Practical Time Series Analysis: Master Time Series Data Processing, Visualization, and Modeling using Python

Cross-sectional data or cross-section of a population is obtained by taking observations from multiple individuals at the same point in time. Cross-sectional data can comprise of observations taken at different points in time, however, in such cases time itself does not play any significant role in the analysis. SAT scores of high school students in a particular year is an example of cross-sectional data. Gross domestic product of countries in a given year is another example of cross-sectional data. Data for customer churn analysis is another example of cross-sectional data. Note that, in case of SAT scores of students and GDP of countries, all the observations have been taken in a single year and this makes the two datasets cross-sectional. In essence, the cross-sectional data represents a snapshot at a given instance of time in both the cases. However, customer data for churn analysis can be obtained from over a span of time such as years and months. But for the purpose of analysis, time might not play an important role and therefore though customer churn data might be sourced from multiple points in time, it may still be considered as a cross-sectional dataset.

Often, analysis of cross-sectional data starts with a plot of the variables to visualize their statistical properties such as central tendency, dispersion, skewness, and kurtosis. The following figure illustrates this with the univariate example of military expenditure as a percentage of Gross Domestic Product of 85 countries in the year 2010. By taking the data from a single year we ensure its cross-sectional nature. The figure combines a normalized histogram and a kernel density plot in order to highlight different statistical properties of the military expense data.

As evident from the plot, military expenditure is slightly left skewed with a major peak at roughly around 1.0 %. A couple of minor peaks can also be observed near 6.0 % and 8.0 %.

Figure 1.1: Example of univariate cross-sectional data

Exploratory data analysis such as the one in the preceding figure can be done for multiple variables as well in order to understand their joint distribution. Let us illustrate a bivariate analysis by considering total debt of the countries' central governments along with their military expenditure in 2010. The following figure shows the joint distributions of these variables as kernel density plots. The bivariate joint distribution shows no clear correlation between the two, except may be for lower values of military expenditure and debt of central government.

Figure 1.2: Example of bi-variate cross-sectional data

The dataset of the example on military expenditures and national debt of countries has been downloaded from the Open Data Catalog of World Bank. You can find the data in the WDIData.csv file under the datasets folder of this book's GitHub repository.

All examples in this book are accompanied by an implementation of the same in Python. So let us now discuss the Python program written to generate the preceding figures. Before we are able to plot the figures, we must read the dataset into Python and familiarize ourselves with the basic structure of the data in terms of columns and rows found in the dataset. Datasets used for the examples and figures, in this book, are in Excel or CSV format. We will use the pandas package to read and manipulate the data. For visualization, matplotlib and seaborn are used. Let us start by importing all the packages to run 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 
import seaborn as sns 

The print_function has been imported from the __future__ package to enable using print as a function for readers who might be using a 2.x version of Python. In Python 3.x, print is by default a function. As this code is written and executed from an IPython notebook, %matplotlib inline ensures that the graphics packages are imported properly and made to work in the HTML environment of the notebook. The os package is used to set the working directory as follows:

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

Now, we read the data from the CSV file and display basic information about it:

data = pd.read_csv('datasets/WDIData.csv') 
print('Column names:', data.columns) 

This gives us the following output showing the column names of the dataset:

Column names: Index([u'Country Name', u'Country Code', u'Indicator Name', 
     u'Indicator Code', u'1960', u'1961', u'1962', u'1963', u'1964',  u'1965', 
       u'1966', u'1967', u'1968', u'1969', u'1970', u'1971', u'1972', u'1973', 
       u'1974', u'1975', u'1976', u'1977', u'1978', u'1979', u'1980', u'1981', 
       u'1982', u'1983', u'1984', u'1985', u'1986', u'1987', u'1988', u'1989', 
       u'1990', u'1991', u'1992', u'1993', u'1994', u'1995', u'1996', u'1997', 
       u'1998', u'1999', u'2000', u'2001', u'2002', u'2003', u'2004', u'2005', 
       u'2006', u'2007', u'2008', u'2009', u'2010', u'2011', u'2012', u'2013', 
       u'2014', u'2015', u'2016'], 
      dtype='object') 

Let us also get a sense of the size of the data in terms of number of rows and columns by running the following line:

print('No. of rows, columns:', data.shape) 

This returns the following output:

No. of rows, columns: (397056, 62) 

This dataset has nearly 400k rows because it captures 1504 world development indicators for 264 different countries. This information about the unique number of indicators and countries can be obtained by running the following four lines:

nb_countries = data['Country Code'].unique().shape[0] 
print('Unique number of countries:', nb_countries) 

As it appears from the structure of the data, every row gives the observations about an indicator that is identified by columns Indicator Name and Indicator Code and for the country, which is indicated by the columns Country Name and Country Code. Columns 1960 through 2016 have the values of an indicator during the same period of time. With this understanding of how the data is laid out in the DataFrame, we are now set to extract the rows and columns that are relevant for our visualization.

Let us start by preparing two other DataFrames that get the rows corresponding to the indicators Total Central Government Debt (as % of GDP) and Military expenditure (% of GDP) for all the countries. This is done by slicing the original DataFrame as follows:

central_govt_debt = data.ix[data['Indicator Name']=='Central government debt, total (% of GDP)'] 
military_exp = data.ix[data['Indicator Name']=='Military expenditure (% of GDP)'] 

The preceding two lines create two new DataFrames, namely central_govt_debt and military_exp. A quick check about the shapes of these DataFrames can be done by running the following two lines:

print('Shape of central_govt_debt:', central_govt_debt.shape) 
print('Shape of military_exp:', military_exp.shape) 

These lines return the following output:

Shape of central_govt_debt: (264, 62) 
Shape of military_exp: (264, 62) 

These DataFrames have all the information we need. In order to plot the univariate and bivariate cross-sectional data in the preceding figure, we need the column 2010. Before we actually run the code for plotting, let us quickly check if the column 2010 has missing. This is done by the following two lines:

central_govt_debt['2010'].describe() 
military_exp['2010'].describe() 

Which generate the following outputs respectively:

count     93.000000 
mean      52.894412 
std       30.866372 
min        0.519274 
25%             NaN 
50%             NaN 
75%             NaN 
max      168.474953 
Name: 2010, dtype: float64 
count    194.000000 
mean       1.958123 
std        1.370594 
min        0.000000 
25%             NaN 
50%             NaN 
75%             NaN 
max        8.588373 
Name: 2010, dtype: float64 

Which tells us that the describe function could not compute the 25^th, 50^th, and 75^th quartiles for either, hence there are missing values to be avoided.

Additionally, we would like the Country Code column to be the row indices. So the following couple of lines are executed:

central_govt_debt.index = central_govt_debt['Country Code'] 
military_exp.index = military_exp['Country Code'] 

Next, we create two pandas.Series by taking non-empty 2010 columns from central_govt_debt and military_exp. The newly created Series objects are then merged into to form a single DataFrame:

central_govt_debt_2010 = central_govt_debt['2010'].ix[~pd.isnull(central_govt_debt['2010'])] 
military_exp_2010 = military_exp['2010'].ix[~pd.isnull(military_exp['2010'])] 
data_to_plot = pd.concat((central_govt_debt_2010, military_exp_2010), axis=1) 
data_to_plot.columns = ['central_govt_debt', 'military_exp'] 
data_to_plot.head() 

The preceding lines return the following table that shows that not all countries have information on both Central Government Debt and Military Expense for the year 2010:

To plot, we have to take only those countries that have both central government debt and military expense. Run the following line, to filter out rows with missing values:

data_to_plot = data_to_plot.ix[(~pd.isnull(data_to_plot.central_govt_debt)) & (~pd.isnull(data_to_plot.military_exp)), :] 

The first five rows of the filtered DataFrame are displayed by running the following line:

data_to_plot.head() 

The preceding table has only non-empty values and we are now ready to generate the plots for the cross-sectional data. The following lines of code generate the plot on the univariate cross-sectional data on military expense:

plt.figure(figsize=(5.5, 5.5)) 
g = sns.distplot(np.array(data_to_plot.military_exp), norm_hist=False) 
g.set_title('Military expenditure (% of GDP) of 85 countries in 2010') 

The plot is saved as a png file under the plots/ch1 folder of this book's GitHub repository. We will also generate the bivariate plot between military expense and central government debt by running the following code:

plt.figure(figsize=(5.5, 5.5)) 
g = sns.kdeplot(data_to_plot.military_exp, data2=data_to_plot.central_govt_debt) 
g.set_title('Military expenditures & Debt of central governments in 2010') 

The example of cross-sectional data discussed earlier is from the year 2010 only. However, instead if we consider only one country, for example United States, and take a look at its military expenses and central government debt for a span of 10 years from 2001 to 2010, that would get two time series - one about the US federal military expenditure and the other about debt of US federal government. Therefore, in essence, a time series is made up of quantitative observations on one or more measurable characteristics of an individual entity and taken at multiple points in time. In this case, the data represents yearly military expenditure and government debt for the United States. Time series data is typically characterized by several interesting internal structures such as trend, seasonality, stationarity, autocorrelation, and so on. These will be conceptually discussed in the coming sections in this chapter.

The internal structures of time series data require special formulation and techniques for its analysis. These techniques will be covered in the following chapters with case studies and implementation of working code in Python.

The following figure plots the couple of time series we have been talking about:

Figure 1.3: Examples of time series data

In order to generate the preceding plots we will extend the code that was developed to get the graphs for the cross-sectional data. We will start by creating two new Series to represent the time series of military expenses and central government debt of the United States from 1960 to 2010:

central_govt_debt_us = central_govt_debt.ix[central_govt_debt['Country Code']=='USA', :].T 
military_exp_us = military_exp.ix[military_exp['Country Code']=='USA', :].T 

The two Series objects created in the preceding code are merged to form a single DataFrame and sliced to hold data for the years 2001 through 2010:

data_us = pd.concat((military_exp_us, central_govt_debt_us), axis=1) 
index0 = np.where(data_us.index=='1960')[0][0] 
index1 = np.where(data_us.index=='2010')[0][0] 
data_us = data_us.iloc[index0:index1+1,:] 
data_us.columns = ['Federal Military Expenditure', 'Debt of Federal  Government'] 
data_us.head(10)  

The data prepared by the preceding code returns the following table:

The preceding table shows that data on federal military expenses and federal debt is not available from several years starting from 1960. Hence, we drop the rows with missing values from the Dataframe data_us before plotting the time series:

data_us.dropna(inplace=True)
print('Shape of data_us:', data_us.shape)

As seen in the output of the print function, the DataFrame has twenty three rows after dropping the missing values:

Shape of data_us: (23, 2)

After dropping rows with missing values, we display the first ten rows of data_us are displayed as follows:

Finally, the time series are generated by executing the following code:

# Two subplots, the axes array is 1-d
f, axarr = plt.subplots(2, sharex=True)
f.set_size_inches(5.5, 5.5)
axarr[0].set_title('Federal Military Expenditure during 1988-2010 (% of GDP)')
data_us['Federal Military Expenditure'].plot(linestyle='-', marker='*', color='b', ax=axarr[0])
axarr[1].set_title('Debt of Federal Government during 1988-2010 (% of GDP)')
data_us['Debt of Federal Government'].plot(linestyle='-', marker='*', color='r', ax=axarr[1]) 

So far, we have seen data taken from multiple individuals but at one point in time (cross-sectional) or taken from an individual entity but over multiple points in time (time series). However, if we observe multiple entities over multiple points in time we get a panel data also known as longitudinal data. Extending our earlier example about the military expenditure, let us now consider four countries over the same period of 1960-2010. The resulting data will be a panel dataset. The figure given below illustrates the panel data in this scenario. Rows with missing values, corresponding to the period 1960 to 1987 have been dropped before plotting the data.

Figure 1.4: Example of panel data

Let us prepare the data that is required to plot the preceding figure. We will continue to expand the code we have used for the cross-sectional and time series data previously in this chapter. We start by creating a DataFrame having the data for the four companies mentioned in the preceding plot. This is done as follows:

chn = data.ix[(data['Indicator Name']=='Military expenditure (% of GDP)')&\
               (data['Country Code']=='CHN'),index0:index1+1
             ]
chn = pd.Series(data=chn.values[0], index=chn.columns)
chn.dropna(inplace=True)

usa = data.ix[(data['Indicator Name']=='Military expenditure (% of GDP)')&\
               (data['Country Code']=='USA'),index0:index1+1
             ]
usa = pd.Series(data=usa.values[0], index=usa.columns)
usa.dropna(inplace=True)

ind = data.ix[(data['Indicator Name']=='Military expenditure (% of GDP)')&\
               (data['Country Code']=='IND'),index0:index1+1
             ]
ind = pd.Series(data=ind.values[0], index=ind.columns)
ind.dropna(inplace=True)

gbr = data.ix[(data['Indicator Name']=='Military expenditure (% of GDP)')&\
               (data['Country Code']=='GBR'),index0:index1+1
             ]
gbr = pd.Series(data=gbr.values[0], index=gbr.columns)
gbr.dropna(inplace=True)

Now that the data is ready for all five countries, we will plot them using the following code:

plt.figure(figsize=(5.5, 5.5))
usa.plot(linestyle='-', marker='*', color='b')
chn.plot(linestyle='-', marker='*', color='r')
gbr.plot(linestyle='-', marker='*', color='g')
ind.plot(linestyle='-', marker='*', color='y')
plt.legend(['USA','CHINA','UK','INDIA'], loc=1)
plt.title('Miltitary expenditure of 5 countries over 10 years')
plt.ylabel('Military expenditure (% of GDP)')
plt.xlabel('Years')s

The discussion about different types of data sets the stage for a closer look at time series. We will start doing that by understanding the special properties of data that can be typically found in a time series or panel data with inherent time series in it.

When a time series exhibits an upward or downward movement in the long run, it is said to have a general trend. A quick way to check the presence of general trend is to plot the time series as in the following figure, which shows CO₂ concentrations in air measured during 1974 through 1987:

Figure 1.5: Time series of CO2 readings with an upward trend

However, general trend might not be evident over a short run of the series. Short run effects such as seasonal fluctuations and irregular variations cause the time series to revisit lower or higher values observed in the past and hence can temporarily obfuscate any general trend. This is evident in the same time series of CO₂ concentrations when zoomed in over the period of 1979 through 1981, as shown in the following figure. Hence to reveal general trend, we need a time series that dates substantially back in the past.

Figure 1.6: Shorter run of CO2 readings time series which is not able to reveal general trend

The general trend in the time series is due to fundamental shifts or systemic changes of the process or system it represents. For example, the upward movement of CO2 concentrations during 1974 through 1987 can be attributed to the gradual rise in automobiles and industrialization over these years.

A general trend is commonly modeled by setting up the time series as a regression against time and other known factors as explanatory variables. The regression or trend line can then be used as a prediction of the long run movement of the time series. Residuals left by the trend line is further analyzed for other interesting properties such as seasonality, cyclical behavior, and irregular variations.

Now, let us go through the code that generated the preceding plots on CO₂ concentrations. We will also show how to build a trend model using linear regression on the time index (which in this case is the index of the year in the data) as explanatory variable and the CO₂ concentration as the dependent variable. But first, let us load the data in a pandas.DataFrame.

We start by importing the required packages as follows:

from __future__ import print_function 
import os 
import pandas as pd 
import numpy as np 
%matplotlib inline 
from matplotlib import pyplot as plt 
import seaborn as sns 
os.chdir('D:\Practical Time Series') 
data = pd.read_excel('datasets/Monthly_CO2_Concentrations.xlsx',                                  converters={'Year': np.int32, 'Month':  np.int32}) 
data.head() 

We have passed the argument converters to the read_excel function in order to make sure that columns Year and Month are assigned the integer (np.int32) datatype. The preceding lines of code will generate the following table:

Before plotting we must remove all columns having missing values. Besides, the DataFrame is sorted in ascending order of Year and Month. These are done as follows:

data = data.ix[(~pd.isnull(data['CO2']))&\ 
               (~pd.isnull(data['Year']))&\ 
               (~pd.isnull(data['Month']))] 
data.sort_values(['Year', 'Month'], inplace=True) 

Finally, the plot for the time period 1974 to 1987 is generated by executing the following lines:

plt.figure(figsize=(5.5, 5.5))
data['CO2'].plot(color='b')
plt.title('Monthly CO2 concentrations')
plt.xlabel('Time')
plt.ylabel('CO2 concentratition')
plt.xticks(rotation=30)

The zoomed-in version of the data for the time period 1980 to 1981 is generated by after the DataFrame for these three years:

plt.figure(figsize=(5.5, 5.5))
data['CO2'].loc[(data['Year']==1980) | (data['Year']==1981)].plot(color='b')
plt.title('Monthly CO2 concentrations')
plt.xlabel('Time')
plt.ylabel('CO2 concentratition')
plt.xticks(rotation=30)

Next, let us fit the trend line. For this we import the LinearRegression class from scikit-learn and fit a linear model on the time index:

from sklearn.linear_model import LinearRegression 
trend_model = LinearRegression(normalize=True, fit_intercept=True) 
trend_model.fit(np.array(data.index).reshape((-1,1)), data['CO2']) 
print('Trend model coefficient={} and intercept={}'.format(trend_model.coef_[0], 
                                                           trend_model.intercept_) 
      ) 

This produces the following output:

Trend model coefficient=0.111822078545 and intercept=329.455422234 

The residuals obtained from the trend line model are shown in the following figure and appear to have seasonal behaviour, which is discussed in the next sub section.

The residuals are calculated and plotted in the preceding by the following lines of code:

residuals = np.array(data['CO2']) - trend_model.predict(np.array(data.index).reshape((-1,1)))
plt.figure(figsize=(5.5, 5.5))
pd.Series(data=residuals, index=data.index).plot(color='b')
plt.title('Residuals of trend model for CO2 concentrations')
plt.xlabel('Time')
plt.ylabel('CO2 concentratition')
plt.xticks(rotation=30)

Figure 1.7: Residuals from a linear model of the general trend in CO2 readings

Seasonality manifests as repetitive and period variations in a time series. In most cases, exploratory data analysis reveals the presence of seasonality. Let us revisit the de-trended time series of the CO₂ concentrations. Though the de-trended line series has constant mean and constant variance, it systematically departs from the trend model in a predictable fashion.

Seasonality is manifested as periodic deviations such as those seen in the de-trended observations of CO₂ emissions. Peaks and troughs in the monthly sales volume of seasonal goods such as Christmas gifts or seasonal clothing is another example of a time series with seasonality.

A practical technique of determining seasonality is through exploratory data analysis through the following plots:

Seasonal sub series plot

For a known periodicity of seasonal variations, seasonal sub series redraws the original series over batches of successive time periods. For example, the periodicity in the CO₂ concentrations is 12 months and based on this a seasonal sub series plots on mean and standard deviation of the residuals are shown in the following figure. To visualize seasonality in the residuals, we create quarterly mean and standard deviations.

A seasonal sub series reveals two properties:

• Variations within seasons as within a batch of successive months
• Variations between seasons as between batches of successive months

Figure 1.8: Quarterly mean of the residuals from a linear model of the general trend in CO2 readings

Figure 1.9: Quarterly standard deviation of the residuals from a linear model of the general trend in CO2 readings

Let us now describe the code used for generating the preceding plots. First, we need to add the residuals and quarter labels to the CO2 concentrations DataFrame. This is done as follows:

data['Residuals'] = residuals 
month_quarter_map = {1: 'Q1', 2: 'Q1', 3: 'Q1', 
                     4: 'Q2', 5: 'Q2', 6: 'Q2', 
                     7: 'Q3', 8: 'Q3', 9: 'Q3', 
                     10: 'Q4', 11: 'Q4', 12: 'Q4' 
                    } 
data['Quarter'] = data['Month'].map(lambda m: month_quarter_map.get(m)) 

Next, the seasonal mean and standard deviations are computed by grouping by the data over Year and Quarter:

seasonal_sub_series_data = data.groupby(by=['Year', 'Quarter'])['Residuals']\ 
                           .aggregate([np.mean, np.std]) 

This creates the new DataFrame as seasonal_sub_series_data, which has quarterly mean and standard deviations over the years. These columns are renamed as follows:

seasonal_sub_series_data.columns = ['Quarterly Mean', 'Quarterly Standard Deviation'] 

Next, the quarterly mean and standard deviations are plotted by running the following lines of code:

#plot quarterly mean of residuals
plt.figure(figsize=(5.5, 5.5))
seasonal_sub_series_data['Quarterly Mean'].plot(color='b')
plt.title('Quarterly Mean of Residuals')
plt.xlabel('Time')
plt.ylabel('CO2 concentratition')
plt.xticks(rotation=30)

#plot quarterly standard deviation of residuals
plt.figure(figsize=(5.5, 5.5))
seasonal_sub_series_data['Quarterly Standard Deviation'].plot(color='b')
plt.title('Quarterly Standard Deviation of Residuals')
plt.xlabel('Time')
plt.ylabel('CO2 concentratition')
plt.xticks(rotation=30)

Multiple box plots

The seasonal sub series plot can be more informative when redrawn with seasonal box plots as shown in the following figure. A box plot displays both central tendency and dispersion within the seasonal data over a batch of time units. Besides, separation between two adjacent box plots reveal the within season variations:

Figure 1.10: Quarterly boxplots of the residuals from a linear model of the general trend in CO2 readings

The code for generating the box plots is as follows:

plt.figure(figsize=(5.5, 5.5)) 
g = sns.boxplot(data=data, y='Residuals', x='Quarter') 
g.set_title('Quarterly Mean of Residuals') 
g.set_xlabel('Time') 
g.set_ylabel('CO2 concentratition') 

Cyclical changes are movements observed after every few units of time, but they occur less frequently than seasonal fluctuations. Unlike seasonality, cyclical changes might not have a fixed period of variations. Besides, the average periodicity for cyclical changes would be larger (most commonly in years), whereas seasonal variations are observed within the same year and corresponds to annual divisions of time such as seasons, quarters, and periods of festivity and holidays and so on.

A long run plot of the time series is required to identify cyclical changes that can occur, for example, every few years and manifests as repetitive crests and troughs. In this regard, time series related to economics and business often show cyclical changes that correspond to usual business and macroeconomic cycles such as periods of recessions followed by every of boom, but are separated by few years of time span. Similar to general trends, identifying cyclical movements might require data that dates significantly back in the past.

The following figure illustrates cyclical changes occurring in inflation of consumer price index (CPI) for India and United States over the period of 1960 through 2016. Both the countries exhibit cyclical patterns in CPI inflation, which is roughly over a period of 2-2.5 years. Moreover, CPI inflation of India has larger variations pre-1990 than that seen after 1990.

Figure 1.11: Example of cyclical movements in time series data

Source: The data for the preceding figure has been downloaded from http://datamarket.com, which maintains data on time series from a wide variety of subjects.

The code written to generate the figure is as follows:

inflation = pd.read_excel('datasets/inflation-consumer-prices-annual.xlsx', parse_dates=['Year']) 
plt.figure(figsize=(5.5, 5.5)) 
plt.plot(range(1960,2017), inflation['India'], linestyle='-', marker='*', color='r') 
plt.plot(range(1960,2017), inflation['United States'], linestyle='-', marker='.', color='b') 
plt.legend(['India','United States'], loc=2) 
plt.title('Inflation in Consumer Price Index') 
plt.ylabel('Inflation') 
plt.xlabel('Years') 

Referring to our model that expresses a time series as a sum of four components, it is noteworthy that in spite of being able to account for the three other components, we might still be left with an irreducible error component that is random and does not exhibit systematic dependency on the time index. This fourth component reflects unexpected variations in the time series. Unexpected variations are stochastic and cannot be framed in a mathematical model for a definitive future prediction. This type of error is due to lack of information about explanatory variables that can model these variations or due to presence of a random noise.

The zero-mean models have a constant mean and constant variance and shows no predictable trends or seasonality. Observations from a zero mean model are assumed to be independent and identically distributed (iid) and represent the random noise around a fixed mean, which has been deducted from the time series as a constant term.

Let us consider that X₁, X₂, ... ,X_n represent the random variables corresponding to n observations of a zero mean model. If x₁, x₂, ... ,x_n are n observations from the zero mean time series, then the joint distribution of the observations is given as a product of probability mass function for every time index as follows:

P(X1 = x1,X2 = x2 , ... , Xn = xn) = f(X1 = x1) f(X2 = x2) ... f(Xn = xn)

Most commonly f(X_t = x_t) is modeled by a normal distribution of mean zero and variance σ ², which is assumed to be the irreducible error of the model and hence treated as a random noise. The following figure shows a zero-mean series of normally distributed random noise of unit variance:

Figure 1.12: Zero-mean time series

The preceding plot is generated by the following code:

import os 
import numpy as np 
%matplotlib inline 
from matplotlib import pyplot as plt 
import seaborn as sns 
os.chdir('D:/Practical Time Series/') 
zero_mean_series = np.random.normal(loc=0.0, scale=1., size=100) 

The zero mean with constant variance represents a random noise that can assume infinitely possible real values and is suited for representing irregular variations in the time series of a continuous variable. However in many cases, the observable state of the system or process might be discrete in nature and confined to a finite number of possible values s₁,s₂, ... , s_m. In such cases, the observed variable (X) is assumed to obey the multinomial distribution, P(X = s₁ )= p₁, P(X = s₂ ) = p₂,…,P(X = s_m) = p_m such that p₁ + p₂ + ... + p_m = 1. Such a time series is a discrete stochastic process.

Multiple throws a dice over time is an example of a discrete stochastic process with six possible outcomes for any throw. A simpler discrete stochastic process is a binary process such as tossing a coin such as only two outcomes namely head and tail. The following figure shows 100 runs from a simulated process of throwing a biased dice for which probability of turning up an even face is higher than that of showing an odd face. Note the higher number of occurrences of even faces, on an average, compared to the number of occurrences of odd faces.

A random walk is given as a sum of n iids, which has zero mean and constant variance. Based on this definition, the realization of a random walk at time index t is given by the sum S = x₁ + x₂ + ... + x_n. The following figure shows the random walk obtained from iids, which vary according to a normal distribution of zero mean and unit variance.

The random walk is important because if such behavior is found in a time series, it can be easily reduced to zero mean model by taking differences of the observations from two consecutive time indices as S_t - S_t-1 = x_t is an iid with zero mean and constant variance.

Figure 1.13: Random walk time series

The random walk in the preceding figure can be generated by taking the cumulative sum of the zero mean model discussed in the previous section. The following code implements this:

random_walk = np.cumsum(zero_mean_series) 
plt.figure(figsize=(5.5, 5.5)) 
g = sns.tsplot(random_walk) 
g.set_title('Random Walk') 
g.set_xlabel('Time index') 

This type of model aims to capture the long run trend in the time series that can be fitted as linear regression of the time index. When the time series does not exhibit any periodic or seasonal fluctuations, it can be expressed just as the sum of the trend and the zero mean model as x_t = μ(t) + y_t, where μ(t) is the time-dependent long run trend of the series.

The choice of the trend model μ(t) is critical to correctly capturing the behavior of the time series. Exploratory data analysis often provides hints for hypothesizing whether the model should be linear or non-linear in t. A linear model is simply μ(t) = wt + b, whereas quadratic model is μ(t) = w₁t + w₂t² + b. Sometimes, the trend can be hypothesized by a more complex relationship in terms of the time index such as μ(t) = w₀t^p + b.

The weights and biases in the trend modes such as the ones discussed previously is obtained by running a regression with t as the explanatory variable and μ as the explained. The residuals x_t - μ(t) of the trend model is considered to the irreducible noise and as realization of the zero mean component y_t.

Seasonality manifests as periodic and repetitive fluctuations in a time series and hence are modelled as sum of weighted sum of sine waves of known periodicity. Assuming that long run trend has been removed by a trend line, the seasonality model can be expressed as x_t = s_t + y_t, where the seasonal variation

Seasonality models are also known as harmonic regression model as they attempt to fit the sum of multiple sin waves.

The four models described here are building blocks of a fully-fledged time series model. As you might have gathered by now, a zero sum model represents irreducible error of the system and all of other three models aim to transform a given time series to the zero sum models through suitable mathematical transformations. To get forecasts in terms of the original time series, relevant inverse transformations are applied.

The upcoming chapters detail the four models discussed here. However, we have reached a point where we can summarize the generic approach of a time series analysis in the following four steps:

These steps are most commonly enough to develop mathematical models for most time series. The individual trend and seasonality models can be simple or complex depending on the original time series and the application.