Time Series Analysis - Time Series Properties

This notebook covers the exploratory analysis of time series data using weather observations.
The goal is to characterise the key statistical properties of a time series before building any forecasting model.

Topics covered:

1. Data loading and resampling

2. Visual exploration

3. Decomposition (trend, seasonality, residual)

4. Stationarity testing (ADF & KPSS)

5. Autocorrelation analysis (ACF / PACF)

6. Distribution and residual diagnostics

Libraries used: pandas, statsmodels, darts, matplotlib, seaborn

Dataset¶

Daily observations from DWD weather station 02932 (Leipzig) covering 1975–2024 (50 years, ~18 000 records).
Variables include temperature, precipitation, humidity, wind, pressure, sunshine, and cloud cover.

Note on missing values:
Several columns have gaps (e.g. wind_gust_max, sunshine_duration). This is typical for long real-world time series due to sensor failures or data gaps.
Many TS analysis methods require a complete, evenly-spaced series. So, missing value handling must come before these steps.

Resampling¶

Daily data often contains high-frequency noise that obscures long-term patterns. Also, Resampling to a lower frequency (here: weekly mean) trades temporal resolution for a cleaner signal.

When to resample:

• Data is very noisy at the original frequency

• The forecasting horizon is longer than the native frequency

• Get a smaller dataset to decrease compute time

Pitfall:
resample().mean() is appropriate for averaged quantities (temperature, humidity) but can distort event-based quantities.
For example, precipitation_height as total rainfall should be summed, not averaged.
For a demo this is acceptable, but be mindful of the aggregation semantics.

Visual Exploration¶

Before any statistics, always plot the raw data. Look for:

• Trend: Is there a long-term upward or downward movement?

• Seasonality: Are there repeating patterns, e.g., within a year?

• Outliers / anomalies: Sudden spikes, flat lines (sensor failure), or level shifts

• Variance changes: Does the spread of values increase over time? (heteroscedasticity)

Pitfall: With 50 years of daily data, individual plots are very dense. Weekly resampling (plotted here) gives a clearer overview of long-term behaviour.

Correlation Between Variables¶

Before modeling, it is useful to understand which variables are linearly related. High correlation can indicate:

• Redundant features (avoid multicollinearity in regression-based models)

• Physically meaningful links (e.g. temperature - vapour pressure)

Pitfall:
Correlation in time series is often driven by a shared trend or seasonality, not a genuine relationship.
Two unrelated variables that both increase in summer will appear highly correlated.
Always verify with partial correlations or by checking the relationship on detrended/deseasonalised data.

Decomposition¶

Time series decomposition separates a signal into interpretable components:

• Trend: Long-term increase or decrease

• Seasonal: Repeating pattern at a fixed period (e.g. yearly temperature cycle)

• Residual: What remains after removing trend and seasonality — ideally pure noise

Understanding these components is a prerequisite for choosing the right forecasting model.
First, we handle missing values via linear interpolation before passing the data to decomposition methods (most of which cannot handle NaNs).

Classical Decomposition (seasonal_decompose)¶

Decomposes a time series into: Trend + Seasonal + Residual

Choose the model type:

• Additive (model='additive'): seasonal amplitude is roughly constant → suitable here for temperature

• Multiplicative (model='multiplicative'): seasonal amplitude grows with the level → common for economic or sales data

The trend is estimated with a centered moving average over one full season (here: 365 days).

Pitfall:
The moving average loses period/2 observations at both ends. With period=365 you lose ~6 months of trend at the start and end of the series.
Also, classical decomposition assumes the seasonal pattern is identical every year, which may not hold over 50 years of climate data.

STL Decomposition (Seasonal-Trend using LOESS)¶

STL is more flexible than classical decomposition:

• Uses LOESS smoothing instead of moving averages, which is more robust to outliers

• The seasonal component can vary over time (e.g. climate trends changing seasonal amplitude)

• Works only for additive models, but you can log-transform first for multiplicative behaviour

Pitfall:
STL has more tuning parameters (seasonal, trend, robust).
The default period is the only required parameter, but the smoothing windows affect how much variation is attributed to trend vs. seasonal vs. residual.

Seasonality Detection¶

check_seasonality runs a statistical test (based on autocorrelation significance) to determine whether a given lag is a dominant seasonal period.
It scans all lags up to max_lag and returns the first significant one. We use max_lag=365 to at least allow detection of the annual cycle.

Pitfall:
check_seasonality returns the first statistically significant lag, not necessarily the dominant annual period.
On daily meteorological data, short weekly patterns (lag 7, 15) are often detected before the annual cycle (lag ~365).
A result like period=7 does not mean there is no yearly seasonality. It means a 7-day pattern was found first.
Always cross-check with decomposition plots.

Stationarity check¶

Stationarity means the statistical properties of the series (mean, variance, autocorrelation) do not change over time.
Most classical forecasting models (ARIMA, exponential smoothing) assume stationarity or require the data to be made stationary first.

Two complementary tests are used because they test opposite null hypotheses:

Using both together avoids false conclusions from either test alone.

ADF and KPSS

• Case 1: Both tests conclude that the series is not stationary - The series is not stationary

• Case 2: Both tests conclude that the series is stationary - The series is stationary

• Case 3: KPSS indicates stationarity and ADF indicates non-stationarity - The series is trend stationary. Trend needs to be removed to make series strict stationary. The detrended series is checked for stationarity.

• Case 4: KPSS indicates non-stationarity and ADF indicates stationarity - The series is difference stationary. Differencing is to be used to make series stationary. The differenced series is checked for stationarity.

Convenience: darts Stationarity Test¶

stationarity_tests() wraps both ADF and KPSS into a single boolean result. It returns True only when both tests agree on stationarity (Case 2 in the table above). Useful for quick programmatic checks, but always inspect the individual p-values for borderline cases.

Autocorrelation¶

ACF (Autocorrelation Function): correlation between the series and its lagged copy at every lag k.
PACF (Partial ACF): correlation at lag k after removing the effect of all shorter lags.

Use these to determine ARMA model orders:

• ACF tails off, PACF cuts off at lag p: AR(p)

• ACF cuts off at lag q, PACF tails off: MA(q)

• Both tail off: ARMA(p, q)

• Slow sinusoidal decay in ACF: strong seasonality present

Pitfall:
With strong seasonality or trend, the ACF decays very slowly and the plot looks non-stationary even if the ADF test says otherwise.
Always decompose or difference before interpreting ACF for model order.

ACF / PACF on Weekly-Aggregated Data¶

Weekly resampling reduces noise and makes the annual seasonal structure more visible.
The m=52 parameter marks multiples of the seasonal period (52 weeks = 1 year) on the plot so you can visually distinguish seasonal autocorrelation from non-seasonal lags.

Pitfall:
Without setting m, the seasonal spikes look like arbitrary autocorrelation and can be misread when estimating ARIMA/SARIMA orders.

Distribution of Variables¶

Histograms reveal the marginal distribution of each variable. Relevant for:

• Spotting skewness or heavy tails that violate normality assumptions in many models

• Identifying multimodal distributions (e.g. precipitation: many zeros + positive tail)

Three variables are shown to contrast two typical cases:

• Temperature: roughly normal (bell-shaped), good candidate for linear models as-is

• Wind speed: moderately right-skewed

• Precipitation: heavily right-skewed (spike at zero + long tail), a log-transform or dedicated count model is usually needed

Pitfall:
Many TS models (ARIMA, linear regression) assume Gaussian errors.
Strongly non-normal variables may benefit from a log-transform or Box-Cox transformation before modeling.

Residuals Analysis¶

After decomposition, the residuals should be pure noise with no structure left. Check for:

• Zero mean and roughly normal distribution (model captured the signal)

• No autocorrelation in the residual ACF. If autocorrelation remains, the model missed patterns

Pitfall:
Non-normal or autocorrelated residuals indicate the decomposition assumptions don’t fully hold.
Consider a different period, model order, or transformation.