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Time Series Analysis: Fundamentals

What is a Time Series?

A time series is a sequence of data points collected or recorded at successive points in time. These data points are typically spaced at uniform intervals such as hourly, daily, monthly, or yearly.

Examples:

  • Daily stock prices
  • Monthly airline passengers
  • Hourly temperature readings
  • Annual GDP of a country

Time series data allow us to:

  • Analyze past trends
  • Forecast future values
  • Understand underlying processes

Components of Time Series

Time series data can often be decomposed into several underlying components. Understanding these components helps in better analysis and forecasting.

1. Trend

The trend component represents the long-term progression of the series. It shows whether the data is increasing, decreasing, or remaining constant over time.

Example: A steady rise in housing prices over several years.

import pandas as pd
import matplotlib.pyplot as plt

# Load sample time series data
url = 'https://raw.githubusercontent.com/jbrownlee/Datasets/master/airline-passengers.csv'
df = pd.read_csv(url, parse_dates=['Month'], index_col='Month')

# Plot trend
df['Passengers'].plot(title='Airline Passengers Trend', figsize=(10, 4))
plt.xlabel('Date')
plt.ylabel('Number of Passengers')
plt.grid(True)
plt.show()

2. Seasonality

Seasonality refers to periodic fluctuations that occur at regular intervals due to seasonal factors such as weather, holidays, or recurring events.

Example: Increase in ice cream sales during summer.

from statsmodels.tsa.seasonal import seasonal_decompose

# Decompose the time series
result = seasonal_decompose(df['Passengers'], model='multiplicative')
result.seasonal.plot(title='Seasonality Component')
plt.show()

3. Cyclicity

Cyclic behavior occurs when data exhibit rises and falls that are not of fixed period and are often related to economic or business cycles.

Example: Economic expansions and recessions over decades.

Cyclic patterns are harder to identify with short datasets, but long-term datasets may reveal cycles distinct from seasonality.

# Plot full decomposition to see cycle trends (if any)
result.plot()
plt.suptitle('Time Series Decomposition', fontsize=16)
plt.show()

4. Noise (Residuals)

Noise is the random variation in the data that cannot be attributed to trend, seasonality, or cycles. It is considered unpredictable and should ideally be minimized.

Example: Unexplained spikes in sales due to random events.

# Plot residual (noise)
result.resid.plot(title='Noise/Residuals Component')
plt.show()

Time Series vs Cross-Sectional Data

Feature Time Series Data Cross-Sectional Data
Definition Observations collected over time Observations collected at a single point in time
Focus Temporal ordering of data Comparison among units (e.g., people, regions)
Example Daily temperature over a year Temperature readings across cities on one day

Key Difference: Time series analysis emphasizes the order and time-dependence of observations, whereas cross-sectional analysis looks at differences between subjects at the same point in time.

Stationarity & White Noise

Stationarity

A time series is stationary if its statistical properties (mean, variance, autocorrelation) are constant over time. Most forecasting methods require the data to be stationary.

Types of Stationarity:

  • Strict stationarity: Distribution does not change over time.
  • Weak stationarity: Mean, variance, and autocovariance are time-invariant.

Why important? Stationary series are easier to model and forecast.

Checking Stationarity:

from statsmodels.tsa.stattools import adfuller

# Perform Augmented Dickey-Fuller test
adf_result = adfuller(df['Passengers'])
print(f"ADF Statistic: {adf_result[0]}")
print(f"p-value: {adf_result[1]}")

Making a Series Stationary:

df['diff'] = df['Passengers'].diff()
df['diff'].dropna().plot(title='Differenced Series')
plt.show()

White Noise

White noise is a series of random variables that:

  • Have constant mean and variance
  • Have zero autocorrelation at all lags

White noise is purely random and unpredictable. It's often used as a benchmark to determine whether a time series model has successfully captured the structure of a time series.

import numpy as np

# Generate white noise series
np.random.seed(42)
white_noise = np.random.normal(loc=0, scale=1, size=100)
plt.plot(white_noise)
plt.title('White Noise Time Series')
plt.show()

📊 Basic Statistics

Basic statistics describe the central tendency and spread of a dataset.

1. Mean (Average)

The mean gives the central value of the time series.

\[ \mu = \frac{1}{n} \sum_{i=1}^{n} x_i \]

import pandas as pd

mean_value = df['Passengers'].mean()
print(f"Mean: {mean_value}")
2. Median

The median is the middle value that separates the higher half from the lower half of the data. 

median_value = df['Passengers'].median()
print(f"Median: {median_value}")
3. Variance

Variance measures the average of the squared deviations from the mean. σ2=1n∑i=1n(xi−μ)2 σ2=n1​i=1∑n​(xi​−μ)2

variance = df['Passengers'].var()
print(f"Variance: {variance}")

🔄 Covariance and Autocorrelation

  1. Covariance

Covariance measures how two variables change together. Cov(X,Y)=1n∑i=1n(xi−xˉ)(yi−yˉ) Cov(X,Y)=n1​i=1∑n​(xi​−xˉ)(yi​−yˉ​)

Covariance between original and lagged series

df['lag1'] = df['Passengers'].shift(1)
cov = df[['Passengers', 'lag1']].cov()
print(cov)
2. Autocorrelation

Autocorrelation (or serial correlation) measures the correlation of a time series with a lagged version of itself. ρk=Cov(Xt,Xt−k)σ2 ρk​=σ2Cov(Xt​,Xt−k​)​

Autocorrelation plot

from pandas.plotting import autocorrelation_plot
autocorrelation_plot(df['Passengers'])
plt.show()

⏱ Lag and Difference Operations

These operations are often used for preprocessing and making a time series stationary. 1. Lag

Creates a shifted version of the time series. 

df['lag1'] = df['Passengers'].shift(1)
2. Difference

Used to remove trends or seasonality and make the series stationary.

df['diff1'] = df['Passengers'].diff()
df['diff1'].dropna().plot(title='First Order Differenced Series')
plt.show()
Next Sections (optional extensions):

  • Measuring Autocorrelation
  • Decomposing Time Series in Python
  • Augmented Dickey-Fuller Test
  • Rolling Windows & Differencing