In many cases, the original time series needs to be transformed into aggregate statistics. For example, observations in the original time series might have been recorded at every second; however, in order to perform any meaningful analysis, data must be aggregated every minute. This would need resampling the observations over periods that are longer than the granular time indices in the original data. The aggregate statistics, such as mean, median, and variance, is calculated for each of the longer periods of time.

Another example of data pre-processing for time series, is computing aggregates over similar segments in the data. Consider the monthly sales of cars manufactured by company X where the data exhibits monthly seasonality, due to which sales during a month of a given year shows patters similar to the sales of the...

The technique of resmapling is illustrated using a time series on chemical concentration readings taken every two hours between 1^st January 1975 and 17^th January 1975. The dataset has been downloaded from http://datamarket.com and is also available in the datasets folder of this book's GitHub repo.

We start by importing the packages required for running this example:

from __future__ import print_function 
import os 
import pandas as pd 
import numpy as np 
%matplotlib inline 
from matplotlib import pyplot as plt 

Then we set the working directory as follows:

os.chdir('D:/Practical Time Series') 

This is followed by reading the data from the CSV file in a pandas.DataFrame and displaying shape and the first 10 rows of the DataFrame:

df = pd.read_csv('datasets/chemical-concentration-readings.csv') 
print('Shape of the dataset...

Properties of data such as central tendency, dispersion, skewness, and kurtosis are called sample statistics. Mean and variance are two of the most commonly used sample statistics. In any analysis, data is collected by gathering information from a sample of the larger population. Mean, variance, and other properties are then estimated based on the sample data. Hence these are referred to as sample statistics.

An important assumption in statistical estimation theory is that, for sample statistics to be reliable, the population does not undergo any fundamental or systemic shifts over the individuals in the sample or over the time during which the data has been collected. This assumption ensures that sample statistics do not alter and will hold for entities that are outside the sample used for their estimation.

This assumption also applies to time series analysis...

The objective of time series decomposition is to model the long-term trend and seasonality and estimate the overall time series as a combination of them. Two popular models for time series decomposition are:

• Additive model
• Multiplicative model

The additive model formulates the original time series (x_t) as the sum of the trend cycle (F_t) and seasonal (S_t) components as follows:

The residuals Є_t obtained after adjusting the trend and seasonal components are the irregular variations. The additive model is usually applied when there is a time-dependent trend cycle component, but independent seasonality that does not change over time.

The multiplicative decomposition model, which gives the time series as product of the trend, seasonal, and irregular components is useful when there is time-varying seasonality:

We started this chapter by discussing advanced data processing techniques such as resampling, group-by, and moving window computations to obtain aggregate statistics from a time series. Next, we described stationary time series and discussed statistical tests of hypothesis such as Ljung-Box test and Augmented Dickey Fuller test to verify stationarity of a time series. Stationarizing non-stationary time series is important for time series forecasting. Therefore, we discussed two different approaches of stationarizing time series.
Firstly, the method of differencing, which covers first, second, and seasonal differencing, has been described for stationarizing a non-stationary time series. Secondly, time series decomposition using the statsmodels.tsa API for additive and multiplicative models has been discussed.
In the next chapter, we delve deeper in techniques of exponential...