vastorbit.machine_learning.model_selection.statistical_tests.tsa.het_arch

vastorbit.machine_learning.model_selection.statistical_tests.tsa.het_arch(input_relation: Annotated[str | VastFrame, ''], eps: str, ts: str, by: Annotated[str | list[str], 'STRING representing one column or a list of columns'] | None = None, p: int = 1) → tuple[float, float, float, float]

Engle’s Test for Autoregressive Conditional Heteroscedasticity (ARCH).

Parameters:

• input_relation (SQLRelation) – Input relation.

• eps (str) – Input residual VastColumn.

• ts (str) – VastColumn used as timeline to to order the data. It can be a numerical or date-like type (date, datetime, timestamp…) VastColumn.

• by (SQLColumns, optional) – VastColumns used in the partition.

• p (int, optional) – Number of lags to consider in the test.

Returns:

Lagrange Multiplier statistic, LM pvalue, F statistic, F pvalue

Return type:

tuple

Examples

Initialization

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

• A value of interest that has noise

• Time-stamp data

Before we begin we can import the necessary libraries:

import vastorbit as vo
import numpy as np

Example 1: Random

Now we can create the dummy dataset:

# Initialization
N = 50 # Number of Rows.
days = list(range(N))
vals = [np.random.normal(5) for x in days]

# VastFrame
vdf = vo.VastFrame(
    {
        "day": days,
        "eps": vals,
    }
)

Let us plot the distribution of noise with respect to time:

vdf.scatter(["day", "eps"])

Test

Now we can apply the Durbin Watson Test:

from vastorbit.machine_learning.model_selection.statistical_tests import het_arch

het_arch(input_relation = vdf, ts = "day", eps = "eps", p = 5)

We can see that there is no relationship with any lag except that which is by chance.

Now let us contrast it with another example where the lags are related:

Example 1: Correlated

We can create an alternate dataset that exhibits some correlation with a specific lag. Below, we intertwine two separate values, one after the other, thereby creating a new value. This new value has the characteristic that every other value is related to the one that is two steps before it, but not to the one immediately before it

# Initialization
N = 50 # Number of Rows
days = list(range(N))
x1 = [2 * -x for x in list(range(40, 40 + 5 * N, 5))]
x2 = [-2 * -x * x * x / 2 for x in list(range(4, 4 + 2 * N, 2))]
vals = []
for elem_1, elem_2 in zip(x1, x2):
    vals.extend([elem_1, elem_2])

# VastFrame
vdf = vo.VastFrame(
    {
        "day": days,
        "eps": vals,
    }
)

Let us plot the distribution of noise with respect to time to observe the trend:

vdf.scatter(["day", "eps"])

Notice that it is a bit hard to see the relationship of certain lags. That is why we need the Engle’s Test for Autoregressive Conditional Heteroscedasticity.

Test

Now we can apply the Durbin Watson Test:

from vastorbit.machine_learning.model_selection.statistical_tests import het_arch

het_arch(input_relation = vdf, ts = "day", eps = "eps", p = 5)

We can see that the lags of multiple of 2 have a very low value of p. This confirms the presence of correaltion with certain lags.