25 area under the curve meaning statistics Ultimate Guide
You are reading about area under the curve meaning statistics. Here are the best content by the team thcsngogiatu.edu.vn synthesize and compile, see more in the section How to.
Classification: ROC Curve and AUC [1]
An ROC curve (receiver operating characteristic curve) is a graph showing the performance of a classification model at all classification thresholds. True Positive Rate (TPR) is a synonym for recall and is therefore defined as follows:
To compute the points in an ROC curve, we could evaluate a logistic regression model many times with different classification thresholds, but this would be inefficient. Fortunately, there’s an efficient, sorting-based algorithm that can provide this information for us, called AUC.
AUC provides an aggregate measure of performance across all possible classification thresholds. One way of interpreting AUC is as the probability that the model ranks a random positive example more highly than a random negative example
Expert Maths Tutoring in the UK [2]
Area under the curve is calculated by different methods, of which the antiderivative method of finding the area is most popular. The area under the curve can be found by knowing the equation of the curve, the boundaries of the curve, and the axis enclosing the curve
The process of integration helps to solve the equation and find the required area.. For finding the areas of irregular plane surfaces the methods of antiderivatives are very helpful
The area under the curve can be calculated through three simple steps. First, we need to know the equation of the curve(y = f(x)), the limits across which the area is to be calculated, and the axis enclosing the area
ROC Analysis: Key Statistical Tool for Evaluating Detection Technologies [3]
A service of the National Library of Medicine, National Institutes of Health.. Institute of Medicine (US) and National Research Council (US) Committee on New Approaches to Early Detection and Diagnosis of Breast Cancer; Joy JE, Penhoet EE, Petitti DB, editors
Washington (DC): National Academies Press (US); 2005.. Saving Women’s Lives: Strategies for Improving Breast Cancer Detection and Diagnosis.Show details
The term receiver operating characteristic (ROC) originates from the use of radar during World War II. Just as American soldiers deciphered a blip on the radar screen as a German bomber, a friendly plane, or just noise, radiologists face the task of identifying abnormal tissue against a complicated background
Area Under the Curve – Definition, Types, and Examples [4]
Area Under the Curve – Definition, Types, and Examples. One of the most useful applications of integral calculus is learning how to calculate the area under the curve
Learning about areas under the curve also makes you appreciate what you’ve learned so far and makes you see how amazing integral calculus is.. Areas under the curve are formed with the function, two vertical lines, and the horizontal axis
By the end of our discussion, you should be able to calculate the following:. – The area of the region completely lying above the $x$-axis.
GraphPad Prism 10 Statistics Guide [5]
The area under the curve is an integrated measurement of a measurable effect or phenomenon. It is used as a cumulative measurement of drug effect in pharmacokinetics and as a means to compare peaks in chromatography.
Start from a data or results table that represents a curve. Click Analyze and choose Area under the curve from the list of XY analyses.
Prism can only do this, however, if the regions are clearly defined: the signal, or graphic representation of the effect or phenomenon, must go below the baseline between regions and the peaks cannot overlap.. For each region, Prism shows the area in units of the X axis times units of the Y axis
Expert Maths Tutoring in the UK [6]
Area under the curve is calculated by different methods, of which the antiderivative method of finding the area is most popular. The area under the curve can be found by knowing the equation of the curve, the boundaries of the curve, and the axis enclosing the curve
The process of integration helps to solve the equation and find the required area.. For finding the areas of irregular plane surfaces the methods of antiderivatives are very helpful
The area under the curve can be calculated through three simple steps. First, we need to know the equation of the curve(y = f(x)), the limits across which the area is to be calculated, and the axis enclosing the area
How to interpret the Area Under the Curve (AUC) stat [7]
One of the questions I often ask in data science interviews is ‘How would you explain the area under the curve statistic to a business person?’. Maybe it is too easy a question even for juniors, as I can’t remember anyone getting it wrong
Only after that do you then even bother to show the ROC curve, and say we calculate the area under the curve (AUC) as a measure of how well the model can discriminate the two classes.. The most recent situation I remember this happened in real life, I actually said to the business rep that the AUC does not directly translate to revenue, but is a good indication that a model is good in an absolute sense (we know others have AUCs typically around 0.7 to 0.8 for this problem, and over 0.5 is better than random)
So while I try to do my best explaining technical statistical content, I often punt to simpler ‘here are the end outcomes we care about’ (which don’t technically answer the question) as opposed to ‘here is how the sausage is made’ explanations.. One alternative and simple explanation of AUC though for binary models is to take the Harrell’s C index interpretation, which for binary predictions is equivalent to the AUC statistic
A Complete Guide to Area Under Curve (AUC) [8]
AUC or ROC curve is a plot of the proportion of true positives (events correctly predicted to be events) versus the proportion of false positives (nonevents wrongly predicted to be events) at different probability cutoffs. Sensitivity is on Y-axis and (1-Specificity) is on X-axis
Cut-off represents minimum threshold after that predicted probability would be classified as ‘event’ (desired outcome). In other words, predictive probability greater than or equal to cut-off would be classified as 1
To generate ROC curve, we calculate Sensitivity and (1-Specificity) at all possible cutoffs and then we plot them.. Diagonal line represents random classification model
ROC curves and Area Under the Curve explained (video) [9]
ROC curves and Area Under the Curve explained (video). While competing in a Kaggle competition this summer, I came across a simple visualization (created by a fellow competitor) that helped me to gain a better intuitive understanding of ROC curves and Area Under the Curve (AUC)
An ROC curve is the most commonly used way to visualize the performance of a binary classifier, and AUC is (arguably) the best way to summarize its performance in a single number. As such, gaining a deep understanding of ROC curves and AUC is beneficial for data scientists, machine learning practitioners, and medical researchers (among others).
If you want to skip to a particular section in the video, simply click one of the time codes listed in the transcript (such as 0:52).. I welcome your feedback and questions in the comments section!
Statistics – ROC Plot and Area under the curve (AUC) [10]
The Area Under Curve (AUC) metric measures the performance of a binary classification.. In a regression classification for a two-class problem using a probability algorithm, you will capture the probability threshold changes in an ROC curve.
False positives (legitimate emails erroneously predicted as spam) are likely to cause more harm than false negatives (spam emails that are not identified as spam), as we might miss an important email, while it is easy to delete a spam message. In this case, we could require a higher threshold (probability) that a message is spam before we move it into a spam folder.
This is a single curve that captures the behaviour of the classification rate when varying the classification threshold.. With a true positive rate of one and a false positive rate of zero, the best curve will right up as far as possible into the top left hand corner.
Area Under the Curve Formula with Solved Example [11]
The area under a curve between two points is found out by doing a definite integral between the two points. To find the area under the curve y = f(x) between x = a & x = b, integrate y = f(x) between the limits of a and b
Question : Calculate the area under the curve of a function, f(x) = 7 – x2, the limit is given as x = -1 to 2.. Given function is, f(x) = 7- x2 and limit is x = -1 to 2
\(\begin{array}{l}\large = \left ( 7x-\frac{1}{3}x^{3}\right)|_{-1}^{2}\end{array} \). \(\begin{array}{l}\large = \left [ 7.2-\frac{1}{3}(8) \right ]-\left [ 7(-1)-\frac{1}{3}(-1)\right ]\end{array} \)
Classification: ROC Curve and AUC [12]
An ROC curve (receiver operating characteristic curve) is a graph showing the performance of a classification model at all classification thresholds. True Positive Rate (TPR) is a synonym for recall and is therefore defined as follows:
To compute the points in an ROC curve, we could evaluate a logistic regression model many times with different classification thresholds, but this would be inefficient. Fortunately, there’s an efficient, sorting-based algorithm that can provide this information for us, called AUC.
AUC provides an aggregate measure of performance across all possible classification thresholds. One way of interpreting AUC is as the probability that the model ranks a random positive example more highly than a random negative example
4.3: Z-scores and the Area under the Curve [13]
\(Z\)-scores and the standard normal distribution go hand-in-hand. A \(z\)-score will tell you exactly where in the standard normal distribution a value is located, and any normal distribution can be converted into a standard normal distribution by converting all of the scores in the distribution into \(z\)-scores, a process known as standardization.
Because \(z\)-scores are in units of standard deviations, this means that 68% of scores fall between \(z\) = -1.0 and \(z\) = 1.0 and so on. We call this 68% (or any percentage we have based on our \(z\)-scores) the proportion of the area under the curve
An important property to point out here is that, by virtue of the fact that the total area under the curve of a distribution is always equal to 1.0 (see section on Normal Distributions at the beginning of this chapter), these areas under the curve can be added together or subtracted from 1 to find the proportion in other areas. For example, we know that the area between \(z\) = -1.0 and \(z\) = 1.0 (i.e
IBM Documentation [14]
The area under the curve represents the probability that the assay result for a randomly chosen positive case will exceed the result for a randomly chosen negative case. The asymptotic significance is less than 0.05, which means that using the assay is better than guessing.
See the coordinates of the curve to compare different cutoffs.
Area Under the Curve: Definition, Applications and Calculations [15]
Curves are ubiquitous in mathematics, appearing in a wide range of fields. One of the key concepts associated with curves is the Area Under the Curve
In this mathematics article, we will explore the fundamentals of Area Under the Curve, delve into its calculation methods, and uncover its wide-ranging applications.
Area under the curve (pharmacokinetics) [16]
In the field of pharmacokinetics, the area under the curve (AUC) is the definite integral of the concentration of a drug in blood plasma as a function of time (this can be done using liquid chromatography–mass spectrometry[1]). In practice, the drug concentration is measured at certain discrete points in time and the trapezoidal rule is used to estimate AUC
The AUC (from zero to infinity) represents the total drug exposure across time. AUC is a useful metric when trying to determine whether two formulations of the same dose (for example a capsule and a tablet) result in equal amounts of tissue or plasma exposure
For example, gentamicin is an antibiotic that can be nephrotoxic (kidney damaging) and ototoxic (hearing damaging); measurement of gentamicin through concentrations in a patient’s plasma and calculation of the AUC is used to guide the dosage of this drug.[3]. AUC becomes useful for knowing the average concentration over a time interval, AUC/t
What is AUC (Area Under the Curve)? [17]
The AUC is the area under the ROC curve (usually, sometimes the precision-recall curve is used, such as when there is class imbalance). Consider this image by BOR at the English language Wikipedia, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=10714489
Note, your cat-dog classifier has only one value for both TPR and FPR and so is only a single point on this curve. However, the classifier is most likely actually a model that outputs the probability of an image being a cat and it only really becomes a classifier after you make a decision by thresholding that probability, e.g
If your threshold is 0, your classifier just decides all images are cats. That results in a TPR and FPR of 1 which is the top right corner of the chart
ROC curve analysis [18]
A ROC curve is a plot of the true positive rate (Sensitivity) in function of the false positive rate (100-Specificity) for different cut-off points of a parameter. Each point on the ROC curve represents a sensitivity/specificity pair corresponding to a particular decision threshold
MedCalc creates a complete sensitivity/specificity report.. The ROC curve is a fundamental tool for diagnostic test evaluation.
ROC curves can also be used to compare the diagnostic performance of two or more laboratory or diagnostic tests (Griner et al., 1981).. When you consider the results of a particular test in two populations, one population with a disease, the other population without the disease, you will rarely observe a perfect separation between the two groups
Area Under the Curve [19]
In the field of pharmacokinetics, the Area Under the Curve (AUC) has a specific meaning. It is the region under a plotted line in a graph of medicine concentration in blood plasma over time
The area under the curve represents the total exposure that the body receives to an active substance, and helps to evaluate and compare bioavailability profiles between medicines. The time at which the highest concentration of the active substance is found in the blood is called Tmax, and the maximum concentration of the active substance found in the blood stream is called Cmax.« Back to Glossary Index
Find the Area Under a Normal Curve [20]
Before you can solve for the area under a normal curve, you must be able to imagine what the area looks like. The best (albeit optional) way do this is to make a sketch
There are seven ways your sketch could look, depending on what z-values you were given. Once you have drawn your sketch, look at the pictures below
The link will take you to a step-by-step guide on how to find the area under a normal curve for that shape. Many of these also have short videos showing the steps.
Inference for the difference in the area under the ROC curve derived from nested binary regression models [21]
Inference for the difference in the area under the ROC curve derived from nested binary regression models. The area under the curve (AUC) statistic is a common measure of model performance in a binary regression model
The regression coefficient estimates used in the AUC statistics are computed using the maximum rank correlation methodology. Typically, inference for the difference in AUC statistics from nested models is derived under asymptotic normality
When none of the new factors are associated with the binary outcome, the asymptotic distribution for the difference in AUC statistics is a linear combination of chi-square random variables. Further, when at least one new factor is associated with the outcome and the population difference is small, a variance stabilizing reparameterization improves the asymptotic normality of the AUC difference statistic
Area Under a Curve – Mathematics A-Level Revision [22]
So far when integrating, there has always been a constant term left. For this reason, such integrals are known as indefinite integrals
The area under a curve between two points can be found by doing a definite integral between the two points.. To find the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b.
This means that you have to be careful when finding an area which is partly above and partly below the x-axis.. You may also be asked to find the area between the curve and the y-axis
Guide to AUC ROC Curve in Machine Learning : What Is Specificity? [23]
Guide to AUC ROC Curve in Machine Learning : What Is Specificity?. You’ve built your machine learning model – so what’s next? You need to evaluate and validate how good (or bad) it is, so you can decide whether to implement it
But don’t worry, we will see what these terms mean in detail, and everything will be a piece of cake!. For now, just know that the AUC-ROC curve helps us visualize how well our machine learning classifier performs
We’ll also cover topics like sensitivity and specificity since these are key topics behind the AUC-ROC curve (or ROC AUC curve in machine learning).. I suggest going through the article on Confusion Matrix as it will introduce some important terms we will use in this article.
Numeracy, Maths and Statistics [24]
When we are dealing with continuous probability distributions, we always look at the probability that a random variable lies within a particular range of values. For example, we might want to know the probability that the weight of a ram of a particular breed is in the range $90-100$kg or the probability that it is greater than $120$kg.
The probability that a continuous random variable lies in a given range is equal to the area under the probability density function curve in that range. The total area under the curve for any pdf is always equal to $1$, this is because the value of a random variable has to lie somewhere in the sample space
The Normal or Gaussian distribution is possibly the best-known and most-used continuous probability distribution. The Normal distribution is a good approximation to many statistics of interest in populations such as height and weight
Introduction to the Normal Distribution (Bell Curve) [25]
The normal distribution is a continuous probability distribution that is symmetrical on both sides of the mean, so the right side of the center is a mirror image of the left side.. The area under the normal distribution curve represents the probability and the total area under the curve sums to one.
The tails are asymptotic, which means that they approach but never quite meet the horizon (i.e., the x-axis).. For a perfectly normal distribution, the mean, median, and mode will be the same value, visually represented by the peak of the curve.
It is also known as called Gaussian distribution, after the German mathematician Carl Gauss who first described it.. A normal distribution is determined by two parameters the mean and the variance
Sources
- https://developers.google.com/machine-learning/crash-course/classification/roc-and-auc#:~:text=AUC%20stands%20for%20%22Area%20under,across%20all%20possible%20classification%20thresholds.
- https://www.cuemath.com/calculus/area-under-the-curve/#:~:text=Area%20under%20the%20curve%20is,the%20axis%20enclosing%20the%20curve.
- https://www.ncbi.nlm.nih.gov/books/NBK22319/#:~:text=The%20ROC%20curve%20maps%20the,various%20correct%20and%20incorrect%20decisions.&text=A%20ROC%20curve%20is%20a,see%20Figure%20C%2D1).
- https://www.storyofmathematics.com/area-under-the-curve/
- https://www.graphpad.com/guides/prism/latest/statistics/stat_area_under_the_curve.htm
- https://www.cuemath.com/calculus/area-under-the-curve/
- https://andrewpwheeler.com/2021/11/19/how-to-interpret-the-area-under-the-curve-auc-stat/
- https://www.listendata.com/2014/08/learn-area-under-curve-auc.html
- https://www.dataschool.io/roc-curves-and-auc-explained/
- https://datacadamia.com/data_mining/roc
- https://byjus.com/area-under-the-curve-formula/
- https://developers.google.com/machine-learning/crash-course/classification/roc-and-auc
- https://stats.libretexts.org/Bookshelves/Applied_Statistics/An_Introduction_to_Psychological_Statistics_(Foster_et_al.)/04%3A_z-scores_and_the_Standard_Normal_Distribution/4.03%3A_Z-scores_and_the_Area_under_the_Curve
- https://www.ibm.com/docs/bs/SSLVMB_23.0.0/spss/tutorials/roc_area_hiv_01.html
- https://testbook.com/maths/area-under-the-curve
- https://en.wikipedia.org/wiki/Area_under_the_curve_(pharmacokinetics)
- https://stats.stackexchange.com/questions/321514/what-is-auc-area-under-the-curve
- https://www.medcalc.org/manual/roc-curves.php
- https://toolbox.eupati.eu/glossary/area-under-the-curve/
- https://www.statisticshowto.com/probability-and-statistics/normal-distributions/find-the-area-under-a-normal-curve/
- https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5965312/
- https://revisionmaths.com/advanced-level-maths-revision/pure-maths/calculus/area-under-curve
- https://www.analyticsvidhya.com/blog/2020/06/auc-roc-curve-machine-learning/
- https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/animal-science/probability/continuous-probability-distributions.html
- https://www.simplypsychology.org/normal-distribution.html
