A Comprehensive Guide to Time Series Analysis

Synopsis of Time Series Analysis

• A Time-Series represents a series of time-based orders. It would be Years, Months, Weeks, Days, Horus, Minutes, and Seconds
• A time series is an observation from the sequence of discrete-time of successive intervals.
• A time series is a running chart.
• The time variable/feature is the independent variable and supports the target variable to predict the results.
• Time Series Analysis (TSA) is used in different fields for time-based predictions – like Weather Forecasting, Financial, Signal processing, Engineering domain – Control Systems, Communications Systems.
• Since TSA involves producing the set of information in a particular sequence, it makes a distinct from spatial and other analyses.
• Using AR, MA, ARMA, and ARIMA models, we could predict the future.

Introduction to Time Series Analysis

Time Series Analysis is the way of studying the characteristics of the response variable with respect to time, as the independent variable. To estimate the target variable in the name of predicting or forecasting, use the time variable as the point of reference. In this article we will discuss in detail TSA Objectives, Assumptions, Components (stationary, and Non- stationary). Along with the TSA algorithm and specific use cases in Python.

Table of Contents

1. What is Time Series Analysis (TSA) and its assumption
2. How to analyze Time Series (TSA)?
3. Time Series Analysis Significance and its types.
4. Components of Time Series
5. What are the limitations of time series?
6. Detailed Study of Time Series Data types.
7. Discussion on stationary and Non- stationary components
8. Conversion of Non- stationary into stationary
9. Why is Time Series Analysis used in Data Science and Machine Learning?
10. Time Series Analysis in Data Science and Machine Learning
11. Implementation of Auto-Regressive model
12. Implementation of Moving Average (WEIGHTS – SIMPLE MOVING AVERAGE)
13. Understanding  ARMA and ARIMA
14. Implementation steps for ARIMA
15. Time Series Analysis – Process flow (Re-gap)

What is Time Series Analysis

Definition: If you see, there are many more definitions for TSA. But make it simple.

A time series is nothing but a sequence of various data points that occurred in a successive order for a given period of time

Objectives:

• To understand how time series works, what factors are affecting a certain variable(s) at different points of time.
• Time series analysis will provide the consequences and insights of features of the given dataset that changes over time.
• Supporting to derive the predicting the future values of the time series variable.
• Assumptions: There is one and the only assumption that 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?

Quick steps here for your reference, anyway. Will see this in detail in this article later.

• 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 and its types

TSA is the backbone for prediction and forecasting analysis, specific to the 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 help of “Time Series” we can prepare numerous time-based analyses and results.

• Forecasting
• Segmentation
• Classification
• Descriptive analysis`
• Intervention analysis

Components of Time Series Analysis

• Trend
• Seasonality
• Cyclical
• Irregularity

• 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?

Data Types of Time Series

Methods to check Stationarity 

Converting Non- stationary into stationary

Yt= Yt – Yt-1

Yt=Value with time

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

Moving Average Methodology

The commonly used time series method is 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: In which MA has calculated by taking averaging 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)

9.1 Simple Moving Average (SMA)

The SMA is the unweighted mean of the previous M or N points. The selection of sliding window data points, depending on the amount of smoothing is preferred since increasing the value of M or N, improves the smoothing at the expense of accuracy.

To understand better, will use the Air-Temperature.

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()

df_temperature.info()

# 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()

# 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)

9.2 Cumulative Moving Average (CMA)

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

# 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)

9.3 Exponential Moving Average (EMA)

EMA is mainly used to identify trends and to 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.

Lets will apply the exponential moving averages with a smoothing factor of 0.1 and 0.3 in the given dataset.

 

# 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)

 

Time Series Analysis in Data Science and Machine Learning

• 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 and Partial Auto-Correlation

plot_acf(df_temperature)
plt.show()

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

 

10.4 Interpret ACF and PACF plots

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

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

Y_t =C+b₁ Y_t-1+ b₂ Y_t-2+……+ b_p Y_t-p+ Er_t

Key Parameters

• p=past values
• Y_t=Function of different past values
• Er_t=errors in time
• C=intercept

Lets’s check, given data-set or time series is random or not

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

Observation: Yes, looks random and scattered.

Implementation of Auto-Regressive model

#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)

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)

 

Understanding ARMA and ARIMA 

ARMA This is a combination of the Auto-Regressive and Moving Average model 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. So ARIMA came in since it supports stationary as well as non-stationary.

AR+I+MA= ARIMA

Step 1: Plot a time series format

Step 2: Difference to make stationary on mean by removing the trend

Step 3: Make stationary by applying log transform.

Step 4: Difference log transform to make as stationary on both statistic mean and variance

Step 5: Plot ACF & PACF, and identify the potential AR and MA model

Step 6: Discovery of best fit ARIMA model

Step 7: Forecast/Predict the value, using the best fit ARIMA model

Step 8: Plot ACF & PACF for residuals of the ARIMA model, and ensure no more information is left.

Implementation of ARIMA

Already we have discussed steps 1-5, let’s focus on the rest here.

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

results_ARIMA.summary()

results_ARIMA.forecast(3)[0]

Output

array([16.47648941, 16.48621826, 16.49594711])

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

Process flow (Re-gap)

 

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

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

Each layer has equal weight and every neuron has to be assigned to fixed time steps. And 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),