In this step by step guide with python code, we will study the details behind a popular technique for interpretable machine learning called LIME (Local Interpretable Model-agnostic Explanations). This technique was proposed by Rubiero et. al. in their paper "Why should I trust you?: Explaining the predictions of any classifier" on 2016 (See paper here).
The example developed in this notebook is for explanations of tabular data. However, it is worth to highlight that this technique can also be applied for explanations with text and image data. For a brief introduction about of the need for interpretability and a short overview of LIME you can check these sildes and/or the YouTube video below.
%%HTML <iframe src="https://www.youtube.com/embed/vz_fkVkoGFM" width="560" height="315" allowfullscreen></iframe>
import pandas as pd import numpy as np import matplotlib.pyplot as plt from matplotlib.colors import LinearSegmentedColormap gray_cmap=LinearSegmentedColormap.from_list('gy',[(.3,.3,.3),(.8,.8,.8)], N=2) np.random.seed(222) #Seed for random numbers generation def set_plot_style(): plt.axis([-2,2,-2,2]) plt.xlabel('x1') plt.ylabel('x2')
An artificial dataset with two explanatory variables as input and one categorical binary variable as output is used here. This dataset was manually designed to intentionally represent an underlying data generation process with non-linear decision boundaries. The first 5 elements of this dataset and a plot of the underlying data generation process are shown below.
df = pd.read_csv("https://arteagac.github.io/blog/lime/artificial_data.csv",header = 'infer') df[0:5]
The explanatory variables are saved in the array
X and the output in
y. To avoid issues due to the units of the explanatory variables the data is standarized using normal standarization. This is a regularly recommended practice in machine learning.
X = df[['x1','x2']].values y = df['y'].values X = (X - np.mean(X,axis=0)) / np.std(X,axis=0) #Standarization of data
The following plot represents the underlying data generation process in the dataset. The axis represent the input variables
x2 and the color (soft and dark gray) represent the class that the element belongs to.
set_plot_style() plt.scatter(X[:,0],X[:,1], c=y, cmap=gray_cmap)
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from sklearn.ensemble import RandomForestClassifier classifier = RandomForestClassifier(n_estimators=100) classifier.fit(X, y)
RandomForestClassifier(bootstrap=True, class_weight=None, criterion='gini', max_depth=None, max_features='auto', max_leaf_nodes=None, min_impurity_decrease=0.0, min_impurity_split=None, min_samples_leaf=1, min_samples_split=2, min_weight_fraction_leaf=0.0, n_estimators=100, n_jobs=None, oob_score=False, random_state=None, verbose=0, warm_start=False)
A mesh grid is created in order to observe the decision boundaries of the trained classifier. If you are not familiar with concept of mesh grid or plotting of decision boundaries you can think of it as sampling a lot of
x2 values in a specific range and use the machine learnig classifier to predict the output and visually observe the decision boundaries fo the trained model. The plot shown below illustrates the decision boundaries of the classifier and it can be noted that they are very similar to the underlying data generation process. Even though we should always control for overfitting, in this case it is not necessarily a problem because we can visually inferr that the decision boundaries of the classifier match the data generation process.
#Function to create a mesh grid def make_meshgrid(x1, x2, h=.02,x1_min=0,x1_max=0,x2_min=0,x2_max=0): if x1_min==0 and x1_max==0 and x2_min==0 and x2_max==0: x1_min, x1_max = x1.min() - 0.1, x1.max() + 0.1 x2_min, x2_max = x2.min() - 0.1, x2.max() + 0.1 xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, h), np.arange(x2_min, x2_max, h)) return np.vstack((xx1.ravel(), xx2.ravel())).T #Create mesh grid and predict class for each element in mesh grid XX = make_meshgrid(X[:,0],X[:,1],h=.07) yy = classifier.predict(XX) set_plot_style() plt.scatter(XX[:,0],XX[:,1], c=yy, cmap=gray_cmap)
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LIME generates local explanations, in other words, explanations for individual instances in a dataset. LIME starts by generating a new dataset of perturbations around the instance to be explained. Then, the trained machine learning classifier is used to predict the class of each instance in the new generated dataset. Finally, a simpler model with intrinsic intepretability, for instance a linear regression model, is fitted and used to explain the prediction of the classifier. Before fitting a simpler model, the instances in the new generated dataset are weighed based on their distance to the original instance being explained. In this way, there is a higher certainty that the model is locally faithful around the explained instance.
In this example, as shown in the plot below, we want to explain the prediction of the blue dot (instance with x1 = 0.8 and x2 = -0.7).
Xi = np.array([0.8,-0.7]) set_plot_style() plt.scatter(XX[:,0],XX[:,1], c=yy, cmap=gray_cmap) plt.scatter(Xi,Xi,c="blue",marker="o",s=70 )
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For the case of tabular data, sampling around the mean and standard deviation of the explanatory variables is recommended. Given that the dataset used in this example was normally standarized in a previous step, a normal random sample with mean 0 and standard deviation 1 is generated and stored in the 2D array
X_lime as shown below. At this point, we still do not know the class for each element in this sample. Such class is predicted in the next step.
num_perturb = 500 X_lime = np.random.normal(0,1,size=(num_perturb,X.shape)) set_plot_style() plt.scatter(X_lime[:,0],X_lime[:,1],s=2,c="black")
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The random forest classifier trained in previous steps is used here to predict the class of each pair (x1,x2) in the new generated dataset.
y_lime = classifier.predict(X_lime) set_plot_style() plt.scatter(X_lime[:,0],X_lime[:,1],s=5, c=y_lime, cmap=gray_cmap)
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classifier.predict(np.array([0.8,-0.7]).reshape(1, -1) )
The distance between each randomly generated instance and the instance being explained is computed using the euclidean distance. For explanations of other types of data such as image or text data the cosine distance can be used. Such distances are then mapped to a value between zero and one (weight) using a kernel function. Depeding on how we set the kernel width it shows how wide we define the "locality" around our instance. For tabular data the definition of a kernel width needs special atention, specially if the data has not been standarized. A more detailed discussion on this topic of kernel width can be found in the book "Interpretable Machine Learning" by Cristoph Molnar in section 5.7. In the code below, the shape of the weights is shown and it can be noted that for each instance in the new randomly generated dataset, there is a corresponding weight or importance value.
kernel_width = 0.2 distances = np.sum((Xi - X_lime)**2,axis=1) #Euclidean distance weights = np.sqrt(np.exp(-(distances**2)/(kernel_width**2))) #Kernel function weights.shape
The plot below shows how these weights would look like around the instance being explained. Green markers represent larger weights or instance with higher importance.
set_plot_style() plt.scatter(XX[:,0],XX[:,1], c=yy, cmap=gray_cmap) plt.scatter(X_lime[:,0],X_lime[:,1],s=10,c= weights,cmap="RdYlGn") plt.scatter(Xi,Xi,c="blue",marker="o",s=70 )
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X_lime), its class predictions (
y_lime) and their importance (
weights) to fit simpler and interpretable (linear) model¶
A linear model is fitted as shown below. This linear model generates new decision boundaries that are locally faitful around the explained instance. This linear decision boundary can be seen with the markers with + and - symbol. It is important to emphasize that this new linear decision boundary is not globally faithful because it is supposed to be a proper discriminator only in the locallity of the instance being explained (blue dot).
from sklearn.linear_model import LinearRegression simpler_model = LinearRegression() simpler_model.fit(X_lime, y_lime, sample_weight=weights) y_linmodel = simpler_model.predict(X_lime) y_linmodel = y_linmodel < 0.5 #Conver to binary class set_plot_style() plt.scatter(XX[:,0],XX[:,1], c=yy, cmap=gray_cmap) plt.scatter(Xi,Xi,c="blue",marker="o",s=70 ) plt.scatter(X_lime[y_linmodel==0,0],X_lime[y_linmodel==0,1],c= weights[y_linmodel==0],cmap="RdYlGn",marker="_",s=80) plt.scatter(X_lime[y_linmodel==1,0],X_lime[y_linmodel==1,1],c= weights[y_linmodel==1],cmap="RdYlGn",marker="+",s=80)
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The coefficients of this estimated linear model can be used to understand how changes in the explanatoriy variables affect the classification output for the instance being explained. For example, as shown below, the estimated coefficients suggest that for the instance being explained, increasing the values of x1 and x2 will cause the prediction to lean towards the prediction of the negative class (darker area).
In this notebook we have studied the basics of how LIME for tabular data works. The concepts covered here are easily transferrable to explanations of other types of data such as text and image data. For an insteresting discussion about some strengths and limitations of LIME please refer to the book "Interpretable Machine Learning" by Cristoph Molnar.