Вивчайте ARIMA Model Structure | Mathematical Foundations of ARIMA

To understand how ARIMA models work, you need to see how three key components—autoregressive (AR), integrated (I), and moving average (MA)—combine to model a wide range of time series data. The ARIMA model is denoted as ARIMA(p, d, q), where p, d, and q are integers that describe the structure of the model. Each parameter has a specific role: p controls the AR part, d determines the number of times the data is differenced to achieve stationarity, and q sets the order of the MA part.

The AR (autoregressive) component models the relationship between an observation and a number of lagged observations. The MA (moving average) component models the relationship between an observation and a residual error from a moving average model applied to lagged observations. The integrated (I) part involves differencing the data, which means subtracting the previous observation from the current one; this helps to stabilize the mean of a time series by removing changes in the level of a time series, and thus eliminating trend and seasonality.

You should use differencing (d > 0) when your time series is non-stationary, meaning its statistical properties change over time. Applying the correct degree of differencing can make the series stationary, which is crucial for ARIMA models to perform effectively. If you over-difference, you may introduce unnecessary complexity and noise; if you under-difference, the model may not capture the underlying structure.

ARIMA Parameter Definitions:

p (AR order): The number of lag observations included in the model; controls how many past values are used to predict the current value.
d (degree of differencing): The number of times the raw observations are differenced to achieve stationarity; helps remove trends and seasonality.
q (MA order): The number of lagged forecast errors in the prediction equation; controls how many past errors are used to predict the current value.


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import numpy as np
import pandas as pd

np.random.seed(42)

# Simulate an ARIMA(1,1,1) process
n = 100
ar_coef = 0.7
ma_coef = 0.5

e = np.random.normal(0, 1, n+1)
y = [0]

for t in range(1, n+1):
    diff = y[-1] if t > 1 else 0
    val = diff + ar_coef * (y[-1] - diff) + e[t] + ma_coef * e[t-1]
    y.append(val)

series = pd.Series(y[1:])
print(series.head())

1. Fill in the blanks to identify the ARIMA parameters from the following model description: A time series model uses the last 2 observations, takes the first difference of the series, and incorporates the last 3 error terms.

2. Which of the following best describes the impact of differencing in an ARIMA model?

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Suggested prompts:

Can you explain how the ARIMA(1,1,1) simulation code works step by step?

What does the output of the simulated series tell us about the ARIMA process?

How would changing the values of p, d, or q affect the simulated time series?

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