Time Series Analysis: Definition, Components, Methods, and Applications

Introduction

Time Series Analysis is a way of studying the characteristics of the response variable concerning time as the independent variable. To estimate the target variable in predicting or forecasting, use the time variable as the reference point. TSA represents a series of time-based orders, it would be Years, Months, Weeks, Days, Horus, Minutes, and Seconds. It is an observation from the sequence of discrete time of successive intervals. Some real-world application of TSA includes weather forecasting models, stock market predictions, signal processing, and control systems. Since TSA involves producing the set of information in a particular sequence, this makes it distinct from spatial and other analyses. We could predict the future using AR, MA, ARMA, and ARIMA models. In this article, we will be decoding time series analysis for you.

Learning Objectives

• We will discuss in detail TSA Objectives, Assumptions, and Components (stationary and non-stationary).
• We will look at the TSA algorithms.
• Finally, we will look at specific use cases in Python.

This article was published as a part of the Data Science Blogathon.

What is Time Series Analysis?

Time series analysis examines data collected at regular intervals over a defined timespan. This method focuses on studying patterns in sequentially ordered observations, rather than analyzing sporadic or arbitrarily gathered data points. By consistently recording measurements at set time intervals, analysts can identify trends, cycles, and other temporal characteristics within the dataset

Objectives of Time Series Analysis

• To understand how time series works and what factors affect a certain variable(s) at different points in time.
• Time series analysis will provide the consequences and insights of the given dataset’s features that change over time.
• Supporting to derive the predicting the future values of the time series variable.
• Assumptions: There is only one assumption in TSA, which is “stationary,” which means that the origin of time does not affect the properties of the process under the statistical factor.

How to Analyze Time Series?

To perform the time series analysis, we have to follow the following steps:

• Collecting the data and cleaning it
• Preparing Visualization with respect to time vs key feature
• Observing the stationarity of the series
• Developing charts to understand its nature.
• Model building – AR, MA, ARMA and ARIMA
• Extracting insights from prediction

Significance of Time Series

TSA is the backbone for prediction and forecasting analysis, specific to time-based problem statements.

• Analyzing the historical dataset and its patterns
• Understanding and matching the current situation with patterns derived from the previous stage.
• Understanding the factor or factors influencing certain variable(s) in different periods.

With the help of “Time Series,” we can prepare numerous time-based analyses and results.

• Forecasting: Predicting any value for the future.
• Segmentation: Grouping similar items together.
• Classification: Classifying a set of items into given classes.
• Descriptive analysis: Analysis of a given dataset to find out what is there in it.
• Intervention analysis: Effect of changing a given variable on the outcome.

Components of Time Series Analysis

Let’s look at the various components of Time Series Analysis:

• Trend: In which there is no fixed interval and any divergence within the given dataset is a continuous timeline. The trend would be Negative or Positive or Null Trend
• Seasonality: In which regular or fixed interval shifts within the dataset in a continuous timeline. Would be bell curve or saw tooth
• Cyclical: In which there is no fixed interval, uncertainty in movement and its pattern
• Irregularity: Unexpected situations/events/scenarios and spikes in a short time span.

What Are the Limitations of Time Series Analysis?

Time series has the below-mentioned limitations; we have to take care of those during our data analysis.

• Similar to other models, the missing values are not supported by TSA
• The data points must be linear in their relationship.
• Data transformations are mandatory, so they are a little expensive.
• Models mostly work on Uni-variate data.

Data Types of Time Series

Let’s discuss the time series’ data types and their influence. While discussing TS data types, there are two major types – stationary and non-stationary.

Stationary: A dataset should follow the below thumb rules without having Trend, Seasonality, Cyclical, and Irregularity components of the time series.

• The mean value of them should be completely constant in the data during the analysis.
• The variance should be constant with respect to the time-frame
• Covariance measures the relationship between two variables.

Non- Stationary: If either the mean-variance or covariance is changing with respect to time, the dataset is called non-stationary.

Methods to Check Stationarity

During the TSA model preparation workflow, we must assess whether the dataset is stationary or not. This is done using Statistical Tests. There are two tests available to test if the dataset is stationary:

• Augmented Dickey-Fuller (ADF) Test
• Kwiatkowski-Phillips-Schmidt-Shin (KPSS) Test

Augmented Dickey-Fuller (ADF) Test or Unit Root Test

The ADF test is the most popular statistical test. It is done with the following assumptions:

• Null Hypothesis (H0): Series is non-stationary
• Alternate Hypothesis (HA): Series is stationary
  • p-value >0.05 Fail to reject (H0)
  • p-value <= 0.05 Accept (H1)

Kwiatkowski–Phillips–Schmidt–Shin (KPSS) Test

These tests are used for testing a NULL Hypothesis (HO) that will perceive the time series as stationary around a deterministic trend against the alternative of a unit root. Since TSA is looking for Stationary Data for its further analysis, we have to ensure that the dataset is stationary.

Converting Non-Stationary Into Stationary

Let’s discuss quickly how to convert non-stationary to stationary for effective time series modeling. There are three methods available for this conversion – detrending, differencing, and transformation.

Detrending

It involves removing the trend effects from the given dataset and showing only the differences in values from the trend. It always allows cyclical patterns to be identified.

Differencing

This is a simple transformation of the series into a new time series, which we use to remove the series dependence on time and stabilize the mean of the time series, so trend and seasonality are reduced during this transformation.

• Yt= Yt – Yt-1
• Yt=Value with time

Transformation

This includes three different methods they are Power Transform, Square Root, and Log Transfer. The most commonly used one is Log Transfer.

Moving Average Methodology

The commonly used time series method is the Moving Average. This method is slick with random short-term variations. Relatively associated with the components of time series.

The Moving Average (MA) (or) Rolling Mean: The value of MA is calculated by taking average data of the time-series within k periods.

Let’s see the types of moving averages:

• Simple Moving Average (SMA),
• Cumulative Moving Average (CMA)
• Exponential Moving Average (EMA)

Simple Moving Average (SMA)

The Simple Moving Average (SMA) calculates the unweighted mean of the previous M or N points. We prefer selecting sliding window data points based on the amount of smoothing, as increasing the value of M or N improves smoothing but reduces accuracy.

To understand better, I will use the air temperature dataset.

Code

import pandas as pd
from matplotlib import pyplot as plt
from statsmodels.graphics.tsaplots import plot_acf
df_temperature = pd.read_csv('temperature_TSA.csv', encoding='utf-8')
df_temperature.head()

Output

Code

df_temperature.info()

Output

Code

# set index for year column
df_temperature.set_index('Any', inplace=True)
df_temperature.index.name = 'year'
# Yearly average air temperature - calculation
df_temperature['average_temperature'] = df_temperature.mean(axis=1)
# drop unwanted columns and resetting the datafreame
df_temperature = df_temperature[['average_temperature']]
df_temperature.head()

Output

Code

# SMA over a period of 10 and 20 years 
df_temperature['SMA_10'] = df_temperature.average_temperature.rolling(10, min_periods=1).mean()
df_temperature['SMA_20'] = df_temperature.average_temperature.rolling(20, min_periods=1).mean()

# Grean = Avg Air Temp, RED = 10 yrs, ORANG colors for the line plot
colors = ['green', 'red', 'orange']
# Line plot 
df_temperature.plot(color=colors, linewidth=3, figsize=(12,6))
plt.xticks(fontsize=14)
plt.yticks(fontsize=14)
plt.legend(labels =['Average air temperature', '10-years SMA', '20-years SMA'], fontsize=14)
plt.title('The yearly average air temperature in city', fontsize=20)
plt.xlabel('Year', fontsize=16)
plt.ylabel('Temperature [°C]', fontsize=16)

Output

Cumulative Moving Average (CMA)

The CMA is the unweighted mean of past values till the current time.

Code

# CMA Air temperature
df_temperature['CMA'] = df_temperature.average_temperature.expanding().mean()

# green -Avg Air Temp and Orange -CMA
colors = ['green', 'orange']
# line plot
df_temperature[['average_temperature', 'CMA']].plot(color=colors, linewidth=3, figsize=(12,6))
plt.xticks(fontsize=14)
plt.yticks(fontsize=14)
plt.legend(labels =['Average Air Temperature', 'CMA'], fontsize=14)
plt.title('The yearly average air temperature in city', fontsize=20)
plt.xlabel('Year', fontsize=16)
plt.ylabel('Temperature [°C]', fontsize=16)

Output

Exponential Moving Average (EMA)

EMA is mainly used to identify trends and filter out noise. The weight of elements is decreased gradually over time. This means It gives weight to recent data points, not historical ones. Compared with SMA, the EMA is faster to change and more sensitive.

α –>Smoothing Factor.

• It has a value between 0,1.
• Represents the weighting applied to the very recent period.

Let’s apply the exponential moving averages with a smoothing factor of 0.1 and 0.3 in the given dataset.

Code

# EMA Air Temperature
# Let's smoothing factor - 0.1
df_temperature['EMA_0.1'] = df_temperature.average_temperature.ewm(alpha=0.1, adjust=False).mean()
# Let's smoothing factor  - 0.3
df_temperature['EMA_0.3'] = df_temperature.average_temperature.ewm(alpha=0.3, adjust=False).mean()

# green - Avg Air Temp, red- smoothing factor - 0.1, yellow - smoothing factor  - 0.3
colors = ['green', 'red', 'yellow']
df_temperature[['average_temperature', 'EMA_0.1', 'EMA_0.3']].plot(color=colors, linewidth=3, figsize=(12,6), alpha=0.8)
plt.xticks(fontsize=14)
plt.yticks(fontsize=14)
plt.legend(labels=['Average air temperature', 'EMA - alpha=0.1', 'EMA - alpha=0.3'], fontsize=14)
plt.title('The yearly average air temperature in city', fontsize=20)
plt.xlabel('Year', fontsize=16)
plt.ylabel('Temperature [°C]', fontsize=16)

Output

Time Series Analysis in Data Science and Machine Learning

When dealing with TSA in Data Science and Machine Learning, there are multiple model options are available. In which the Autoregressive–Moving-Average (ARMA) models with [p, d, and q].

• P==> autoregressive lags
• q== moving average lags
• d==> difference in the order

Before we get to know about Arima, first, you should understand the below terms better.

• Auto-Correlation Function (ACF)
• Partial Auto-Correlation Function (PACF)

Auto-Correlation Function (ACF)

ACF indicates how similar a value is within a given time series and the previous value. (OR) It measures the degree of the similarity between a given time series and the lagged version of that time series at the various intervals we observed.

Python Statsmodels library calculates autocorrelation. It identifies a set of trends in the given dataset and the influence of former observed values on the currently observed values.

Partial Auto-Correlation (PACF)

PACF is similar to Auto-Correlation Function and is a little challenging to understand. It always shows the correlation of the sequence with itself with some number of time units per sequence order in which only the direct effect has been shown, and all other intermediary effects are removed from the given time series.

Auto-Correlation and Partial Auto-Correlation

Code

plot_acf(df_temperature)
plt.show()

plot_acf(df_temperature, lags=30)
plt.show()

Output

Observation

The previous temperature influences the current temperature, but the significance of that influence decreases and slightly increases from the above visualization along with the temperature with regular time intervals.

Types of Auto-Correlation

Interpret ACF and PACF Plots

Remember that both ACF and PACF require stationary time series for analysis.

What is an Auto-Regressive Model?

An auto-regressive model is a simple model that predicts future performance based on past performance. It is mainly used for forecasting when there is some correlation between values in a given time series and those that precede and succeed (back and forth).

An AR is a Linear Regression model that uses lagged variables as input. By indicating the input, the Linear Regression model can be easily built using the scikit-learn library. Statsmodels library provides autoregression model-specific functions where you must specify an appropriate lag value and train the model. It is provided in the AutoTeg class to get the results using simple steps.

• Creating the model AutoReg()
• Call fit() to train it on our dataset.
• Returns an AutoRegResults object.
• Once fit, make a prediction by calling the predict () function

The equation for the AR model (Let’s compare Y=mX+c)

Yt =C+b1 Yt-1+ b2 Yt-2+……+ bp Yt-p+ Ert

Key Parameters

• p=past values
• Yt=Function of different past values
• Ert=errors in time
• C=intercept

Lets’s check whether the given data set or time series is random or not.

Code

from matplotlib import pyplot
from pandas.plotting import lag_plot
lag_plot(df_temperature)
pyplot.show()

Output

Observation

Yes, it looks random and scattered.

Implementation of Auto-Regressive Model

Code

#import libraries
from matplotlib import pyplot
from statsmodels.tsa.ar_model import AutoReg
from sklearn.metrics import mean_squared_error
from math import sqrt
# load csv as dataset
#series = read_csv('daily-min-temperatures.csv', header=0, index_col=0, parse_dates=True, squeeze=True)
# split dataset for test and training
X = df_temperature.values
train, test = X[1:len(X)-7], X[len(X)-7:]
# train autoregression
model = AutoReg(train, lags=20)
model_fit = model.fit()
print('Coefficients: %s' % model_fit.params)
# Predictions
predictions = model_fit.predict(start=len(train), end=len(train)+len(test)-1, dynamic=False)
for i in range(len(predictions)):
    print('predicted=%f, expected=%f' % (predictions[i], test[i]))
rmse = sqrt(mean_squared_error(test, predictions))
print('Test RMSE: %.3f' % rmse)
# plot results
pyplot.plot(test)
pyplot.plot(predictions, color='red')
pyplot.show()

Output

predicted=15.893972, expected=16.275000
predicted=15.917959, expected=16.600000
predicted=15.812741, expected=16.475000
predicted=15.787555, expected=16.375000
predicted=16.023780, expected=16.283333
predicted=15.940271, expected=16.525000
predicted=15.831538, expected=16.758333
Test RMSE: 0.617

Observation

Expected (blue) Against Predicted (red). The forecast looks good on the 4th and the deviation on the 6th day.

Implementation of Moving Average (Weights – Simple Moving Average)

Code

import numpy as np
alpha= 0.3
n = 10
w_sma = np.repeat(1/n, n)
colors = ['green', 'yellow']
# weights - exponential moving average alpha=0.3 adjust=False
w_ema = [(1-ALPHA)**i if i==N-1 else alpha*(1-alpha)**i for i in range(n)]
pd.DataFrame({'w_sma': w_sma, 'w_ema': w_ema}).plot(color=colors, kind='bar', figsize=(8,5))
plt.xticks([])
plt.yticks(fontsize=10)
plt.legend(labels=['Simple moving average', 'Exponential moving average (α=0.3)'], fontsize=10)
# title and labels
plt.title('Moving Average Weights', fontsize=10)
plt.ylabel('Weights', fontsize=10)

Output

Understanding ARMA and ARIMA

ARMA is a combination of the Auto-Regressive and Moving Average models for forecasting. This model provides a weakly stationary stochastic process in terms of two polynomials, one for the Auto-Regressive and the second for the Moving Average.

ARMA is best for predicting stationary series. ARIMA was thus developed to support both stationary as well as non-stationary series.

• AR ==> Uses past values to predict the future.
• MA ==> Uses past error terms in the given series to predict the future.
• I==> Uses the differencing of observation and makes the stationary data.

AR+I+MA= ARIMA

Understand the signature of ARIMA

• p==> log order => No of lag observations.
• d==> degree of differencing => No of times that the raw observations are differenced.
• q==>order of moving average => the size of the moving average window

Implementation Steps for ARIMA

• Plot a time series format
• Difference to make stationary on mean by removing the trend
• Make stationary by applying log transform.
• Difference log transform to make as stationary on both statistic mean and variance
• Plot ACF & PACF, and identify the potential AR and MA model
• Discovery of best fit ARIMA model
• Forecast/Predict the value using the best fit ARIMA model
• Plot ACF & PACF for residuals of the ARIMA model, and ensure no more information is left.

Implementation of ARIMA in Python

We have already discussed steps 1-5 which will remain the same; let’s focus on the rest here.

Code

from statsmodels.tsa.arima_model import ARIMA
model = ARIMA(df_temperature, order=(0, 1, 1)) 
results_ARIMA = model.fit()

results_ARIMA.summary()

Output

Code

results_ARIMA.forecast(3)[0]

Output

array([16.47648941, 16.48621826, 16.49594711])

Code

results_ARIMA.plot_predict(start=200)
plt.show()

Output

Process Flow (Re-Gap)

In recent years, the use of Deep Learning for Time Series Analysis and Forecasting has increased to resolve problem statements that couldn’t be handled using Machine Learning techniques. Let’s discuss this briefly.

Recurrent Neural Networks (RNN) is the most traditional and accepted architecture fitment for Time-Series forecasting-based problems.

RNN is organized into successive layers and divided into

• Input
• Hidden
• Output

Each layer has equal weight, and every neuron has to be assigned to fixed time steps. Do remember that every one of them is fully connected with a hidden layer (Input and Output) with the same time steps, and the hidden layers are forwarded and time-dependent in direction.

Components of RNN

• Input: The function vector of x(t)​ is the input at time step t.
• Hidden:
  • The function vector h(t)​ is the hidden state at time t,
  • This is a kind of memory of the established network;
  • This has been calculated based on the current input x(t) and the previous-time step’s hidden-state h(t-1):
• Output: The function vector y(t) ​is the output at time step t.
• Weights : Weights: In the RNNs, the input vector connected to the hidden layer neurons at time t is by a weight matrix of U (Please refer to the above picture),

internally weight matrix W is formed by the hidden layer neurons of time t-1 and t+1. Following this, the hidden layer with to the output vector y(t) of time t by a V (weight matrix); all the weight matrices U, W, and V are constant for each time step.

Advantages of RNN

• It has the special feature that it remembers every piece of information, so RNN is much useful for time series prediction
• Perfect for creating complex patterns from the input time series dataset
• Fast in prediction/forecasting
• Not affected by missing values, so the cleansing process can be limited.

Disadvantages of RNN

• The big challenge is during the training period
• Expensive computation cost.

Conclusion

A time series is constructed by data that is measured over time at evenly spaced intervals. I hope this comprehensive guide has helped you all understand the time series, its flow, and how it works. Although the TSA is widely used to handle data science problems, it has certain limitations, such as not supporting missing values. Note that the data points must be linear in their relationship for Time Series Analysis to be done.

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Key Takeaways

• Time series is a sequence of various data points that occurred in a successive order for a given period of time.
• Trend, Seasonality, Cyclical, and Irregularity are components of TSA.

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Shanthababu Pandian 24 Aug, 2024

Shanthababu has 20+ years of experience in Information Technology (IT). Expertise in ML & Data Architect and Program Delivery. Analytics and AIML Solutions Lead, Data Architecting, Modelling, Data Quality, Data Governance, Data Privacy and Data Integration, Data Platform, Data analysis and Data Science program.