---
title: Explained Variance
description: Compute explained variance.
canonical: 'https://lightning.ai/docs/torchmetrics/stable/regression/explained_variance'
---
# Explained Variance {#explained-variance}

## Module Interface {#module-interface}

### `class torchmetrics.ExplainedVariance(multioutput='uniform_average', **kwargs)` {#torchmetrics.ExplainedVariance}

[\[source\]](https://github.com/Lightning-AI/torchmetrics/blob/v1.9.0/src/torchmetrics/regression/explained_variance.py#L33-L178)

Compute [explained variance](https://en.wikipedia.org/wiki/Explained_variation).

```math
\text{ExplainedVariance} = 1 - \frac{\text{Var}(y - \hat{y})}{\text{Var}(y)}
```

Where $`y`$ is a tensor of target values, and $`\hat{y}`$ is a tensor of predictions.

As input to `forward` and `update` the metric accepts the following input:

*   `preds` ([`Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor "(in PyTorch v2.15)")): Predictions from model in float tensor with shape `(N,)` or `(N, ...)` (multioutput)
    
*   `target` ([`Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor "(in PyTorch v2.15)")): Ground truth values in long tensor with shape `(N,)` or `(N, ...)` (multioutput)
    

As output of `forward` and `compute` the metric returns the following output:

*   `explained_variance` ([`Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor "(in PyTorch v2.15)")): A tensor with the explained variance(s)
    

In the case of multioutput, as default the variances will be uniformly averaged over the additional dimensions. Please see argument `multioutput` for changing this behavior.

Parameters:

*   **multioutput** ([`Literal`](https://docs.python.org/3/library/typing.html#typing.Literal "(in Python v3.15)")\[`'raw_values'`, `'uniform_average'`, `'variance_weighted'`\]) –
    
    Defines aggregation in the case of multiple output scores. Can be one of the following strings (default is `'uniform_average'`.):
    
    *   `'raw_values'` returns full set of scores
        
    *   `'uniform_average'` scores are uniformly averaged
        
    *   `'variance_weighted'` scores are weighted by their individual variances
        
    
*   **kwargs** ([`Any`](https://docs.python.org/3/library/typing.html#typing.Any "(in Python v3.15)")) – Additional keyword arguments, see [Advanced metric settings](/docs/torchmetrics/1.9.0/user-guide/overview.md#metric-kwargs) for more info.
    

Raises:

[**ValueError**](https://docs.python.org/3/builtins/exceptions.html#ValueError "(in Python v3.15)") – If `multioutput` is not one of `"raw_values"`, `"uniform_average"` or `"variance_weighted"`.

Example

```python
>>> from torch import tensor
>>> from torchmetrics.regression import ExplainedVariance
>>> target = tensor([3, -0.5, 2, 7])
>>> preds = tensor([2.5, 0.0, 2, 8])
>>> explained_variance = ExplainedVariance()
>>> explained_variance(preds, target)
tensor(0.9572)
```

```python
>>> target = tensor([[0.5, 1], [-1, 1], [7, -6]])
>>> preds = tensor([[0, 2], [-1, 2], [8, -5]])
>>> explained_variance = ExplainedVariance(multioutput='raw_values')
>>> explained_variance(preds, target)
tensor([0.9677, 1.0000])
```

#### `plot(val=None, ax=None)` {#torchmetrics.ExplainedVariance.plot}

[\[source\]](https://github.com/Lightning-AI/torchmetrics/blob/v1.9.0/src/torchmetrics/regression/explained_variance.py#L138-L178)

Plot a single or multiple values from the metric.

Parameters:

*   **val** (`Union`\[[`Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor "(in PyTorch v2.15)"), [`Sequence`](https://docs.python.org/3/library/collections.abc.html#collections.abc.Sequence "(in Python v3.15)")\[[`Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor "(in PyTorch v2.15)")\], [`None`](https://docs.python.org/3/builtins/constants.html#None "(in Python v3.15)")\]) – Either a single result from calling metric.forward or metric.compute or a list of these results. If no value is provided, will automatically call metric.compute and plot that result.
    
*   **ax** ([`Optional`](https://docs.python.org/3/library/typing.html#typing.Optional "(in Python v3.15)")\[[`Axes`](https://matplotlib.org/stable/api/_as_gen/matplotlib.axes.Axes.html#matplotlib.axes.Axes "(in Matplotlib v3.11.2)")\]) – An matplotlib axis object. If provided will add plot to that axis
    

Return type:

[`tuple`](https://docs.python.org/3/builtins/stdtypes.html#tuple "(in Python v3.15)")\[[`Figure`](https://matplotlib.org/stable/api/_as_gen/matplotlib.figure.Figure.html#matplotlib.figure.Figure "(in Matplotlib v3.11.2)"), `Union`\[[`Axes`](https://matplotlib.org/stable/api/_as_gen/matplotlib.axes.Axes.html#matplotlib.axes.Axes "(in Matplotlib v3.11.2)"), [`ndarray`](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray "(in NumPy v2.5)")\]\]

Returns:

Figure and Axes object

Raises:

[**ModuleNotFoundError**](https://docs.python.org/3/builtins/exceptions.html#ModuleNotFoundError "(in Python v3.15)") – If matplotlib is not installed

```python
>>> from torch import randn
>>> # Example plotting a single value
>>> from torchmetrics.regression import ExplainedVariance
>>> metric = ExplainedVariance()
>>> metric.update(randn(10,), randn(10,))
>>> fig_, ax_ = metric.plot()
```

[![../../\_images/explained\_variance-1.png](/docs/torchmetrics/1.9.0/static/_images/explained_variance-1.png)](/docs/torchmetrics/1.9.0/static/_images/explained_variance-1.png)

```python
>>> from torch import randn
>>> # Example plotting multiple values
>>> from torchmetrics.regression import ExplainedVariance
>>> metric = ExplainedVariance()
>>> values = []
>>> for _ in range(10):
...     values.append(metric(randn(10,), randn(10,)))
>>> fig, ax = metric.plot(values)
```

[![../../\_images/explained\_variance-2.png](/docs/torchmetrics/1.9.0/static/_images/explained_variance-2.png)](/docs/torchmetrics/1.9.0/static/_images/explained_variance-2.png)

## Functional Interface {#functional-interface}

### `torchmetrics.functional.explained_variance(preds, target, multioutput='uniform_average')` {#torchmetrics.functional.explained_variance}

[\[source\]](https://github.com/Lightning-AI/torchmetrics/blob/v1.9.0/src/torchmetrics/functional/regression/explained_variance.py#L102-L142)

Compute explained variance.

Parameters:

*   **preds** ([`Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor "(in PyTorch v2.15)")) – estimated labels
    
*   **target** ([`Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor "(in PyTorch v2.15)")) – ground truth labels
    
*   **multioutput** ([`Literal`](https://docs.python.org/3/library/typing.html#typing.Literal "(in Python v3.15)")\[`'raw_values'`, `'uniform_average'`, `'variance_weighted'`\]) –
    
    Defines aggregation in the case of multiple output scores. Can be one of the following strings):
    
    *   `'raw_values'` returns full set of scores
        
    *   `'uniform_average'` scores are uniformly averaged
        
    *   `'variance_weighted'` scores are weighted by their individual variances
        
    

Return type:

`Union`\[[`Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor "(in PyTorch v2.15)"), [`Sequence`](https://docs.python.org/3/library/collections.abc.html#collections.abc.Sequence "(in Python v3.15)")\[[`Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor "(in PyTorch v2.15)")\]\]

Example

```python
>>> from torchmetrics.functional.regression import explained_variance
>>> target = torch.tensor([3, -0.5, 2, 7])
>>> preds = torch.tensor([2.5, 0.0, 2, 8])
>>> explained_variance(preds, target)
tensor(0.9572)
```

```python
>>> target = torch.tensor([[0.5, 1], [-1, 1], [7, -6]])
>>> preds = torch.tensor([[0, 2], [-1, 2], [8, -5]])
>>> explained_variance(preds, target, multioutput='raw_values')
tensor([0.9677, 1.0000])
```

---

**Navigation**

- Parent: [Regression](/docs/torchmetrics/1.9.0/regression.md)