# PyCM is a multi-class confusion matrix library written in Python that supports both input data vectors and direct matrix, and a proper tool for post-classification model evaluation that supports most classes and overall statistics parameters.
# PyCM is the swiss-army knife of confusion matrices, targeted mainly at data scientists that need a broad array of metrics for predictive models and accurate evaluation of a large variety of classifiers.
#
#
#
#
#
Fig1. ConfusionMatrix Block Diagram
#
# ## Installation
# ⚠️ PyCM 3.9 is the last version to support **Python 3.5**
# ⚠️ PyCM 2.4 is the last version to support **Python 2.7** & **Python 3.4**
# ⚠️ Plotting capability requires **Matplotlib (>= 3.0.0)** or **Seaborn (>= 0.9.1)**
# ### Source code
# - Download [Version 4.0](https://github.com/sepandhaghighi/pycm/archive/v4.0.zip) or [Latest Source](https://github.com/sepandhaghighi/pycm/archive/dev.zip)
# - Run `pip install -r requirements.txt` or `pip3 install -r requirements.txt` (Need root access)
# - Run `python3 setup.py install` or `python setup.py install` (Need root access)
# ### PyPI
#
#
# - Check [Python Packaging User Guide](https://packaging.python.org/installing/)
# - Run `pip install pycm==4.0` or `pip3 install pycm==4.0` (Need root access)
# ### Conda
#
# - Check [Conda Managing Package](https://conda.io/docs/user-guide/tasks/manage-pkgs.html#installing-packages-from-anaconda-org)
# - `conda install -c sepandhaghighi pycm` (Need root access)
# ### Easy install
#
# - Run `easy_install --upgrade pycm` (Need root access)
# ### MATLAB
# - Download and install [MATLAB](https://www.mathworks.com/products/matlab.html) (>=8.5, 64/32 bit)
# - Download and install [Python3.x](https://www.python.org/downloads/) (>=3.6, 64/32 bit)
# - Select `Add to PATH` option
# - Select `Install pip` option
# - Run `pip install pycm` or `pip3 install pycm` (Need root access)
# - Configure Python interpreter
# ```
# >> pyversion PYTHON_EXECUTABLE_FULL_PATH
# ```
# - Visit [MATLAB Examples](https://github.com/sepandhaghighi/pycm/tree/master/MATLAB)
# ## Usage
# ### Environment check
# Checking that the notebook is running on Google Colab or not.
# In[1]:
import sys
try:
import google.colab
get_ipython().system('{sys.executable} -m pip -q -q install pycm')
except:
pass
# ### From vector
# In[2]:
from pycm import *
# In[3]:
y_actu = [2, 0, 2, 2, 0, 1, 1, 2, 2, 0, 1, 2]
y_pred = [0, 0, 2, 1, 0, 2, 1, 0, 2, 0, 2, 2]
# In[4]:
cm = ConfusionMatrix(y_actu, y_pred,digit=5)
#
#
Notice : digit (the number of digits to the right of the decimal point in a number) is new in version 0.6 (default value : 5)
Notice : confusion matrices input in array format is new in version 3.6
#
# In[39]:
cm.table
# In[40]:
cm.matrix
# In[41]:
cm.normalized_matrix
# In[42]:
cm.normalized_table
# In[43]:
cm.print_matrix()
# In[44]:
import numpy
# In[45]:
array = numpy.array([[1, 2, 3], [4, 6, 1], [1, 2, 3]])
# In[46]:
cm = ConfusionMatrix(matrix=array)
# In[47]:
cm
# In[48]:
cm = ConfusionMatrix(matrix=array, classes=["L1", "L2", "L3"])
# In[49]:
cm
# ### Iterating and casting
# From `version 3.5`, `ConfusionMatrix` is an **iterator** object.
# In[50]:
for row, col in cm:
print(row, col)
# In[51]:
cm_iter = iter(cm)
next(cm_iter)
# In[52]:
cm_dict = dict(cm)
cm_dict
# In[53]:
cm_list = list(cm)
cm_list
#
#
Notice : new in version 3.5
#
# ### Activation threshold
# `threshold` is added in `version 0.9` for real value prediction.
#
# For more information visit Example 3
#
#
#
Notice : new in version 0.9
#
#
# ### Load from file
# `file` is added in `version 0.9.5` in order to load saved confusion matrix with `.obj` format generated by `save_obj` method.
#
# For more information visit Example 4
#
#
#
#
Notice : new in version 0.9.5
#
# ### Sample weights
# `sample_weight` is added in `version 1.2`
#
# For more information visit Example 5
#
#
Notice : new in version 1.2
#
# ### Transpose
# `transpose` is added in `version 1.2` in order to transpose input matrix (only in `Direct CM` mode)
# In[54]:
cm = ConfusionMatrix(matrix={0: {0: 3, 1: 0, 2: 0}, 1: {0: 0, 1: 1, 2: 2}, 2: {0: 2, 1: 1, 2: 3}}, digit=5, transpose=True)
# In[55]:
cm.print_matrix()
#
#
Notice : new in version 1.2
#
# ### Metrics off
# `metrics_off` is added in `version 3.9` in order to bypass metrics calculation.
# In[56]:
cm3 = ConfusionMatrix(y_actu, y_pred, metrics_off=True)
# In[57]:
cm3.class_stat
# In[58]:
cm3.overall_stat
#
#
Notice : new in version 3.9
#
# ### Relabel
# `relabel` method is added in `version 1.5` in order to change ConfusionMatrix class names.
# In[59]:
cm.relabel(mapping={0:"L1",1:"L2",2:"L3"}, sort=True)
# In[60]:
cm
# #### Parameters
# 1. `mapping` : mapping dictionary (type : `dict`)
# 2. `sort` : flag for sorting new classes (type : `bool`, default : `False`)
#
#
Notice : new in version 1.5
#
#
#
Notice : sort added in version 3.9
#
# ### Position
# `position` method is added in `version 2.8` in order to find the indexes of observations in `predict_vector` which made TP, TN, FP, FN.
# In[61]:
cm3 = ConfusionMatrix(y_actu, y_pred, digit=5)
cm3.position()
#
#
Notice : new in version 2.8
#
#
#
Notice : only works in vector mode
#
# ### To array
# `to_array` method is added in `version 2.9` in order to returns the confusion matrix in the form of a NumPy array. This can be helpful to apply different operations over the confusion matrix for different purposes such as aggregation, normalization, and combination.
# In[62]:
cm.to_array()
# In[63]:
cm.to_array(normalized=True)
# In[64]:
cm.to_array(normalized=True, one_vs_all=True, class_name="L1")
# #### Parameters
# 1. `normalized` : a flag for getting normalized confusion matrix (type : `bool`, default : `False`)
# 2. `one_vs_all` : one-vs-all mode flag (type : `bool`, default : `False`)
# 3. `class_name` : target class name for one-vs-all mode (type : `any valid type`, default : `None`)
# #### Output
# `Confusion Matrix in NumPy array format`
#
#
Notice : new in version 2.9
#
# ### Combine
# `combine` method is added in `version 3.0` in order to merge two confusion matrices. This option will be useful in mini-batch learning.
# In[65]:
cm_combined = cm2.combine(cm3)
cm_combined.print_matrix()
# #### Parameters
# 1. `other` : the other matrix that is going to be combined (type : `ConfusionMatrix`)
# #### Output
# New `ConfusionMatrix`
#
#
Notice : new in version 3.0
#
# ### Plot
# `plot` method is added in `version 3.0` in order to plot a confusion matrix using Matplotlib or Seaborn.
# In[66]:
import sys
get_ipython().system('{sys.executable} -m pip -q -q install matplotlib;')
get_ipython().system('{sys.executable} -m pip -q -q install seaborn;')
import matplotlib.pyplot as plt
# In[67]:
cm.plot()
# In[68]:
cm.plot(cmap=plt.cm.Greens, number_label=True, normalized=True)
# In[69]:
cm.plot(plot_lib="seaborn", number_label=True)
# In[70]:
cm.plot(cmap=plt.cm.Blues, number_label=True, one_vs_all=True, class_name="L1")
# In[71]:
cm.plot(cmap=plt.cm.Reds, number_label=True, normalized=True, one_vs_all=True, class_name="L3")
# #### Parameters
# 1. `normalized` :normalized flag for matrix (type : `bool`, default : `False`)
# 2. `one_vs_all` : one-vs-all mode flag (type : `bool`, default : `False`)
# 3. `class_name` : target class name for one-vs-all mode (type : `any valid type`, default : `None`)
# 4. `title` : plot title (type : `str`, default : `Confusion Matrix`)
# 5. `number_label` : number label flag (type : `bool`, default : `False`)
# 6. `cmap` : color map (type : `matplotlib.colors.ListedColormap`, default : `None`)
# 7. `plot_lib` : plotting library (type : `str`, default : `matplotlib`)
# #### Output
# Plot axes
#
# ### Parameter recommender
# This option has been added in `version 1.9` to recommend the most related parameters considering the characteristics of the input dataset. The suggested parameters are selected according to some characteristics of the input such as being balance/imbalance and binary/multi-class. All suggestions can be categorized into three main groups: imbalanced dataset, binary classification for a balanced dataset, and multi-class classification for a balanced dataset. The recommendation lists have been gathered according to the respective paper of each parameter and the capabilities which had been claimed by the paper.
#
#
#
#
Fig2. Parameter Recommender Block Diagram
# For determining if the dataset is imbalanced, we use the following strategy:
# $$R=\frac{Max(P)}{Min(P)}$$
# $$State=\begin{cases}Balance & R\leq 3\\Imbalance & R > 3\end{cases}$$
# In[72]:
cm.imbalance
# In[73]:
cm.binary
# In[74]:
cm.recommended_list
# `is_imbalanced` parameter has been added in `version 3.3`, so the user can indicate whether the concerned dataset is imbalanced or not. As long as the user does not provide any information in this regard, the automatic detection algorithm will be used.
# In[75]:
cm4 = ConfusionMatrix(y_actu, y_pred, is_imbalanced=True)
cm4.imbalance
# In[76]:
cm4 = ConfusionMatrix(y_actu, y_pred, is_imbalanced=False)
cm4.imbalance
#
#
Notice : also available in HTML report
#
#
#
Notice : The recommender system assumes that the input is the result of classification over the whole data rather than just a part of it. If the confusion matrix is the result of test data classification, the recommendation is not valid.
#
#
#
#
Notice : is_imbalanced , new in version 3.3
#
# ### Compare
# In `version 2.0`, a method for comparing several confusion matrices is introduced. This option is a combination of several overall and class-based benchmarks. Each of the benchmarks evaluates the performance of the classification algorithm from good to poor and give them a numeric score. The score of good and poor performances are 1 and 0, respectively.
#
# After that, two scores are calculated for each confusion matrices, overall and class-based. The overall score is the average of the score of seven overall benchmarks which are Landis & Koch, Cramer, Matthews, Goodman-Kruskal's Lambda A, Goodman-Kruskal's Lambda B, Krippendorff's Alpha, and Pearson's C. In the same manner, the class-based score is the average of the score of six class-based benchmarks which are Positive Likelihood Ratio Interpretation, Negative Likelihood Ratio Interpretation, Discriminant Power Interpretation, AUC value Interpretation, Matthews Correlation Coefficient Interpretation and Yule's Q Interpretation. It should be noticed that if one of the benchmarks returns none for one of the classes, that benchmarks will be eliminated in total averaging. If the user sets weights for the classes, the averaging over the value of class-based benchmark scores will transform to a weighted average.
#
# If the user sets the value of `by_class` boolean input `True`, the best confusion matrix is the one with the maximum class-based score. Otherwise, if a confusion matrix obtains the maximum of both overall and class-based scores, that will be reported as the best confusion matrix, but in any other case, the compared object doesn’t select the best confusion matrix.
#
#
#
#
Fig3. Compare Block Diagram
# This is how the overall and class-based scores are determined for each confusion matrix. Note that here $|Set|$ shows the cardinality of the set and the cardinality of each benchmark is equal to the maximum possible score for that benchmark.
# $$S_{Overall}=\sum_{i=1}^{|B_O|}\frac{W_{OB}(i)}{\sum W_{OB}}\times\frac{R_O(i)}{|B_O(i)|}$$
# $$S_{Class}=\sum_{i=1}^{|B_C|}\sum_{j=1}^{|C|}\frac{W_{CB}(i)}{\sum W_{CB}}\times\frac{W_C(j)}{\sum W_{C}}\times\frac{R_C(i,j)}{|B_C(i)|}$$
# $$0\leq S_{Overall},S_{Class}\leq1$$
# - $B_C$ : Class benchmarks
# - $B_O$ : Overall benchmarks
# - $C$ : Classes
# - $W_{CB}$ : Class benchmark weights
# - $W_{OB}$ : Overall benchmark weights
# - $W_{C}$ : Class weights
# - $R_C$ : Class benchmark result
# - $R_O$ : Overall benchmark result
# In[77]:
cm2 = ConfusionMatrix(matrix={0: {0:2, 1:50, 2:6}, 1:{0:5, 1:50, 2:3}, 2:{0:1, 1:7, 2:50}})
cm3 = ConfusionMatrix(matrix={0: {0:50, 1:2, 2:6}, 1:{0:50, 1:5, 2:3}, 2:{0:1, 1:55, 2:2}})
# In[78]:
cp = Compare({"cm2":cm2, "cm3":cm3})
# In[79]:
print(cp)
# In[80]:
cp.scores
# In[81]:
cp.sorted
# In[82]:
cp.best
# In[83]:
cp.best_name
# In[84]:
cp2 = Compare({"cm2":cm2, "cm3":cm3}, by_class=True, class_weight={0:5, 1:1, 2:1})
# In[85]:
print(cp2)
# In[86]:
cp3 = Compare({"cm2":cm2, "cm3":cm3}, class_benchmark_weight={"PLRI":0, "NLRI":0, "DPI":0, "AUCI":1, "MCCI":0, "QI":0})
# In[87]:
print(cp3)
# In[88]:
cp4 = Compare(
{"cm2":cm2, "cm3":cm3},
overall_benchmark_weight={"SOA1":1, "SOA2":0, "SOA3":0, "SOA4":0, "SOA5":0, "SOA6":1, "SOA7":0, "SOA8":0, "SOA9":0, "SOA10":0})
# In[89]:
print(cp4)
# Overall and class benchmark lists are available in `CLASS_BENCHMARK_LIST` and `OVERALL_BENCHMARK_LIST`
# In[90]:
from pycm import CLASS_BENCHMARK_LIST, OVERALL_BENCHMARK_LIST
print(CLASS_BENCHMARK_LIST)
print(OVERALL_BENCHMARK_LIST)
#
#
Notice : overall_benchmark_weight and class_benchmark_weight, new in version 3.3
#
#
#
Notice : From version 3.8, Goodman-Kruskal's Lambda A, Goodman-Kruskal's Lambda B, Krippendorff's Alpha, and Pearson's C benchmarks are considered in the overall score and default weights of the overall benchmarks are modified accordingly.
#
# ### ROC curve
# `ROCCurve`, added in `version 3.7`, is devised to compute the Receiver Operating Characteristic (ROC) or simply ROC curve. In ROC curves, the Y axis represents the True Positive Rate, and the X axis represents the False Positive Rate. Thus, the ideal point is located at the top left of the curve, and a larger area under the curve represents better performance. ROC curve is a graphical representation of binary classifiers' performance. In PyCM, `ROCCurve` binarizes the output based on the "One vs. Rest" strategy to provide an extension of ROC for multi-class classifiers. By getting the actual labels vector and the target probability estimates of the positive classes, this method is able to compute and plot TPR-FPR pairs for different discrimination thresholds and compute the area under the ROC curve.
# The thresholds for which the TPR-FPR pairs are calculated can be either specified by users (by setting `thresholds` input) or calculated automatically. Furthermore, sample weights can be adjusted via `sample_weight` as an input; otherwise, they are assumed to be 1. `ROCCurve` has two methods named `area()` and `plot()`. `area()` provides the user with the value of area under curve, which can be calculated using either `trapezoidal` (default method) or `midpoint` numerical integral technique. `plot()` is also provided to plot the given curve.
# In[91]:
from pycm import ROCCurve
crv = ROCCurve(
actual_vector=numpy.array([1, 1, 2, 2]),
probs=numpy.array([[0.1, 0.9], [0.4, 0.6], [0.35, 0.65], [0.8, 0.2]]),
classes=[2, 1])
crv.thresholds
auc_trp = crv.area()
auc_trp[1]
auc_trp[2]
# In[92]:
crv.plot(area=True, classes=[2])
#
#
Notice : new in version 3.7
#
# ### Precision-Recall curve
# `PRCurve`, added in `version 3.7`, is devised to compute the Precision-Recall curve in which the Y axis represents the Precision, and the X axis represents the Recall of a classifier. Thus, the ideal point is located at the top right of the curve, and a larger area under the curve represents better performance. Precision-Recall curve is a graphical representation of binary classifiers' performance. In PyCM, `PRCurve` binarizes the output based on the "One vs. Rest" strategy to provide an extension of this curve for multi-class classifiers. By getting the actual labels vector and the target probability estimates of the positive classes, this method is able to compute and plot Precision-Recall pairs for different discrimination thresholds and compute the area under the curve.
# The thresholds for which the Precision-Recall pairs are calculated can be either specified by users (by setting `thresholds` input) or calculated automatically. Furthermore, sample weights can be adjusted via `sample_weight` as an input; otherwise, they are assumed to be 1. `PRCurve` has two methods named `area()` and `plot()`. `area()` provides the user with the value of area under curve, which can be calculated using either `trapezoidal` (default method) or `midpoint` numerical integral technique. `plot()` is also provided to plot the given curve.
# In[93]:
from pycm import PRCurve
crv = PRCurve(
actual_vector=numpy.array([1, 1, 2, 2]),
probs=numpy.array([[0.1, 0.9], [0.4, 0.6], [0.35, 0.65], [0.8, 0.2]]),
classes=[2, 1])
crv.thresholds
auc_trp = crv.area()
auc_trp[1]
auc_trp[2]
# In[94]:
crv.plot(area=True, classes=[2])
#
#
Notice : new in version 3.7
#
# ### Multilabel confusion matrix
# From `version 4.0`, `MultiLabelCM` has been added to calculate class-wise or sample-wise multilabel confusion matrices. In class-wise mode, confusion matrices are calculated for each class, and in sample-wise mode, they are generated per sample. All generated confusion matrices are binarized with a one-vs-rest transformation.
# In[95]:
mlcm = MultiLabelCM(actual_vector=[{"cat", "bird"}, {"dog"}],
predict_vector=[{"cat"}, {"dog", "bird"}],
classes=["cat", "dog", "bird"])
print(mlcm.actual_vector_multihot)
print(mlcm.predict_vector_multihot)
mlcm.get_cm_by_class("cat").print_matrix()
mlcm.get_cm_by_sample(0).print_matrix()
#
#
Notice : new in version 4.0
#
# ### Online help
# `online_help` function is added in `version 1.1` in order to open each statistics definition in web browser.
# ```python
#
# >>> from pycm import online_help
# >>> online_help("J")
# >>> online_help("J", alt_link=True)
# >>> online_help("SOA1(Landis & Koch)")
# >>> online_help(2)
#
# ```
# * List of items are available by calling `online_help()` (without argument)
# * If PyCM website is not available, set `alt_link = True`
# In[96]:
online_help()
# #### Parameters
# 1. `param` : input parameter (type : `int or str`, default : `None`)
# 2. `alt_link` : alternative link for document flag (type : `bool`, default : `False`)
#
#
Notice : alt_link , new in version 2.4
#
# ### Acceptable data types
# ConfusionMatrix
# 1. `actual_vector` : python `list` or numpy `array` of any stringable objects
# 2. `predict_vector` : python `list` or numpy `array` of any stringable objects
# 3. `matrix` : `dict`
# 4. `digit`: `int`
# 5. `threshold` : `FunctionType (function or lambda)`
# 6. `file` : `File object`
# 7. `sample_weight` : python `list` or numpy `array` of numbers
# 8. `transpose` : `bool`
# 9. `classes` : python `list`
# 10. `is_imbalanced` : `bool`
# 11. `metrics_off` : `bool`
# * run `help(ConfusionMatrix)` for more information
#
#
Notice : metrics_off, new in version 3.9
#
# Compare
# 1. `cm_dict` : python `dict` of `ConfusionMatrix` object (`str` : `ConfusionMatrix`)
# 2. `by_class` : `bool`
# 3. `class_weight` : python `dict` of class weights (`class_name` : `float`)
# 4. `class_benchmark_weight`: python `dict` of class benchmark weights (`class_benchmark_name` : `float`)
# 5. `overall_benchmark_weight`: python `dict` of overall benchmark weights (`overall_benchmark_name` : `float`)
# 6. `digit`: `int`
# * run `help(Compare)` for more information
#
#
Notice : weight renamed to class_weight in version 3.3
#
#
#
Notice : overall_benchmark_weight and class_benchmark_weight, new in version 3.3
#
# ROCCurve
# 1. `actual_vector` : python `list` or numpy `array` of any stringable objects
# 2. `probs` : python `list` or numpy `array`
# 3. `classes` : python `list`
# 4. `thresholds`: python `list` or numpy `array`
# 5. `sample_weight`: python `list` or numpy `array`
# * run `help(ROCCurve)` for more information
# PRCurve
# 1. `actual_vector` : python `list` or numpy `array` of any stringable objects
# 2. `probs` : python `list` or numpy `array`
# 3. `classes` : python `list`
# 4. `thresholds`: python `list` or numpy `array`
# 5. `sample_weight`: python `list` or numpy `array`
# * run `help(PRCurve)` for more information
# MultiLabelCM
# 1. `actual_vector` : python `list` or numpy array of `sets`
# 2. `predict_vector` : python `list` or numpy array of `sets`
# 3. `sample_weight`: python `list` or numpy `array`
# 4. `classes` : python `list`
# * run `help(MultiLabelCM)` for more information
# ## Basic parameters
# ### TP (True positive)
# A true positive test result is one that detects the condition when the
# condition is present (correctly identified) [[3]](#ref3).
# In[97]:
cm.TP
# ### TN (True negative)
# A true negative test result is one that does not detect the condition when
# the condition is absent (correctly rejected) [[3]](#ref3).
# In[98]:
cm.TN
# ### FP (False positive)
# A false positive test result is one that detects the condition when the
# condition is absent (incorrectly identified) [[3]](#ref3).
# In[99]:
cm.FP
# ### FN (False negative)
# A false negative test result is one that does not detect the condition when
# the condition is present (incorrectly rejected) [[3]](#ref3).
# In[100]:
cm.FN
# ### P (Condition positive)
# Number of positive samples.
# Also known as support (the number of occurrences of each class in y_true) [[3]](#ref3).
# $$P=TP+FN$$
# In[101]:
cm.P
# ### N (Condition negative)
# Number of negative samples [[3]](#ref3).
# $$N=TN+FP$$
# In[102]:
cm.N
# ### TOP (Test outcome positive)
# Number of positive outcomes [[3]](#ref3).
# $$TOP=TP+FP$$
# In[103]:
cm.TOP
# ### TON (Test outcome negative)
# Number of negative outcomes [[3]](#ref3).
# $$TON=TN+FN$$
# In[104]:
cm.TON
# ### POP (Population)
# Total sample size [[3]](#ref3).
# $$POP=TP+TN+FN+FP$$
# In[105]:
cm.POP
# * Wikipedia page
# ## Class statistics
# ### TPR (True positive rate)
# Sensitivity (also called the true positive rate, the recall, or probability of detection in some fields) measures the proportion of positives that are correctly identified as such (e.g. the percentage of sick people who are correctly identified as having the condition) [[3]](#ref3).
#
# Wikipedia page
# $$TPR=\frac{TP}{P}=\frac{TP}{TP+FN}$$
# In[106]:
cm.TPR
# ### TNR (True negative rate)
# Specificity (also called the true negative rate) measures the proportion of negatives that are correctly identified as such (e.g. the percentage of healthy people who are correctly identified as not having the condition) [[3]](#ref3).
#
# Wikipedia page
# $$TNR=\frac{TN}{N}=\frac{TN}{TN+FP}$$
# In[107]:
cm.TNR
# ### PPV (Positive predictive value)
# Positive predictive value (PPV) is the proportion of positives that correspond to
# the presence of the condition [[3]](#ref3).
#
# Wikipedia page
# $$PPV=\frac{TP}{TP+FP}$$
# In[108]:
cm.PPV
# ### NPV (Negative predictive value)
# Negative predictive value (NPV) is the proportion of negatives that correspond to
# the absence of the condition [[3]](#ref3).
#
# Wikipedia page
# $$NPV=\frac{TN}{TN+FN}$$
# In[109]:
cm.NPV
# ### FNR (False negative rate)
# The false negative rate is the proportion of positives which yield negative test outcomes with the test, i.e., the conditional probability of a negative test result given that the condition being looked for is present [[3]](#ref3).
#
# Wikipedia page
# $$FNR=\frac{FN}{P}=\frac{FN}{FN+TP}=1-TPR$$
# In[110]:
cm.FNR
# ### FPR (False positive rate)
# The false positive rate is the proportion of all negatives that still yield positive test outcomes, i.e., the conditional probability of a positive test result given an event that was not present [[3]](#ref3).
#
# The false positive rate is equal to the significance level. The specificity of the test is equal to $ 1 $ minus the false positive rate.
#
# Wikipedia page
# $$FPR=\frac{FP}{N}=\frac{FP}{FP+TN}=1-TNR$$
# In[111]:
cm.FPR
# ### FDR (False discovery rate)
# The false discovery rate (FDR) is a method of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. FDR-controlling procedures are designed to control the expected proportion of "discoveries" (rejected null hypotheses) that are false (incorrect rejections) [[3]](#ref3).
#
# Wikipedia page
# $$FDR=\frac{FP}{FP+TP}=1-PPV$$
# In[112]:
cm.FDR
# ### FOR (False omission rate)
# False omission rate (FOR) is a statistical method used in multiple hypothesis testing to correct for multiple comparisons and it is the complement of the negative predictive value. It measures the proportion of false negatives which are incorrectly rejected [[3]](#ref3).
#
# Wikipedia page
# $$FOR=\frac{FN}{FN+TN}=1-NPV$$
# In[113]:
cm.FOR
# ### ACC (Accuracy)
# The accuracy is the number of correct predictions from all predictions made [[3]](#ref3).
#
# Wikipedia page
# $$ACC=\frac{TP+TN}{P+N}=\frac{TP+TN}{TP+TN+FP+FN}$$
# In[114]:
cm.ACC
# ### ERR (Error rate)
# The error rate is the number of incorrect predictions from all predictions made [[3]](#ref3).
# $$ERR=\frac{FP+FN}{P+N}=\frac{FP+FN}{TP+TN+FP+FN}=1-ACC$$
# In[115]:
cm.ERR
#
#
Notice : new in version 0.4
#
# ### FBeta-Score
# In statistical analysis of classification, the F1 score (also F-score or F-measure) is a measure of a test's accuracy. It considers both the precision $ p $ and the recall $ r $ of the test to compute the score.
# The F1 score is the harmonic average of the precision and recall, where F1 score reaches its best value at $ 1 $ (perfect precision and recall) and worst at $ 0 $ [[3]](#ref3).
#
# Wikipedia page
# $$F_{\beta}=(1+\beta^2)\times \frac{PPV\times TPR}{(\beta^2 \times PPV)+TPR}=\frac{(1+\beta^2) \times TP}{(1+\beta^2)\times TP+FP+\beta^2 \times FN}$$
# In[116]:
cm.F1
# In[117]:
cm.F05
# In[118]:
cm.F2
# In[119]:
cm.F_beta(beta=4)
# #### Parameters
# 1. `beta` : beta parameter (type : `float`)
# #### Output
# `{class1: FBeta-Score1, class2: FBeta-Score2, ...}`
#
#
Notice : new in version 0.4
#
# ### MCC (Matthews correlation coefficient)
# The Matthews correlation coefficient is used in machine learning as a measure of the quality of binary (two-class) classifications, introduced by biochemist Brian W. Matthews in 1975. It takes into account true and false positives and negatives and is generally regarded as a balanced measure that can be used even if the classes are of very different sizes. The MCC is, in essence, a correlation coefficient between the observed and predicted binary classifications; it returns a value between $ −1 $ and $ +1 $. A coefficient of $ +1 $ represents a perfect prediction, $ 0 $ no better than random prediction and $ −1 $ indicates total disagreement between prediction and observation [[27]](#ref27).
#
# Interpretation
#
# Wikipedia page
# $$MCC=\frac{TP \times TN-FP \times FN}{\sqrt{(TP+FP)\times (TP+FN)\times (TN+FP)\times (TN+FN)}}$$
# In[120]:
cm.MCC
# ### BM (Bookmaker informedness)
# The informedness of a prediction method as captured by a contingency matrix is defined as the probability that the prediction method will make a correct decision as opposed to guessing and is calculated using the bookmaker algorithm [[2]](#ref2).
#
# Equals to Youden Index
# $$BM=TPR+TNR-1$$
# In[121]:
cm.BM
# ### MK (Markedness)
# In statistics and psychology, the social science concept of markedness is quantified as a measure of how much one variable is marked as a predictor or possible cause of another and is also known as $ \triangle P $ in simple two-choice cases [[2]](#ref2).
# $$MK=PPV+NPV-1$$
# In[122]:
cm.MK
# ### PLR (Positive likelihood ratio)
# Likelihood ratios are used for assessing the value of performing a diagnostic test. They use the sensitivity and specificity of the test to determine whether a test result usefully changes the probability that a condition (such as a disease state) exists. The first description of the use of likelihood ratios for decision rules was made at a symposium on information theory in 1954 [[28]](#ref28).
#
# Interpretation
#
# Wikipedia page
# $$LR_+=PLR=\frac{TPR}{FPR}$$
# In[123]:
cm.PLR
#
#
Notice : LR+ renamed to PLR in version 1.5
#
# ### NLR (Negative likelihood ratio)
# Likelihood ratios are used for assessing the value of performing a diagnostic test. They use the sensitivity and specificity of the test to determine whether a test result usefully changes the probability that a condition (such as a disease state) exists. The first description of the use of likelihood ratios for decision rules was made at a symposium on information theory in 1954 [[28]](#ref28).
#
# Interpretation
#
# Wikipedia page
# $$LR_-=NLR=\frac{FNR}{TNR}$$
# In[124]:
cm.NLR
#
#
Notice : LR- renamed to NLR in version 1.5
#
# ### DOR (Diagnostic odds ratio)
# The diagnostic odds ratio is a measure of the effectiveness of a diagnostic test. It is defined as the ratio of the odds of the test being positive if the subject has a disease relative to the odds of the test being positive if the subject does not have the disease [[28]](#ref28).
#
# Wikipedia page
# $$DOR=\frac{LR_+}{LR_-}$$
# In[125]:
cm.DOR
# ### PRE (Prevalence)
# Prevalence is a statistical concept referring to the number of cases of a disease that are present in a particular population at a given time (Reference Likelihood) [[14]](#ref14).
#
# Wikipedia page
# $$Prevalence=\frac{P}{POP}$$
# In[126]:
cm.PRE
# ### G (G-measure)
# The geometric mean of precision and sensitivity, also known as Fowlkes–Mallows index [[3]](#ref3).
#
# Wikipedia page
# $$G=\sqrt{PPV\times TPR}$$
# In[127]:
cm.G
# ### RACC (Random accuracy)
# The expected accuracy from a strategy of randomly guessing categories according to reference and response distributions [[24]](#ref24).
# $$RACC=\frac{TOP \times P}{POP^2}$$
# In[128]:
cm.RACC
#
#
Notice : new in version 0.3
#
# ### RACCU (Random accuracy unbiased)
# The expected accuracy from a strategy of randomly guessing categories according to the average of the reference and response distributions [[25]](#ref25).
# $$RACCU=(\frac{TOP+P}{2 \times POP})^2$$
# In[129]:
cm.RACCU
#
#
Notice : new in version 0.8.1
#
# ### J (Jaccard index)
# The Jaccard index, also known as Intersection over Union and the Jaccard similarity coefficient (originally coined coefficient de communauté by Paul Jaccard), is a statistic used for comparing the similarity and diversity of sample sets [[29]](#ref29).
#
# Wikipedia page
#
# Some articles also named it as the F* (An Interpretable Transformation of the F-measure) [[77]](#ref77).
# $$J=\frac{TP}{TOP+P-TP}$$
# In[130]:
cm.J
#
#
Notice : new in version 0.9
#
# ### IS (Information score)
# The amount of information needed to correctly classify an example into
# class C, whose prior probability is $ p(C) $, is defined as $ -\log_2(p(C)) $ [[18]](#ref18) [[39]](#ref39).
# $$IS=-log_2(\frac{TP+FN}{POP})+log_2(\frac{TP}{TP+FP})$$
# In[131]:
cm.IS
#
#
Notice : new in version 1.3
#
# ### CEN (Confusion entropy)
# CEN based upon the concept of entropy for evaluating classifier performances. By exploiting the misclassification information of confusion matrices, the measure evaluates the confusion level of the class distribution of
# misclassified samples. Both theoretical analysis and statistical results show that the proposed measure is more discriminating than accuracy and RCI while it remains relatively consistent with the two measures. Moreover, it is more capable of measuring how the samples of different classes have been separated from each
# other. Hence the proposed measure is more precise than the two measures and can substitute for them to evaluate classifiers in classification applications [[17]](#ref17).
# $$P_{i,j}^{j}=\frac{Matrix(i,j)}{\sum_{k=1}^{|C|}\Big(Matrix(j,k)+Matrix(k,j)\Big)}$$
# $$P_{i,j}^{i}=\frac{Matrix(i,j)}{\sum_{k=1}^{|C|}\Big(Matrix(i,k)+Matrix(k,i)\Big)}$$
# $$CEN_j=-\sum_{k=1,k\neq j}^{|C|}\Bigg(P_{j,k}^jlog_{2(|C|-1)}\Big(P_{j,k}^j\Big)+P_{k,j}^jlog_{2(|C|-1)}\Big(P_{k,j}^j\Big)\Bigg)$$
# In[132]:
cm.CEN
#
# ### AUC (Area under the ROC curve)
# The area under the curve (often referred to as simply the AUC) is equal to the probability that a classifier will rank a randomly chosen positive instance higher than a randomly chosen negative one (assuming 'positive' ranks higher than 'negative').
# Thus, AUC corresponds to the arithmetic mean of sensitivity and specificity values of each class [[23]](#ref23).
#
# Interpretation
# $$AUC=\frac{TNR+TPR}{2}$$
# In[134]:
cm.AUC
#
#
Notice : new in version 1.4
#
Notice : this is an approximate calculation of AUC
#
# ### dInd (Distance index)
# Euclidean distance of a ROC point from the top left corner of the ROC space, which can take values between 0 (perfect classification) and $ \sqrt{2} $ [[23]](#ref23).
# $$dInd=\sqrt{(1-TNR)^2+(1-TPR)^2}$$
# In[135]:
cm.dInd
#
#
Notice : new in version 1.4
#
# ### sInd (Similarity index)
# sInd is comprised between $ 0 $ (no correct classifications) and $ 1 $ (perfect classification) [[23]](#ref23).
# $$sInd = 1 - \sqrt{\frac{(1-TNR)^2+(1-TPR)^2}{2}}$$
# In[136]:
cm.sInd
#
#
Notice : new in version 1.4
#
# ### DP (Discriminant power)
# Discriminant power (DP) is a measure that summarizes sensitivity and specificity.
# The DP has been used mainly in feature selection over imbalanced data [[33]](#ref33).
#
# Interpretation
# $$X=\frac{TPR}{1-TPR}$$
# $$Y=\frac{TNR}{1-TNR}$$
# $$DP=\frac{\sqrt{3}}{\pi}(log_{10}X+log_{10}Y)$$
# In[137]:
cm.DP
#
#
Notice : new in version 1.5
#
# ### Y (Youden index)
# Youden’s index evaluates the algorithm’s ability to avoid failure; it’s derived from sensitivity and
# specificity and denotes a linear correspondence balanced accuracy.
# As Youden’s index is a linear transformation of the mean sensitivity and specificity, its values are difficult to
# interpret, we retain that a higher value of Y indicates better ability to avoid failure.
# Youden’s index has been conventionally used to evaluate tests diagnostic, improve the efficiency of
# Telemedical prevention [[33]](#ref33) [[34]](#ref34).
#
# Wikipedia page
#
# Equals to Bookmaker Informedness
# $$Y=BM=TPR+TNR-1$$
# In[138]:
cm.Y
#
#
Notice : new in version 1.5
#
# ### PLRI (Positive likelihood ratio interpretation)
# For more information visit [[33]](#ref33).
#
#
#
PLR
#
Model contribution
#
#
#
1 >
#
Negligible
#
#
#
1 - 5
#
Poor
#
#
#
5 - 10
#
Fair
#
#
#
> 10
#
Good
#
#
#
#
# In[139]:
cm.PLRI
#
#
Notice : new in version 1.5
#
# ### NLRI (Negative likelihood ratio interpretation)
# For more information visit [[48]](#ref48).
#
#
#
NLR
#
Model contribution
#
#
#
0.5 - 1
#
Negligible
#
#
#
0.2 - 0.5
#
Poor
#
#
#
0.1 - 0.2
#
Fair
#
#
#
0.1 >
#
Good
#
#
#
#
# In[140]:
cm.NLRI
#
#
Notice : new in version 2.2
#
# ### DPI (Discriminant power interpretation)
# For more information visit [[33]](#ref33).
#
#
#
DP
#
Model contribution
#
#
#
1 >
#
Poor
#
#
#
1 - 2
#
Limited
#
#
#
2 - 3
#
Fair
#
#
#
> 3
#
Good
#
#
#
#
# In[141]:
cm.DPI
#
#
Notice : new in version 1.5
#
# ### AUCI (AUC value interpretation)
# For more information visit [[33]](#ref33).
#
#
#
AUC
#
Model performance
#
#
#
0.5 - 0.6
#
Poor
#
#
#
0.6 - 0.7
#
Fair
#
#
#
0.7 - 0.8
#
Good
#
#
#
0.8 - 0.9
#
Very Good
#
#
#
0.9 - 1.0
#
Excellent
#
#
#
# In[142]:
cm.AUCI
#
#
Notice : new in version 1.6
#
# ### MCCI (Matthews correlation coefficient interpretation)
# MCC is a confusion matrix method of calculating the Pearson product-moment correlation coefficient (not to be confused with Pearson's C). Therefore, it has the same interpretation [[2]](#ref2).
#
# For more information visit [[49]](#ref49).
#
#
#
MCC
#
Interpretation
#
#
#
0.3 >
#
Negligible
#
#
#
0.3 - 0.5
#
Weak
#
#
#
0.5 - 0.7
#
Moderate
#
#
#
0.7 - 0.9
#
Strong
#
#
#
0.9 - 1.0
#
Very Strong
#
#
#
# In[143]:
cm.MCCI
#
#
Notice : new in version 2.2
#
#
#
Notice : only positive values are considered
#
# ### QI (Yule's Q interpretation)
# For more information visit [[67]](#ref67).
#
#
#
Q
#
Interpretation
#
#
#
0.25 >
#
Negligible
#
#
#
0.25 - 0.5
#
Weak
#
#
#
0.5 - 0.75
#
Moderate
#
#
#
> 0.75
#
Strong
#
#
#
#
# In[144]:
cm.QI
#
#
Notice : new in version 2.6
#
# ### GI (Gini index)
# A chance-standardized variant of the AUC is given by Gini coefficient, taking values between $ 0 $ (no difference
# between the score distributions of the two classes) and $ 1 $ (complete separation between the two distributions).
# Gini coefficient is widespread use metric in imbalanced data learning [[33]](#ref33).
#
# Wikipedia page
# $$GI=2\times AUC-1$$
# In[145]:
cm.GI
#
#
Notice : new in version 1.7
#
# ### LS (Lift score)
# In the context of classification, lift compares model predictions to randomly generated predictions. Lift is often used in marketing research combined with gain and lift charts as a visual aid [[35]](#ref35) [[36]](#ref36).
# $$LS=\frac{PPV}{PRE}$$
# In[146]:
cm.LS
#
#
Notice : new in version 1.8
#
# ### AM (Automatic/Manual)
# Difference between automatic and manual classification i.e., the difference between positive outcomes and of positive samples.
# $$AM=TOP-P=(TP+FP)-(TP+FN)$$
# In[147]:
cm.AM
#
#
Notice : new in version 1.9
#
# ### BCD (Bray-Curtis dissimilarity)
# In ecology and biology, the Bray–Curtis dissimilarity, named after J. Roger Bray and John T. Curtis, is a statistic used to quantify the compositional dissimilarity between two different sites, based on counts at each site [[37]](#ref37).
#
# Wikipedia page
# $$BCD=\frac{|AM|}{\sum_{i=1}^{|C|}\Big(TOP_i+P_i\Big)}=\frac{|AM|}{2\times POP}$$
# In[148]:
cm.BCD
#
#
Notice : new in version 1.9
#
# ### OP (Optimized precision)
# Optimized precision is a type of hybrid threshold metric and has been proposed as a
# discriminator for building an optimized heuristic classifier. This metric is a combination of
# accuracy, sensitivity and specificity metrics. The sensitivity and specificity metrics were used for
# stabilizing and optimizing the accuracy performance when dealing with an imbalanced class of two-class problems [[40]](#ref40) [[42]](#ref42).
# $$OP = ACC - \frac{|TNR-TPR|}{|TNR+TPR|}$$
# In[149]:
cm.OP
#
#
Notice : new in version 2.0
#
# ### IBA (Index of balanced accuracy)
# The method combines an unbiased index of its overall accuracy and a measure about
# how dominant is the class with the highest individual accuracy rate [[41]](#ref41) [[42]](#ref42).
# $$IBA_{\alpha}=(1+\alpha \times(TPR-TNR))\times TNR \times TPR$$
# In[150]:
cm.IBA
# In[151]:
cm.IBA_alpha(0.5)
# In[152]:
cm.IBA_alpha(0.1)
# #### Parameters
# 1. `alpha` : alpha parameter (type : `float`)
# #### Output
# `{class1: IBA1, class2: IBA2, ...}`
#
#
Notice : new in version 2.0
#
# ### GM (G-mean)
# Geometric mean of specificity and sensitivity [[3]](#ref3) [[41]](#ref41) [[42]](#ref42).
# $$GM=\sqrt{TPR \times TNR}$$
# In[153]:
cm.GM
#
#
Notice : new in version 2.0
#
# ### Q (Yule's Q)
# In statistics, Yule's Q, also known as the coefficient of colligation, is a measure of association between two binary variables [[45]](#ref45).
#
# Interpretation
#
# Wikipedia page
# $$OR = \frac{TP\times TN}{FP\times FN}$$
# $$Q = \frac{OR-1}{OR+1}$$
# In[154]:
cm.Q
#
#
Notice : new in version 2.1
#
# ### AGM (Adjusted G-mean)
# An adjusted version of the geometric mean of specificity and sensitivity [[46]](#ref46).
# $$N_n=\frac{N}{POP}$$
# $$AGM=\frac{GM+TNR\times N_n}{1+N_n};TPR>0$$
# $$AGM=0;TPR=0$$
# In[155]:
cm.AGM
#
#
Notice : new in version 2.1
#
# ### AGF (Adjusted F-score)
# The F-measures used only three of the four elements of the confusion matrix and hence two classifiers with different TNR values may have the same F-score. Therefore, the AGF metric is introduced to use all elements of the confusion matrix and provide more weights to samples which are correctly classified in the minority class [[50]](#ref50).
# $$AGF=\sqrt{F_2 \times InvF_{0.5}}$$
# $$F_{2}=5\times \frac{PPV\times TPR}{(4 \times PPV)+TPR}$$
# $$InvF_{0.5}=(1+0.5^2)\times \frac{NPV\times TNR}{(0.5^2 \times NPV)+TNR}$$
# In[156]:
cm.AGF
#
#
Notice : new in version 2.3
#
# ### OC (Overlap coefficient)
# The overlap coefficient, or Szymkiewicz–Simpson coefficient, is a similarity measure that measures the overlap between two finite sets. It is defined as the size of the intersection divided by the smaller of the size of the two sets [[52]](#ref52).
#
# Wikipedia page
# $$OC=\frac{TP}{min(TOP,P)}=max(PPV,TPR)$$
# In[157]:
cm.OC
#
#
Notice : new in version 2.3
#
# ### BB (Braun-Blanquet similarity)
# The Braun-Blanquet coefficient is a similarity measure that is mostly used in botany. It is defined as the size of the intersection divided by the larger of the size of the two sets [[82]](#ref82) [[83]](#ref83).
# $$BB=\frac{TP}{max(TOP,P)}=min(PPV,TPR)$$
# In[158]:
cm.BB
#
#
Notice : new in version 3.6
#
# ### OOC (Otsuka-Ochiai coefficient)
# In biology, there is a similarity index, known as the Otsuka-Ochiai coefficient named after Yanosuke Otsuka and Akira Ochiai, also known as the Ochiai-Barkman or Ochiai coefficient. If sets are represented as bit vectors, the Otsuka-Ochiai coefficient can be seen to be the same as the cosine similarity [[53]](#ref53).
#
# Wikipedia page
# $$OOC=\frac{TP}{\sqrt{TOP\times P}}$$
# In[159]:
cm.OOC
#
#
Notice : new in version 2.3
#
# ### TI (Tversky index)
# The Tversky index, named after Amos Tversky, is an asymmetric similarity measure on sets that compares a variant to a prototype. The Tversky index can be seen as a generalization of Dice's coefficient and Tanimoto coefficient [[54]](#ref54).
#
# Wikipedia page
# $$TI(\alpha,\beta)=\frac{TP}{TP+\alpha FN+\beta FP}$$
# In[160]:
cm.TI(2,3)
# #### Parameters
# 1. `alpha` : alpha coefficient (type : `float`)
# 2. `beta` : beta coefficient (type : `float`)
# #### Output
# `{class1: TI1, class2: TI2, ...}`
#
#
Notice : new in version 2.4
#
# ### AUPR (Area under the PR curve)
# A PR curve is plotting precision against recall. The precision recall area under curve (AUPR) is just the area under the PR curve. The higher it is, the better the model is [[55]](#ref55) [[56]](#ref56).
#
#
# $$AUPR=\frac{TPR+PPV}{2}$$
# In[161]:
cm.AUPR
#
#
Notice : new in version 2.4
#
Notice : this is an approximate calculation of AUPR
#
# ### ICSI (Individual classification success index)
# The Individual Classification Success Index (ICSI), is a
# class-specific symmetric measure defined for classification
# assessment purpose. ICSI is hence $ 1 $ minus the sum of type I and type II errors.
# It ranges from $ -1 $ (both errors are maximal, i.e. $ 1 $) to $ 1 $ (both
# errors are minimal, i.e. $ 0 $), but the value $ 0 $ does not have any
# clear meaning. The measure is symmetric, and linearly related
# to the arithmetic mean of TPR and PPV [[58]](#ref58).
# $$ICSI=PPV+TPR-1$$
# In[162]:
cm.ICSI
#
#
Notice : new in version 2.5
#
# ### CI (Confidence interval)
# In statistics, a confidence interval (CI) is a type of interval estimate (of a population parameter) that is computed from the observed data. The confidence level is the frequency (i.e., the proportion) of possible confidence intervals that contain the true value of their corresponding parameter. In other words, if confidence intervals are constructed using a given confidence level in an infinite number of independent experiments, the proportion of those intervals that contain the true value of the parameter will match the confidence level [[31]](#ref31).
#
# Supported statistics : `ACC`,`AUC`,`PRE`,`Overall ACC`,`Kappa`,`TPR`,`TNR`,`PPV`,`NPV`,`PLR`,`NLR`
#
# Supported alpha values (two-sided) : 0.001, 0.002, 0.01, 0.02, 0.05, 0.1, 0.2
#
# Supported alpha values (one-sided) : 0.0005, 0.001, 0.005, 0.01, 0.05, 0.1
# Confidence intervals for `TPR`,`TNR`,`PPV`,`NPV`,`ACC`,`PRE` and `Overall ACC` are calculated using the normal approximation to the binomial distribution [[59]](#ref59), Wilson score [[62]](#ref62) and Agresti-Coull method [[63]](#ref63):
#
# #### Normal approximation
#
# $$SE=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$
#
# $$CI=\hat{p}\pm z\times SE$$
#
# $$n=\begin{cases}P & \hat{p} == TPR/FNR\\N & \hat{p} == TNR/FPR\\TOP & \hat{p} == PPV\\TON & \hat{p} ==NPV \\POP& \hat{p} == ACC/ACC_{Overall}\end{cases}$$
# #### Wilson score
#
# $$CI=\frac{\hat{p}+\frac{z^2}{2n}}{1+\frac{z^2}{n}}\pm\frac{z}{1+\frac{z^2}{n}}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}+\frac{z^2}{4n^2}}$$
#
# #### Agresti-Coull
#
# $$\hat{p}=\frac{x}{n}$$
#
# $$\tilde{p}=\frac{x+\frac{z^2}{2}}{n+z^2}$$
#
# $$CI =\tilde{p}\pm\sqrt{\frac{\tilde{p}(1-\tilde{p})}{n+z^2}}$$
# Confidence interval for `Kappa` are calculated using Fleiss formula [[24]](#ref24) [[38]](#ref38) :
#
# $$SE_{Kappa}=\sqrt{\frac{ACC_{Overall}\times (1-RACC_{Overall})}{(1-RACC_{Overall})^2}}$$
#
# $$CI_{Kappa}=Kappa\pm z\times SE_{Kappa}$$
# Confidence intervals for `NLR` and `PLR` are calculated using the log method [[60]](#ref60) :
#
# $$SE_{LR}=\sqrt{\frac{1}{a}-\frac{1}{b}+\frac{1}{c}-\frac{1}{d}}$$
#
# $$CI_{LR}=e^{ln(LR)\pm z\times SE_{LR}}$$
#
# $$PLR:\begin{cases}a=TP\\b=P\\c=FP\\d=N\end{cases}$$
#
# $$NLR:\begin{cases}a=FN\\b=P\\c=TN\\d=N\end{cases}$$
#
# Confidence interval for `AUC` is calculated using Hanley and McNeil formula [[61]](#ref61) :
#
# $$SE_{AUC}=\sqrt{\frac{q_0+(N-1)q_1+(P-1)q_2}{N\times P}}$$
#
# $$q_0=AUC(1-AUC)$$
#
# $$q_1=\frac{AUC}{2-AUC}-AUC^2$$
#
# $$q_2=\frac{2AUC^2}{1+AUC}-AUC^2$$
#
# $$CI_{AUC}=AUC\pm z\times SE_{AUC}$$
# In[163]:
cm.CI("TPR")
# In[164]:
cm.CI("FNR", alpha=0.001, one_sided=True)
# In[165]:
cm.CI("PRE", alpha=0.05, binom_method="wilson")
# In[166]:
cm.CI("Overall ACC", alpha=0.02, binom_method="agresti-coull")
# In[167]:
cm.CI("Overall ACC", alpha=0.05)
# #### Parameters
# 1. `param` : input parameter (type : `str`)
# 2. `alpha` : type I error (type : `float`, default : `0.05`)
# 3. `one_sided` : one-sided mode flag (type : `bool`, default : `False`)
# 4. `binom_method` : binomial confidence intervals method (type : `str`, default : `normal-approx`)
# #### Output
# 1. Two-sided : `{class1: [SE1, (Lower CI, Upper CI)], ...}`
# 2. One-sided : `{class1: [SE1, (Lower one-sided CI, Upper one-sided CI)], ...}`
#
# ### NB (Net benefit)
# NB is a weighted sum of true positive
# classifications with compensation for false positive classifications by giving
# these a weight $ w $ [[64]](#ref64) [[65]](#ref65).
# $$NB=\frac{TP-w\times FP}{POP}$$
# Vickers and Elkin (2006) suggested considering a range of thresholds and
# calculating the NB across these thresholds. The results can be plotted in a
# decision curve [[66]](#ref66).
# $$p_t=threshold$$
# $$w=\frac{p_t}{1-p_t}$$
# In[168]:
cm.NB(w=0.059)
# #### Parameters
# 1. `w` : weight
# #### Output
# `{class1: NB1, class2: NB2, ...}`
#
#
Notice : new in version 2.6
#
# ### Average
# Here "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged.
#
# Wikipedia page
# In[169]:
cm.average("PPV")
# In[170]:
cm.average("F1")
# In[171]:
cm.average("DOR", none_omit=True)
# #### Parameters
# 1. `param` : input parameter (type : `str`)
# 2. `none_omit` : none items omitting flag (type : `bool`, default : `False`)
# #### Output
# `Average`
#
#
Notice : new in version 2.7
#
# ### Weighted average
# The weighted average is similar to an ordinary average, except that instead of each of the data points contributing equally to the final average, some data points contribute more than others.
#
# Default weight is **condition positive** (number of positive samples).
#
# Wikipedia page
# In[172]:
cm.weighted_average("PPV")
# In[173]:
cm.weighted_average("F1")
# In[174]:
cm.weighted_average("DOR", none_omit=True)
# In[175]:
cm.weighted_average("F1", weight={"L1": 23, "L2": 2, "L3": 1})
# #### Parameters
# 1. `param` : input parameter (type : `str`)
# 2. `weight` : explicitly passes weights (type : `dict`, default : `None`)
# 3. `none_omit` : none items omitting flag (type : `bool`, default : `False`)
# #### Output
# `Weighted average`
#
#
Notice : new in version 2.7
#
# ### Sensitivity index
# The sensitivity index or d′ is a statistic used in signal detection theory. It provides the separation between the means of the signal and the noise distributions, compared against the standard deviation of the signal or noise distribution.
# d′ can be estimated from the observed hit rate and false-alarm rate, as follows [[76]](#ref76):
# $$d^{\prime}=Z(TPR) - Z(FPR)$$
# Function Z(p), p ∈ [0,1], is the inverse of the cumulative distribution function of the Gaussian distribution.
#
# Wikipedia page
# In[176]:
cm.sensitivity_index()
# #### Output
# `{class1: SI1, class2: SI2, ...}`
#
#
Notice : new in version 3.1
#
# ### HD (Hamming distance)
# In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of substitutions required to change one string into the other, or the minimum number of errors that could have transformed one string into the other. In a more general context, the Hamming distance is one of several string metrics for measuring the edit distance between two sequences. It is named after the American mathematician Richard Hamming [[80]](#ref80) [[81]](#ref81).
#
# A major application is in coding theory, more specifically to block codes, in which the equal-length strings are vectors over a finite field.
# Wikipedia page
# $$HD = FN + FP$$
# In[177]:
cm.HD
#
#
Notice : new in version 3.6
#
# ## Overall statistics
# ### Kappa
# Kappa is a statistic that measures inter-rater agreement for qualitative (categorical) items. It is generally thought to be a more robust measure than simple percent agreement calculation, as kappa takes into account the possibility of the agreement occurring by chance [[24]](#ref24).
#
# Benchmark1
# Benchmark2
# Benchmark3
# Benchmark4
#
# Wikipedia page
# $$Kappa=\frac{ACC_{Overall}-RACC_{Overall}}{1-RACC_{Overall}}$$
# In[178]:
cm.Kappa
#
#
Notice : new in version 0.3
#
# ### Kappa unbiased
# The unbiased kappa value is defined in terms of total accuracy and a slightly different computation of expected likelihood that averages the reference and response probabilities [[25]](#ref25).
#
# Equals to [Scott's Pi](#Scott's-Pi)
# $$Kappa_{Unbiased}=\frac{ACC_{Overall}-RACCU_{Overall}}{1-RACCU_{Overall}}$$
# In[179]:
cm.KappaUnbiased
#
#
Notice : new in version 0.8.1
#
# ### Kappa no prevalence
# The kappa statistic adjusted for prevalence [[14]](#ref14).
# $$Kappa_{NoPrevalence}=2 \times ACC_{Overall}-1$$
# In[180]:
cm.KappaNoPrevalence
#
#
Notice : new in version 0.8.1
#
# ### Weighted kappa
# The weighted kappa allows disagreements to be weighted differently and is especially useful when codes are ordered. Three matrices are involved, the matrix of observed scores, the matrix of expected scores based on chance agreement, and the weight matrix [[70]](#ref70) [[71]](#ref71).
# $$v_{ij}=1-\frac{w_{ij}}{max(w)}$$
# $$P_e=\sum_{i,j=1}^{|C|}\frac{TOP_i \times P_j}{POP^2}\times v_{ij}$$
# $$P_a=\sum_{i,j=1}^{|C|}\frac{Matrix(i,j)}{POP}\times v_{ij}$$
# $$Kappa_{Weighted}=\frac{P_a-P_e}{1-P_e}$$
# In[181]:
cm.weighted_kappa(
weight={
"L1": {"L1": 0, "L2": 1, "L3": 2},
"L2": {"L1": 1, "L2": 0, "L3": 1},
"L3": {"L1": 2, "L2": 1, "L3": 0}})
# In[182]:
cm.weighted_kappa()
# #### Parameters
# 1. `weight` : weight matrix (type : `dict`, default : `None`)
# #### Output
# `Weighted kappa`
#
#
Notice : new in version 2.7
#
# ### Kappa standard error
# The standard error(s) of the Kappa coefficient was obtained by Fleiss (1969) [[24]](#ref24) [[38]](#ref38).
# $$SE_{Kappa}=\sqrt{\frac{ACC_{Overall}\times (1-RACC_{Overall})}{(1-RACC_{Overall})^2}}$$
# In[183]:
cm.Kappa_SE
#
# ### Chi-squared
# Pearson's chi-squared test is a statistical test applied to sets of categorical data to evaluate how likely it is that any observed difference between the sets arose by chance. It is suitable for unpaired data from large samples [[10]](#ref10).
#
# Wikipedia page
# $$\chi^2=\sum_{i=1}^{|C|}\sum_{j=1}^{|C|}\frac{\Big(Matrix(i,j)-E(i,j)\Big)^2}{E(i,j)}$$
# $$E(i,j)=\frac{TOP_j\times P_i}{POP}$$
# In[185]:
cm.Chi_Squared
#
#
Notice : new in version 0.7
#
# ### Chi-squared DF
# Number of degrees of freedom of this confusion matrix for the chi-squared statistic [[10]](#ref10).
# $$DF=(|C|-1)^2$$
# In[186]:
cm.DF
#
#
Notice : new in version 0.7
#
# ### Phi-squared
# In statistics, the phi coefficient (or mean square contingency coefficient) is a measure of association for two binary variables. Introduced by Karl Pearson, this measure is similar to the Pearson correlation coefficient in its interpretation. In fact, a Pearson correlation coefficient estimated for two binary variables will return the phi coefficient [[10]](#ref10).
#
# Wikipedia page
# $$\phi^2=\frac{\chi^2}{POP}$$
# In[187]:
cm.Phi_Squared
#
#
Notice : new in version 0.7
#
# ### Cramer's V
# In statistics, Cramér's V (sometimes referred to as Cramér's phi) is a measure of association between two nominal variables, giving a value between $ 0 $ and $ +1 $ (inclusive). It is based on Pearson's chi-squared statistic and was published by Harald Cramér in 1946 [[26]](#ref26).
#
# Benchmark
#
# Wikipedia page
# $$V=\sqrt{\frac{\phi^2}{|C|-1}}$$
# In[188]:
cm.V
#
#
Notice : new in version 0.7
#
# ### Standard error
# The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation [[31]](#ref31).
#
# Wikipedia page
# $$SE_{ACC}=\sqrt{\frac{ACC\times (1-ACC)}{POP}}$$
# In[189]:
cm.SE
#
#
Notice : new in version 0.7
#
# ### 95% CI
# In statistics, a confidence interval (CI) is a type of interval estimate (of a population parameter) that is computed from the observed data. The confidence level is the frequency (i.e., the proportion) of possible confidence intervals that contain the true value of their corresponding parameter. In other words, if confidence intervals are constructed using a given confidence level in an infinite number of independent experiments, the proportion of those intervals that contain the true value of the parameter will match the confidence level [[31]](#ref31).
#
# Wikipedia page
# $$CI=ACC \pm 1.96\times SE_{ACC}$$
# In[190]:
cm.CI95
#
#
Notice : new in version 0.7
#
#
#
Notice : CI renamed to CI95 in version 2.5
#
# ### Bennett's S
# Bennett, Alpert & Goldstein’s S is a statistical measure of inter-rater agreement. It was created by Bennett et al. in 1954.
# Bennett et al. suggested adjusting inter-rater reliability to accommodate the percentage of rater agreement that might be expected by chance was a better measure than a simple agreement between raters [[8]](#ref8).
#
# Wikipedia Page
# $$p_c=\frac{1}{|C|}$$
# $$S=\frac{ACC_{Overall}-p_c}{1-p_c}$$
# In[191]:
cm.S
#
#
Notice : new in version 0.5
#
# ### Scott's Pi
# Scott's pi (named after William A. Scott) is a statistic for measuring inter-rater reliability for nominal data in communication studies. Textual entities are annotated with categories by different annotators, and various measures are used to assess the extent of agreement between the annotators, one of which is Scott's pi. Since automatically annotating text is a popular problem in natural language processing, and the goal is to get the computer program that is being developed to agree with the humans in the annotations it creates, assessing the extent to which humans agree with each other is important for establishing a reasonable upper limit on computer performance [[7]](#ref7).
#
# Wikipedia page
#
#
# Equals to [Kappa Unbiased](#Kappa-unbiased)
# $$p_c=\sum_{i=1}^{|C|}(\frac{TOP_i + P_i}{2\times POP})^2$$
# $$\pi=\frac{ACC_{Overall}-p_c}{1-p_c}$$
# In[192]:
cm.PI
#
#
Notice : new in version 0.5
#
# ### Gwet's AC1
# AC1 was originally introduced by Gwet in 2001 (Gwet, 2001). The interpretation of AC1 is similar to generalized kappa (Fleiss, 1971), which is used to assess inter-rater reliability when there are multiple raters. Gwet (2002) demonstrated that AC1 can overcome the limitations that kappa is sensitive to trait prevalence and rater's classification probabilities (i.e., marginal probabilities), whereas AC1 provides more robust measure of inter-rater reliability [[6]](#ref6).
# $$\pi_i=\frac{TOP_i + P_i}{2\times POP}$$
# $$p_c=\frac{1}{|C|-1}\sum_{i=1}^{|C|}\Big(\pi_i\times (1-\pi_i)\Big)$$
# $$AC_1=\frac{ACC_{Overall}-p_c}{1-p_c}$$
# In[193]:
cm.AC1
#
#
Notice : new in version 0.5
#
# ### Reference entropy
# The entropy of the decision problem itself as defined by the counts for the reference. The entropy of a distribution is the average negative log probability of outcomes [[30]](#ref30).
# $$Likelihood_{Reference}=\frac{P_i}{POP}$$
# $$Entropy_{Reference}=-\sum_{i=1}^{|C|}Likelihood_{Reference}(i)\times\log_{2}{Likelihood_{Reference}(i)}$$
# $$0\times\log_{2}{0}\equiv0$$
# In[194]:
cm.ReferenceEntropy
#
#
Notice : new in version 0.8.1
#
# ### Response entropy
# The entropy of the response distribution. The entropy of a distribution is the average negative log probability of outcomes [[30]](#ref30).
# $$Likelihood_{Response}=\frac{TOP_i}{POP}$$
# $$Entropy_{Response}=-\sum_{i=1}^{|C|}Likelihood_{Response}(i)\times\log_{2}{Likelihood_{Response}(i)}$$
# $$0\times\log_{2}{0}\equiv0$$
# In[195]:
cm.ResponseEntropy
#
#
Notice : new in version 0.8.1
#
# ### Cross entropy
# The cross-entropy of the response distribution against the reference distribution. The cross-entropy is defined by the negative log probabilities of the response distribution weighted by the reference distribution [[30]](#ref30).
#
# Wikipedia page
# $$Likelihood_{Reference}=\frac{P_i}{POP}$$
# $$Likelihood_{Response}=\frac{TOP_i}{POP}$$
# $$Entropy_{Cross}=-\sum_{i=1}^{|C|}Likelihood_{Reference}(i)\times\log_{2}{Likelihood_{Response}(i)}$$
# $$0\times\log_{2}{0}\equiv0$$
# In[196]:
cm.CrossEntropy
#
#
Notice : new in version 0.8.1
#
# ### Joint entropy
# The entropy of the joint reference and response distribution as defined by the underlying matrix [[30]](#ref30).
# $$P^{'}(i,j)=\frac{Matrix(i,j)}{POP}$$
# $$Entropy_{Joint}=-\sum_{i=1}^{|C|}\sum_{j=1}^{|C|}P^{'}(i,j)\times\log_{2}{P^{'}(i,j)}$$
# $$0\times\log_{2}{0}\equiv0$$
# In[197]:
cm.JointEntropy
#
#
Notice : new in version 0.8.1
#
# ### Conditional entropy
# The entropy of the distribution of categories in the response given that the reference category was as specified [[30]](#ref30).
#
# Wikipedia page
# $$P^{'}(j|i)=\frac{Matrix(j,i)}{P_i}$$
# $$Entropy_{Conditional}=\sum_{i=1}^{|C|}\Bigg(Likelihood_{Reference}(i)\times\Big(-\sum_{j=1}^{|C|}P^{'}(j|i)\times\log_{2}{P^{'}(j|i)}\Big)\Bigg)$$
# $$0\times\log_{2}{0}\equiv0$$
# In[198]:
cm.ConditionalEntropy
#
#
Notice : new in version 0.8.1
#
# ### Kullback-Leibler divergence
# In mathematical statistics, the Kullback–Leibler divergence (also called relative entropy) is a measure of how one probability distribution diverges from a second, expected probability distribution [[11]](#ref11) [[30]](#ref30).
#
# Wikipedia Page
# $$Likelihood_{Response}=\frac{TOP_i}{POP}$$
# $$Likelihood_{Reference}=\frac{P_i}{POP}$$
# $$Divergence=-\sum_{i=1}^{|C|}Likelihood_{Reference}\times\log_{2}{\frac{Likelihood_{Reference}}{Likelihood_{Response}}}$$
# In[199]:
cm.KL
#
#
Notice : new in version 0.8.1
#
# ### Mutual information
# Mutual information is defined as Kullback-Leibler divergence, between the product of the individual distributions and the joint distribution.
# Mutual information is symmetric. We could also subtract the conditional entropy of the reference given the response from the reference entropy to get the same result [[11]](#ref11) [[30]](#ref30).
#
# Wikipedia Page
# $$P^{'}(i,j)=\frac{Matrix(i,j)}{POP}$$
# $$Likelihood_{Reference}=\frac{P_i}{POP}$$
# $$Likelihood_{Response}=\frac{TOP_i}{POP}$$
# $$MI=-\sum_{i=1}^{|C|}\sum_{j=1}^{|C|}P^{'}(i,j)\times\log_{2}\Big({\frac{P^{'}(i,j)}{Likelihood_{Reference}(i)\times Likelihood_{Response}(i) }\Big)}$$
# $$MI=Entropy_{Response}-Entropy_{Conditional}$$
# In[200]:
cm.MutualInformation
#
#
Notice : new in version 0.8.1
#
# ### Goodman & Kruskal's lambda A
# In probability theory and statistics, Goodman & Kruskal's lambda is a measure of proportional reduction in error in cross tabulation analysis [[12]](#ref12).
#
# Wikipedia page
# $$\lambda_A=\frac{\sum_{j=1}^{|C|}Max\Big(Matrix(-,j)\Big)-Max(P)}{POP-Max(P)}$$
# In[201]:
cm.LambdaA
#
#
Notice : new in version 0.8.1
#
# ### Goodman & Kruskal's lambda B
# In probability theory and statistics, Goodman & Kruskal's lambda is a measure of proportional reduction in error in cross tabulation analysis [[13]](#ref13).
#
# Wikipedia Page
# $$\lambda_B=\frac{\sum_{i=1}^{|C|}Max\Big(Matrix(i,-)\Big)-Max(TOP)}{POP-Max(TOP)}$$
# In[202]:
cm.LambdaB
#
#
Notice : new in version 0.8.1
#
# ### SOA1 (Landis & Koch's benchmark)
# For more information visit [[1]](#ref1).
#
#
#
Kappa
#
Strength of Agreement
#
#
#
0 >
#
Poor
#
#
#
0 - 0.2
#
Slight
#
#
#
0.2 – 0.4
#
Fair
#
#
#
0.4 – 0.6
#
Moderate
#
#
#
0.6 – 0.8
#
Substantial
#
#
#
0.8 – 1.0
#
Almost perfect
#
#
#
#
# In[203]:
cm.SOA1
#
#
Notice : new in version 0.3
#
# ### SOA2 (Fleiss' benchmark)
# For more information visit [[4]](#ref4).
#
#
#
Kappa
#
Strength of Agreement
#
#
#
0.40 >
#
Poor
#
#
#
0.40 - 0.75
#
Intermediate to Good
#
#
#
More than 0.75
#
Excellent
#
#
#
#
# In[204]:
cm.SOA2
#
#
Notice : new in version 0.4
#
# ### SOA3 (Altman's benchmark)
# For more information visit [[5]](#ref5).
#
#
#
Kappa
#
Strength of Agreement
#
#
#
0.2 >
#
Poor
#
#
#
0.2 – 0.4
#
Fair
#
#
#
0.4 – 0.6
#
Moderate
#
#
#
0.6 – 0.8
#
Good
#
#
#
0.8 – 1.0
#
Very Good
#
#
#
#
# In[205]:
cm.SOA3
#
#
Notice : new in version 0.4
#
# ### SOA4 (Cicchetti's benchmark)
# For more information visit [[9]](#ref9).
#
#
#
Kappa
#
Strength of Agreement
#
#
#
0.40 >
#
Poor
#
#
#
0.40 – 0.59
#
Fair
#
#
#
0.59 – 0.74
#
Good
#
#
#
0.74 – 1.00
#
Excellent
#
#
#
#
# In[206]:
cm.SOA4
#
#
Notice : new in version 0.7
#
# ### SOA5 (Cramer's benchmark)
# For more information visit [[47]](#ref47).
#
#
#
Cramer's V
#
Strength of Association
#
#
#
0.1 >
#
Negligible
#
#
#
0.1 – 0.2
#
Weak
#
#
#
0.2 – 0.4
#
Moderate
#
#
#
0.4 – 0.6
#
Relatively Strong
#
#
#
0.6 – 0.8
#
Strong
#
#
#
0.8 – 1.0
#
Very Strong
#
#
#
#
# In[207]:
cm.SOA5
#
#
Notice : new in version 2.2
#
# ### SOA6 (Matthews's benchmark)
# MCC is a confusion matrix method of calculating the Pearson product-moment correlation coefficient (not to be confused with Pearson's C). Therefore, it has the same interpretation [[2]](#ref2).
#
# For more information visit [[49]](#ref49).
#
#
#
Overall MCC
#
Strength of Association
#
#
#
0.3 >
#
Negligible
#
#
#
0.3 - 0.5
#
Weak
#
#
#
0.5 - 0.7
#
Moderate
#
#
#
0.7 - 0.9
#
Strong
#
#
#
0.9 - 1.0
#
Very Strong
#
#
#
# In[208]:
cm.SOA6
#
#
Notice : new in version 2.2
#
#
#
Notice : only positive values are considered
#
# ### SOA7 (Goodman & Kruskal's lambda A benchmark)
# For more information visit [[84]](#ref84).
#
#
#
Lambda A
#
Strength of Association
#
#
#
0 - 0.2
#
Very Weak
#
#
#
0.2 - 0.4
#
Weak
#
#
#
0.4 - 0.6
#
Moderate
#
#
#
0.6 - 0.8
#
Strong
#
#
#
0.8 - 1.0
#
Very Strong
#
#
#
1.0
#
Perfect
#
#
#
# In[209]:
cm.SOA7
#
#
Notice : new in version 3.8
#
# ### SOA8 (Goodman & Kruskal's lambda B benchmark)
# For more information visit [[84]](#ref84).
#
#
#
Lambda B
#
Strength of Association
#
#
#
0 - 0.2
#
Very Weak
#
#
#
0.2 - 0.4
#
Weak
#
#
#
0.4 - 0.6
#
Moderate
#
#
#
0.6 - 0.8
#
Strong
#
#
#
0.8 - 1.0
#
Very Strong
#
#
#
1.0
#
Perfect
#
#
#
# In[210]:
cm.SOA8
#
#
Notice : new in version 3.8
#
# ### SOA9 (Krippendorff's alpha benchmark)
# For more information visit [[85]](#ref85).
#
#
#
Alpha
#
Strength of Agreement
#
#
#
0.667 >
#
Low
#
#
#
0.667 - 0.8
#
Tentative
#
#
#
0.8 <
#
High
#
#
# In[211]:
cm.SOA9
#
#
Notice : new in version 3.8
#
# ### SOA10 (Pearson's C benchmark)
# For more information visit [[86]](#ref86).
#
#
#
C
#
Strength of Association
#
#
#
0 - 0.1
#
Not Appreciable
#
#
#
0.1 - 0.2
#
Weak
#
#
#
0.2 - 0.3
#
Medium
#
#
#
0.3 <
#
Strong
#
#
# In[212]:
cm.SOA10
#
#
Notice : new in version 3.8
#
# ### Overall_ACC
# For more information visit [[3]](#ref3).
# $$ACC_{Overall}=\frac{\sum_{i=1}^{|C|}TP_i}{POP}$$
# Equals to [TPR Micro](#TPR_Micro), [F1 Micro](#F1_Micro) and [PPV Micro](#PPV_Micro)
# In[213]:
cm.Overall_ACC
#
#
Notice : new in version 0.4
#
# ### Overall_RACC
# For more information visit [[24]](#ref24).
# $$RACC_{Overall}=\sum_{i=1}^{|C|}RACC_i$$
# In[214]:
cm.Overall_RACC
#
#
Notice : new in version 0.4
#
# ### Overall_RACCU
# For more information visit [[25]](#ref25).
# $$RACCU_{Overall}=\sum_{i=1}^{|C|}RACCU_i$$
# In[215]:
cm.Overall_RACCU
#
#
Notice : new in version 0.8.1
#
# ### PPV_Micro
# For more information visit [[3]](#ref3).
# $$PPV_{Micro}=\frac{\sum_{i=1}^{|C|}TP_i}{\sum_{i=1}^{|C|}TP_i+FP_i}$$
# Equals to [TPR Micro](#TPR_Micro), [F1 Micro](#F1_Micro) and [Overall ACC](#Overall_ACC)
# In[216]:
cm.PPV_Micro
#
#
Notice : new in version 0.4
#
# ### NPV_Micro
# For more information visit [[3]](#ref3).
# $$NPV_{Micro}=\frac{\sum_{i=1}^{|C|}TN_i}{\sum_{i=1}^{|C|}TN_i+FN_i}$$
# In[217]:
cm.NPV_Micro
#
#
Notice : new in version 3.9
#
# ### TPR_Micro
# For more information visit [[3]](#ref3).
# $$TPR_{Micro}=\frac{\sum_{i=1}^{|C|}TP_i}{\sum_{i=1}^{|C|}TP_i+FN_i}$$
# Equals to [PPV Micro](#PPV_Micro), [F1 Micro](#F1_Micro) and [Overall ACC](#Overall_ACC)
# In[218]:
cm.TPR_Micro
#
#
Notice : new in version 0.4
#
# ### TNR_Micro
# For more information visit [[3]](#ref3).
# $$TNR_{Micro}=\frac{\sum_{i=1}^{|C|}TN_i}{\sum_{i=1}^{|C|}TN_i+FP_i}$$
# In[219]:
cm.TNR_Micro
#
#
Notice : new in version 2.6
#
# ### FPR_Micro
# For more information visit [[3]](#ref3).
# $$FPR_{Micro}=\frac{\sum_{i=1}^{|C|}FP_i}{\sum_{i=1}^{|C|}TN_i+FP_i}$$
# In[220]:
cm.FPR_Micro
#
#
Notice : new in version 2.6
#
# ### FNR_Micro
# For more information visit [[3]](#ref3).
# $$FNR_{Micro}=\frac{\sum_{i=1}^{|C|}FN_i}{\sum_{i=1}^{|C|}TP_i+FN_i}$$
# In[221]:
cm.FNR_Micro
#
#
Notice : new in version 2.6
#
# ### F1_Micro
# For more information visit [[3]](#ref3).
# $$F_{1_{Micro}}=2\frac{\sum_{i=1}^{|C|}TPR_i\times PPV_i}{\sum_{i=1}^{|C|}TPR_i+PPV_i}$$
# Equals to [PPV Micro](#PPV_Micro), [TPR Micro](#TPR_Micro) and [Overall ACC](#Overall_ACC)
# In[222]:
cm.F1_Micro
#
#
Notice : new in version 2.2
#
# ### PPV_Macro
# For more information visit [[3]](#ref3).
# $$PPV_{Macro}=\frac{1}{|C|}\sum_{i=1}^{|C|}\frac{TP_i}{TP_i+FP_i}$$
# In[223]:
cm.PPV_Macro
#
#
Notice : new in version 0.4
#
# ### NPV_Macro
# For more information visit [[3]](#ref3).
# $$NPV_{Macro}=\frac{1}{|C|}\sum_{i=1}^{|C|}\frac{TN_i}{TN_i+FN_i}$$
# In[224]:
cm.NPV_Macro
#
#
Notice : new in version 3.9
#
# ### TPR_Macro
# For more information visit [[3]](#ref3).
# $$TPR_{Macro}=\frac{1}{|C|}\sum_{i=1}^{|C|}\frac{TP_i}{TP_i+FN_i}$$
# In[225]:
cm.TPR_Macro
#
#
Notice : new in version 0.4
#
# ### TNR_Macro
# For more information visit [[3]](#ref3).
# $$TNR_{Macro}=\frac{1}{|C|}\sum_{i=1}^{|C|}\frac{TN_i}{TN_i+FP_i}$$
# In[226]:
cm.TNR_Macro
#
#
Notice : new in version 2.6
#
# ### FPR_Macro
# For more information visit [[3]](#ref3).
# $$FPR_{Macro}=\frac{1}{|C|}\sum_{i=1}^{|C|}\frac{FP_i}{TN_i+FP_i}$$
# In[227]:
cm.FPR_Macro
#
#
Notice : new in version 2.6
#
# ### FNR_Macro
# For more information visit [[3]](#ref3).
# $$FNR_{Macro}=\frac{1}{|C|}\sum_{i=1}^{|C|}\frac{FN_i}{TP_i+FN_i}$$
# In[228]:
cm.FNR_Macro
#
#
Notice : new in version 2.6
#
# ### F1_Macro
# For more information visit [[3]](#ref3).
# $$F_{1_{Macro}}=\frac{2}{|C|}\sum_{i=1}^{|C|}\frac{TPR_i\times PPV_i}{TPR_i+PPV_i}$$
# In[229]:
cm.F1_Macro
#
#
Notice : new in version 2.2
#
# ### ACC_Macro
# For more information visit [[3]](#ref3).
# $$ACC_{Macro}=\frac{1}{|C|}\sum_{i=1}^{|C|}{ACC_i}$$
# In[230]:
cm.ACC_Macro
#
#
Notice : new in version 2.2
#
# ### Overall_J
# For more information visit [[29]](#ref29).
# $$J_{Mean}=\frac{1}{|C|}\sum_{i=1}^{|C|}J_i$$
# $$J_{Sum}=\sum_{i=1}^{|C|}J_i$$
# $$J_{Overall}=(J_{Sum},J_{Mean})$$
# In[231]:
cm.Overall_J
#
#
Notice : new in version 0.9
#
# ### Hamming loss
# The average Hamming loss or Hamming distance between two sets of samples [[31]](#ref31).
# $$L_{Hamming}=\frac{1}{POP}\sum_{i=1}^{POP}1(y_i \neq \widehat{y}_i)$$
# In[232]:
cm.HammingLoss
#
#
Notice : new in version 1.0
#
# ### Zero-one loss
# Zero-one loss is a common loss function used with classification learning. It assigns $ 0 $ to loss for a correct classification and $ 1 $ for an incorrect classification [[31]](#ref31).
# $$L_{0-1}=\sum_{i=1}^{POP}1(y_i \neq \widehat{y}_i)$$
# In[233]:
cm.ZeroOneLoss
#
#
Notice : new in version 1.1
#
# ### NIR (No information rate)
# Largest class percentage in the data [[57]](#ref57).
# $$NIR=\frac{1}{POP}Max(P)$$
# In[234]:
cm.NIR
#
#
Notice : new in version 1.2
#
# ### P-Value
# In statistical hypothesis testing, the p-value or probability value is, for a given statistical model, the probability that, when the null hypothesis is true, the statistical summary (such as the absolute value of the sample mean difference between two compared groups) would be greater than or equal to the actual observed results [[31]](#ref31) .
# Here a one-sided binomial test to see if the accuracy is better than the no information rate [[57]](#ref57).
#
#
#
#
# Wikipedia Page
# $$x=\sum_{i=1}^{|C|}TP_{i}$$
# $$p=NIR$$
# $$n=POP$$
# $$P-Value_{(ACC > NIR)}=1-\sum_{i=1}^{x}\left(\begin{array}{c}n\\ i\end{array}\right)p^{i}(1-p)^{n-i}$$
# In[235]:
cm.PValue
#
#
Notice : new in version 1.2
#
# ### Overall_CEN
# For more information visit [[17]](#ref17).
# $$P_j=\frac{\sum_{k=1}^{|C|}\Big(Matrix(j,k)+Matrix(k,j)\Big)}{2\sum_{k,l=1}^{|C|}Matrix(k,l)}$$
# $$CEN_{Overall}=\sum_{j=1}^{|C|}P_jCEN_j$$
# In[236]:
cm.Overall_CEN
#
#
Notice : new in version 1.3
#
# ### Overall_MCEN
# For more information visit [[19]](#ref19).
# $$\alpha=\begin{cases}1 & |C| > 2\\0 & |C| = 2\end{cases}$$
# $$P_j=\frac{\sum_{k=1}^{|C|}\Big(Matrix(j,k)+Matrix(k,j)\Big)-Matrix(j,j)}{2\sum_{k,l=1}^{|C|}Matrix(k,l)-\alpha \sum_{k=1}^{|C|}Matrix(k,k)}$$
# $$MCEN_{Overall}=\sum_{j=1}^{|C|}P_jMCEN_j$$
# In[237]:
cm.Overall_MCEN
#
# ### RR (Global performance index)
# For more information visit [[21]](#ref21).
# $$RR=\frac{1}{|C|}\sum_{i,j=1}^{|C|}Matrix(i,j)$$
# In[239]:
cm.RR
#
#
Notice : new in version 1.4
#
# ### CBA (Class balance accuracy)
# As an evaluation tool, CBA creates an overall assessment of
# model predictive power by scrutinizing measures simultaneously across each class in a conservative manner that guarantees that a model’s ability to recall observations from each class and
# its ability to do so efficiently won’t fall below the bound [[22]](#ref22) [[51]](#ref51).
# $$CBA=\frac{\sum_{i=1}^{|C|}\frac{Matrix(i,i)}{Max(TOP_i,P_i)}}{|C|}$$
# In[240]:
cm.CBA
#
#
Notice : new in version 1.4
#
# ### AUNU
# When dealing with multiclass problems, a global measure of classification performances based on the ROC approach (AUNU) has been proposed as the average of single-class measures [[23]](#ref23).
# $$AUNU=\frac{\sum_{i=1}^{|C|}AUC_i}{|C|}$$
# In[241]:
cm.AUNU
#
#
Notice : new in version 1.4
#
# ### AUNP
# Another option (AUNP) is that of averaging the $ AUC_i $ values with weights proportional to the number of samples experimentally belonging to each class, that is, the a priori class distribution [[23]](#ref23).
# $$AUNP=\sum_{i=1}^{|C|}\frac{P_i}{POP}AUC_i$$
# In[242]:
cm.AUNP
#
#
Notice : new in version 1.4
#
# ### RCI (Relative classifier information)
# Performance of different classifiers on the same domain can be measured by
# comparing relative classifier information while classifier information (mutual information) can be used for comparison across different decision problems [[32]](#ref32) [[22]](#ref22).
# $$H_d=-\sum_{i=1}^{|C|}\Big(\frac{\sum_{l=1}^{|C|}Matrix(i,l)}{\sum_{h,k=1}^{|C|}Matrix(h,k)}log_2\frac{\sum_{l=1}^{|C|}Matrix(i,l)}{\sum_{h,k=1}^{|C|}Matrix(h,k)}\Big)=Entropy_{Reference}$$
# $$H_o=\sum_{j=1}^{|C|}\Big(\frac{\sum_{k=1}^{|C|}Matrix(k,j)}{\sum_{h,l=0}^{|C|}Matrix(h,l)}H_{oj}\Big)=Entropy_{Conditional}$$
# $$H_{oj}=-\sum_{i=1}^{|C|}\Big(\frac{Matrix(i,j)}{\sum_{k=1}^{|C|}Matrix(k,j)}log_2\frac{Matrix(i,j)}{\sum_{k=1}^{|C|}Matrix(k,j)}\Big)$$
# $$RCI=\frac{H_d-H_o}{H_d}=\frac{MI}{Entropy_{Reference}}$$
# In[243]:
cm.RCI
#
#
Notice : new in version 1.5
#
# ### Pearson's C
# The contingency coefficient is a coefficient of association that tells whether two variables or data sets are independent or dependent of/on each other. It is also known as Pearson’s coefficient (not to be confused with Pearson’s coefficient of skewness) [[43]](#ref43) [[44]](#ref44).
# $$C=\sqrt{\frac{\chi^2}{\chi^2+POP}}$$
# In[244]:
cm.C
#
#
Notice : new in version 2.0
#
# ### CSI (Classification success index)
# The Classification Success Index (CSI) is an overall
# measure defined by averaging ICSI over all classes [[58]](#ref58).
# $$CSI=\frac{1}{|C|}\sum_{i=1}^{|C|}{ICSI_i}$$
# In[245]:
cm.CSI
#
#
Notice : new in version 2.5
#
# ### ARI (Adjusted Rand index)
# The Rand index or Rand measure (named after William M. Rand) in statistics, and in particular in data clustering, is a measure of the similarity between two data clusterings. A form of the Rand index may be defined that is adjusted for the chance grouping of elements, this is the adjusted Rand index. From a mathematical standpoint, Rand index is related to the accuracy, but is applicable even when class labels are not used [[68]](#ref68).
#
# The Adjusted Rand Index (ARI) is frequently used in cluster validation since it is a measure of agreement between two partitions: one given by the clustering process and the other defined by external criteria, but it can also be used in supervised learning [[69]](#ref69).
#
# Wikipedia Page
# $$X=\frac{\sum_{i}C_{2}^{P_i}\times \sum_{j}C_{2}^{TOP_j}}{C_2^{POP}}$$
# $$ARI=\frac{\sum_{i,j}C_{2}^{Matrix(i,j)}-X}{\frac{1}{2}[\sum_{i}C_{2}^{P_i} + \sum_{j}C_{2}^{TOP_j}]-X}$$
# In[246]:
cm.ARI
#
#
Notice : $ C_{r}^{n} $ is the number of combinations of $ n $ objects taken $ r $
#
#
#
Notice : new in version 2.6
#
# ### Bangdiwala's B
# Bangdiwala's B statistic was created by Shrikant Bangdiwala in 1985 and is a measure of inter-rater agreement. While not as commonly used as the kappa statistic the B test has been used by various workers [[72]](#ref72) [[73]](#ref73).
#
# Wikipedia Page
# $$B=\frac{\sum_{i=1}^{|C|}TP_i^2}{\sum_{i=1}^{|C|}TOP_i\times P_i}$$
# In[247]:
cm.B
#
#
Notice : new in version 2.7
#
# ### Krippendorff's alpha
# Krippendorff's alpha coefficient, named after academic Klaus Krippendorff, is a statistical measure of the agreement achieved when coding a set of units of analysis in terms of the values of a variable.
# Krippendorff's alpha generalizes several known statistics, often called measures of inter-coder agreement, inter-rater reliability, reliability of coding given sets of units (as distinct from unitizing) but it also distinguishes itself from statistics that are called reliability coefficients but are unsuitable to the particulars of coding data generated for subsequent analysis [[74]](#ref74).
#
# Wikipedia Page
# $$\epsilon = \frac{1}{2\times POP}$$
# $$P_a=(1-\epsilon)\times ACC_{Overall}+\epsilon$$
# $$P_e=RACCU_{Overall}$$
# $$\alpha=\frac{P_a-P_e}{1-P_e}$$
# In[248]:
cm.Alpha
#
# ### Brier score
# The Brier Score is a strictly proper score function or strictly proper scoring rule that measures the accuracy of probabilistic predictions. For unidimensional predictions, it is strictly equivalent to the mean squared error as applied to predicted probabilities [[78]](#ref78) [[79]](#ref79).
#
# Wikipedia Page
# $$BS=\frac{1}{N} \sum_{t=1}^{N}(f_t-o_t)^2$$
# in which $f_t$ is the probability that was forecast, $o_t$ the actual outcome of the event at instance $t$ ($0$ if it does not happen and $1$ if it does happen) and $N$ is the number of forecasting instances.
# In[253]:
cm_test = ConfusionMatrix([0, 1, 1, 0], [0.1, 0.9, 0.8, 0.3], threshold=lambda x: 1)
cm_test.brier_score()
# In[254]:
cm_test.brier_score(pos_class=0)
# #### Parameters
# 1. `pos_class` : positive class name (type : `int/str`, default : `None`)
# #### Output
# `Brier score`
#
#
Notice : new in version 3.4
#
#
#
Notice : This option only works in binary probability mode
#
#
#
Notice : pos_class always defaults to the greater class name (i.e. max(classes)), unless, the actual_vector contains string. In that case, pos_class does not have any default value, and it must be explicitly specified or else an error will result.
#
# ### Log loss
# In information theory, the cross-entropy between two probability distributions
# $p$ and $q$ over the same underlying set of events measures the average number of bits needed to identify an event drawn from the set if a coding scheme used for the set is optimized for an estimated probability distribution $q$, rather than the true distribution $p$.
# This is also known as the log loss (logarithmic loss or logistic loss); the terms "log loss" and "cross-entropy loss" are used interchangeably. [[30]](#ref30).
#
# Wikipedia Page
# $$L_{\log}(y, p) = -(y \log (p) + (1 - y) \log (1 - p))$$
# In[255]:
cm_test.log_loss()
# In[256]:
cm_test.log_loss(pos_class=0)
# #### Parameters
# 1. `pos_class` : positive class name (type : `int/str`, default : `None`)
# 2. `normalize` : normalization flag (type : `bool`, default : `True`)
# #### Output
# `Log loss`
#
#
Notice : new in version 3.9
#
#
#
Notice : This option only works in binary probability mode
#
#
#
Notice : pos_class always defaults to the greater class name (i.e. max(classes)), unless, the actual_vector contains string. In that case, pos_class does not have any default value, and it must be explicitly specified or else an error will result.
# ## Examples
#
# ### Example-1 (Comparison of three different classifiers)
#
# - [Jupyter Notebook](https://nbviewer.jupyter.org/github/sepandhaghighi/pycm/blob/master/Document/Example1.ipynb)
# - [HTML](http://www.pycm.io/doc/Example1.html)
#
# ### Example-2 (How to plot via matplotlib)
#
# - [Jupyter Notebook](https://nbviewer.jupyter.org/github/sepandhaghighi/pycm/blob/master/Document/Example2.ipynb)
# - [HTML](http://www.pycm.io/doc/Example2.html)
#
# ### Example-3 (Activation threshold)
#
# - [Jupyter Notebook](https://nbviewer.jupyter.org/github/sepandhaghighi/pycm/blob/master/Document/Example3.ipynb)
# - [HTML](http://www.pycm.io/doc/Example3.html)
#
# ### Example-4 (File)
#
# - [Jupyter Notebook](https://nbviewer.jupyter.org/github/sepandhaghighi/pycm/blob/master/Document/Example4.ipynb)
# - [HTML](http://www.pycm.io/doc/Example4.html)
#
# ### Example-5 (Sample weights)
#
# - [Jupyter Notebook](https://nbviewer.jupyter.org/github/sepandhaghighi/pycm/blob/master/Document/Example5.ipynb)
# - [HTML](http://www.pycm.io/doc/Example5.html)
#
# ### Example-6 (Unbalanced data)
#
# - [Jupyter Notebook](https://nbviewer.jupyter.org/github/sepandhaghighi/pycm/blob/master/Document/Example6.ipynb)
# - [HTML](http://www.pycm.io/doc/Example6.html)
#
# ### Example-7 (How to plot via seaborn+pandas)
#
# - [Jupyter Notebook](https://nbviewer.jupyter.org/github/sepandhaghighi/pycm/blob/master/Document/Example7.ipynb)
# - [HTML](http://www.pycm.io/doc/Example7.html)
#
# ### Example-8 (Confidence interval)
#
# - [Jupyter Notebook](https://nbviewer.jupyter.org/github/sepandhaghighi/pycm/blob/master/Document/Example8.ipynb)
# - [HTML](http://www.pycm.io/doc/Example8.html)
# ## Cite
# If you use PyCM in your research, we would appreciate citations to the following paper :
#
Haghighi, S., Jasemi, M., Hessabi, S. and Zolanvari, A. (2018). PyCM: Multiclass confusion matrix library in Python. Journal of Open Source Software, 3(25), p.729.
#
# @article{Haghighi2018,
# doi = {10.21105/joss.00729},
# url = {https://doi.org/10.21105/joss.00729},
# year = {2018},
# month = {may},
# publisher = {The Open Journal},
# volume = {3},
# number = {25},
# pages = {729},
# author = {Sepand Haghighi and Masoomeh Jasemi and Shaahin Hessabi and Alireza Zolanvari},
# title = {{PyCM}: Multiclass confusion matrix library in Python},
# journal = {Journal of Open Source Software}
# }
#
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