[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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