Time Series Analysis

Overview

Time series analysis examines data points collected over time to identify patterns, trends, and seasonality for forecasting and understanding temporal dynamics.

When to Use

Forecasting future values based on historical trends

Detecting seasonality and cyclical patterns in data

Analyzing trends over time in sales, stock prices, or website traffic

Understanding autocorrelation and temporal dependencies

Making time-based predictions with confidence intervals

Decomposing data into trend, seasonal, and residual components

Core Components

Trend

Long-term directional movement

Seasonality

Repeating patterns at fixed intervals

Cyclicity

Long-term oscillations (non-fixed periods)

Stationarity

Constant mean, variance over time

Autocorrelation

Correlation with past values

Key Techniques

Decomposition

Separating trend, seasonal, residual components

Differencing

Making data stationary

ARIMA

AutoRegressive Integrated Moving Average models

Exponential Smoothing

Weighted average of past values

SARIMA

Seasonal ARIMA models Implementation with Python import pandas as pd import numpy as np import matplotlib . pyplot as plt from statsmodels . tsa . seasonal import seasonal_decompose from statsmodels . tsa . stattools import adfuller , acf , pacf from statsmodels . graphics . tsaplots import plot_acf , plot_pacf from statsmodels . tsa . arima . model import ARIMA from statsmodels . tsa . holtwinters import ExponentialSmoothing

Create sample time series data

dates

pd . date_range ( '2020-01-01' , periods = 365 , freq = 'D' ) values = 100 + np . sin ( np . arange ( 365 ) * 2 * np . pi / 365 ) * 20 + np . random . normal ( 0 , 5 , 365 ) ts = pd . Series ( values , index = dates )

Visualize time series

fig , axes = plt . subplots ( 2 , 2 , figsize = ( 14 , 8 ) ) axes [ 0 , 0 ] . plot ( ts ) axes [ 0 , 0 ] . set_title ( 'Original Time Series' ) axes [ 0 , 0 ] . set_ylabel ( 'Value' )

Decomposition

decomposition

seasonal_decompose ( ts , model = 'additive' , period = 30 ) axes [ 0 , 1 ] . plot ( decomposition . trend ) axes [ 0 , 1 ] . set_title ( 'Trend Component' ) axes [ 1 , 0 ] . plot ( decomposition . seasonal ) axes [ 1 , 0 ] . set_title ( 'Seasonal Component' ) axes [ 1 , 1 ] . plot ( decomposition . resid ) axes [ 1 , 1 ] . set_title ( 'Residual Component' ) plt . tight_layout ( ) plt . show ( )

Test for stationarity (Augmented Dickey-Fuller)

result

adfuller ( ts ) print ( f"ADF Test Statistic: { result [ 0 ] : .6f } " ) print ( f"P-value: { result [ 1 ] : .6f } " ) print ( f"Critical Values: { result [ 4 ] } " ) if result [ 1 ] <= 0.05 : print ( "Time series is stationary" ) else : print ( "Time series is non-stationary - differencing needed" )

First differencing for stationarity

ts_diff

ts . diff ( ) . dropna ( ) result_diff = adfuller ( ts_diff ) print ( f"\nAfter differencing - ADF p-value: { result_diff [ 1 ] : .6f } " )

Autocorrelation and Partial Autocorrelation

fig , axes = plt . subplots ( 1 , 2 , figsize = ( 12 , 4 ) ) plot_acf ( ts_diff , lags = 40 , ax = axes [ 0 ] ) axes [ 0 ] . set_title ( 'ACF' ) plot_pacf ( ts_diff , lags = 40 , ax = axes [ 1 ] ) axes [ 1 ] . set_title ( 'PACF' ) plt . tight_layout ( ) plt . show ( )

ARIMA Model

arima_model

ARIMA ( ts , order = ( 1 , 1 , 1 ) ) arima_result = arima_model . fit ( ) print ( arima_result . summary ( ) )

Forecast

forecast_steps

30 forecast = arima_result . get_forecast ( steps = forecast_steps ) forecast_df = forecast . conf_int ( ) forecast_mean = forecast . predicted_mean

Plot forecast

fig , ax = plt . subplots ( figsize = ( 12 , 5 ) ) ax . plot ( ts . index [ - 90 : ] , ts [ - 90 : ] , label = 'Historical' ) ax . plot ( forecast_df . index , forecast_mean , label = 'Forecast' , color = 'red' ) ax . fill_between ( forecast_df . index , forecast_df . iloc [ : , 0 ] , forecast_df . iloc [ : , 1 ] , color = 'red' , alpha = 0.2 ) ax . set_title ( 'ARIMA Forecast with Confidence Interval' ) ax . legend ( ) ax . grid ( True , alpha = 0.3 ) plt . show ( )

Exponential Smoothing

exp_smooth

ExponentialSmoothing ( ts , seasonal_periods = 30 , trend = 'add' , seasonal = 'add' , initialization_method = 'estimated' ) exp_result = exp_smooth . fit ( )

Model diagnostics

fig

exp_result . plot_diagnostics ( figsize = ( 12 , 8 ) ) plt . tight_layout ( ) plt . show ( )

Custom moving average analysis

window_sizes

[ 7 , 30 , 90 ] fig , ax = plt . subplots ( figsize = ( 12 , 5 ) ) ax . plot ( ts . index , ts . values , label = 'Original' , alpha = 0.7 ) for window in window_sizes : ma = ts . rolling ( window = window ) . mean ( ) ax . plot ( ma . index , ma . values , label = f'MA( { window } )' ) ax . set_title ( 'Moving Averages' ) ax . legend ( ) ax . grid ( True , alpha = 0.3 ) plt . show ( )

Seasonal subseries plot

fig , axes = plt . subplots ( 2 , 2 , figsize = ( 12 , 8 ) ) for i , month in enumerate ( range ( 1 , 5 ) ) : month_data = ts [ ts . index . month == month ] axes [ i // 2 , i % 2 ] . plot ( month_data . values ) axes [ i // 2 , i % 2 ] . set_title ( f'Month { month } Pattern' ) plt . tight_layout ( ) plt . show ( )

Forecast accuracy metrics

def calculate_forecast_metrics ( actual , predicted ) : mae = np . mean ( np . abs ( actual - predicted ) ) rmse = np . sqrt ( np . mean ( ( actual - predicted ) ** 2 ) ) mape = np . mean ( np . abs ( ( actual - predicted ) / actual ) ) * 100 return { 'MAE' : mae , 'RMSE' : rmse , 'MAPE' : mape } metrics = calculate_forecast_metrics ( ts [ - 30 : ] , forecast_mean [ : 30 ] ) print ( f"\nForecast Metrics:\n { metrics } " )

Additional analysis techniques

Step 10: Seasonal subseries plots

fig , axes = plt . subplots ( 2 , 2 , figsize = ( 12 , 8 ) ) for i , season in enumerate ( [ 1 , 2 , 3 , 4 ] ) : seasonal_ts = ts [ ts . index . month % 4 == season % 4 ] axes [ i // 2 , i % 2 ] . plot ( seasonal_ts . values ) axes [ i // 2 , i % 2 ] . set_title ( f'Season { season } ' ) plt . tight_layout ( ) plt . show ( )

Step 11: Granger causality (for multiple series)

from statsmodels . tsa . stattools import grangercausalitytests

Create another series for testing

ts2

ts . shift ( 1 ) . fillna ( method = 'bfill' ) try : print ( "\nGranger Causality Test:" ) print ( f"Test whether ts2 Granger-causes ts:" ) gc_result = grangercausalitytests ( np . column_stack ( [ ts . values , ts2 . values ] ) , maxlag = 3 ) except Exception as e : print ( f"Granger causality not performed: { str ( e ) [ : 50] } " )

Step 12: Autocorrelation and partial autocorrelation analysis

from statsmodels . graphics . tsaplots import plot_acf , plot_pacf acf_values = acf ( ts . dropna ( ) , nlags = 20 ) pacf_values = pacf ( ts . dropna ( ) , nlags = 20 )

Step 13: Seasonal strength

def seasonal_strength ( series , seasonal_period = 30 ) : seasonal = seasonal_decompose ( series , model = 'additive' , period = seasonal_period ) var_residual = np . var ( seasonal . resid . dropna ( ) ) var_seasonal = np . var ( seasonal . seasonal ) return 1 - ( var_residual / ( var_residual + var_seasonal ) ) if ( var_residual + var_seasonal )

0 else 0 ss = seasonal_strength ( ts ) print ( f"\nSeasonal Strength: { ss : .3f } " )

Step 14: Forecasting with uncertainty

fig , ax = plt . subplots ( figsize = ( 12 , 5 ) ) ax . plot ( ts . index [ - 60 : ] , ts . values [ - 60 : ] , label = 'Historical' , linewidth = 2 )

Multiple horizon forecasts

for steps_ahead in [ 10 , 20 , 30 ] : try : fc = arima_result . get_forecast ( steps = steps_ahead ) fc_mean = fc . predicted_mean ax . plot ( pd . date_range ( ts . index [ - 1 ] , periods = steps_ahead + 1 ) [ 1 : ] , fc_mean . values , marker = 'o' , label = f'Forecast (+ { steps_ahead } )' ) except : pass ax . set_title ( 'Multi-step Ahead Forecasts' ) ax . set_xlabel ( 'Date' ) ax . set_ylabel ( 'Value' ) ax . legend ( ) ax . grid ( True , alpha = 0.3 ) plt . tight_layout ( ) plt . show ( )

Step 15: Model comparison summary

print

(

"\nTime Series Analysis Complete!"

)

print

(

f"Original series length:

{

len

(

ts

)

}

"

)

print

(

f"Trend strength:

{

1

-

np

.

var

(

decomposition

.

resid

.

dropna

(

)

)

/

np

.

var

(

(

ts

-

ts

.

mean

(

)

)

.

dropna

(

)

)

:

.3f

}

"

)

print

(

f"Seasonal strength:

{

ss

:

.3f

}

"

)

Stationarity

Stationary

Mean, variance, autocorrelation constant over time

Non-stationary

Trend or seasonal patterns present

Solution

Differencing, log transformation, or detrending

Model Selection

ARIMA

Good for univariate forecasting

SARIMA

Includes seasonal components

Exponential Smoothing

Simpler, good for trends

Prophet

Handles holidays and changepoints

Evaluation Metrics

MAE

Mean Absolute Error

RMSE

Root Mean Squared Error

MAPE

Mean Absolute Percentage Error Deliverables Decomposition analysis charts Stationarity test results ACF/PACF plots Fitted models with diagnostics Forecast with confidence intervals Accuracy metrics comparison