# [Developing] Lecture Notes on Time Series Analysis

Note: This is assuming the reader already knows basic statistics.

**Time Series** is a collection of observations made sequentially through time.

**Definition of Terms**

**continuous time series**
are time series whose observations are made continuously through time. This type can also be used when the measured variable can only take a discrete set of values.

**discrete time series**
are time series whose observations are taken only at specific times, usually equally spaced. This type can also be used when the measured variable is a continuous variable.

**point process**
are series of events occuring ‘randomly’ through time.

**sampled time series**
are digitized version of continuous time series sampled at equal intervals of time to give a discrete time series.

When successive observations are dependent, future values may be predicted from past observations.

**deterministic time series**
are time series that can be *predicted exactly* from precious observations.

**stochastic time series**
are time series that can be *predicted only partly* from precious observations. This entails thinking of future values as having a probability distribution.

**outliers**
observations that do not appear to be consistent with the rest of the data. (may be valid or just a freak observation from data acquisition mishap)

**robust**
insensitivity to outliers.

**objective of time series analysis**

- description
**Time plot**observations against time. First step in analyzing a time series to obtain a simple descriptive measures of the main properties of the series.

- explanation
**Linear system**converts an input series into an output series by a linear operation.

- prediction
- used to describe SUBJECTIVE methods in inferring possible future values
*forecasting*on the other hand is used to describe OBJCTIVE methods.

- control
- improve or govern some physical or economic system such as keeping a power plant operation process at a high level or on statistical modelling working out an optimal control strategy.

**Usual Techniques used**

- Simple Descriptive techniques
- Autocorrelation
- Analysis in the time domain
- Spectral Density Function
- Linear Systems
- State-Space Models
- Kalman Filter
- Nonlinear and Multivariate Time Series Models

**Contents**

- Simple Descriptive Techniques
- Some Time Series Models
- Fitting Time Series Models in Time Domain
- Forecasting
- Stationary processes in the Frequency Domain
- Spectral Analysis
- Bivariate Processes
- Linear Systems
- State-Space Models and the Kalman Filter
- Nonlinear Models
- Multivariate Time Series Modelling
- Fourier, Laplace and z-Transforms
- My Beloved Dirac delta <3

**Simple Descriptive techniques**
Typical surface level statistics methods in analyzing the data include getting the mean, median, or mode and the standard deviation to quantify location and dispersion of the data set. *Time series analysis is different*.

The previously mentioned summary statistics can be misleading when the time series contains trend, seasonality, inherent systematic components, and correlations to observables.

**Types of Variations**

- Seasonal Variations
- Other Cyclic Variations
- Trend
- Other Irregular Fluctuations

**Types of Variations**

- Stationary Time Series

**The Time Plot**

**Transformations**

- To stabilize the variance.
- To make the seasonal effect additive.
- To make the data normally distributed.

to be continued :)

sample on ARIMA principles, derivations and applications.

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