FullCausalContextExplicitEquationBivarLinSVAR

Seed 1

Given are variables X_0 and X_1, where X_0 is a covariate and X_1 is the variable to forecast. Variables are generated from a linear Structural Vector Autoregressive (SVAR) model with additive gauss noise and a noise scale of 5.533e-04, with lag = 3.

The task is to forecast the value of the variable X_1 at time t, given the values of the covariate X_0 and the variable X_1 itself at times t-1, ... t-3.
For the first 128 days, the covariate X_0 takes a value of 8 from 2023-11-28 to 2024-01-13, 20 from 2024-01-14 to 2024-02-29, 8 from 2024-03-01 to 2024-04-03.
For the next 32 days, the covariate X_0 takes a value of 10 from 2024-04-04 to 2024-04-19, 20 from 2024-04-20 to 2024-04-21, 24 from 2024-04-22 to 2024-05-05. Each day can be treated as a timestep for the forecasting task. The causal parents affect the child variables at different lags.
The mathematical equations describing the cause-effect relationship for each variable is given below:
X_0^{t} = \epsilon_0^{t}
X_1^{t} = -0.925 * X_0^{t-1} + 1.223 * X_1^{t-1} + -0.990 * X_0^{t-2} + -1.181 * X_1^{t-2} + -0.720 * X_0^{t-3} + 0.751 * X_1^{t-3} + \epsilon_1^{t}
\epsilon_{i}^{t} given in the equations corresponds to the noise variable with the given noise scale, for X_i^{t}.

Types of context: ['Covariate information', 'Causal information']

Capabilities: ['Reasoning: Math', 'Reasoning: Causal']


Figure

Seed 2

Given are variables X_0 and X_1, where X_0 is a covariate and X_1 is the variable to forecast. Variables are generated from a linear Structural Vector Autoregressive (SVAR) model with additive gauss noise and a noise scale of 5.127e-04, with lag = 3.

The task is to forecast the value of the variable X_1 at time t, given the values of the covariate X_0 and the variable X_1 itself at times t-1, ... t-3.
For the first 128 days, the covariate X_0 takes a value of 2 from 2027-12-17 to 2027-12-18, 12 from 2027-12-19 to 2028-02-07, 2 from 2028-02-08 to 2028-03-26, 12 from 2028-03-27 to 2028-04-22.
For the next 32 days, the covariate X_0 takes a value of 40 from 2028-04-23 to 2028-05-05, 30 from 2028-05-06 to 2028-05-08, 60 from 2028-05-09 to 2028-05-17, 20 from 2028-05-18 to 2028-05-24. Each day can be treated as a timestep for the forecasting task. The causal parents affect the child variables at different lags.
The mathematical equations describing the cause-effect relationship for each variable is given below:
X_0^{t} = \epsilon_0^{t}
X_1^{t} = 0.834 * X_0^{t-1} + 0.656 * X_1^{t-1} + -0.580 * X_0^{t-2} + 0.572 * X_1^{t-2} + 0.952 * X_0^{t-3} + -0.701 * X_1^{t-3} + \epsilon_1^{t}
\epsilon_{i}^{t} given in the equations corresponds to the noise variable with the given noise scale, for X_i^{t}.

Types of context: ['Covariate information', 'Causal information']

Capabilities: ['Reasoning: Math', 'Reasoning: Causal']


Figure

Seed 3

Given are variables X_0 and X_1, where X_0 is a covariate and X_1 is the variable to forecast. Variables are generated from a linear Structural Vector Autoregressive (SVAR) model with additive gauss noise and a noise scale of 4.700e-04, with lag = 3.

The task is to forecast the value of the variable X_1 at time t, given the values of the covariate X_0 and the variable X_1 itself at times t-1, ... t-3.
For the first 128 days, the covariate X_0 takes a value of 12 from 2026-03-14 to 2026-04-12, 20 from 2026-04-13 to 2026-06-10, 12 from 2026-06-11 to 2026-07-19.
For the next 32 days, the covariate X_0 takes a value of 40 from 2026-07-20 to 2026-08-04, 80 from 2026-08-05 to 2026-08-04, 60 from 2026-08-05 to 2026-08-17, 20 from 2026-08-18 to 2026-08-20. Each day can be treated as a timestep for the forecasting task. The causal parents affect the child variables at different lags.
The mathematical equations describing the cause-effect relationship for each variable is given below:
X_0^{t} = \epsilon_0^{t}
X_1^{t} = 0.862 * X_0^{t-1} + 0.674 * X_1^{t-1} + -0.881 * X_0^{t-2} + 0.837 * X_1^{t-2} + -0.624 * X_0^{t-3} + -0.735 * X_1^{t-3} + \epsilon_1^{t}
\epsilon_{i}^{t} given in the equations corresponds to the noise variable with the given noise scale, for X_i^{t}.

Types of context: ['Covariate information', 'Causal information']

Capabilities: ['Reasoning: Math', 'Reasoning: Causal']


Figure

Seed 4

Given are variables X_0 and X_1, where X_0 is a covariate and X_1 is the variable to forecast. Variables are generated from a linear Structural Vector Autoregressive (SVAR) model with additive gauss noise and a noise scale of 1.487e-03, with lag = 3.

The task is to forecast the value of the variable X_1 at time t, given the values of the covariate X_0 and the variable X_1 itself at times t-1, ... t-3.
For the first 128 days, the covariate X_0 takes a value of 8 from 2024-02-21 to 2024-03-11, 12 from 2024-03-12 to 2024-05-06, 12 from 2024-05-07 to 2024-06-27.
For the next 32 days, the covariate X_0 takes a value of 30 from 2024-06-28 to 2024-07-13, 60 from 2024-07-14 to 2024-07-14, 60 from 2024-07-15 to 2024-07-29. Each day can be treated as a timestep for the forecasting task. The causal parents affect the child variables at different lags.
The mathematical equations describing the cause-effect relationship for each variable is given below:
X_0^{t} = \epsilon_0^{t}
X_1^{t} = 0.527 * X_0^{t-1} + -0.895 * X_1^{t-1} + 1.380 * X_0^{t-2} + -0.758 * X_1^{t-2} + -0.661 * X_0^{t-3} + -0.793 * X_1^{t-3} + \epsilon_1^{t}
\epsilon_{i}^{t} given in the equations corresponds to the noise variable with the given noise scale, for X_i^{t}.

Types of context: ['Covariate information', 'Causal information']

Capabilities: ['Reasoning: Math', 'Reasoning: Causal']


Figure

Seed 5

Given are variables X_0 and X_1, where X_0 is a covariate and X_1 is the variable to forecast. Variables are generated from a linear Structural Vector Autoregressive (SVAR) model with additive gauss noise and a noise scale of 2.782e-04, with lag = 3.

The task is to forecast the value of the variable X_1 at time t, given the values of the covariate X_0 and the variable X_1 itself at times t-1, ... t-3.
For the first 128 days, the covariate X_0 takes a value of 8 from 2027-03-05 to 2027-03-20, 2 from 2027-03-21 to 2027-04-29, 2 from 2027-04-30 to 2027-06-22, 2 from 2027-06-23 to 2027-07-10.
For the next 32 days, the covariate X_0 takes a value of 10 from 2027-07-11 to 2027-07-25, 40 from 2027-07-26 to 2027-07-26, 80 from 2027-07-27 to 2027-08-11. Each day can be treated as a timestep for the forecasting task. The causal parents affect the child variables at different lags.
The mathematical equations describing the cause-effect relationship for each variable is given below:
X_0^{t} = \epsilon_0^{t}
X_1^{t} = 1.322 * X_0^{t-1} + -0.604 * X_1^{t-1} + 0.926 * X_0^{t-2} + 0.763 * X_1^{t-2} + -0.851 * X_0^{t-3} + 0.623 * X_1^{t-3} + \epsilon_1^{t}
\epsilon_{i}^{t} given in the equations corresponds to the noise variable with the given noise scale, for X_i^{t}.

Types of context: ['Covariate information', 'Causal information']

Capabilities: ['Reasoning: Math', 'Reasoning: Causal']


Figure