vastorbit.machine_learning.model_selection.statistical_tests.ols.het_goldfeldquandt

vastorbit.machine_learning.model_selection.statistical_tests.ols.het_goldfeldquandt(input_relation: Annotated[str | VastFrame, ''], y: str, X: Annotated[str | list[str], 'STRING representing one column or a list of columns'], idx: int = 0, split: float = 0.5, alternative: Literal['increasing', 'decreasing', 'two-sided'] = 'increasing') → tuple[float, float]

Goldfeld-Quandt Homoscedasticity test.

Parameters:

• input_relation (SQLRelation) – Input relation.

• y (str) – Response Column.

• X (SQLColumns) – Exogenous Variables.

• idx (int, optional) – Column index of variable according to which observations are sorted for the split.

• split (float, optional) – Float to indicate where to split (Example: 0.5 to split on the median).

• alternative (str, optional) – Specifies the alternative hypothesis for the p-value calculation, one of the following variances: “increasing”, “decreasing”, “two-sided”.

Returns:

statistic, p_value

Return type:

tuple

Examples

Initialization

Let’s try this test on a dummy dataset that has the following elements:

• x (a predictor)

• y (the response)

• Random noise

Before we begin we can import the necessary libraries:

import vastorbit as vo
import numpy as np

Example 1: Homoscedasticity

Next, we can create some values with random noise:

N = 50 # Number of rows
x_val = list(range(N))
y_val = [x * 2 for x in x_val] + np.random.normal(0, 0.4, N)

We can use those values to create the VastFrame:

vdf = vo.VastFrame(
    {
        "x": x_val,
        "y": y_val,
    }
)

We can plot the values to see the trend:

vdf.scatter(["x", "y"])

Notice the randomness of the data with respect to x. This shows that the noise is homoscedestic.

To test its score, we can import the test function:

from vastorbit.machine_learning.model_selection.statistical_tests import het_goldfeldquandt

And simply apply it on the VastFrame:

statistic, pvalue = het_goldfeldquandt(vdf, y = "y", X = "x")

print(statistic, pvalue)

As the noise was not heteroscedestic, we got higher p_value scores and lower statistics score.

Note

A p_value in statistics represents the probability of obtaining results as extreme as, or more extreme than, the observed data, assuming the null hypothesis is true. A smaller p-value typically suggests stronger evidence against the null hypothesis i.e. the test data does not have a heteroscedestic noise in the current case.

However, small is a relative term. And the choice for the threshold value which determines a “small” should be made before analyzing the data.

Generally a p-value less than 0.05 is considered the threshold to reject the null hypothesis. But it is not always the case - read more

Note

F-statistics tests the overall significance of a model, while LM statistics tests the validity of linear restrictions on model parameters. High values indicate heterescedestic noise in this case.

Example 2: Heteroscedasticity

We can contrast the above result with a dataset that has heteroscedestic noise below:

# y values
x_val = list(range(N))
y_val = [x * 2 for x in x_val]

# Adding some heteroscedestic noise
y_val = [x + np.random.normal() for x in y_val]

vdf = vo.VastFrame(
    {
        "x": x_val,
        "y": y_val,
    }
)

We can plot the data to see the trend:

vdf.scatter(["x", "y"])

Notice the relationship of the residuals with respect to x. This shows that the noise is heteroscedestic.

Now we can perform the test on this dataset:

statistic, pvalue = het_goldfeldquandt(vdf, y = "y", X = "x")

print(statistic, pvalue)

Note

Notice the contrast of the two test results. In this dataset, the noise was heteroscedestic so we got very low p_value scores and higher statistics score. Thus confirming that the noise was in fact heteroscedestic.