Time Series

Overview


A time series is a series of measured values with a timestamp attached to each measurement. For example, a series of equity asset prices may be taken at the close of the market every day. Then you would have a series of data points, each with a date and price.


The Standard Model


the standard approach to time series modeling is to find a function that transforms a time series into a white noise series.
{% X_1,X_2,...,X_n \rightarrow W_1, W_2,...,W_n %}
A white noise series is one in which

  • {% \mathbb{E}(W_i) = 0 %}
  • {% \mathbb{E}(W_i * W_i) = constant %}
  • {% \mathbb{E}(W_i * W_j) = 0 %}


The transformation that accomplishes this goal contains all the predictable structure of the time series. In the context of information theory, a white noise series is such that there is no information relevant to the realization of the nth in the first (n-1) items.

Examples

Single Variable Time Series Frameworks


Time series analysis is complicated by the fact that sample points in a time series are not independent from a statistical perspective. This means that a number of statistical tools that rely on random sampling may not be valid. Time series frameworks are developed to present a method to deal with the issue of non-independence.

Topics


  • Multivariable extends the basic ideas of time series analysis to cases where there are multiple variables present. Whereas a single variable time series can only use past values of the time series in analysis, a multi-variable time series can explore relationships among the variables.
  • Diagnostics - is the process of analyzing the results of the modeling process to determine the fit of the model to the available data. Is used to help pick the best model among a set of candidates, as well as to assess the quality of the selected model.
  • Forecasting
  • Simulating Time Series - uses Monte Carlo style techniques to generate random simulations a given time series, which can be used to simplify model analysis.
  • Spectral analysis of a time series is the method of applying Fourier series analysis to a time series. It is complicated by the notion of randomness in the signal, or likewise and signal and noise considerations.
  • Standard Process
  • Recurrent Neural Networks

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