Oppiskele Autoregressive (AR) Models | Mathematical Foundations of ARIMA

Autoregressive (AR) models form the foundation of many time series forecasting techniques by capturing how past values in a sequence influence its future values. The key intuition is that observations in a time series are often correlated with their own previous values—meaning the past can help predict the future. In an AR model, you use a linear combination of previous observations, known as lagged variables, to estimate the current value. The simplest form, an $AR(1)$ model, predicts the current value based solely on the immediately preceding value. More generally, an $AR(p)$ model uses the previous p values.

Mathematically, an $AR(p)$ model is represented as:

X_t = c + \phi_1 X_{t-1} + \phi_2 X_{t-2} + \cdots + \phi_p X_{t-p} + \epsilon_t

where:

$X_t$ is the value at time $t$ ;
$c$ is a constant;
$\phi₁, ..., \phiₚ$ are the model coefficients;
$ε_t$ is white noise (random error at time $t$ ).

This approach allows you to model the persistence and memory within a time series, which is especially useful in fields like finance, economics, and signal processing.

Definition

Lag refers to how many time steps back you look in the series. For instance, a lag of 1 means using the value from one time point before the current one.

Definition

Order in AR models, denoted as $p$ in $AR(p)$ , specifies how many lagged values are included. An $AR(1)$ model uses one lag, $AR(2)$ uses two, and so on.


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import numpy as np
import matplotlib.pyplot as plt

# Set random seed for reproducibility
np.random.seed(42)

# Parameters for AR(1): X_t = 0.7 * X_{t-1} + noise
phi = 0.7
n = 100
noise = np.random.normal(0, 1, n)
X = np.zeros(n)

# Simulate AR(1) process
for t in range(1, n):
    X[t] = phi * X[t-1] + noise[t]

# Plot the simulated series
plt.figure(figsize=(10, 4))
plt.plot(X, label="Simulated AR(1) Process")
plt.xlabel("Time")
plt.ylabel("Value")
plt.title("Simulation of an AR(1) Time Series")
plt.legend()
plt.tight_layout()
plt.show()

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Pyyhkäise näyttääksesi valikon

Mathematically, an $AR(p)$ model is represented as:

X_t = c + \phi_1 X_{t-1} + \phi_2 X_{t-2} + \cdots + \phi_p X_{t-p} + \epsilon_t

where:

$X_t$ is the value at time $t$ ;
$c$ is a constant;
$\phi₁, ..., \phiₚ$ are the model coefficients;
$ε_t$ is white noise (random error at time $t$ ).

This approach allows you to model the persistence and memory within a time series, which is especially useful in fields like finance, economics, and signal processing.

Definition

Lag refers to how many time steps back you look in the series. For instance, a lag of 1 means using the value from one time point before the current one.

Definition

Order in AR models, denoted as $p$ in $AR(p)$ , specifies how many lagged values are included. An $AR(1)$ model uses one lag, $AR(2)$ uses two, and so on.


              12345678910111213141516171819202122232425
            
import numpy as np
import matplotlib.pyplot as plt

# Set random seed for reproducibility
np.random.seed(42)

# Parameters for AR(1): X_t = 0.7 * X_{t-1} + noise
phi = 0.7
n = 100
noise = np.random.normal(0, 1, n)
X = np.zeros(n)

# Simulate AR(1) process
for t in range(1, n):
    X[t] = phi * X[t-1] + noise[t]

# Plot the simulated series
plt.figure(figsize=(10, 4))
plt.plot(X, label="Simulated AR(1) Process")
plt.xlabel("Time")
plt.ylabel("Value")
plt.title("Simulation of an AR(1) Time Series")
plt.legend()
plt.tight_layout()
plt.show()

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Kiitos palautteestasi!

Osio 2. Luku 1