TSA: Introduction
Introduction
The Concepts of Process, Realization and Model
Figure 1: The interconnection of process, realization, and model.
A Process pertains to the true mechanism or system that generates data set. The data generated from the
process is called realization. This realization is then used as the basis in constructing the (mathematical)
model. This model represents or approximate the system that generates the data. Of course, it doesn’t end
in the model. The model can now be used as a basis for intervention in the process. The updated or new
process will have a new realization. New realization will result to a new model.
Statistical Modeling
Statistical modeling aims at providing more flexible tools for data analysis.
Objectives for Modeling a Structure of a System
• Understanding and describing the generating mechanism: One goal of statistical modeling
is to approximate the reality or the system where the data is observed or generated. The derived
mathematical equation (called statistical model) is used to describe the processes involve in the system.
1
, • Forecasting of future values: The statistical model that approximate the reality can be used to
predict events (or forecast future events).
• Optimal control of a system: By understanding the system and (optionally) predict the future
events, one can make intervention or plan a optimal control in the system to either prevent or make
changes in the system.
Areas in Statistical Modeling
• Time Series Modeling – describes how a particular value is influenced by its past values.
• Spatial Modeling – describes how a particular value is influenced by its “neighboring” value
• Space-Time Modeling – describes how a particular value is influenced by its “neighboring” and
its past value. Regression Modeling – describes how a particular value is influenced by independent
variables or covariates.
Time Series
A time series is a sequence of observations that are arranged according to the time of their outcome. Simply,
it pertains to a series of data points ordered in time.
Common directions/reasons in doing time series analysis
• to study the dynamic structure of a process;
• to investigate the dynamic relationship between variables;
• to perform seasonal adjustment of economic data; and
• to improve regression analysis when the errors are serially correlated
Time Series Data and Cross Section Data
(Discrete) Time Series
• a sequence of values of some variables taken at successively equally spaced time periods like a day, a
week, a quarter, a month, or a year.
• assumed data are measured at equally spaced, discrete time intervals
• a set of observations generated sequentially in time. Hence, they are dependent (correlated) to each
other
Some examples of Time Series:
• Daily exchange rate of Philippine Peso to U.S Dollar from January 1, 2000- December 31, 2019
• Monthly deaths due to respiratory diseases in the Philippines from 1990-2019
• Monthly sales of rice from 1990-2019
• Daily COVId-19 infected individuals from February 2020 to November 1, 2020.
Cross Section Data
• a sequence of values of some variables taken for a specific period of time and for different entities.
• usually uncorrelated observations are observed in cross section data.
Some examples:
• This day exchange rate of national currency of Asian countries to U.S Dollar.
• Grade point average of the freshmen MSU-IIT students. Sentiment of 500 customers about the service
of the company.
Time Series Decomposition
Time Series components
A time series can be decomposed in to four major components:
2
, Trend Tt
• refers to the long term upward or downward movement that characterizes a time series over a period of
time.
• reflects the long run growth or decline in the time series.
• for example, possible causes of sales movement are increase in total population, market growth or
changes in consumer tastes.
• not necessarily linear and can be estimated by performing a linear or nonlinear regression of the
seasonally adjusted series on time.
plot(co2,ylab= "CO2 Concentration (ppm)")
360
CO2 Concentration (ppm)
350
340
330
320
1960 1970 1980 1990
Time
Figure 2: Monthly Mauna Loa Atmospheric CO2 Concentration from 1959 to 1997(Box, Jenkins and Reinsel,
1976).
plot(decompose(co2)$trend,ylab= "CO2 Concentration (ppm)")
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Introduction
The Concepts of Process, Realization and Model
Figure 1: The interconnection of process, realization, and model.
A Process pertains to the true mechanism or system that generates data set. The data generated from the
process is called realization. This realization is then used as the basis in constructing the (mathematical)
model. This model represents or approximate the system that generates the data. Of course, it doesn’t end
in the model. The model can now be used as a basis for intervention in the process. The updated or new
process will have a new realization. New realization will result to a new model.
Statistical Modeling
Statistical modeling aims at providing more flexible tools for data analysis.
Objectives for Modeling a Structure of a System
• Understanding and describing the generating mechanism: One goal of statistical modeling
is to approximate the reality or the system where the data is observed or generated. The derived
mathematical equation (called statistical model) is used to describe the processes involve in the system.
1
, • Forecasting of future values: The statistical model that approximate the reality can be used to
predict events (or forecast future events).
• Optimal control of a system: By understanding the system and (optionally) predict the future
events, one can make intervention or plan a optimal control in the system to either prevent or make
changes in the system.
Areas in Statistical Modeling
• Time Series Modeling – describes how a particular value is influenced by its past values.
• Spatial Modeling – describes how a particular value is influenced by its “neighboring” value
• Space-Time Modeling – describes how a particular value is influenced by its “neighboring” and
its past value. Regression Modeling – describes how a particular value is influenced by independent
variables or covariates.
Time Series
A time series is a sequence of observations that are arranged according to the time of their outcome. Simply,
it pertains to a series of data points ordered in time.
Common directions/reasons in doing time series analysis
• to study the dynamic structure of a process;
• to investigate the dynamic relationship between variables;
• to perform seasonal adjustment of economic data; and
• to improve regression analysis when the errors are serially correlated
Time Series Data and Cross Section Data
(Discrete) Time Series
• a sequence of values of some variables taken at successively equally spaced time periods like a day, a
week, a quarter, a month, or a year.
• assumed data are measured at equally spaced, discrete time intervals
• a set of observations generated sequentially in time. Hence, they are dependent (correlated) to each
other
Some examples of Time Series:
• Daily exchange rate of Philippine Peso to U.S Dollar from January 1, 2000- December 31, 2019
• Monthly deaths due to respiratory diseases in the Philippines from 1990-2019
• Monthly sales of rice from 1990-2019
• Daily COVId-19 infected individuals from February 2020 to November 1, 2020.
Cross Section Data
• a sequence of values of some variables taken for a specific period of time and for different entities.
• usually uncorrelated observations are observed in cross section data.
Some examples:
• This day exchange rate of national currency of Asian countries to U.S Dollar.
• Grade point average of the freshmen MSU-IIT students. Sentiment of 500 customers about the service
of the company.
Time Series Decomposition
Time Series components
A time series can be decomposed in to four major components:
2
, Trend Tt
• refers to the long term upward or downward movement that characterizes a time series over a period of
time.
• reflects the long run growth or decline in the time series.
• for example, possible causes of sales movement are increase in total population, market growth or
changes in consumer tastes.
• not necessarily linear and can be estimated by performing a linear or nonlinear regression of the
seasonally adjusted series on time.
plot(co2,ylab= "CO2 Concentration (ppm)")
360
CO2 Concentration (ppm)
350
340
330
320
1960 1970 1980 1990
Time
Figure 2: Monthly Mauna Loa Atmospheric CO2 Concentration from 1959 to 1997(Box, Jenkins and Reinsel,
1976).
plot(decompose(co2)$trend,ylab= "CO2 Concentration (ppm)")
3
