partial_dependence

A python library for plotting partial dependence patterns of machine learning classifiers. The technique is a black box approach to recognize sets of instances where the model makes similar decisions.

Partial dependence measures the prediction change when changing one or more input features. We will focus only on 1D and 2D partial dependence plots. For each instance in the data we can plot the prediction change as we change one or two features in defined sample ranges. Then we cluster similar plots or heatmaps, e.g., instances reacting similarly when a feature value changes, to reduce clutter.

You can install partial_dependence via

pip install partial_dependence

and import it in python using:

import partial_dependence as pdp_plot

1. Plotting clustering of partial dependence

Following we will show how the pipeline of functions works. Please refer to the inline documentation of the methods for full information.

You can also run the Jupyter notebook file to have a running example.

The visualization we are using as example are coming from a Random Forest model trained on the UCI Wine Quality Data Set. The prediction is towards the class "good wine".

1.1 Initialization

Required arguments:

• df_test: a pandas.DataFrame containing only the features values for each instance in the test-set.

• model: trained classifier as an object with the following properties.

  The object must have a method predict_proba(X) which takes a numpy.array of shape (n, num_feat) as input and returns a numpy.array of shape (n, len(class_array)).

• class_array: a list of strings with all the classes name in the same order as the predictions returned by predict_proba(X).

• class_focus: a string with the class name of the desired partial dependence.

Optional arguments:

• num_samples: number of desired samples. Sampling a feature is done with:

  numpy.linspace(min_value, max_value, num_samples)

  where the bounds are related to min and max value for that feature in the test-set. Default value is 100.

• scale: scale parameter vector for normalization.

• shift: shift parameter vector for normalization.

If you need to provide your data to the model in normalized form, you have to define scale and shift such that:

transformed_data = (original_data + shift)*scale

where shift and scale are both numpy.array of shape (1,num_feat).

If the model uses directly the raw data in df_test without any transformation, do not insert any scale and shift parameters.

If our model does not use normalization, we can initialize the tool this way:

my_pdp_plot = pdp_plot.PartialDependence( my_df_test,
                                          my_model,
                                          my_labels_name,
                                          my_labels_focus )

1.2 Creating the PdpCurves object

By choosing a feature and changing it in the sample range, for each row in the test-set we can create num_samples different versions of the original instance.

Then we are able to compute prediction values for each of the different vectors.

pdp() initialize and returns a python object from the class PdpCurves() containing such predictions values.

Required argument:

• fix: string with name of the chosen feature as reported in a column of df_test.

curves = my_pdp_plot.pdp( chosen_feature )

1.3 Getting an overview of the partial dependence

It is already possible to plot something with the function plot().

Whenever you have a PdpCurves object available, you can plot something. Here you can find a first example. The visualization is automatically saved in a png file in the same folder of the script.

my_pdp_plot.plot( curves, local_curves = True, plot_full_curves = True )

1.4 Clustering 1D partial dependence

To call compute_clusters(), we define the integer number of desired clusters with the n_clusters argument and we provide curves.

The function returns a list of PdpCurves objects. Each element of the list is a different cluster.

curves_list_RF = my_pdp_plot.compute_clusters( curves, chosen_cluster_number )

1.5 Plotting the clustering results

Without customization, plotting the clustering is quite straightforward.

my_pdp_plot.plot( curves_list_RF )

1.6 2D partial dependence heatmaps

It is possible to visualize the increase/decrease in prediction of instances when changing two features at the same time. For a single instance the samples vary around the original pair of values. You can specify the desired instance by providing the row index integer from df_test. In this case we are taking the instance with index 88.

instance_heatmap = my_pdp_plot.pdp_2D("alcohol", "density", instances = 88)
my_pdp_plot.plot_heatmap(instance_heatmap)

In case you want to visualize the average 2D partial dependence over a set of instances, just provide a list of integers. The color will resemble the average increase/decrease across all instances and the samples will vary from min to max values of the set. If you want to visualize the average 2D partial dependence across the entire test-set instead..

all_inst = my_pdp_plot.pdp_2D("alcohol", "density")
my_pdp_plot.plot_heatmap(all_inst)

1.7 Clustering 2D partial dependence

With same function my_pdp_plot.compute_clusters() of Section 1.4, it is also possible to cluster heatmaps.

An heatmap object from the command my_pdp_plot.pdp_2D(feat_y, feat_x, instances) contains: num_samples X num_samples X len(instances) prediction values.

It is possible to cluster all the test instances (using the RMSE metric) and to display an heatmaps for each cluster with the following code:

all_inst = my_pdp_plot.pdp_2D("alcohol", "density")
list_clust_heats = my_pdp_plot.compute_clusters(all_inst, n_clusters = 16)
my_pdp_plot.plot_heatmap(list_clust_heats)

1.8 2D partial dependence SPLOMs

We can combine all the possible heatmaps in a single visualization. The SPLOM will show the patterns describing all possible pairs of features partial dependence.

The code to visualize the SPLOM for that same instance 88 is quite simple:

sploms_objs = my_pdp_plot.get_data_splom(88)
my_pdp_plot.plot_splom(sploms_objs)

A stripe of blue/red over a column and row of a feature determines an increase/decrease of prediction when that feature is changed, no matter what other feature varies. For example for this particular instance, when changing just two features, an increase in alcohol or decrease in volatile acidity would generally bring an increase in prediction towards the class good wine.

The SPLOM can give you a hint of average prediction change also over the entire test-set. The visualization combines the 2D scatter plots with the average change in prediction.

The user can detect global patterns when a same color disposition is present across row and columns of a same feature. For example this model generally has an average increase in prediction towards the class good wine when the alcohol increases with any other feature. Dark orange areas and blue areas show where there is an average decrease/increase in prediction. For example there is an enclaved blue area within the heatmap cell for pH and total sulfur dioxide where the prediction generally increases.

sploms_objs = my_pdp_plot.get_data_splom()
my_pdp_plot.plot_splom(sploms_objs)

1.9 Clustering SPLOMs

Each instance SPLOM can be represented by a long vector of prediction values. The vector is created by appending the data from each unique heatmap in a SPLOM. We can measure the distance among different instances SPLOMs by computing RMSE among such vectors. By building an RMSE distance matrix and clustering the instances we are able to represent a SPLOM for each cluster set. With the following code we can cluster the SPLOMs of the entire test-set.

sploms_objs = my_pdp_plot.get_data_splom()
list_clust_sploms = my_pdp_plot.compute_clusters(sploms_objs, n_clusters = 16)

To have an overview over the entire set of clusters:

my_pdp_plot.plot_splom(list_clust_sploms)

We can now plot the first cluster (cluster with label "#8" in the left top corner of the last viz)

my_pdp_plot.plot_splom(list_clust_sploms[0])

The distance matrix is stored, so it is less time consuming to change the number of clusters and plot again.

list_clust_sploms = my_pdp_plot.compute_clusters(sploms_objs, n_clusters = 49)
my_pdp_plot.plot_splom(list_clust_sploms)

2. Customization and extra functions

2.1 Computing predictions in chunks

When using pdp(), sometimes the amount of data to process is too large and it is necessary to divide it in chunks so that we don't run out of memory. To do so, just set the optional argument batch_size to the desired integer number.

batch_size cannot be lower than num_samples or higher than num_samples * len(df_test). If batch_size is 0, then the computation of prediction will take place in a single chunk, which is much faster if you have enough memory.

curves = my_pdp_plot.pdp( chosen_feature, batch_size = 1000 )

2.2 Using your own matplotlib figure

If you really like to hand yourself matplotlib and be free to customize the visualization this is how it works:

curves_list_RF = my_pdp_plot.compute_clusters(curves, chosen_cluster_number)

cluster_7 = curves_list_RF[7]
cluster_0 = curves_list_RF[0]
cluster_9 = curves_list_RF[9]

fig, ax = plt.subplots(figsize=(16, 9), dpi=100)

my_pdp_plot.plot(cluster_7,
                   color_plot="red",
                   plot_object=ax)

my_pdp_plot.plot(cluster_0,
                   color_plot="blue",
                   plot_object=ax)

my_pdp_plot.plot(cluster_9,
                   color_plot="green",
                   plot_object=ax)

plt.show()
plt.close("all")

2.3 Comparing different models

There might be scenarios in which you want to compare clusters from different models. For example let's compare the Random Forest model we had so far with a Support Vector Machine model.

wine_pdp_plot_SVM = pdp_plot.PartialDependence(df_test,
                                                model_SVM,
                                                labels_name,
                                                labels_focus,
                                                num_samples,
                                                scale_SVM,
                                                shift_SVM)

curves = wine_pdp_plot_SVM.pdp(chosen_feature)
curves_list_SVM = wine_pdp_plot_SVM.compute_clusters(curves, chosen_cluster_number)
wine_pdp_plot_SVM.plot(curves_list_SVM)

2.4 Clustering with DTW distance

To cluster together the partial dependence plots, we measure the distance among each pair. By default this distance is measured with RMSE. Another option for 1D partial dependence clustering is LB Keogh distance, an approximation of Dynamic Time Warping (DTW) distance. By setting the curves.r_param parameter of the formula to a value different from None, you are able to compute the clustering with the LB Keogh. The method get_optimal_keogh_radius() gives you a quick way to automatically compute an optimal value for curves.r_param. To set the distance back to RMSE just set curves.set_keogh_radius(None) before recomputing the clustering.

The first time you compute the clustering, a distance matrix is computed. Especially when using DTW distance, this might get time consuming. After the first time you call compute_clusters() on the curves object, the distance matrix will be stored in memory and the computation will be then much faster. Anyway if you change the radius with curves.set_keogh_radius(), you will need to recompute again the distance matrix.

curves.set_keogh_radius( my_pdp_plot.get_optimal_keogh_radius() )
keogh_curves_list = my_pdp_plot.compute_clusters( curves, chosen_cluster_number )

2.5 An example of the visualization customizations

my_pdp_plot.plot( keogh_curves_list, local_curves = False, plot_full_curves = True )

curves_list_RF = my_pdp_plot.compute_clusters( curves_RF, 5 )

my_pdp_plot.plot( curves_list_RF, cell_view = True )

curves_list_SVM = my_pdp_plot_SVM.compute_clusters( curves_SVM, 25 )

my_pdp_plot_SVM.plot( curves_list_SVM,
                        cell_view = True,
                        plot_full_curves = True,
                        local_curves = False,
                        path="plot_alcohol.png" )

2.6 Highlighting a custom vector

In case you want to highlight the partial dependence of a particular vector custom_vect, this is how it works..

curves, custom_preds = my_pdp_plot.pdp( chosen_feature, chosen_row = custom_vect )

my_pdp_plot.compute_clusters( curves, chosen_cluster_number )

my_pdp_plot.plot( curves, local_curves = False,
                   chosen_row_preds_to_plot = custom_preds )