precision.

What is the primary purpose of the 'Original research' notice at the beginning of the article?

The 'Original research' notice serves as a flag indicating that the article may contain unsourced claims or information not adequately verified by published sources. It prompts editors and readers to improve the article by verifying statements and adding inline citations to ensure adherence to Wikipedia's policies on verifiability.

What does the hatnote 'Circular error redirects here' signify?

The hatnote clarifies that the term 'Circular error' is a redirect to this article on 'Circular error probable'. It also directs readers to specific articles on 'pendulum' and 'pendulum (mathematics)' if they are interested in the circular error concept related to pendulums.

How is CEP defined as a 50% confidence interval?

Defining CEP as a 50% confidence interval means that there is a 50% probability that an actual impact point will fall within the circle defined by the CEP radius around the aimpoint. Conversely, there is an equal 50% probability that the impact point will fall outside this radius.

What is the statistical meaning of the 'median error radius' definition of CEP?

Defining CEP as the 'median error radius' means that this radius value divides the distribution of impact points exactly in half. Half of the projectiles are expected to land within this radius, and the other half are expected to land beyond it, making it the 50th percentile of the radial error.

What is the relationship between CEP and the standard deviation (σ) for a circular bivariate normal distribution?

For a circular bivariate normal distribution, the CEP is approximately 1.18 times the standard deviation (σ) of the individual error components (CEP ≈ 1.18 * σ). This indicates that the radius containing 50% of impacts is slightly larger than the standard deviation of the errors along the primary axes.

What is the statistical meaning of the value 63.213... associated with DRMS in the table?

The value 63.213... associated with DRMS in the table represents the percentage of impacts expected to fall within the radius defined by the DRMS. This means DRMS is the radius that encompasses approximately 63.2% of the projectiles in a circular bivariate normal distribution.

What is the relationship between R95 and the standard deviation (σ)?

The relationship between R95 (the radius containing 95% of impacts) and the standard deviation (σ) for a circular bivariate normal distribution is approximately R95 ≈ 2.45 * σ. This shows that the radius needed to cover 95% of impacts is significantly larger than the standard deviation of the individual error components.

What is the primary difference between CEP and DRMS as measures of dispersion?

The primary difference lies in the probability coverage: CEP defines the radius for 50% of impacts (median error), while DRMS corresponds to approximately 63.2% of impacts and is directly related to the standard deviation of the error components. Generally, DRMS is larger than CEP for the same distribution.

What does the term 'munitions' refer to in the context of CEP?

In the context of CEP, 'munitions' refers to projectiles or weapon systems, such as artillery shells, missiles, or bombs, whose precision of delivery is being measured. CEP quantifies the accuracy with which these munitions can be expected to strike their intended target area.

What is the purpose of the 'See also' section in the article?

The 'See also' section lists related articles that may provide additional context or information on the topic. For this article, it points to 'Probable error', a related statistical concept concerning dispersion.

What is the statistical concept mentioned in the 'See also' section that is related to CEP?

The related statistical concept mentioned in the 'See also' section is 'Probable error'. Probable error is another measure of dispersion, historically linked to CEP, representing the value such that there is a 50% chance of the error being less than it.

What kind of information is found in the 'References' section?

The 'References' section lists the specific sources cited within the article, providing details like author, title, publication date, and publisher. This allows readers to verify the information presented and explore the original research.

What is the definition of 'bias' in the context of CEP estimation?

Bias in CEP estimation refers to a systematic error where the average impact point consistently deviates from the intended aimpoint. It indicates a predictable miss pattern rather than random dispersion, and it needs to be accounted for in comprehensive accuracy assessments, often through measures like Mean Square Error (MSE).

How does CEP relate to the concept of accuracy versus precision?

CEP is primarily a measure of precision, quantifying the consistency or spread of impacts. Accuracy, however, refers to how close the impacts are to the true target value, encompassing both precision and the absence of bias. While CEP measures the tightness of the grouping, it doesn't guarantee accuracy if bias is present.

What does the R99.7 measure signify?

R99.7 signifies the radius of a circle within which 99.7% of the impact points are expected to fall. This represents a very high probability coverage, indicating the outer boundary for the vast majority of expected hits from a weapon system, often corresponding to approximately three standard deviations in a normal distribution.

What is the statistical meaning of the 'median error radius' definition of CEP?

The 'median error radius' definition of CEP means that this radius value divides the distribution of impact points into two equal halves: 50% of projectiles land within this radius, and 50% land outside it. It represents the 50th percentile of the radial error.

What is the role of the 'date-container' element in the original research notice?

The 'date-container' element, displaying 'June 2024' in this article, indicates the month and year when the 'Original research' notice was added. This timestamp helps track the maintenance status of the article and signals how long it has been flagged for review.

What is the relationship between the standard deviation (σ) and the R95 measure?

For a circular bivariate normal distribution, the R95 (radius containing 95% of impacts) is approximately 2.45 times the standard deviation (σ) of the individual error components (R95 ≈ 2.45 * σ). This means the radius for 95% probability is significantly larger than the standard deviation.

What does the term 'ballistics' refer to in the context of CEP?

In the context of CEP, 'ballistics' refers to the science dealing with the motion, behavior, and effects of projectiles like bullets, shells, missiles, and bombs. CEP is a key metric used within ballistics to evaluate and compare the precision of these systems.

What is the statistical significance of the formula Q(F, σ) = σ * sqrt(-2 * ln(1 - F/100%))?

This formula is statistically significant because it allows the calculation of the radius (Q) needed to achieve any desired probability (F) of impact within a circular bivariate normal distribution, given the standard deviation (σ). It provides a method to determine accuracy measures for various confidence levels beyond the standard 50% CEP.

What is the purpose of the 'hide-when-compact' class in the original research notice?

The 'hide-when-compact' class is a technical design element likely used to control the visibility of certain parts of the notice based on screen size or layout. It helps manage clutter by potentially hiding less critical information on smaller displays or condensed views.

What is the relationship between DRMS and the probability of impact?

DRMS represents the radius within which approximately 63.2% of projectiles are expected to land, assuming a circular bivariate normal distribution. It serves as a measure of dispersion that covers a higher probability than CEP (50%) but less than R95 (95%).

What does the conversion table suggest about the relationship between R95 and 2DRMS?

The conversion table indicates that R95 is approximately 0.409 times the value of 2DRMS. Conversely, 2DRMS is about 2.45 times larger than R95, showing that R95 represents a tighter bound (95% probability) compared to the radius associated with 2DRMS.

What is the statistical meaning of the 'circular bivariate normal distribution'?

The 'circular bivariate normal distribution' is a probability distribution used to model errors in two dimensions where the errors are normally distributed, uncorrelated, and have equal variances. This results in a symmetrical, bell-shaped probability density function where contours of equal probability are circles centered on the mean.

What is the role of the 'cite-bracket' notation in the source text?

The 'cite-bracket' notation, such as `[1]` or `[5]`, signifies inline citations within the article. These brackets link specific pieces of information in the text to the corresponding reference entry listed at the end, enabling readers to verify the source of the information.

How is the concept of CEP applied in navigation systems like GPS?

When applied to navigation systems like GPS, CEP quantifies the precision of the position data. A stated CEP value indicates the radius around the reported location within which the true position is expected to lie with 50% probability, helping users assess the reliability of the navigation information.

What is the statistical significance of the formula Q(F, σd) = σd * (sqrt(-2 * ln(1 - F/100%)) / sqrt(2))?

This formula is significant as it allows the calculation of the radius (Q) for a given probability (F) using the Distance Root Mean Square (DRMS, denoted σd). It adapts the general percentile calculation formula for DRMS, enabling conversions between different accuracy measures based on DRMS.

What does the image 'Circular error probable - percentage.png' likely depict?

The image 'Circular error probable - percentage.png' likely visually represents the concept of CEP and its associated hit probabilities. It probably illustrates concentric circles around an aimpoint, showing how different percentages of rounds fall within radii of varying sizes, potentially highlighting the 50% CEP and the small percentage (0.2%) falling far outside.

What is the purpose of the 'Further reading' section?

The 'Further reading' section suggests additional resources, such as academic papers and technical documents, that readers can consult for a more in-depth understanding of Circular Error Probable and related topics beyond the scope of the main article.

Circular Error Probable Wiki: Precision Under Scrutiny: Decoding Circular Error Probable (CEP)

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What is CEP?

Defining Precision

Circular Error Probable (CEP), also known as circular error probability or circle of equal probability, is a fundamental measure of a weapon system's precision within the field of ballistics. It is defined as the radius of a circle, centered on the intended aimpoint, within which 50% of the rounds are expected to land. Essentially, it represents the median error radius, serving as a 50% confidence interval for the impact points.

Statistical Significance

If a specific munitions design exhibits a CEP of 10 meters, it implies that when 100 rounds are fired at the same target point, an average of 50 rounds will fall within a circle of 10 meters radius around that point. This metric is crucial for evaluating and comparing the accuracy of different weapon systems and delivery methods.

Beyond Ballistics

The concept of CEP extends beyond traditional ballistics. It is also employed when assessing the accuracy of positions determined by navigation systems, including modern technologies like GPS (Global Positioning System) and older systems such as LORAN and Loran-C.

The Core Concept

Ideal Distribution

The original formulation of CEP was predicated on a circular bivariate normal (CBN) distribution. In this idealized scenario, munitions impacts cluster symmetrically around the mean impact point. The distribution follows a pattern where most impacts are close to the mean, with progressively fewer impacts occurring at greater distances.

Limitations and Real-World Scenarios

CEP's utility diminishes when the actual distribution of impacts deviates from the ideal CBN model. This can occur if the standard deviation of range errors differs significantly from the standard deviation of azimuth (deflection) errors, leading to an elliptical confidence region rather than a circular one. Furthermore, the mean impact point may not precisely align with the intended aimpoint, introducing bias. In such cases, CEP might not fully capture the complex error characteristics.

Mathematical Foundations

Error Distribution Models

The horizontal position error is often modeled as a 2D vector composed of two orthogonal Gaussian random variables, each with a standard deviation denoted by $\sigma$ . When these variables are uncorrelated, the magnitude of this error vector follows a Rayleigh distribution with a scale factor of $\sigma$ .

The DRMS, a measure related to the standard deviation, is given by $\sigma _{d}={\sqrt {2}}\sigma$ . Errors within this DRMS value account for approximately 63% of the sample in a bivariate circular distribution.

Percentile Conversions

The relationship between different measures of error (like DRMS, CEP, R95) and their corresponding probability levels (F) can be complex. The following table illustrates these relationships and provides coefficients for converting between different percentile levels, particularly useful when dealing with Rayleigh distributions.

The radius Q for a given probability F can be calculated using the scale factor $\sigma$ or DRMS $\sigma _{d}$ :

$Q(F,\sigma )=\sigma {\sqrt {-2\ln(1-F/100\%)}}$

$Q(F,\sigma _{d})=\sigma _{d}{\frac {\sqrt {-2\ln(1-F/100\%)}}}{\sqrt {2}}}$

Measure of $Q$	Probability $F\,(\%)$
DRMS	63.213...
CEP	50
2DRMS	98.169...
R95	95
R99.7	99.7

The following table shows conversion coefficients ( $\alpha$ ) to convert a measure X into another measure Y = α · X:

From $X\downarrow$ to $Y\rightarrow$	RMS ( $\sigma$ )	CEP	DRMS	R95	2DRMS	R99.7
RMS ( $\sigma$ )	1.00	1.18	1.41	2.45	2.83	3.41
CEP	0.849	1.00	1.20	2.08	2.40	2.90
DRMS	0.707	0.833	1.00	1.73	2.00	2.41
R95	0.409	0.481	0.578	1.00	1.16	1.39
2DRMS	0.354	0.416	0.500	0.865	1.00	1.21
R99.7	0.293	0.345	0.415	0.718	0.830	1.00

For instance, a GPS receiver with a DRMS of 1.25 meters would exhibit a 95% radius (R95) of approximately 1.25 m × 1.73 = 2.16 meters.

Applications & Related Concepts

Navigation Systems

CEP is a vital metric for evaluating the accuracy of navigation systems. It provides a clear, statistically grounded understanding of how precisely a system, such as GPS, can determine a location. This is critical for applications ranging from military targeting to civilian logistics and surveying.

Statistical Measures

Beyond CEP, other statistical measures are used to characterize error distributions. These include the Distance Root Mean Square (DRMS), which is related to the standard deviation of the radial error, and R95, indicating the radius encompassing 95% of impacts. Understanding these different metrics provides a more complete picture of system performance.

Further Study

The rigorous analysis of CEP and related error metrics involves advanced statistical concepts. Research into estimators for quantiles of circular error, Bayesian approaches, and maximum likelihood methods continues to refine how we measure and understand precision in complex systems.

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References

Circular Error Probable (CEP), Air Force Operational Test and Evaluation Center Technical Paper 6, Ver 2, July 1987, p. 1

Frank van Diggelen, "GPS Accuracy: Lies, Damn Lies, and Statistics", GPS World, Vol 9 No. 1, January 1998

A full list of references for this article are available at the Circular error probable Wikipedia page

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This page was generated by an Artificial Intelligence and is intended for informational and educational purposes only. The content is based on a snapshot of publicly available data from Wikipedia and may not be entirely accurate, complete, or up-to-date.

This is not professional advice. The information provided on this website is not a substitute for professional consultation in fields such as ballistics, engineering, navigation, or statistics. Always refer to official documentation and consult with qualified professionals for specific applications or critical decisions.

The creators of this page are not responsible for any errors or omissions, or for any actions taken based on the information provided herein.

Precision Under Scrutiny

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What is CEP? ℹ️

🎯 Defining Precision

📊 Statistical Significance

🛰️ Beyond Ballistics

The Core Concept 💡

⚖️ Ideal Distribution

⚠️ Limitations and Real-World Scenarios

Mathematical Foundations 🧮

📈 Error Distribution Models

🧮 Percentile Conversions

Applications & Related Concepts 🔗

🧭 Navigation Systems

⚖️ Statistical Measures

📚 Further Study

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📜 Important Notice

Dive in with Flashcard Learning!

What is CEP?

Defining Precision

Statistical Significance

Beyond Ballistics

The Core Concept

Ideal Distribution

Limitations and Real-World Scenarios

Mathematical Foundations

Error Distribution Models

Percentile Conversions

Applications & Related Concepts

Navigation Systems

Statistical Measures

Further Study

Teacher's Corner

True or False?

Test Your Knowledge!

Gamer's Corner

Discover other topics to study!

References

Feedback & Support

Disclaimer

Important Notice