Inverse Distance Weighting (IDW)

The Inverse Distance Weighting (IDW) is a deterministic interpolation method.

The assigned values to unknown points are calculated with a weighted average of the values available at the known points. The weights are calculated as a function of the distance between the known and unknown points. The function used is the inverse of the distance, elevated to a given power, to each known point.

The model can be tuned by the following parameters: - power: the power of the inverse distance function

Mathematical formulation

Given a set of sample points:

\(\{ \mathbf{x}_i, u_i | \text{for } \mathbf{x}_i \in \mathbb{R}^n, u_i \in \mathbb{R}\}_{i=1}^N\)

the IDW interpolation function \(u(\mathbf{x}): \mathbb{R}^n \to \mathbb{R}\) is defined as:

\(u(\mathbf{x}) = \begin{cases} \dfrac{\sum_{i = 1}^{N}{ w_i(\mathbf{x}) u_i } }{ \sum_{i = 1}^{N}{ w_i(\mathbf{x}) } }, & \text{if } d(\mathbf{x},\mathbf{x}_i) \neq 0 \text{ for all } i, \\ u_i, & \text{if } d(\mathbf{x},\mathbf{x}_i) = 0 \text{ for some } i, \end{cases}\)

where

\(w_i(\mathbf{x}) = \frac{1}{d(\mathbf{x},\mathbf{x}_i)^p}\)

Examples

[21]:

import py3dinterpolations as p3i
import pandas as pd

The autoreload extension is already loaded. To reload it, use:
  %reload_ext autoreload

[22]:

df = pd.read_csv(
    "../../../tests/fixtures/griddata_default_colnames.csv"
)
df.tail()

[22]:

	ID	X	Z	V
278	ID00	15.194	12.0	9.047969
279	ID00	15.194	10.0	10.077271
280	ID00	15.194	8.0	20.082454
281	ID00	15.194	6.0	19.042223
282	ID00	15.194	4.0	12.889411

[23]:

gd = GridData(df)
gd.data

[23]:

				V
ID	X	Y	Z
ID30	62.163	14.336	20.0	7.523950
			18.0	7.504403
			16.0	12.431670
			14.0	12.653931
			12.0	17.956143
...	...	...	...	...
ID00	15.194	0.000	12.0	9.047969
			10.0	10.077271
			8.0	20.082454
			6.0	19.042223
			4.0	12.889411

283 rows × 1 columns

[24]:

interpolated, model = p3i.interpolate(
    gd,
    model_name = "idw",
    model_params = {
        "power": 2,
    },
    grid_resolution=5,
    preprocess_kwags={
        "normalize_xyz": True,
        "standardize_v": True,
    },
    return_model=True,
)

[25]:

fig = p3i.plot_3d_model(
    model,
    plot_points=True,
    scale_points=10,
    volume_kwargs={
        "surface_count": 10,
    }
)
fig.show()

Data type cannot be displayed: application/vnd.plotly.v1+json