# Quantities of Interest¶

A Quantity of Interest (QoI) is a function of the output that determines the network output behavior that the attributions describe.

The quantity of interest lets us specify what we want to explain. Often, this is the output of the network corresponding to a particular class, addressing, e.g., "Why did the model classify a given image as a car?" However, we could also consider various combinations of outputs, allowing us to ask more specific questions, such as, "Why did the model classify a given image as a sedan and not a convertible?" The former may highlight general “car features,” such as tires, while the latter (called a comparative explanation) might focus on the roof of the car, a “car feature” not shared by convertibles.

## ClassQoI¶

Bases: QoI

Quantity of interest for attributing output towards a specified class.

Source code in trulens_explain/trulens/nn/quantities.py
 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 class ClassQoI(QoI): """ Quantity of interest for attributing output towards a specified class. """ def __init__(self, cl: int): """ Parameters: cl: The index of the class the QoI is for. """ self.cl = cl def __str__(self): return render_object(self, ["cl"]) def __call__(self, y: TensorLike) -> TensorLike: self._assert_cut_contains_only_one_tensor(y) return y[:, self.cl] 

### __init__(cl)¶

Parameters:

Name Type Description Default
cl int

The index of the class the QoI is for.

required
Source code in trulens_explain/trulens/nn/quantities.py
 243 244 245 246 247 248 249 def __init__(self, cl: int): """ Parameters: cl: The index of the class the QoI is for. """ self.cl = cl 

## ClassSeqQoI¶

Bases: QoI

Quantity of interest for attributing output towards a sequence of classes for each input.

Source code in trulens_explain/trulens/nn/quantities.py
 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 class ClassSeqQoI(QoI): """ Quantity of interest for attributing output towards a sequence of classes for each input. """ def __init__(self, seq_labels: List[int]): """ Parameters: seq_labels: A sequence of classes corresponding to each input. """ self.seq_labels = seq_labels def __call__(self, y): self._assert_cut_contains_only_one_tensor(y) assert get_backend().shape(y) == len(self.seq_labels) return y[:, self.seq_labels] 

### __init__(seq_labels)¶

Parameters:

Name Type Description Default
seq_labels List[int]

A sequence of classes corresponding to each input.

required
Source code in trulens_explain/trulens/nn/quantities.py
 395 396 397 398 399 400 401 def __init__(self, seq_labels: List[int]): """ Parameters: seq_labels: A sequence of classes corresponding to each input. """ self.seq_labels = seq_labels 

## ComparativeQoI¶

Bases: QoI

Quantity of interest for attributing network output towards a given class, relative to another.

Source code in trulens_explain/trulens/nn/quantities.py
 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 class ComparativeQoI(QoI): """ Quantity of interest for attributing network output towards a given class, relative to another. """ def __init__(self, cl1: int, cl2: int): """ Parameters: cl1: The index of the class the QoI is for. cl2: The index of the class to compare against. """ self.cl1 = cl1 self.cl2 = cl2 def __str__(self): return render_object(self, ["cl1", "cl2"]) def __call__(self, y: TensorLike) -> TensorLike: self._assert_cut_contains_only_one_tensor(y) return y[:, self.cl1] - y[:, self.cl2] 

### __init__(cl1, cl2)¶

Parameters:

Name Type Description Default
cl1 int

The index of the class the QoI is for.

required
cl2 int

The index of the class to compare against.

required
Source code in trulens_explain/trulens/nn/quantities.py
 266 267 268 269 270 271 272 273 274 275 def __init__(self, cl1: int, cl2: int): """ Parameters: cl1: The index of the class the QoI is for. cl2: The index of the class to compare against. """ self.cl1 = cl1 self.cl2 = cl2 

## InternalChannelQoI¶

Bases: QoI

Quantity of interest for attributing output towards the output of an internal convolutional layer channel, aggregating using a specified operation.

Also works for non-convolutional dense layers, where the given neuron's activation is returned.

Source code in trulens_explain/trulens/nn/quantities.py
 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 class InternalChannelQoI(QoI): """ Quantity of interest for attributing output towards the output of an internal convolutional layer channel, aggregating using a specified operation. Also works for non-convolutional dense layers, where the given neuron's activation is returned. """ @staticmethod def _batch_sum(x): """ Sums batched 2D channels, leaving the batch dimension unchanged. """ return get_backend().sum(x, axis=(1, 2)) def __init__( self, channel: Union[int, List[int]], channel_axis: Optional[int] = None, agg_fn: Optional[Callable] = None ): """ Parameters: channel: Channel to return. If a list is provided, then the quantity sums over each of the channels in the list. channel_axis: Channel dimension index, if relevant, e.g., for 2D convolutional layers. If channel_axis is None, then the channel axis of the relevant backend will be used. This argument is not used when the channels are scalars, e.g., for dense layers. agg_fn: Function with which to aggregate the remaining dimensions (except the batch dimension) in order to get a single scalar value for each channel. If agg_fn is None then a sum over each neuron in the channel will be taken. This argument is not used when the channels are scalars, e.g., for dense layers. """ if channel_axis is None: channel_axis = get_backend().channel_axis if agg_fn is None: agg_fn = InternalChannelQoI._batch_sum self._channel_ax = channel_axis self._agg_fn = agg_fn self._channels = channel if isinstance(channel, list) else [channel] def __call__(self, y: TensorLike) -> TensorLike: B = get_backend() self._assert_cut_contains_only_one_tensor(y) if len(B.int_shape(y)) == 2: return sum([y[:, ch] for ch in self._channels]) elif len(B.int_shape(y)) == 3: return sum([self._agg_fn(y[:, :, ch]) for ch in self._channel]) elif len(B.int_shape(y)) == 4: if self._channel_ax == 1: return sum([self._agg_fn(y[:, ch]) for ch in self._channels]) elif self._channel_ax == 3: return sum( [self._agg_fn(y[:, :, :, ch]) for ch in self._channels] ) else: raise ValueError( 'Unsupported channel axis for convolutional layer: {}'. format(self._channel_ax) ) else: raise QoiCutSupportError( 'Unsupported tensor rank for InternalChannelQoI: {}'.format( len(B.int_shape(y)) ) ) 

### __init__(channel, channel_axis=None, agg_fn=None)¶

Parameters:

Name Type Description Default
channel Union[int, List[int]]

Channel to return. If a list is provided, then the quantity sums over each of the channels in the list.

required
channel_axis Optional[int]

Channel dimension index, if relevant, e.g., for 2D convolutional layers. If channel_axis is None, then the channel axis of the relevant backend will be used. This argument is not used when the channels are scalars, e.g., for dense layers.

None
agg_fn Optional[Callable]

Function with which to aggregate the remaining dimensions (except the batch dimension) in order to get a single scalar value for each channel. If agg_fn is None then a sum over each neuron in the channel will be taken. This argument is not used when the channels are scalars, e.g., for dense layers.

None
Source code in trulens_explain/trulens/nn/quantities.py
 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 def __init__( self, channel: Union[int, List[int]], channel_axis: Optional[int] = None, agg_fn: Optional[Callable] = None ): """ Parameters: channel: Channel to return. If a list is provided, then the quantity sums over each of the channels in the list. channel_axis: Channel dimension index, if relevant, e.g., for 2D convolutional layers. If channel_axis is None, then the channel axis of the relevant backend will be used. This argument is not used when the channels are scalars, e.g., for dense layers. agg_fn: Function with which to aggregate the remaining dimensions (except the batch dimension) in order to get a single scalar value for each channel. If agg_fn is None then a sum over each neuron in the channel will be taken. This argument is not used when the channels are scalars, e.g., for dense layers. """ if channel_axis is None: channel_axis = get_backend().channel_axis if agg_fn is None: agg_fn = InternalChannelQoI._batch_sum self._channel_ax = channel_axis self._agg_fn = agg_fn self._channels = channel if isinstance(channel, list) else [channel] 

## LambdaQoI¶

Bases: QoI

Generic quantity of interest allowing the user to specify a function of the model's output as the QoI.

Source code in trulens_explain/trulens/nn/quantities.py
 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 class LambdaQoI(QoI): """ Generic quantity of interest allowing the user to specify a function of the model's output as the QoI. """ def __init__(self, function: Callable): """ Parameters: function: A callable that takes a single argument representing the model's tensor output and returns a differentiable batched scalar tensor representing the QoI. """ if len(signature(function).parameters) != 1: raise ValueError( 'QoI function must take exactly 1 argument, but provided ' 'function takes {} arguments'.format( len(signature(function).parameters) ) ) self.function = function def __call__(self, y: TensorLike) -> TensorLike: return self.function(y) 

### __init__(function)¶

Parameters:

Name Type Description Default
function Callable

A callable that takes a single argument representing the model's tensor output and returns a differentiable batched scalar tensor representing the QoI.

required
Source code in trulens_explain/trulens/nn/quantities.py
 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 def __init__(self, function: Callable): """ Parameters: function: A callable that takes a single argument representing the model's tensor output and returns a differentiable batched scalar tensor representing the QoI. """ if len(signature(function).parameters) != 1: raise ValueError( 'QoI function must take exactly 1 argument, but provided ' 'function takes {} arguments'.format( len(signature(function).parameters) ) ) self.function = function 

## MaxClassQoI¶

Bases: QoI

Quantity of interest for attributing output towards the maximum-predicted class.

Source code in trulens_explain/trulens/nn/quantities.py
 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 class MaxClassQoI(QoI): """ Quantity of interest for attributing output towards the maximum-predicted class. """ def __init__( self, axis: int = 1, activation: Union[Callable, str, None] = None ): """ Parameters: axis: Output dimension over which max operation is taken. activation: Activation function to be applied to the output before taking the max. If activation is a string, use the corresponding named activation function implemented by the backend. The following strings are currently supported as shorthands for the respective standard activation functions: - 'sigmoid' - 'softmax' If activation is None, no activation function is applied to the input. """ self._axis = axis self.activation = activation def __str__(self): return render_object(self, ["_axis", "activation"]) def __call__(self, y: TensorLike) -> TensorLike: self._assert_cut_contains_only_one_tensor(y) if self.activation is not None: if isinstance(self.activation, str): self.activation = self.activation.lower() if self.activation in ['sigmoid', 'softmax']: y = getattr(get_backend(), self.activation)(y) else: raise NotImplementedError( 'This activation function is not currently supported ' 'by the backend' ) else: y = self.activation(y) return get_backend().max(y, axis=self._axis) 

### __init__(axis=1, activation=None)¶

Parameters:

Name Type Description Default
axis int

Output dimension over which max operation is taken.

1
activation Union[Callable, str, None]

Activation function to be applied to the output before taking the max. If activation is a string, use the corresponding named activation function implemented by the backend. The following strings are currently supported as shorthands for the respective standard activation functions:

• 'sigmoid'
• 'softmax'

If activation is None, no activation function is applied to the input.

None
Source code in trulens_explain/trulens/nn/quantities.py
 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 def __init__( self, axis: int = 1, activation: Union[Callable, str, None] = None ): """ Parameters: axis: Output dimension over which max operation is taken. activation: Activation function to be applied to the output before taking the max. If activation is a string, use the corresponding named activation function implemented by the backend. The following strings are currently supported as shorthands for the respective standard activation functions: - 'sigmoid' - 'softmax' If activation is None, no activation function is applied to the input. """ self._axis = axis self.activation = activation 

## QoI¶

Bases: AbstractBaseClass

Interface for quantities of interest. The Quantity of Interest (QoI) is a function of the output specified by the slice that determines the network output behavior that the attributions describe.

Source code in trulens_explain/trulens/nn/quantities.py
 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 class QoI(AbstractBaseClass): """ Interface for quantities of interest. The *Quantity of Interest* (QoI) is a function of the output specified by the slice that determines the network output behavior that the attributions describe. """ def __str__(self): return render_object(self, []) # TODO: Need to give a seperate value of y at target instance here since # these are values are interventions. Cannot presently define a QoI that says: # logits of the predicted class for each instance. # Issue GH-72 . Task MLNN-415 . def _wrap_public_call(self, y: Outputs[Tensor]) -> Outputs[Tensor]: """ Wrap a public call that may result in one or more tensors. Signature of this class is not specific while public calls are flexible. """ return many_of_om(self.__call__(om_of_many(y))) @abstractmethod def __call__(self, y: OM[Outputs, Tensor]) -> OM[Outputs, Tensor]: """ Computes the distribution of interest from an initial point. Parameters: y: Output point from which the quantity is derived. Must be a differentiable tensor. Returns: A differentiable batched scalar tensor representing the QoI. """ raise NotImplementedError def _assert_cut_contains_only_one_tensor(self, x): if isinstance(x, DATA_CONTAINER_TYPE): raise QoiCutSupportError( 'Cut provided to quantity of interest was comprised of ' 'multiple tensors, but {} is only defined for cuts comprised ' 'of a single tensor (received a list of {} tensors).\n' '\n' 'Either (1) select a slice where the to_cut corresponds to a ' 'single tensor, or (2) implement/use a QoI object that ' 'supports lists of tensors, i.e., where the parameter, x, to ' '__call__ is expected/allowed to be a list of {} tensors.'. format(self.__class__.__name__, len(x), len(x)) ) elif not get_backend().is_tensor(x): raise ValueError( '{} expected to receive an instance of Tensor, but ' 'received an instance of {}'.format( self.__class__.__name__, type(x) ) ) 

### __call__(y) abstractmethod ¶

Computes the distribution of interest from an initial point.

Parameters:

Name Type Description Default
y OM[Outputs, Tensor]

Output point from which the quantity is derived. Must be a differentiable tensor.

required

Returns:

Type Description
OM[Outputs, Tensor]

A differentiable batched scalar tensor representing the QoI.

Source code in trulens_explain/trulens/nn/quantities.py
 63 64 65 66 67 68 69 70 71 72 73 74 75 76 @abstractmethod def __call__(self, y: OM[Outputs, Tensor]) -> OM[Outputs, Tensor]: """ Computes the distribution of interest from an initial point. Parameters: y: Output point from which the quantity is derived. Must be a differentiable tensor. Returns: A differentiable batched scalar tensor representing the QoI. """ raise NotImplementedError 

## QoiCutSupportError¶

Bases: ValueError

Exception raised if the quantity of interest is called on a cut whose output is not supported by the quantity of interest.

Source code in trulens_explain/trulens/nn/quantities.py
 32 33 34 35 36 37 class QoiCutSupportError(ValueError): """ Exception raised if the quantity of interest is called on a cut whose output is not supported by the quantity of interest. """ pass 

## ThresholdQoI¶

Bases: QoI

Quantity of interest for attributing network output toward the difference between two regions seperated by a given threshold. I.e., the quantity of interest is the "high" elements minus the "low" elements, where the high elements have activations above the threshold and the low elements have activations below the threshold.

Use case: bianry segmentation.

Source code in trulens_explain/trulens/nn/quantities.py
 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 class ThresholdQoI(QoI): """ Quantity of interest for attributing network output toward the difference between two regions seperated by a given threshold. I.e., the quantity of interest is the "high" elements minus the "low" elements, where the high elements have activations above the threshold and the low elements have activations below the threshold. Use case: bianry segmentation. """ def __init__( self, threshold: float, low_minus_high: bool = False, activation: Union[Callable, str, None] = None ): """ Parameters: threshold: A threshold to determine the element-wise sign of the input tensor. The elements with activations higher than the threshold will retain their sign, while the elements with activations lower than the threshold will have their sign flipped (or vice versa if low_minus_high is set to True). low_minus_high: If True, substract the output with activations above the threshold from the output with activations below the threshold. If False, substract the output with activations below the threshold from the output with activations above the threshold. activation: str or function, optional Activation function to be applied to the quantity before taking the threshold. If activation is a string, use the corresponding activation function implemented by the backend (currently supported: 'sigmoid' and 'softmax'). Otherwise, if activation is not None, it will be treated as a callable. If activation is None, do not apply an activation function to the quantity. """ # TODO(klas):should this support an aggregation function? By default # this is a sum, but it could, for example, subtract the greatest # positive element from the least negative element. self.threshold = threshold self.low_minus_high = low_minus_high self.activation = activation def __call__(self, x: TensorLike) -> TensorLike: B = get_backend() self._assert_cut_contains_only_one_tensor(x) if self.activation is not None: if isinstance(self.activation, str): self.activation = self.activation.lower() if self.activation in ['sigmoid', 'softmax']: x = getattr(B, self.activation)(x) else: raise NotImplementedError( 'This activation function is not currently supported ' 'by the backend' ) else: x = self.activation(x) # TODO(klas): is the clone necessary here? Not sure why it was # included. mask = B.sign(B.clone(x) - self.threshold) if self.low_minus_high: mask = -mask non_batch_dimensions = tuple(range(len(B.int_shape(x)))[1:]) return B.sum(mask * x, axis=non_batch_dimensions) 

### __init__(threshold, low_minus_high=False, activation=None)¶

Parameters:

Name Type Description Default
threshold float

A threshold to determine the element-wise sign of the input tensor. The elements with activations higher than the threshold will retain their sign, while the elements with activations lower than the threshold will have their sign flipped (or vice versa if low_minus_high is set to True).

required
low_minus_high bool

If True, substract the output with activations above the threshold from the output with activations below the threshold. If False, substract the output with activations below the threshold from the output with activations above the threshold.

False
activation Union[Callable, str, None]

str or function, optional Activation function to be applied to the quantity before taking the threshold. If activation is a string, use the corresponding activation function implemented by the backend (currently supported: 'sigmoid' and 'softmax'). Otherwise, if activation is not None, it will be treated as a callable. If activation is None, do not apply an activation function to the quantity.

None
Source code in trulens_explain/trulens/nn/quantities.py
 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 def __init__( self, threshold: float, low_minus_high: bool = False, activation: Union[Callable, str, None] = None ): """ Parameters: threshold: A threshold to determine the element-wise sign of the input tensor. The elements with activations higher than the threshold will retain their sign, while the elements with activations lower than the threshold will have their sign flipped (or vice versa if low_minus_high is set to True). low_minus_high: If True, substract the output with activations above the threshold from the output with activations below the threshold. If False, substract the output with activations below the threshold from the output with activations above the threshold. activation: str or function, optional Activation function to be applied to the quantity before taking the threshold. If activation is a string, use the corresponding activation function implemented by the backend (currently supported: 'sigmoid' and 'softmax'). Otherwise, if activation is not None, it will be treated as a callable. If activation is None, do not apply an activation function to the quantity. """ # TODO(klas):should this support an aggregation function? By default # this is a sum, but it could, for example, subtract the greatest # positive element from the least negative element. self.threshold = threshold self.low_minus_high = low_minus_high self.activation = activation