Matrices de confusión y otros valores estadísticos

Introducción

Hemos realizado esta página para resumir y concretar algunos coeficientes estadísticos que siempre aparecen en muchos de nuestros experimentos. Estos coeficientes nos ayudan a medir la calidad de nuestros resultados al compararlos con la verdad del terreno.

El trabajo en el que se basa esta página es sobre evaluación de algunos resultados de segmentación de imágenes de resonancia magnética, entonces oiréis hablar sobre vóxeles, aunque estos coeficientes son aplicables a otras materias.

Definiciones

In statistics, the terms Type I error (also, α error, or false positive) and type II error (β error, or a false negative) are used to describe possible errors made in a statistical decision process. In 1928, Jerzy Neyman (1894-1981) and Egon Pearson (1895-1980), both eminent statisticians, discussed the problems associated with "deciding whether or not a particular sample may be judged as likely to have been randomly drawn from a certain population" (1928/1967, p.1): and identified "two sources of error", namely:

   (α) the error of rejecting a "correct" null hypothesis, and
   (β) the error of not rejecting a "false" null hypothesis

La matriz de confusión donde las filas corresponden a la verdad del terreno y las columnas a los resultados:

TP FP

FN TN

En muchas referencias las columnas corresponden a la verdad del terreno y las filas a los resultados, entonces la matriz de confusión es la traspuesta de la descrita arriba.

Por otro lado, nombraremos como N el número total de vóxeles de cada imagen.

N = número total de vóxeles de la imagen

Valores estadísticos

En principio tenemos que: TP + FP + FN + TN = N

Misclassification Ratio or Mean error: MCR = (FP + FN) / N

Jaccard Similarity: Jaccard = TP / (TP + FN + FP)

Dice's Coefficient: DC = (2*TP) / (2*TP + FN + FP)

Dice is similar to Jaccard but gives twice the weight to agreements.

It is also known as "overlap ratio" when used for segmentation validation. This sensitive measure varies in the range [0,1]...

For medical images, values greater than 0.6 for smaller structures and 0.8 for larger structures are considered good by some authors (Avants et al., 2007).

Precision: is the proportion of the predicted positive cases that were correct, as calculated using the equation: precision = TP / (TP+FP)

True Positive Rate or Sensitivity or Recall: TPF = TP / (TP + FN) = sensitivity

Puede ser multiplicada por 100 para expresarlo como un porcentaje.

F-score: Precision x Recall

True Negative Rate or Specificity: TNF = TN / (TN + FP) = specificity

Puede ser multiplicada por 100 para expresarlo como un porcentaje.

False Positive Rate: FPF = FP / (FP + TN) = 1 - specificity

Puede ser multiplicada por 100 para expresarlo como un porcentaje.

Segmentation Accuracy : SA = ((TP + TN) / N) (x 100)

Accurate Segmentation Ratio :

The ASR is only meaningful when we are dealing with multiple class problems: Gray Matter, White Matter and CSF. Then we will have specific contigency matrices, one for each class of the problem. So, in our problem we have three contingency matrices, one for each brain tissue class:

TP_c, FP_c, TN_c, FN_c, N_c where c = GM, WM or CSF

Then, ASR = (sum_c TP_c) / (sum_c (TP_c + FP_c ))

where sum_c means the sum over all different classes.

The ASR provides the overall evaluation based on the entire image volume, while TPF and FPF are used to measure the performance of different algorithm for a specific tissue.

Nota

The difference between ratio and fraction in the literature seems to be that the ratio is a value between 0 and 1, while the fraction seems to be a percentage (ratio x 100).