Over a million developers have joined DZone.
{{announcement.body}}
{{announcement.title}}

DZone's Guide to

# Time Series Analysis vs. DSP Terminology

### Digital signal processing and time series analysis are very similar — but the terminology that they use is very different. This article is here to clarify the confusion.

· AI Zone ·
Free Resource

Comment (1)

Save
{{ articles[0].views | formatCount}} Views

Did you know that 50- 80% of your enterprise business processes can be automated with AssistEdge?  Identify processes, deploy bots and scale effortlessly with AssistEdge.

Time series analysis and digital signal processing are closely related. Unfortunately, the two fields use different terms to refer to the same things.

Suppose you have a sequence of inputs x[n] and a sequence of outputs y[n] for integers n.

## Moving Average/FIR

If each output depends on a linear combination of a finite number of previous inputs:

...then time series analysis would call this a moving average (MA) model of order q, provided b0 = 1. Note that this might not really be an average, i.e. the bs are not necessarily positive and don't necessarily sum to 1.

Digital signal processing would call this a finite impulse response (FIR) filter of order q.

## Autoregressive/IIR

If each output depends on a linear combination of a finite number of previous outputs:

...then time series analysis would call this an autoregressive (AR) model of order p.

Digital signal processing would call this an infinite impulse response (IIR) filter of order p.

Sometimes, you'll see the opposite sign convention on the as.

## ARMA/IIR

If each output depends on a linear combination of a finite number of previous inputs and outputs:

...then time series analysis would call this an autoregressive moving average (ARMA) model of order (p, q), i.e. p AR terms and q MA terms.

Digital signal processing would call this an infinite impulse response (IIR) filter with q feedforward coefficients and p feedback coefficients. Also, as above, you may see the opposite sign convention on the as.

## ARMA Notation

Box and Jenkins use as for input and zs for output. We'll stick with xs and ys to make the comparison to DSP easier.

Using the backward shift operator B that takes a sample at n to the sample at n-1, the ARMA system can be written as:

...where φ and θ are polynomials, and:

## System Function Notation

In DSP, filters are described by their system function, the z-transform of the impulse response. In this notation (as in Oppenheim and Shafer, for example), we have:

## Related

Consuming AI in byte sized applications is the best way to transform digitally. #BuiltOnAI, EdgeVerve’s business application, provides you with everything you need to plug & play AI into your enterprise.  Learn more.

Topics:
ai ,time series ,data analytics ,dsp ,statistics

Comment (1)

Save
{{ articles[0].views | formatCount}} Views

Published at DZone with permission of

Opinions expressed by DZone contributors are their own.