Calculate Area Under Curve Excel

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Receiver operating characteristic (ROC) curves are a popular way to visualize the trade-off between sensitivity and specificity in a binary classifier. In a previous post, I described a simple “turtle view” of these plots: a classifier is used to sort items into those most likely to be positive, and a logo-like turtle along This sequence of cases moves. Turtle considers all approved items as positive. Depending on their actual class, they are false positives (FP) or true positives (TP). This is related to the point threshold setting. When the turtle crosses the TP, it moves up one step on the y-axis, and when it crosses the FP, it moves one step to the right on the x-axis. The size of the steps is inversely proportional to the number of real positives (in the y direction) or negatives (in the x direction), so the path always ends at the coordinate (1, 1). The result is a plot of the true positive rate (TPR or sensitivity) versus the false positive rate (FPR or 1-specificity), which is the entire ROC curve.

Calculate Area Under Curve Excel

Calculate Area Under Curve Excel

Calculating the area under the curve is one way to condense it into a single value. This metric is so common that if data scientists say “area under the curve” or “AUC,” you can assume they mean the ROC curve unless otherwise stated.

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Accuracy is probably the simplest and most intuitive measure of the performance of classifiers. Unfortunately, there are situations where simple accuracy does not work well. For example, for a disease that affects only one in a million people, a completely inaccurate screening test that always reports negative is 99.9999% accurate. Unlike accuracy, ROC curves are not sensitive to class imbalance. A false positive screening test would have an AUC of 0.5 as if no test had been performed at all.

In this post, I’ll go through the geometry exercise of calculating area and create a compact vector function that uses this method. We then consider another way of looking at AUC that leads to a possible interpretation.

The franchise owner is ersatz. Usually these are assigned by the classifier, but here we have just assigned numbers such that the descending order of the scores matches the order given by the class labels. Scores 9 and 10, one representing a positive case and the other a negative case, are replaced by their mean so that the data have links without disturbing the order.

To plot the ROC curve, we need to calculate the true positive and false positive rates. In the previous paper, we did this by using a positive (or negative) cumulative sum along the sorted binary labels. But here we use

Ex 8.1, 1

The function returns an object with graph methods and other niceties, but for our purposes we only want vectors of TPR and FPR values. TPR is the sensitivity and FPR is the feature 1 (see confusion matrix on Wikipedia). Unfortunately

The function reports these values ​​in ascending order. We’re going to start at the bottom left, so I’ll reverse the order. By

The function from the pROC package gives an AUC of 0.825 on our simulated class and prediction data. We compare other attempts to calculate AUC with this value.

Calculate Area Under Curve Excel

If the ROC curve were a perfect step function, we could find the area under it by adding a set of vertical bars with widths equal to the spacing of points on the FPR axis and heights equal to the step height on the TPR axis. Since real ROC curves may contain segments that represent sets of values ​​with finite points that are not quadratic steps, we need to adjust the area of ​​these segments. In the image below, we use green bars to show the areas under the steps. Settings for limited values ​​are shown as blue rectangles. Half of the area of ​​each blue rectangle is under the sloping part of the curve.

Ex 8.1, 2

The function to draw polygons in base R takes vectors of x and y values. We start by defining a

A function that uses a simpler and more specialized syntax. It takes the x and y coordinates of the bottom left corner of the rectangle as well as the height and width. Sets some default display settings and passes any other parameters we may have specified (such as colors).

. Since this results in a position vector shorter than the original data, we end each difference vector with zero:

For this figure, we’ll draw the ROC curve last to place it on top of the other elements, so we’ll start by drawing a blank graph (

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) which opens from 0 to 1 in each axis. Since there are exactly ten positive and ten negative states in the dataset, the TPR and FPR values ​​are all multiples of 1/10, and the points on the ROC curve all lie on a regularly spaced grid. We draw the grid using light blue horizontal and vertical lines that are separated by a tenth of a unit. Now we can transfer the values ​​we calculated above using the rectangle function

) to repeat all cases and draw all green and blue rectangles. Finally, we plot the ROC curve (ie TPR vs. FPR) on everything in red.

The area under the red curve is the entire green area plus half of the blue area. When adding regions, we only care about the height and width of each rectangle, not its (x,y) position. The heights of the green rectangles, all starting at 0, are in the TPR column and the widths are in the dFPR column, so the total area of ​​the green rectangles is the dot product of TPR and dFPR. Note that the vector approach calculates a rectangle for each data point, even if the height or width is zero (in which case it’s okay to add them). Similarly, the height and width of the blue rectangles (if any) are in the dTPR and dFPR columns, so their total area is the dot product of these vectors. In the areas of the graph that make up the square steps, each of these values ​​is zero, so you only get blue rectangles (non-zero area) if both TPR and FPR change in the same step. Only half of the area of ​​each blue rectangle is below the portion of its ROC curve (which is the diagonal of the blue rectangle). Remember “real”.

Calculate Area Under Curve Excel

Now let’s try a completely different approach. Here we create a matrix that shows all possible combinations of positive and negative states. Each row represents a positive case in order from the positive case with the highest score at the bottom to the positive case with the lowest score at the top. Likewise, the columns show the negatives, sorted by highest score on the left. Each cell represents a comparison between a given positive case and a given negative case, and we label the cell based on whether its positive case has a higher score (or higher total value) than its negative case. If your classifier is good, most positives outnumber most negatives, except for all cases in the upper left corner where low-ranking positives are compared to high-ranking negatives.

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The blue line is the ROC curve calculated in the usual way (slightly sliding and stretching so that the coordinates align with the corners of the matrix cells). This makes it clear that the ROC curve defines the boundary of the region where positive cases precede negative cases. AUC can be calculated by adjusting the values ​​of the matrix in such a way that the cells where the positive state is more than the negative state are obtained.

The function for the score difference gives these values ​​1, -1, and 0. We put them in your desired range by adding one and dividing by two.) By averaging these values, we find the AUC.

A possible interpretation is that if you randomly select one positive and one negative case, the AUC gives the probability that the positive case outperforms the negative case by classification. This is evident from the figure, where the total area of ​​the curve is normalized to one, the cells of the matrix list all possible combinations of positive and negative cases, and the fraction under the curve contains the cells over which the positive case is greater. negative

Now let’s try our new AUC functions on a larger data set. I am using the simulated data set from the previous post which is labels

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This data does not have connection points, so a modified version with connection points was made for testing. We draw a black line that represents the original data. Since each point has a unique score, the ROC curve is a

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