Note

Go to the end to download the full example code.

# Create a linear least squares model¶

In this example we are going to create a global approximation of a model
response based on a linear function using the `LinearLeastSquares`

class.

We consider the function is defined by:

for any . Since the output is a dimension 2 vector, the model has vector coefficients. We use the linear model:

for any where are the vector coefficients and are the basis functions. This implies that each marginal output is approximated by the linear model:

for any and any where is the -th coefficient of the -th output marginal:

for . We consider the basis functions:

for any .

```
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)
```

## Define the model¶

Prepare an input sample. Each point is a pair of coordinates .

```
inputTrain = [[0.5, 0.5], [-0.5, -0.5], [-0.5, 0.5], [0.5, -0.5]]
inputTrain += [[0.25, 0.25], [-0.25, -0.25], [-0.25, 0.25], [0.25, -0.25]]
inputTrain = ot.Sample(inputTrain)
inputTrain.setDescription(["x1", "x2"])
inputTrain
```

Compute the output sample from the input sample and a function.

```
formulas = ["cos(x1 + x2)", "(x2 + 1) * exp(x1 - 2 * x2)"]
model = ot.SymbolicFunction(["x1", "x2"], formulas)
model.setOutputDescription(["y1", "y2"])
outputTrain = model(inputTrain)
outputTrain
```

## Linear least squares¶

Create a linear least squares model.

```
algo = ot.LinearLeastSquares(inputTrain, outputTrain)
algo.run()
```

Get the linear term.

```
algo.getLinear()
```

Get the constant term.

```
algo.getConstant()
```

Get the metamodel.

```
responseSurface = algo.getMetaModel()
```

Plot the second output of our model with .

```
graph = ot.ParametricFunction(model, [0], [0.5]).getMarginal(1).draw(-0.5, 0.5)
graph.setLegends(["Model"])
curve = (
ot.ParametricFunction(responseSurface, [0], [0.5])
.getMarginal(1)
.draw(-0.5, 0.5)
.getDrawable(0)
)
curve.setLineStyle("dashed")
curve.setLegend("Linear L.S.")
graph.add(curve)
graph.setLegendPosition("upper right")
graph.setColors(ot.Drawable.BuildDefaultPalette(2))
view = viewer.View(graph)
plt.show()
```