Stationarity and differencing in ML Python - Model Metrics & Evaluation

Metrics & Evaluation - Stationarity and differencing

Which metric matters for Stationarity and differencing and WHY

For stationarity, the key metric is the Augmented Dickey-Fuller (ADF) test statistic and its p-value. This test tells us if a time series is stationary or not. A low p-value (usually below 0.05) means the series is stationary, which is important because many forecasting models assume stationarity.

For differencing, the metric is the order of differencing needed to achieve stationarity. We want to find the smallest number of differences that make the series stationary without losing important information.

Confusion matrix or equivalent visualization

Stationarity is not about classification, so no confusion matrix applies. Instead, we use the ADF test result table like this:

    +----------------------+----------------+
    | Statistic            | -3.45          |
    | p-value              | 0.01           |
    | Critical Values (5%)  | -2.86          |
    +----------------------+----------------+
    

If the test statistic is less than the critical value and p-value < 0.05, the series is stationary.

Precision vs Recall tradeoff (or equivalent)

Here, the tradeoff is between under-differencing and over-differencing.

• Under-differencing: The series remains non-stationary. Models may give biased or poor forecasts because trends or seasonality remain.
• Over-differencing: The series becomes too noisy and loses meaningful patterns, making forecasts unstable.

The goal is to find the just right differencing order that makes the series stationary but keeps useful information.

What "good" vs "bad" metric values look like for this use case

Good:

• ADF test p-value < 0.05, indicating stationarity.
• Differencing order is as low as possible (often 0 or 1).
• Time series plots show stable mean and variance after differencing.

Bad:

• ADF test p-value > 0.05, series is non-stationary.
• High differencing order (2 or more) causing noisy data.
• Time series plots show trends or changing variance after differencing.

Metrics pitfalls

• Ignoring stationarity: Using models that assume stationarity on non-stationary data leads to bad forecasts.
• Over-differencing: Differencing too many times can remove important signals and increase noise.
• Misinterpreting ADF test: A p-value slightly above 0.05 does not always mean non-stationary; consider domain knowledge and plots.
• Data leakage: Differencing using future data points can leak information and bias results.

Self-check question

Your time series model uses first-order differencing and the ADF test p-value is 0.07. Is your series stationary? Should you difference more?

Answer: Since p-value is 0.07 > 0.05, the series is likely still non-stationary. You may need to difference one more time or try other transformations. But also check plots and domain knowledge before deciding.

Key Result

Use the Augmented Dickey-Fuller test p-value to check stationarity; choose the smallest differencing order that achieves stationarity without over-differencing.

Practice

1. What does it mean when a time series is stationary?

easy

A. It has missing values that need to be filled

B. It has a clear upward or downward trend

C. It contains seasonal patterns repeating over fixed intervals

D. Its statistical properties like mean and variance do not change over time

Solution

1. Step 1: Understand stationarity definition

   Stationarity means the data's mean, variance, and other statistics stay constant over time.

2. Step 2: Compare options to definition

   Only Its statistical properties like mean and variance do not change over time describes constant statistical properties; others describe trends, seasonality, or missing data.

3. Final Answer:

   Its statistical properties like mean and variance do not change over time -> Option D

4. Quick Check:

   Stationary = constant mean/variance [OK]

Hint: Stationary means stats don't change over time [OK]

Common Mistakes:

• Confusing stationarity with trend presence
• Thinking seasonality means stationarity
• Assuming missing data affects stationarity

2. Which Python code correctly applies first-order differencing to a pandas Series data?

easy

A. data.dropna()

B. data.diff(1)

C. data.cumsum()

D. data.shift(1)

Solution

1. Step 1: Recall differencing method in pandas

   The diff(1) method calculates the difference between current and previous values, performing first-order differencing.

2. Step 2: Check other options

   shift(1) shifts data, cumsum() sums cumulatively, and dropna() removes missing values, none perform differencing.

3. Final Answer:

   data.diff(1) -> Option B

4. Quick Check:

   First difference = diff(1) [OK]

Hint: Use diff(1) for first-order differencing in pandas [OK]

Common Mistakes:

• Using shift instead of diff for differencing
• Confusing cumulative sum with differencing
• Dropping NaNs instead of differencing

3. Given this code snippet:

import pandas as pd
series = pd.Series([10, 12, 15, 20, 25])
diff_series = series.diff(1).dropna()
print(diff_series.tolist())

What is the output?

medium

A. [0, 2, 3, 5, 5]

B. [10, 12, 15, 20, 25]

C. [2.0, 3.0, 5.0, 5.0]

D. [nan, 2, 3, 5, 5]

Solution

1. Step 1: Calculate first differences

   Differences: 12-10=2, 15-12=3, 20-15=5, 25-20=5.

2. Step 2: Drop NaN and print list

   The first difference is NaN, dropped by dropna(), so output is [2.0, 3.0, 5.0, 5.0].

3. Final Answer:

   [2.0, 3.0, 5.0, 5.0] -> Option C

4. Quick Check:

   Diff values = [2.0,3.0,5.0,5.0] [OK]

Hint: First diff drops first NaN, output is differences list [OK]

Common Mistakes:

• Including NaN in output list
• Printing original series instead of differences
• Confusing shift with diff output

4. You applied first-order differencing to a time series but it still shows a trend. What is the likely issue?

medium

A. The series needs second-order differencing to remove the trend

B. You should use cumulative sum instead of differencing

C. The series is already stationary and differencing added noise

D. You forgot to normalize the data before differencing

Solution

1. Step 1: Understand differencing orders

   First-order differencing removes linear trends; if trend remains, higher order differencing may be needed.

2. Step 2: Evaluate other options

   Cumulative sum adds trend, normalization doesn't remove trend, and differencing adding noise means series was not stationary before.

3. Final Answer:

   The series needs second-order differencing to remove the trend -> Option A

4. Quick Check:

   Trend remains -> try second differencing [OK]

Hint: If trend remains, increase differencing order [OK]

Common Mistakes:

• Using cumulative sum instead of differencing
• Assuming normalization removes trend
• Stopping at first differencing without checking stationarity

5. You have a monthly sales time series with a yearly seasonal pattern and an upward trend. Which differencing approach should you apply to make it stationary?

hard

A. Apply first-order differencing followed by seasonal differencing with lag 12

B. Apply only first-order differencing

C. Apply only seasonal differencing with lag 12

D. Apply logarithm transformation without differencing

Solution

1. Step 1: Identify components to remove

   The series has both trend and yearly seasonality, so both need to be removed for stationarity.

2. Step 2: Choose differencing methods

   First-order differencing removes trend; seasonal differencing with lag 12 removes yearly seasonality.

3. Step 3: Combine differencing steps

   Applying first-order differencing then seasonal differencing is the correct approach to achieve stationarity.

4. Final Answer:

   Apply first-order differencing followed by seasonal differencing with lag 12 -> Option A

5. Quick Check:

   Trend + seasonality -> first + seasonal differencing [OK]

Hint: Remove trend then seasonality with two differencing steps [OK]

Common Mistakes:

• Applying only one differencing type ignoring trend or seasonality
• Using log transform alone to fix non-stationarity
• Confusing seasonal lag with differencing order