• Faithfulness Correlation (Bhatt et al., 2020): iteratively replaces a random subset of given attributions with a baseline value and then measuring the correlation between the sum of this attribution subset and the difference in function output
• Faithfulness Estimate (Alvarez-Melis et al., 2018): computes the correlation between probability drops and attribution scores on various points
• Monotonicity Metric (Arya et al. 2019): starts from a reference baseline to then incrementally replace each feature in a sorted attribution vector, measuring the effect on model performance
• Monotonicity Metric (Nguyen et al, 2020): measures the spearman rank correlation between the absolute values of the attribution and the uncertainty in the probability estimation
• Pixel Flipping (Bach et al., 2015): captures the impact of perturbing pixels in descending order according to the attributed value on the classification score
• Region Perturbation (Samek et al., 2015): is an extension of Pixel-Flipping to flip an area rather than a single pixel
• Selectivity (Montavon et al., 2018): measures how quickly an evaluated prediction function starts to drop when removing features with the highest attributed values
• SensitivityN (Ancona et al., 2019): computes the correlation between the sum of the attributions and the variation in the target output while varying the fraction of the total number of features, averaged over several test samples
• IROF (Rieger at el., 2020): computes the area over the curve per class for sorted mean importances of feature segments (superpixels) as they are iteratively removed (and prediction scores are collected), averaged over several test samples
• Infidelity (Chih-Kuan, Yeh, et al., 2019): represents the expected mean square error between 1) a dot product of an attribution and input perturbation and 2) difference in model output after significant perturbation
• ROAD (Rong, Leemann, et al., 2022): measures the accuracy of the model on the test set in an iterative process of removing k most important pixels, at each step k most relevant pixels (MoRF order) are replaced with noisy linear imputations
• Sufficiency (Dasgupta et al., 2022): measures the extent to which similar explanations have the same prediction label

• Local Lipschitz Estimate (Alvarez-Melis et al., 2018): tests the consistency in the explanation between adjacent examples
• Max-Sensitivity (Yeh et al., 2019): measures the maximum sensitivity of an explanation using a Monte Carlo sampling-based approximation
• Avg-Sensitivity (Yeh et al., 2019): measures the average sensitivity of an explanation using a Monte Carlo sampling-based approximation
• Continuity (Montavon et al., 2018): captures the strongest variation in explanation of an input and its perturbed version
• Consistency (Dasgupta et al., 2022): measures the probability that the inputs with the same explanation have the same prediction label
• Relative Input Stability (RIS) (Agarwal, et. al., 2022): measures the relative distance between explanations e_x and e_x' with respect to the distance between the two inputs x and x'
• Relative Representation Stability (RRS) (Agarwal, et. al., 2022): measures the relative distance between explanations e_x and e_x' with respect to the distance between internal models representations L_x and L_x' for x and x' respectively
• Relative Output Stability (ROS) (Agarwal, et. al., 2022): measures the relative distance between explanations e_x and e_x' with respect to the distance between output logits h(x) and h(x') for x and x' respectively

• Pointing Game (Zhang et al., 2018): checks whether attribution with the highest score is located within the targeted object
• Attribution Localization (Kohlbrenner et al., 2020): measures the ratio of positive attributions within the targeted object towards the total positive attributions
• Top-K Intersection (Theiner et al., 2021): computes the intersection between a ground truth mask and the binarized explanation at the top k feature locations
• Relevance Rank Accuracy (Arras et al., 2021): measures the ratio of highly attributed pixels within a ground-truth mask towards the size of the ground truth mask
• Relevance Mass Accuracy (Arras et al., 2021): measures the ratio of positively attributed attributions inside the ground-truth mask towards the overall positive attributions
• AUC (Fawcett et al., 2006): compares the ranking between attributions and a given ground-truth mask
• Focus (Arias et al., 2022): quantifies the precision of the explanation by creating mosaics of data instances from different classes

• Sparseness (Chalasani et al., 2020): uses the Gini Index for measuring, if only highly attributed features are truly predictive of the model output
• Complexity (Bhatt et al., 2020): computes the entropy of the fractional contribution of all features to the total magnitude of the attribution individually
• Effective Complexity (Nguyen at el., 2020): measures how many attributions in absolute values are exceeding a certain threshold

• MPRT (Model Parameter Randomisation Test) (Adebayo et. al., 2018): randomises the parameters of single model layers in a cascading or independent way and measures the distance of the respective explanation to the original explanation
• Smooth MPRT (Hedström et. al., 2023): adds a "denoising" preprocessing step to the original MPRT, where the explanations are averaged over N noisy samples before the similarity between the original- and fully random model's explanations is measured
• Efficient MPRT (Hedström et. al., 2023): reinterprets MPRT by evaluating the rise in explanation complexity (discrete entropy) before and after full model randomisation, asking for increased explanation complexity post-randomisation
• Random Logit Test (Sixt et al., 2020): computes for the distance between the original explanation and the explanation for a random other class

• Completeness (Sundararajan et al., 2017): evaluates whether the sum of attributions is equal to the difference between the function values at the input x and baseline x' (and referred to as Summation to Delta (Shrikumar et al., 2017), Sensitivity-n (slight variation, Ancona et al., 2018) and Conservation (Montavon et al., 2018))
• Non-Sensitivity (Nguyen at el., 2020): measures whether the total attribution is proportional to the explainable evidence at the model output
• Input Invariance (Kindermans et al., 2017): adds a shift to input, asking that attributions should not change in response (assuming the model does not)

Let's first load the data and model. In this example, a pre-trained LeNet available from Quantus for the purpose of this tutorial is loaded, but generally, you might use any Pytorch (or TensorFlow) model instead. To follow this example, one needs to have quantus and torch installed, by e.g., pip install 'quantus[torch]'.

import quantus
from quantus.helpers.model.models import LeNet
import torch
import torchvision
from torchvision import transforms
  
# Enable GPU.
device = torch.device("cuda:0" if torch.cuda.is_available() else "cpu")

# Load a pre-trained LeNet classification model (architecture at quantus/helpers/models).
model = LeNet()
if device.type == "cpu":
    model.load_state_dict(torch.load("tests/assets/mnist", map_location=torch.device('cpu')))
else: 
    model.load_state_dict(torch.load("tests/assets/mnist"))

# Load datasets and make loaders.
test_set = torchvision.datasets.MNIST(root='./sample_data', download=True, transform=transforms.Compose([transforms.ToTensor()]))
test_loader = torch.utils.data.DataLoader(test_set, batch_size=24)

# Load a batch of inputs and outputs to use for XAI evaluation.
x_batch, y_batch = iter(test_loader).next()
x_batch, y_batch = x_batch.cpu().numpy(), y_batch.cpu().numpy()

We still need some explanations to evaluate. For this, there are two possibilities in Quantus. You can provide either:

1. a set of re-computed attributions (np.ndarray)
2. any arbitrary explanation function (callable), e.g., the built-in method quantus.explain or your own customised function

We show the different options below.

Quantus allows you to evaluate explanations that you have pre-computed, assuming that they match the data you provide in x_batch. Let's say you have explanations for Saliency and Integrated Gradients already pre-computed.

In that case, you can simply load these into corresponding variables a_batch_saliency and a_batch_intgrad:

a_batch_saliency = load("path/to/precomputed/saliency/explanations")
a_batch_intgrad = load("path/to/precomputed/intgrad/explanations")

Another option is to simply obtain the attributions using one of many XAI frameworks out there, such as Captum, Zennit, tf.explain, or iNNvestigate. The following code example shows how to obtain explanations (Saliency and Integrated Gradients, to be specific) using Captum:

import captum
from captum.attr import Saliency, IntegratedGradients

# Generate Integrated Gradients attributions of the first batch of the test set.
a_batch_saliency = Saliency(model).attribute(inputs=x_batch, target=y_batch, abs=True).sum(axis=1).cpu().numpy()
a_batch_intgrad = IntegratedGradients(model).attribute(inputs=x_batch, target=y_batch, baselines=torch.zeros_like(x_batch)).sum(axis=1).cpu().numpy()

# Save x_batch and y_batch as numpy arrays that will be used to call metric instances.
x_batch, y_batch = x_batch.cpu().numpy(), y_batch.cpu().numpy()

# Quick assert.
assert [isinstance(obj, np.ndarray) for obj in [x_batch, y_batch, a_batch_saliency, a_batch_intgrad]]

If you don't have a pre-computed set of explanations but rather want to pass an arbitrary explanation function that you wish to evaluate with Quantus, this option exists.

For this, you can for example rely on the built-in quantus.explain function to get started, which includes some popular explanation methods (please run quantus.available_methods() to see which ones). Examples of how to use quantus.explain or your own customised explanation function are included in the next section.

As seen in the above image, the qualitative aspects of explanations may look fairly uninterpretable --- since we lack ground truth of what the explanations should be looking like, it is hard to draw conclusions about the explainable evidence. To gather quantitative evidence for the quality of the different explanation methods, we can apply Quantus.

Quantus implements XAI evaluation metrics from different categories, e.g., Faithfulness, Localisation and Robustness etc which all inherit from the base quantus.Metric class. To apply a metric to your setting (e.g., Max-Sensitivity) it first needs to be instantiated:

metric = quantus.MaxSensitivity(nr_samples=10,
                                lower_bound=0.2,
                                norm_numerator=quantus.fro_norm,
                                norm_denominator=quantus.fro_norm,
                                perturb_func=quantus.uniform_noise,
                                similarity_func=quantus.difference,
                                abs=True,
                                normalise=True)

and then applied to your model, data, and (pre-computed) explanations:

scores = metric(
    model=model,
    x_batch=x_batch,
    y_batch=y_batch,
    a_batch=a_batch_saliency,
    device=device,
    explain_func=quantus.explain,
    explain_func_kwargs={"method": "Saliency"},
)

Since a re-computation of the explanations is necessary for robustness evaluation, in this example, we also pass an explanation function (explain_func) to the metric call. Here, we rely on the built-in quantus.explain function to recompute the explanations. The hyperparameters are set with the explain_func_kwargs dictionary. Please find more details on how to use quantus.explain at API documentation.

You can alternatively use your own customised explanation function (assuming it returns an np.ndarray in a shape that matches the input x_batch). This is done as follows:

def your_own_callable(model, models, targets, **kwargs) -> np.ndarray
  """Logic goes here to compute the attributions and return an 
  explanation  in the same shape as x_batch (np.array), 
  (flatten channels if necessary)."""
  return explanation(model, x_batch, y_batch)

scores = metric(
    model=model,
    x_batch=x_batch,
    y_batch=y_batch,
    device=device,
    explain_func=your_own_callable
)

Quantus also provides high-level functionality to support large-scale evaluations, e.g., multiple XAI methods, multifaceted evaluation through several metrics, or a combination thereof. To utilise quantus.evaluate(), you simply need to define two things:

1. The Metrics you would like to use for evaluation (each __init__ parameter configuration counts as its own metric):

   metrics = {
       "max-sensitivity-10": quantus.MaxSensitivity(nr_samples=10),
       "max-sensitivity-20": quantus.MaxSensitivity(nr_samples=20),
       "region-perturbation": quantus.RegionPerturbation(),
   }

2. The XAI methods you would like to evaluate, e.g., a dict with pre-computed attributions:

   xai_methods = {
       "Saliency": a_batch_saliency,
       "IntegratedGradients": a_batch_intgrad
   }

You can then simply run a large-scale evaluation as follows (this aggregates the result by np.mean averaging):

import numpy as np
results = quantus.evaluate(
      metrics=metrics,
      xai_methods=xai_methods,
      agg_func=np.mean,
      model=model,
      x_batch=x_batch,
      y_batch=y_batch,
      **{"softmax": False,}
)