This past week while waiting for the renewal of our Matlab license I did some reading on time series analysis. With the aid of technology I will apply some basic time series analysis. I would like time series analysis and applying statistical methods to data to be the primary focus of my internship moving forward. Time series analysis has application in solving financial problems. After graduate school in mathematics I plan to specialize in the application of mathematical and statistical methods to financial problems. This blog post will focus on time series analysis definitions, characteristics, objectives, and approaches. Time series is a sequence or collection of random variables with some similarity in terms of the probability distribution, called a stochastic process. Stochastic process is a collection of random variables, defined on a probability space. Note: the term "time series" is also used to refer to the realization of such a process (observed time series). Here are some important characteristics of time series.
1. Trend: long term increase or decrease in the data over time
2. Seasonality: influenced by seasonal factors (e.g. quarter of the year, month, or day of the week)
3. Periodicity: exact repetition in a regular pattern (seasonal series often called periodic, although they do not exactly repeat themselves)
4. Cyclical trend: data exhibit rises and falls that are not of a fixed period
5. Heteroskedasticity: varying variance with time
6. Dependence: positive (successive observations are similar) or negative (successive observations are dissimilar)
Here are the objectives of time series analysis.
1. Describe the data. Plot the data and obtain simple descriptive measures of the main properties of the series.
2. Explain the data. Find a model to describe the time dependence in data. The model relies heavily on the first step, exploratory data analysis can provide insight on the type of dependence in the data.
3. Forecasting is the third objective. Given a finite sample from the series(observations), forecast the next value of the next several values.
4. The final step is control/tuning. After forecasting, adjust various control/tune parameters.
Here are the basic time series approaches.
1. Time domain approach: assume that correlation between adjacent points in time can be explained through dependence of the current value on past values
2. Frequency domain approach: characteristics of interest relate to periodic(systematic) sinusoidal variations in the data, often caused by biological, physical, or environmental phenomena
 Georgia Tech Time Series Analysis Mooc https://courses.edx.org/courses/course-v1:GTx+ISYE6402+2T2018/course/