# Logistic Regression with Python in Machine Learning | KGP Talkie

## What is Logistic Regression?

`Logistic Regression`

is a Machine Learning algorithm which is used for the `classification problems`

, it is a predictive analysis algorithm and based on the concept of `probability`

. `Logistic regression`

is basically a supervised classification algorithm. In a classification problem, the target variable(or output),`y`

, can take only discrete values for given set of features(or inputs),`X`

. `Logistic regression`

becomes a classification technique only when a `decision threshold`

is brought into the picture. The setting of the threshold value is a very important aspect of `Logistic regression`

and is dependent on the classification problem itself.

Some of the examples of classification problems are Email spam or not spam, Online transactions Fraud or not Fraud, Tumor Malignant or Benign. `Logistic regression`

transforms its output using the logistic sigmoid function to return a `probability value`

.

We can call a `Logistic Regression`

as a Linear Regression model but the `Logistic Regression`

uses a more `complex cost function`

, this `cost function`

can be defined as the `Sigmoid function`

or also known as the `logistic function`

instead of a `linear function`

.

The hypothesis of `Logistic regression`

tends it to limit the `cost function`

between `0`

and `1`

. Therefore linear functions fail to represent it as it can have a value greater than `1`

or less than `0`

which is not possible as per the hypothesis of `Logistic regression`

.

### What are the types of Logistic regression

Based on the number of categories, Logistic regression can be classified as:

Target variable can have only`Binomial`

:`2`

possible types: “`0`

” or`“1”`

which may represent “win” vs “loss”, “pass” vs “fail”, “dead” vs “alive”, etc.Target variable can have`Multinomial`

:`3`

or more possible types which are not ordered(i.e. types have no quantitative significance) like “disease A” vs “disease B” vs “disease C”.It deals with target variables with`Ordinal`

:`ordered categories`

. For example, a test score can be categorized as:“very poor”, “poor”, “good”, “very good”. Here, each category can be given a score like 0, 1, 2, 3.

### What is the Sigmoid Function?

In order to map predicted values to probabilities, we use the `Sigmoid function`

. The function maps any real value into another value between `0`

and `1`

. In machine learning, we use sigmoid to map predictions to `probabilities`

.

In the curve it shown that any small changes in the values of `X`

in that region will cause values of `Y`

to change significantly. That means this function has a tendency to bring the `Y`

values to either end of the curve. Looks like it’s good for a classifier considering its property? Yes ! It indeed is. It tends to bring the activations to either side of the curve.Making clear distinction on prediction.

Another advantage of this `activation function`

is, unlike linear function, the output of the `activation function`

is always going to be in range `(0,1)`

compared to `(-inf, inf)`

of linear function. So we have our activations bound in a range. Nice, it won’t blow up the activations then.

This is great. `Sigmoid functions`

are one of the most widely used activation functions today. Then what are the problems with this?

If you notice, towards either end of the sigmoid function, the `Y`

values tend to respond very less to changes in `X`

. What does that mean? The `gradient`

at that region is going to be small. It gives rise to a problem of `“vanishing gradients”`

. Hmm. So what happens when the activations reach near the `“near-horizontal”`

part of the curve on either sides?

`Gradient`

is small or has vanished (cannot make significant change because of the extremely small value). The network refuses to learn further or is drastically slow ( depending on use case and until gradient /computation gets hit by floating point value limits ). There are ways to work around this problem and sigmoid is still very popular in classification problems.

In order to map predicted values to probabilities, we use the Sigmoid function. The function maps any real value into another value between `0`

and `1`

. In machine learning, we use sigmoid to map predictions to `probabilities`

.

### Decision Boundary

We expect our classifier to give us a set of outputs or classes based on `probability`

when we pass the inputs through a prediction function and returns a probability score between `0`

and `1`

.

### Cost function

We learned about the `cost function`

in the Linear regression, the cost function represents `optimization objective`

i.e. we create a `cost function`

and minimize it so that we can develop an accurate model with `minimum error`

. If we try to use the `cost function`

of the linear regression in `Logistic Regression`

then it would be of no use as it would end up being a `non-convex function`

with many local minimums, in which it would be very difficult to minimize the `cost value`

and find the `global minimum`

.

$$ \text cost function, [\large \text{cost}(h(\theta )) , x) =

\begin{cases}

-log(h_{\theta}(x) , y) & if y = 1\\\\

-log(1- h_{\theta}(x) , y)& if y =0

\end{cases}$$

The RMS `Titanic`

was a British passenger liner that sank in the`North Atlantic Ocean`

in the early morning hours of `15 April 1912`

, after it collided with an `iceber`

g during its maiden voyage from `Southampton to New York City`

. There were an estimated `2,224`

passengers and crew aboard the ship, and more than `1,500`

died.

## Lets go ahead and build a model which can predict if a passenger is gonna survive

import numpy as np import pandas as pd import seaborn as sns import matplotlib.pyplot as plt %matplotlib inline

from sklearn.linear_model import LogisticRegression

titanic = sns.load_dataset('titanic') titanic.head(10)

survived | pclass | sex | age | sibsp | parch | fare | embarked | class | who | adult_male | deck | embark_town | alive | alone | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | 3 | male | 22.0 | 1 | 0 | 7.2500 | S | Third | man | True | NaN | Southampton | no | False |

1 | 1 | 1 | female | 38.0 | 1 | 0 | 71.2833 | C | First | woman | False | C | Cherbourg | yes | False |

2 | 1 | 3 | female | 26.0 | 0 | 0 | 7.9250 | S | Third | woman | False | NaN | Southampton | yes | True |

3 | 1 | 1 | female | 35.0 | 1 | 0 | 53.1000 | S | First | woman | False | C | Southampton | yes | False |

4 | 0 | 3 | male | 35.0 | 0 | 0 | 8.0500 | S | Third | man | True | NaN | Southampton | no | True |

5 | 0 | 3 | male | NaN | 0 | 0 | 8.4583 | Q | Third | man | True | NaN | Queenstown | no | True |

6 | 0 | 1 | male | 54.0 | 0 | 0 | 51.8625 | S | First | man | True | E | Southampton | no | True |

7 | 0 | 3 | male | 2.0 | 3 | 1 | 21.0750 | S | Third | child | False | NaN | Southampton | no | False |

8 | 1 | 3 | female | 27.0 | 0 | 2 | 11.1333 | S | Third | woman | False | NaN | Southampton | yes | False |

9 | 1 | 2 | female | 14.0 | 1 | 0 | 30.0708 | C | Second | child | False | NaN | Cherbourg | yes | False |

titanic.describe()

survived | pclass | age | sibsp | parch | fare | |
---|---|---|---|---|---|---|

count | 891.000000 | 891.000000 | 714.000000 | 891.000000 | 891.000000 | 891.000000 |

mean | 0.383838 | 2.308642 | 29.699118 | 0.523008 | 0.381594 | 32.204208 |

std | 0.486592 | 0.836071 | 14.526497 | 1.102743 | 0.806057 | 49.693429 |

min | 0.000000 | 1.000000 | 0.420000 | 0.000000 | 0.000000 | 0.000000 |

25% | 0.000000 | 2.000000 | 20.125000 | 0.000000 | 0.000000 | 7.910400 |

50% | 0.000000 | 3.000000 | 28.000000 | 0.000000 | 0.000000 | 14.454200 |

75% | 1.000000 | 3.000000 | 38.000000 | 1.000000 | 0.000000 | 31.000000 |

max | 1.000000 | 3.000000 | 80.000000 | 8.000000 | 6.000000 | 512.329200 |

## Data understanding

titanic.isnull().sum()

survived 0 pclass 0 sex 0 age 177 sibsp 0 parch 0 fare 0 embarked 2 class 0 who 0 adult_male 0 deck 688 embark_town 2 alive 0 alone 0 dtype: int64

## Heatmap

A `Heatmap`

is a `two-dimensional`

graphical representation of data where the individual values that are contained in a `matrix`

are represented as `colors`

. The `seaborn`

python package allows the creation of `annotated heatmaps`

which can be tweaked using `Matplotlib`

tools as per the creator’s requirement.

sns.heatmap(titanic.isnull(), cbar = False, cmap = 'viridis') plt.title('Number of people in the ship with respect their features ') plt.show()

titanic['age'].isnull().sum()/titanic.shape[0]*100

19.865319865319865

## Histogram

A `Histogram`

shows the `frequency`

on the vertical axis and the horizontal axis is another dimension. Usually it has `bins`

, where every bin has a `minimum`

and `maximum`

value. Each bin also has a frequency between `x`

and `infinite`

.

ax = titanic['age'].hist(bins = 30, density = True, stacked = True, color = 'teal', alpha = 0.7, figsize = (16, 5)) titanic['age'].plot(kind = 'density', color = 'teal') ax.set_xlabel('Age') plt.title('Percentage of the people with respect to their age ') plt.show()

In this part of the code, we plotted `female survived`

& `not survived`

passengers and aslo `male survived`

& `not survived`

passengers with respect to their age.

### distplot

Flexibly plot a `univariate distribution`

of observations.

survived = 'survived' not_survived = 'not survived' fig, axes = plt.subplots(nrows = 1, ncols = 2, figsize = (20, 4)) women = titanic[titanic['sex'] == 'female'] men = titanic[titanic['sex'] == 'male'] ax = sns.distplot(women[women[survived]==1].age.dropna(), bins = 18, label = survived, ax = axes[0], kde = False) ax = sns.distplot(women[women[survived]==0].age.dropna(), bins = 40, label = not_survived, ax = axes[0], kde = False) ax.legend() ax.set_title('Number of female passenger whether survived or not with respect to their age ') ax = sns.distplot(men[men[survived]==1].age.dropna(), bins = 18, label = survived, ax = axes[1], kde = False) ax = sns.distplot(men[men[survived]==0].age.dropna(), bins = 40, label = not_survived, ax = axes[1], kde = False) ax.legend() ax.set_title('Number of male passenger whether survived or not with respect to their age') plt.ylabel('No. of people') plt.show()

titanic['sex'].value_counts()

male 577 female 314 Name: sex, dtype: int64

Let's observe the people with respect to thier age who are in pclass. We will get this by using catplot() function.

sns.catplot(x = 'pclass', y = 'age', data = titanic, kind = 'box') plt.title('Age of the people who are in pclass') plt.show()

sns.catplot(x = 'pclass', y = 'fare', data = titanic, kind = 'box') plt.title('Fare for the different classes of the pclass') plt.show()

titanic[titanic['pclass'] == 1]['age'].mean()

38.233440860215055

titanic[titanic['pclass'] == 2]['age'].mean()

29.87763005780347

titanic[titanic['pclass'] == 3]['age'].mean()

25.14061971830986

### Imputation

In the following function,We are dealing with `missing values`

by using `imputation`

.`Imputation`

is the process of replacing `missing data`

with substituted values.The missing values are substituted by another `computed value`

.

def impute_age(cols): age = cols[0] pclass = cols[1] if pd.isnull(age): if pclass == 1: return titanic[titanic['pclass'] == 1]['age'].mean() elif pclass == 2: return titanic[titanic['pclass'] == 2]['age'].mean() elif pclass == 3: return titanic[titanic['pclass'] == 3]['age'].mean() else: return age

titanic['age'] = titanic[['age', 'pclass']].apply(impute_age, axis = 1)

Let's see the resultant plot :

sns.heatmap(titanic.isnull(), cbar = False, cmap = 'viridis') plt.title('Number of people with respect to their features') plt.show()

## Analysing Embarked

Analyzing number of `passengers`

get into the ship.

**Facegrid:** `Multi-plot`

grid for plotting `conditional relationships`

.

f = sns.FacetGrid(titanic, row = 'embarked', height = 2.5, aspect= 3) f.map(sns.pointplot, 'pclass', 'survived', 'sex', order = None, hue_order = None) f.add_legend() plt.show()

titanic['embarked'].isnull().sum()

2

titanic['embark_town'].value_counts()

Southampton 644 Cherbourg 168 Queenstown 77 Name: embark_town, dtype: int64

common_value = 'S' titanic['embarked'].fillna(common_value, inplace = True) titanic['embarked'].isnull().sum()

0

sns.heatmap(titanic.isnull(), cbar = False, cmap = 'viridis') plt.title('Number of people with respect to their features') plt.show()

We will try to print the output by removing labes: deck, embark_town, alive. We will do this by using `drop()`

function.`The drop()`

function is used to `drop`

specified labels from `rows`

or `columns`

. Let's look into the script:

titanic.drop(labels=['deck', 'embark_town', 'alive'], inplace = True, axis = 1)

Here is the resultant output :

sns.heatmap(titanic.isnull(), cbar = False, cmap = 'viridis') plt.title('Number of people with respect to their features') plt.show()

Here we can observe the difference between two figures.

titanic.info()

<class 'pandas.core.frame.DataFrame'> RangeIndex: 891 entries, 0 to 890 Data columns (total 12 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 survived 891 non-null int64 1 pclass 891 non-null int64 2 sex 891 non-null object 3 age 891 non-null float64 4 sibsp 891 non-null int64 5 parch 891 non-null int64 6 fare 891 non-null float64 7 embarked 891 non-null object 8 class 891 non-null category 9 who 891 non-null object 10 adult_male 891 non-null bool 11 alone 891 non-null bool dtypes: bool(2), category(1), float64(2), int64(4), object(3) memory usage: 65.5+ KB

titanic.head()

survived | pclass | sex | age | sibsp | parch | fare | embarked | class | who | adult_male | alone | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | 3 | male | 22.0 | 1 | 0 | 7.2500 | S | Third | man | True | False |

1 | 1 | 1 | female | 38.0 | 1 | 0 | 71.2833 | C | First | woman | False | False |

2 | 1 | 3 | female | 26.0 | 0 | 0 | 7.9250 | S | Third | woman | False | True |

3 | 1 | 1 | female | 35.0 | 1 | 0 | 53.1000 | S | First | woman | False | False |

4 | 0 | 3 | male | 35.0 | 0 | 0 | 8.0500 | S | Third | man | True | True |

titanic['fare'] = titanic['fare'].astype('int') titanic['age'] = titanic['age'].astype('int') titanic['pclass'] = titanic['pclass'].astype('int') titanic.info()

<class 'pandas.core.frame.DataFrame'> RangeIndex: 891 entries, 0 to 890 Data columns (total 12 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 survived 891 non-null int64 1 pclass 891 non-null int32 2 sex 891 non-null object 3 age 891 non-null int32 4 sibsp 891 non-null int64 5 parch 891 non-null int64 6 fare 891 non-null int32 7 embarked 891 non-null object 8 class 891 non-null category 9 who 891 non-null object 10 adult_male 891 non-null bool 11 alone 891 non-null bool dtypes: bool(2), category(1), int32(3), int64(3), object(3) memory usage: 55.0+ KB

## Convert categorical data into numerical data

In this part `categorical data`

( male , female) converted into `numeriacl data`

(0,1,2..).

genders = {'male': 0, 'female': 1} titanic['sex'] = titanic['sex'].map(genders) who = {'man': 0, 'women': 1, 'child': 2} titanic['who'] = titanic['who'].map(who) adult_male = {True: 1, False: 0} titanic['adult_male'] = titanic['adult_male'].map(adult_male) alone = {True: 1, False: 0} titanic['alone'] = titanic['alone'].map(alone) ports = {'S': 0, 'C': 1, 'Q': 2} titanic['embarked'] = titanic['embarked'].map(ports) titanic.head()

survived | pclass | sex | age | sibsp | parch | fare | embarked | class | who | adult_male | alone | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | 3 | 0 | 22 | 1 | 0 | 7 | 0 | Third | 0.0 | 1 | 0 |

1 | 1 | 1 | 1 | 38 | 1 | 0 | 71 | 1 | First | NaN | 0 | 0 |

2 | 1 | 3 | 1 | 26 | 0 | 0 | 7 | 0 | Third | NaN | 0 | 1 |

3 | 1 | 1 | 1 | 35 | 1 | 0 | 53 | 0 | First | NaN | 0 | 0 |

4 | 0 | 3 | 0 | 35 | 0 | 0 | 8 | 0 | Third | 0.0 | 1 | 1 |

titanic.drop(labels = ['class', 'who'], axis = 1, inplace= True) titanic.head()

survived | pclass | sex | age | sibsp | parch | fare | embarked | adult_male | alone | |
---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | 3 | 0 | 22 | 1 | 0 | 7 | 0 | 1 | 0 |

1 | 1 | 1 | 1 | 38 | 1 | 0 | 71 | 1 | 0 | 0 |

2 | 1 | 3 | 1 | 26 | 0 | 0 | 7 | 0 | 0 | 1 |

3 | 1 | 1 | 1 | 35 | 1 | 0 | 53 | 0 | 0 | 0 |

4 | 0 | 3 | 0 | 35 | 0 | 0 | 8 | 0 | 1 | 1 |

### Build Logistic Regression Model

In this part of code,we try to buid a `Logistic model`

for the data values.

from sklearn.model_selection import train_test_split from sklearn.metrics import accuracy_score

X = titanic.drop('survived', axis = 1) y = titanic['survived'] X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.33, random_state = 42) model = LogisticRegression(solver= 'lbfgs', max_iter = 400) model.fit(X_train, y_train) y_predict = model.predict(X_test) model.score(X_test, y_test)

0.8271186440677966

### Let's go ahead with age and fare grouping

#### Recursive Feature Elimination :

Given an `external estimator`

that assigns weights to features, `recursive`

feature elimination (RFE) is to select features by recursively considering smaller and smaller sets of features. First, the estimator is trained on the `initial set`

of features and the importance of each feature is obtained either through a `coef_ attribute`

or through a feature*importances* attribute. Then, the least important features are pruned from `current set of features`

.That procedure is `recursively`

repeated on the pruned set until the desired number of features to select is eventually reached.

from sklearn.feature_selection import RFE

model = LogisticRegression(solver='lbfgs', max_iter=500) rfe = RFE(model, 5, verbose=1) rfe = rfe.fit(X, y) rfe.support_

E:\callme_conda\lib\site-packages\sklearn\utils\validation.py:68: FutureWarning: Pass n_features_to_select=5 as keyword args. From version 0.25 passing these as positional arguments will result in an error warnings.warn("Pass {} as keyword args. From version 0.25 "

Fitting estimator with 9 features. Fitting estimator with 8 features. Fitting estimator with 7 features. Fitting estimator with 6 features.

array([ True, False, False, True, True, False, False, True, True])

titanic.head(3)

survived | pclass | sex | age | sibsp | parch | fare | embarked | adult_male | alone | |
---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | 3 | 0 | 22 | 1 | 0 | 7 | 0 | 1 | 0 |

1 | 1 | 1 | 1 | 38 | 1 | 0 | 71 | 1 | 0 | 0 |

2 | 1 | 3 | 1 | 26 | 0 | 0 | 7 | 0 | 0 | 1 |

X.head()

pclass | sex | age | sibsp | parch | fare | embarked | adult_male | alone | |
---|---|---|---|---|---|---|---|---|---|

0 | 3 | 0 | 22 | 1 | 0 | 7 | 0 | 1 | 0 |

1 | 1 | 1 | 38 | 1 | 0 | 71 | 1 | 0 | 0 |

2 | 3 | 1 | 26 | 0 | 0 | 7 | 0 | 0 | 1 |

3 | 1 | 1 | 35 | 1 | 0 | 53 | 0 | 0 | 0 |

4 | 3 | 0 | 35 | 0 | 0 | 8 | 0 | 1 | 1 |

XX = X[X.columns[rfe.support_]]

XX.head()

pclass | sibsp | parch | adult_male | alone | |
---|---|---|---|---|---|

0 | 3 | 1 | 0 | 1 | 0 |

1 | 1 | 1 | 0 | 0 | 0 |

2 | 3 | 0 | 0 | 0 | 1 |

3 | 1 | 1 | 0 | 0 | 0 |

4 | 3 | 0 | 0 | 1 | 1 |

X_train, X_test, y_train, y_test = train_test_split(XX, y, test_size = 0.2, random_state = 8, stratify = y)

model = LogisticRegression(solver= 'lbfgs', max_iter = 500) model.fit(X_train, y_train) y_predict = model.predict(X_test) model.score(X_test, y_test)

0.8547486033519553

#### Accuracy, F1-Score, P, R, AUC_ROC curve

Let's have some discussions on Precision, Recall, Accuracy .**Accuracy** tells the fraction of predictions our model got right .**Recall** tells us amount of actual positives are identified correctly .**Precision** tells us amount of positive proportion identifications are actually correct .

Have a look at following formulae:

from sklearn.model_selection import train_test_split, cross_val_score from sklearn.metrics import accuracy_score, classification_report, precision_score, recall_score from sklearn.metrics import confusion_matrix, precision_recall_curve, roc_auc_score, roc_curve, auc, log_loss

model = LogisticRegression(solver= 'lbfgs', max_iter = 500) model.fit(X_train, y_train) y_predict = model.predict(X_test)

### Predict_proba ( )

The function `predict_proba()`

returns a numpy array of two columns. The first column is the probability that `target=0`

and the second column is the probability that `target=1`

. That is why we add `[:,1]`

after `predict_proba()`

in order to get the probabilities of `target=1`

.

Let's find out `y_predict_prob`

by using `predict_proba()`

.Now have a look into the script:

y_predict_prob = model.predict_proba(X_test)[:, 1] y_predict_prob[: 5]

array([0.55566832, 0.87213996, 0.09376084, 0.09376084, 0.37996908])

We will compute the `Compute Receiver operating characteristic (ROC)`

by using the function `roc_curve()`

.Since the `thresholds`

are sorted from `low to high`

values, they are reversed upon returning them to ensure they correspond to both `fpr`

and `tpr`

, which are sorted in `reversed order`

during their calculation.

Let's see into the script:

[fpr, tpr, thr] = roc_curve(y_test, y_predict_prob) [fpr, tpr, thr][: 2]

[array([0. , 0. , 0. , 0. , 0.00909091, 0.00909091, 0.00909091, 0.00909091, 0.00909091, 0.00909091, 0.03636364, 0.03636364, 0.03636364, 0.06363636, 0.09090909, 0.12727273, 0.12727273, 0.13636364, 0.21818182, 0.23636364, 0.24545455, 0.27272727, 0.29090909, 0.43636364, 0.45454545, 0.47272727, 0.52727273, 0.92727273, 1. ]), array([0. , 0.07246377, 0.20289855, 0.24637681, 0.33333333, 0.39130435, 0.44927536, 0.55072464, 0.60869565, 0.63768116, 0.63768116, 0.65217391, 0.69565217, 0.7826087 , 0.7826087 , 0.7826087 , 0.79710145, 0.79710145, 0.86956522, 0.88405797, 0.88405797, 0.88405797, 0.88405797, 0.89855072, 0.91304348, 0.91304348, 0.92753623, 1. , 1. ])]

Now we will calculate accuracy of predicted data ,log loss and auc.To get this we will use functions accuracy_score(), log_loss(), auc().

### accuracy_score( )

In multilabel classification, this function can compute subset accuracy: the set of labels predicted for a sample must exactly match the corresponding set of labels in `y_true`

.

### log_loss( )

It is defined as the negative log-likelihood of a logistic model that returns `y_pred`

probabilities for its training data `y_true`

.

### auc( )

Compute Area Under the Curve (AUC),ROC using the trapezoidal rule.

Let's look into the script:

print('Accuracy: ', accuracy_score(y_test, y_predict)) print('log loss: ', log_loss(y_test, y_predict_prob)) print('auc: ', auc(fpr, tpr))

Accuracy: 0.8547486033519553 log loss: 0.36597373727139876 auc: 0.9007246376811595

idx = np.min(np.where(tpr>0.95))

### Plot for True Positive Rate (recall) vs False Positive Rate (1 - specificity) i.e. Receiver operating characteristic (ROC) curve.

A `receiver operating characteristic curve, ROC`

curve, is a graphical plot that illustrates the `diagnostic`

ability of a `binary classifier`

system as its discrimination threshold is varied. The `ROC`

curve is created by plotting the `True Positive Rate (TPR)`

against the `False Positive Rate (FPR)`

at various threshold settings.

True positive rate = correctly identified(For example : Sick people correctly identified as sick)

False positive rate = Incorrectly identified(For example : Healthy people incorrectly identified as sick) .

Now we will try to plot the Receiver Operating Characteristics curve.Let's have a look at the following code:

plt.figure() plt.plot(fpr, tpr, color = 'coral', label = "ROC curve area: " + str(auc(fpr, tpr))) plt.plot([0, 1], [0, 1], 'k--') plt.plot([0, fpr[idx]], [tpr[idx], tpr[idx]], 'k--', color = 'blue') plt.plot([fpr[idx],fpr[idx]], [0,tpr[idx]], 'k--', color='blue') plt.xlim([0.0, 1.0]) plt.ylim([0.0, 1.05]) plt.xlabel('False Positive Rate (1 - specificity)', fontsize=14) plt.ylabel('True Positive Rate (recall)', fontsize=14) plt.title('Receiver operating characteristic (ROC) curve') plt.legend(loc="lower right") plt.show() print("Using a threshold of %.3f " % thr[idx] + "guarantees a sensitivity of %.3f " % tpr[idx] + "and a specificity of %.3f" % (1-fpr[idx]) + ", i.e. a false positive rate of %.2f%%." % (np.array(fpr[idx])*100))

Using a threshold of 0.094 guarantees a sensitivity of 1.000 and a specificity of 0.073, i.e. a false positive rate of 92.73%.

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