Stationarity and Its Importance
Understanding stationarity is fundamental to time series analysis and especially to building ARIMA models. A time series is said to be stationary if its statistical properties — such as mean, variance, and autocorrelation — do not change over time. Stationarity ensures that relationships observed in the past persist into the future, which is a core assumption behind most forecasting models, including ARIMA.
There are two main types of stationarity:
- Strict stationarity requires that the entire distribution of the series is unchanged by shifts in time, meaning all moments (mean, variance, higher-order moments) remain constant;
- Weak (or second-order) stationarity only requires the mean and variance to be constant over time, and that covariance between values depends only on the lag between them, not on the actual time at which the covariance is computed.
For ARIMA models to provide reliable forecasts, the input time series must be stationary. If a series is not stationary—perhaps due to trends, seasonality, or changing variance—the model’s assumptions are violated, often resulting in poor predictions. Non-stationary data often need to be transformed before modeling, with differencing being a common technique to remove trends and stabilize the mean.
Stationarity refers to a time series whose statistical properties do not change over time.
Key indicators of non-stationarity include:
- Changing mean (
trend); - Changing variance (
heteroscedasticity); - Seasonal patterns.
1234567891011121314151617181920212223242526import numpy as np import pandas as pd import matplotlib.pyplot as plt # Generate a stationary time series (white noise) np.random.seed(0) stationary_series = np.random.normal(loc=0, scale=1, size=100) # Generate a non-stationary time series (random walk) non_stationary_series = np.cumsum(np.random.normal(loc=0, scale=1, size=100)) plt.figure(figsize=(12, 5)) plt.subplot(1, 2, 1) plt.plot(stationary_series, color='blue') plt.title('Stationary Series (White Noise)') plt.xlabel('Time') plt.ylabel('Value') plt.subplot(1, 2, 2) plt.plot(non_stationary_series, color='red') plt.title('Non-Stationary Series (Random Walk)') plt.xlabel('Time') plt.ylabel('Value') plt.tight_layout() plt.show()
12345678910111213141516import numpy as np import pandas as pd import matplotlib.pyplot as plt np.random.seed(0) non_stationary_series = np.cumsum(np.random.normal(loc=0, scale=1, size=100)) # Assume 'non_stationary_series' from previous code diff_series = np.diff(non_stationary_series) plt.figure(figsize=(8, 4)) plt.plot(diff_series, color='green') plt.title('Differenced Series (First Order)') plt.xlabel('Time') plt.ylabel('Value') plt.show()
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Understanding stationarity is fundamental to time series analysis and especially to building ARIMA models. A time series is said to be stationary if its statistical properties — such as mean, variance, and autocorrelation — do not change over time. Stationarity ensures that relationships observed in the past persist into the future, which is a core assumption behind most forecasting models, including ARIMA.
There are two main types of stationarity:
- Strict stationarity requires that the entire distribution of the series is unchanged by shifts in time, meaning all moments (mean, variance, higher-order moments) remain constant;
- Weak (or second-order) stationarity only requires the mean and variance to be constant over time, and that covariance between values depends only on the lag between them, not on the actual time at which the covariance is computed.
For ARIMA models to provide reliable forecasts, the input time series must be stationary. If a series is not stationary—perhaps due to trends, seasonality, or changing variance—the model’s assumptions are violated, often resulting in poor predictions. Non-stationary data often need to be transformed before modeling, with differencing being a common technique to remove trends and stabilize the mean.
Stationarity refers to a time series whose statistical properties do not change over time.
Key indicators of non-stationarity include:
- Changing mean (
trend); - Changing variance (
heteroscedasticity); - Seasonal patterns.
1234567891011121314151617181920212223242526import numpy as np import pandas as pd import matplotlib.pyplot as plt # Generate a stationary time series (white noise) np.random.seed(0) stationary_series = np.random.normal(loc=0, scale=1, size=100) # Generate a non-stationary time series (random walk) non_stationary_series = np.cumsum(np.random.normal(loc=0, scale=1, size=100)) plt.figure(figsize=(12, 5)) plt.subplot(1, 2, 1) plt.plot(stationary_series, color='blue') plt.title('Stationary Series (White Noise)') plt.xlabel('Time') plt.ylabel('Value') plt.subplot(1, 2, 2) plt.plot(non_stationary_series, color='red') plt.title('Non-Stationary Series (Random Walk)') plt.xlabel('Time') plt.ylabel('Value') plt.tight_layout() plt.show()
12345678910111213141516import numpy as np import pandas as pd import matplotlib.pyplot as plt np.random.seed(0) non_stationary_series = np.cumsum(np.random.normal(loc=0, scale=1, size=100)) # Assume 'non_stationary_series' from previous code diff_series = np.diff(non_stationary_series) plt.figure(figsize=(8, 4)) plt.plot(diff_series, color='green') plt.title('Differenced Series (First Order)') plt.xlabel('Time') plt.ylabel('Value') plt.show()
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