# Precision, recall and f-measure
### George Tzanetakis, University of Victoria

In this notebook we go over the terminology of retrieval metrics: precision, recall and f-measure and how they are used in a variety of Music Information Retrieval (MIR) tasks. We first examine their classic original use as measures for the effectiveness of text information retrieval systems. Then we explore how they are used for evaluating classification systems, auto-tagging systems, as well as systems that estimate time boundaries (such as beat tracking and structure segmentation).

## Text Retrieval

The original scenario for which retrieval metrics were proposed is topic-based text retrieval. Today we are all familiar with this concept from search engines. The idea is that the user submits a query and the text retrieval system returns a set of what the system thinks are relevant documents. To evaluate the system an expert rates each returned document as relevant or not.

```python
import numpy as np

# suppose that are query is beatles and we are interested in retrieving documents
# about the corresponding insect species
query = 'beatles'

# Let's say a set of 10 documents is returned - their exact representation is not important
# and could be a bag-of-words representation, text, a list of keywords etc

# an expert/user goes over these automatically retrieved documents and marks
# them as relevant or not relevant.
# The documents that are about the insects are marked with a 1 and the
# ones that are not relevant (perhaps about the music group) are marked
# with a zero.
retrieved = np.array([1,1,0,1,1,1,1,0,1,0])

```

Precision is defined as the number of relevant retrieved documents divded by the number of returned documents. So for our particular example the precision is 0.70 or 70%.

```python
import numpy as np

def precision(retrieved):
    retrieved_relevant = np.count_nonzero(retrieved == 1)
    return (retrieved_relevant / len(retrieved))

print("Precision = %.2f\n" % (precision(retrieved)))
```

    Precision = 0.70

Notice that we can improve the precision in this case by returning less items. For example the precision when returning the first two items is 1 or 100%.

```python
less_retrieved = np.array([1,1])
print("Precision = %.2f\n" % (precision(retrieved)))
```

    Precision = 0.70

Now suppose that in the set of documents we are considering there are 15 documents about beatles (the insect species). Our set of retrieved results only contains 7 of them and therefore the recall in this case is 7/15=0.47

```python
def recall(retrieved, num_relevant):
    retrieved_relevant = np.count_nonzero(retrieved == 1)
    return (retrieved_relevant / num_relevant)
```

```python
retrieved = np.array([1,1,0,1,1,1,1,0,1,0])
print("Recall = %.2f\n" % (recall(retrieved,15)))
```

    Recall = 0.47

We can trivially increase recall by returning more items with the extreme cases of returning all documents as relevant at the expense of diminishing precision. Alternatively we can only return less and less document increasing precision at the expense of recall. An effective retrieval system should achieve both high precision and recall. The most common measure that balances these two metrics is the f1-score or f-measure defined as the
harmonic mean of precision and recall.

\begin{equation}
F_{1} = 2 * \frac{precision * recall}{precision + recall}
\end{equation}

```python
def f1_score(precision, recall):
    return 2 * ((precision * recall) / (precision+recall))

precision = 0.7
recall = 0.47
f1 = f1_score(precision,recall)
print("F1 = %.2f\n" % f1)
```

    F1 = 0.56

## Binary Classification

For a binary classification problem we can consider the set of predictions as the retrieved documents and the ground truth as the annotations of relevance. For example suppose that we have a music/speech classification system that predicts 100 instances and lets say that 50 of them are labeled as music and 50 of them are labeled as speech. A system that would predict everything as music would have high recall (i.e all music instances according to the ground truth would be predicted correctly) but 0.5 precision as half of the predicted instances
(the ones labeled speech according to the ground truth) would not be correctly predicted. A system that predicted a single instance of music correctly as music and all the other instances as speech would have a precision of 1 but really bad recall 1/50. Notice that in any binary classification system that outputs a class probability we can trade precision and recall by adjusting the threhold for classification. F1-score is defined similarly as with text retrieval as the harmonic mean of precision and recall.

Another way to view precision and recall is through the terminology used for binary classification problems in medical tests which is in terms of true positives (TP), true negatives (TN), false positives (FP) ad false negatives (FN). It is easy to see that using this terminology precision can be defined as:

\begin{equation}
Precision = \frac{TP}{TP+FP}
\end{equation}

\begin{equation}
Recall = \frac{TP}{TP+FN}
\end{equation}

## Multi-class and multi-label classification

For a multi-class classification problem with K classes we can not directly calculate retrieval metrics as described. There are three common approaches that are used. In macro-averaging the metrics are computed
independetly for each class and then the average is taken (this means that each class is treated equally even
though some classes might have more instances/support than others). A micro-average aggregates the contributions of each class and computes the metric overall which means that classes that are over-represented will effect the metrics more. Finally weighted average returns support-weighted metrics.

In multi-label classification (auto-tagging) each instances can be assigned more than one label. The approach to calculating retrieval metrics is similar with macro-averaging meaning that retrieval metrics are calculated separately for each tag column and then averagedm and micro-averaging meaning that retrieval metrics are calculating over the entiure matrix of predictions.

Let's look at a particular example from musical genre classification. First audio features are extracted for the GTZAN dataset.

```python

import glob
import librosa
import numpy as np

fnames = glob.glob("/Users/georgetzanetakis/data/sound/genres/*/*.wav")

genres = ['classical', 'country', 'disco', 'hiphop', 'jazz', 'rock', 'blues', 'reggae', 'pop', 'metal']

# allocate matrix for audio features and target
audio_features = np.zeros((len(fnames), 40))
target = np.zeros(len(fnames))

# compute the features
for (i,fname) in enumerate(fnames):
    #print("Processing %d %s" % (i, fname))
    for (label,genre) in enumerate(genres):
        if genre in fname:
            audio, srate = librosa.load(fname)
            mfcc_matrix = librosa.feature.mfcc(y=audio, sr=srate)
            mean_mfcc = np.mean(mfcc_matrix,axis=1)
            std_mfcc = np.std(mfcc_matrix, axis=1)
            audio_fvec = np.hstack([mean_mfcc, std_mfcc])
            audio_features[i] = audio_fvec
            target[i] = label

print(audio_features.shape)

```

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    (1000, 40)

We can view the confusion matrix and classificaton report with micro and macro average retrieval metrics. You can observe from the confusion matrix that classical is the easiest genre to classify with 89/100 instances classified correctly. This is reflected on the corresponding f1-score of 0.89. Because the classes are balanced there is no difference between the macro and micro averaging.

```python

from sklearn.model_selection import cross_val_predict
from sklearn import svm, metrics
clf = svm.SVC(gamma='scale', kernel='linear')

# perform 10-fold cross-validation to calculate accuracy and confusion matrix
predicted = cross_val_predict(clf, audio_features, target, cv=10)

print("Confusion matrix:\n%s" % metrics.confusion_matrix(target, predicted))
print("Classification report for classifier %s:\n%s\n"
      % (clf, metrics.classification_report(target, predicted, target_names=genres)))
```

    Confusion matrix:
    [[89  1  1  0  8  0  0  1  0  0]
     [ 2 61  4  0  4  8 11  5  4  1]
     [ 1  9 52  9  0 12  3  7  6  1]
     [ 0  2  9 62  0  3  3 16  1  4]
     [ 7  6  2  0 76  6  2  1  0  0]
     [ 0  7 17  4  3 41 17  4  1  6]
     [ 0 18  8  1  9  8 46  2  0  8]
     [ 1  8  9 22  1  5  7 40  6  1]
     [ 0  4 10  5  2  2  0  4 73  0]
     [ 0  2  4  7  0  9  1  1  0 76]]
    Classification report for classifier SVC(C=1.0, cache_size=200, class_weight=None, coef0=0.0,
      decision_function_shape='ovr', degree=3, gamma='scale', kernel='linear',
      max_iter=-1, probability=False, random_state=None, shrinking=True,
      tol=0.001, verbose=False):
                  precision    recall  f1-score   support

       classical       0.89      0.89      0.89       100
         country       0.52      0.61      0.56       100
           disco       0.45      0.52      0.48       100
          hiphop       0.56      0.62      0.59       100
            jazz       0.74      0.76      0.75       100
            rock       0.44      0.41      0.42       100
           blues       0.51      0.46      0.48       100
          reggae       0.49      0.40      0.44       100
             pop       0.80      0.73      0.76       100
           metal       0.78      0.76      0.77       100

       micro avg       0.62      0.62      0.62      1000
       macro avg       0.62      0.62      0.62      1000
    weighted avg       0.62      0.62      0.62      1000

```python
# merge rock and pop to a single genre with more support (300 instances) than the other genres (100 instanes)
new_genres = ['classsical_jazz', 'country', 'disco', 'hiphop','rock','pop', 'blues','reggage', 'metal']
new_target = [0 if k==4 else k for k in target]
print(new_genres)
```

    ['classsical_jazz', 'country', 'disco', 'hiphop', 'rock', 'pop', 'blues', 'reggage', 'metal']

```python
# perform 10-fold cross-validation to calculate accuracy and confusion matrix
predicted = cross_val_predict(clf, audio_features, new_target, cv=10)

print("Confusion matrix:\n%s" % metrics.confusion_matrix(target, predicted))
print("Classification report for classifier %s:\n%s\n"
      % (clf, metrics.classification_report(new_target, predicted, target_names=new_genres)))
```

    Confusion matrix:
    [[95  1  1  0  0  1  0  2  0  0]
     [ 7 59  3  0  0  9 11  6  4  1]
     [ 2  8 52  9  0 12  3  7  6  1]
     [ 0  2  9 62  0  3  3 16  1  4]
     [83  5  2  0  0  5  4  1  0  0]
     [ 7  6 17  4  0 36 17  4  3  6]
     [ 6 15 10  1  0  8 48  5  0  7]
     [ 2  7 10 23  0  4  7 40  6  1]
     [ 2  4 10  5  0  2  0  4 73  0]
     [ 0  2  4  7  0  9  1  1  0 76]]
    Classification report for classifier SVC(C=1.0, cache_size=200, class_weight=None, coef0=0.0,
      decision_function_shape='ovr', degree=3, gamma='scale', kernel='linear',
      max_iter=-1, probability=False, random_state=None, shrinking=True,
      tol=0.001, verbose=False):
                     precision    recall  f1-score   support

    classsical_jazz       0.87      0.89      0.88       200
            country       0.54      0.59      0.56       100
              disco       0.44      0.52      0.48       100
             hiphop       0.56      0.62      0.59       100
               rock       0.40      0.36      0.38       100
                pop       0.51      0.48      0.49       100
              blues       0.47      0.40      0.43       100
            reggage       0.78      0.73      0.76       100
              metal       0.79      0.76      0.78       100

          micro avg       0.62      0.62      0.62      1000
          macro avg       0.60      0.59      0.59      1000
       weighted avg       0.62      0.62      0.62      1000

Notice that because of the unbalanced support of the different classes after merging classical and jazz the values of the micro-average f1-score and macro-average f1-score are different.

## MIR tasks with time markers

Finally let's look at a last usage of retrieval metrics (precision, recall and f-measure for tasks where the predicted output and the ground truth is a set of time markers corresponding to things like structure boundaries,
beat locations, downbeats etc.

In this case typically a time tolerance window is defined so that if a time marker is predicted near (within that window) a ground truth marker it is consider correct or relevant.

One can easily see how over-segmemntation will affect precision. For example consider a predicted beat track output that is twice as fast as the ground truth. All the ground truth markers will be matched (recall will be 1)
but only half of them will be correct (precision 0.5). Alternatively if the predicted beat track output is every two ground truth beats then the precision will be 1 (all the predicted beats are correct) but the recall will be 0.5 (half of the ground truth beat markers are predicted).

Again the f1-score provides a balance between precision and recall. As an example if the marker detection depends on some parameters, we can perform a grid search over their values and select the one that provides the best f1-measure.

Depending on the application domain we might be more interested in precision or recall. For example in a automatic video generation system for music that takes into account segmentation boundaries precision might be more important as we want to make sure that if a boundary is detected it is a true boundary with high probability. At the samne time we might not care as much about recall as if a boundary is missed occassionally the video will still look ok.

```python
# Let's consider a simple toy example for calculating the retrieval metrics based on a time markers
# To make things simple the tolerance is +/-1 i.e an estimated time marker is consider correct if it is
# within +/-1 of the reference ground truth.

import numpy as np

ref_times = np.array([0, 4, 8, 12, 16])
est_times = np.array([0, 9, 11, 18])

# precision is the number of "correct" estimated times divided by the number of estimates
precision = 0
for e in est_times:
    diff = np.abs(e - ref_times)
    for d in diff:
        if (d <= 1):
            precision += 1
precision /= len(est_times)
print(precision)

# Note that the precision is 0.75 because the 4th estimated time marker - 18 is not part of the ground truth

# recall is the percentage of the reference time markers that are returned as estimates
recall = 0
for r in ref_times:
    diff = np.abs(r - est_times)
    for d in diff:
        if (d <=1):
            recall += 1
recall /= len(ref_times)
print(recall)

# Not that the recall is 0.6 because out of the 5 reference time markers only 3 are estimated correctly
# (within +/-1 of a reference marker) |

def f1_score(precision, recall):
    return 2 * ((precision * recall) / (precision+recall))

f1 = f1_score(precision,recall)
print("F1 = %.2f\n" % f1)
```

    0.75
    0.6
    F1 = 0.67

```python
def segment_metrics(ref_times, est_times):
    precision = 0
    for e in est_times:
        diff = np.abs(e - ref_times)
        for d in diff:
            if (d <= 1):
                precision += 1
    precision /= len(est_times)

    recall = 0
    for r in ref_times:
        diff = np.abs(r - est_times)
        for d in diff:
            if (d <=1):
                recall += 1
    recall /= len(ref_times)
    f1_score = 2 * ((precision * recall) / (precision+recall))
    return (precision, recall, f1_score)

segment_metrics(ref_times, est_times)
```

    (0.75, 0.6, 0.6666666666666665)

```python
# high precision by undersegmentation
ref_times = np.array([0, 4, 8, 12, 16])
est_times = np.array([9, 11])
segment_metrics(ref_times, est_times)
```

    (1.0, 0.4, 0.5714285714285715)

```python
# high recall by oversegmentation
ref_times = np.array([0, 4, 8, 12, 16])
est_times = np.array([0,2,4,6,8,10,12,14,16])
segment_metrics(ref_times, est_times)
```

    (0.5555555555555556, 1.0, 0.7142857142857143)

## MIR tasks with pair-wise labels

Another use of retrieval metrics is when there are cluster/segment labels per frame as in chord detection or structure segmentation. This evaluation combines information from the precision of the boundaries as well as ensuring that corrresponding segments are marked by the same label.

The input consists of two sequences of labels (one corresponding to the reference ground truth and one
corresponding to the estimated labels) using the same time granularity - typically frames or beats. The vocabulary of the labels can be different between the two segmentations. The retrieval metrics are computed over all possible pairs with a pair considered valid/correct/relevant if both items share a label.

```python

```