by Alejandro Correa Bahnsen and Jesus Solano
version 1.5, January 2019
This notebook is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. Special thanks goes to Rick Muller, Sandia National Laboratories(https://github.com/justmarkham)
In [1]:
# glass identification dataset
import pandas as pd
import numpy as np
url = 'http://archive.ics.uci.edu/ml/machine-learning-databases/glass/glass.data'
col_names = ['id','ri','na','mg','al','si','k','ca','ba','fe','glass_type']
glass = pd.read_csv(url, names=col_names, index_col='id')
glass.sort_values('al', inplace=True)
glass.head()
Out[1]:
Question: Pretend that we want to predict ri, and our only feature is al. How could we do it using machine learning?
Answer: We could frame it as a regression problem, and use a linear regression model with al as the only feature and ri as the response.
Question: How would we visualize this model?
Answer: Create a scatter plot with al on the x-axis and ri on the y-axis, and draw the line of best fit.
In [2]:
%matplotlib inline
import matplotlib.pyplot as plt
plt.style.use('bmh')
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# scatter plot using Pandas
glass.plot(kind='scatter', x='al', y='ri')
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In [4]:
# equivalent scatter plot using Matplotlib
plt.scatter(glass.al, glass.ri)
plt.xlabel('al')
plt.ylabel('ri')
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In [5]:
# fit a linear regression model
from sklearn.linear_model import LinearRegression
linreg = LinearRegression()
feature_cols = ['al']
X = glass[feature_cols]
y = glass.ri
linreg.fit(X, y)
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In [6]:
# make predictions for all values of X
glass['ri_pred'] = linreg.predict(X)
glass.head()
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In [7]:
# put the plots together
plt.scatter(glass.al, glass.ri)
plt.plot(glass.al, glass.ri_pred, color='red')
plt.xlabel('al')
plt.ylabel('ri')
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Linear regression equation: $y = \beta_0 + \beta_1x$
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# compute prediction for al=2 using the equation
linreg.intercept_ + linreg.coef_ * 2
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In [9]:
# compute prediction for al=2 using the predict method
test = np.array(2)
test = test.reshape(-1,1)
linreg.predict(test)
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In [10]:
# examine coefficient for al
print(feature_cols, linreg.coef_)
Interpretation: A 1 unit increase in 'al' is associated with a 0.0025 unit decrease in 'ri'.
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# increasing al by 1 (so that al=3) decreases ri by 0.0025
1.51699012 - 0.0024776063874696243
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In [12]:
# compute prediction for al=3 using the predict method
test = np.array(3)
test = test.reshape(-1,1)
linreg.predict(test)
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In [13]:
# examine glass_type
glass.glass_type.value_counts().sort_index()
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In [14]:
# types 1, 2, 3 are window glass
# types 5, 6, 7 are household glass
glass['household'] = glass.glass_type.map({1:0, 2:0, 3:0, 5:1, 6:1, 7:1})
glass.head()
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Let's change our task, so that we're predicting household using al. Let's visualize the relationship to figure out how to do this:
In [15]:
plt.scatter(glass.al, glass.household)
plt.xlabel('al')
plt.ylabel('household')
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Let's draw a regression line, like we did before:
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# fit a linear regression model and store the predictions
feature_cols = ['al']
X = glass[feature_cols]
y = glass.household
linreg.fit(X, y)
glass['household_pred'] = linreg.predict(X)
In [17]:
# scatter plot that includes the regression line
plt.scatter(glass.al, glass.household)
plt.plot(glass.al, glass.household_pred, color='red')
plt.xlabel('al')
plt.ylabel('household')
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If al=3, what class do we predict for household? 1
If al=1.5, what class do we predict for household? 0
We predict the 0 class for lower values of al, and the 1 class for higher values of al. What's our cutoff value? Around al=2, because that's where the linear regression line crosses the midpoint between predicting class 0 and class 1.
Therefore, we'll say that if household_pred >= 0.5, we predict a class of 1, else we predict a class of 0.
In [18]:
# understanding np.where
import numpy as np
nums = np.array([5, 15, 8])
# np.where returns the first value if the condition is True, and the second value if the condition is False
np.where(nums > 10, 'big', 'small')
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In [19]:
# transform household_pred to 1 or 0
glass['household_pred_class'] = np.where(glass.household_pred >= 0.5, 1, 0)
glass.head()
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In [20]:
# plot the class predictions
plt.scatter(glass.al, glass.household)
plt.plot(glass.al, glass.household_pred_class, color='red')
plt.xlabel('al')
plt.ylabel('household')
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In [21]:
# fit a logistic regression model and store the class predictions
from sklearn.linear_model import LogisticRegression
logreg = LogisticRegression(solver='liblinear',C=1e9)
feature_cols = ['al']
X = glass[feature_cols]
y = glass.household
logreg.fit(X, y)
glass['household_pred_class'] = logreg.predict(X)
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# plot the class predictions
plt.scatter(glass.al, glass.household)
plt.plot(glass.al, glass.household_pred_class, color='red')
plt.xlabel('al')
plt.ylabel('household')
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What if we wanted the predicted probabilities instead of just the class predictions, to understand how confident we are in a given prediction?
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# store the predicted probabilites of class 1
glass['household_pred_prob'] = logreg.predict_proba(X)[:, 1]
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# plot the predicted probabilities
plt.scatter(glass.al, glass.household)
plt.plot(glass.al, glass.household_pred_prob, color='red')
plt.xlabel('al')
plt.ylabel('household')
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In [25]:
# examine some example predictions
print(logreg.predict_proba([[1]]))
print(logreg.predict_proba([[2]]))
print(logreg.predict_proba([[3]]))
The first column indicates the predicted probability of class 0, and the second column indicates the predicted probability of class 1.
Examples:
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# create a table of probability versus odds
table = pd.DataFrame({'probability':[0.1, 0.2, 0.25, 0.5, 0.6, 0.8, 0.9]})
table['odds'] = table.probability/(1 - table.probability)
table
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What is e? It is the base rate of growth shared by all continually growing processes:
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# exponential function: e^1
np.exp(1)
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What is a (natural) log? It gives you the time needed to reach a certain level of growth:
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# time needed to grow 1 unit to 2.718 units
np.log(2.718)
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It is also the inverse of the exponential function:
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np.log(np.exp(5))
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# add log-odds to the table
table['logodds'] = np.log(table.odds)
table
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Linear regression: continuous response is modeled as a linear combination of the features:
$$y = \beta_0 + \beta_1x$$Logistic regression: log-odds of a categorical response being "true" (1) is modeled as a linear combination of the features:
$$\log \left({p\over 1-p}\right) = \beta_0 + \beta_1x$$This is called the logit function.
Probability is sometimes written as pi:
$$\log \left({\pi\over 1-\pi}\right) = \beta_0 + \beta_1x$$The equation can be rearranged into the logistic function:
$$\pi = \frac{e^{\beta_0 + \beta_1x}} {1 + e^{\beta_0 + \beta_1x}}$$In other words:
The logistic function has some nice properties:
We have covered how this works for binary classification problems (two response classes). But what about multi-class classification problems (more than two response classes)?
In [31]:
# plot the predicted probabilities again
plt.scatter(glass.al, glass.household)
plt.plot(glass.al, glass.household_pred_prob, color='red')
plt.xlabel('al')
plt.ylabel('household')
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In [32]:
# compute predicted log-odds for al=2 using the equation
logodds = logreg.intercept_ + logreg.coef_[0] * 2
logodds
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In [33]:
# convert log-odds to odds
odds = np.exp(logodds)
odds
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In [34]:
# convert odds to probability
prob = odds/(1 + odds)
prob
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In [35]:
# compute predicted probability for al=2 using the predict_proba method
logreg.predict_proba([[2]])[:, 1]
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In [36]:
# examine the coefficient for al
feature_cols, logreg.coef_[0]
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Interpretation: A 1 unit increase in 'al' is associated with a 4.18 unit increase in the log-odds of 'household'.
In [37]:
# increasing al by 1 (so that al=3) increases the log-odds by 4.18
logodds = 0.64722323 + 4.1804038614510901
odds = np.exp(logodds)
prob = odds/(1 + odds)
prob
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In [38]:
# compute predicted probability for al=3 using the predict_proba method
logreg.predict_proba([[3]])[:, 1]
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Bottom line: Positive coefficients increase the log-odds of the response (and thus increase the probability), and negative coefficients decrease the log-odds of the response (and thus decrease the probability).
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# examine the intercept
logreg.intercept_
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Interpretation: For an 'al' value of 0, the log-odds of 'household' is -7.71.
In [40]:
# convert log-odds to probability
logodds = logreg.intercept_
odds = np.exp(logodds)
prob = odds/(1 + odds)
prob
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That makes sense from the plot above, because the probability of household=1 should be very low for such a low 'al' value.
Changing the $\beta_0$ value shifts the curve horizontally, whereas changing the $\beta_1$ value changes the slope of the curve.
Advantages of logistic regression:
Disadvantages of logistic regression: