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 $h : \Rset^2 \rightarrow \Rset^2$ is defined by:

$\vect{h}(x_1, x_2) = \Tr{\left(\cos(x_1 + x_2), (x_2 + 1) e^{x_1 - 2 x_2} \right)}$

for any $\vect{x} \in \Rset^2$ . Since the output is a dimension 2 vector, the model has vector coefficients. We use the linear model:

$\vect{y} \, \approx \, \widehat{\vect{h}}(\vect{x}) \, = \, \sum_{k=0}^2 \; \vect{a}_k \; \psi_k(\vect{x})$

for any $\vect{x} \in \Rset^2$ where $\left\{\vect{a}_k \in \Rset^2\right\}_{k = 0,..., 2}$ are the vector coefficients and $\left\{\psi_k : \Rset^2 \rightarrow \Rset\right\}_{k = 0, ..., 2}$ are the basis functions. This implies that each marginal output $\widehat{h}_i(\vect{x})$ is approximated by the linear model:

$\widehat{h}_i(\vect{x}) \, = \, \sum_{k=0}^2 \; a_{ki} \; \psi_k(\vect{x})$

for any $\vect{x} \in \Rset^2$ and any $i = 1, 2$ where $a_{ki}$ is the $k$ -th coefficient of the $i$ -th output marginal:

$\vect{a}_k = \begin{pmatrix}a_{k1} \\ a_{k2} \end{pmatrix}$

for $k = 0, 1, 2$ . We consider the basis functions:

$\psi_0(\vect{x}) & = 1 \\ \psi_1(\vect{x}) & = x_1 \\ \psi_2(\vect{x}) & = x_2 \\$

for any $\vect{x} \in \Rset^2$ .

import openturns as ot
import openturns.viewer as viewer

ot.Log.Show(ot.Log.NONE)

Define the model¶

Prepare an input sample. Each point is a pair of coordinates $(x_1, x_2)$ .

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

	x1	x2
0	0.5	0.5
1	-0.5	-0.5
2	-0.5	0.5
3	0.5	-0.5
4	0.25	0.25
5	-0.25	-0.25
6	-0.25	0.25
7	0.25	-0.25

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

	y1	y2
0	0.5403023	0.909796
1	0.5403023	0.8243606
2	1	0.3346952
3	1	2.240845
4	0.8775826	0.973501
5	0.8775826	0.9630191
6	1	0.5904582
7	1	1.58775

Linear least squares¶

Create a linear least squares model.

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

Get the linear term.

algo.getLinear()

[[ 9.93014e-17 0.998189 ]
[ 4.96507e-17 -0.925648 ]]

Get the constant term.

algo.getConstant()

class=Point name=Unnamed dimension=2 values=[0.854471,1.05305]

Get the metamodel.

responseSurface = algo.getMetaModel()

Plot the second output of our model with $x_1=0.5$ .

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")
view = viewer.View(graph)

view.ShowAll()

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Create a linear least squares model¶

Define the model¶

Linear least squares¶