Statistical Analysis Stack Exchange
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Questions tagged [moving-average]

In time series analysis, the moving-average (MA) model is a common approach for modeling univariate time series. The moving-average model specifies that the output variable depends linearly on the current and various past values of a stochastic (imperfectly predictable) term.

$\{X_t\}$ is a Moving Average Process of order $q$, MA($q$), if we can write

$$X_t = Z_t + \theta_1Z_{t-1} + \dots + \theta_qZ_{t-q},$$

where $Z_t \sim N(0,\sigma^2)$ and $\theta_i$'s are scalars.

  • $X_t$ is a linear combination of i.i.d. mean 0 random variables
  • $X_t$ can also be thought of as a weighted sum of the past $q$ forecast errors and a contemporanous error term.

References:

https://en.wikipedia.org/wiki/Moving-average_model http://www.maths.qmul.ac.uk/~bb/TS_Chapter4_3&4.pdf

311 questions
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Real-life examples of moving average processes

Can you give some real-life examples of time series for which a moving average process of order $q$, i.e. $$ y_t = \sum_{i=1}^q \theta_i \varepsilon_{t-i} + \varepsilon_t, \text{ where } \varepsilon_t \sim \mathcal{N}(0, \sigma^2) $$ has some a…
weez13
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Why are MA(q) time series models called "moving averages"?

When I read "moving average" in relation to a time series, I think something like $\frac{(x_{t-1} + x_{t-2} + x_{t-3})}3$, or perhaps a weighted average like $0.5x_{t-1} + 0.3x_{t-2} + 0.2x_{t-3}$. (I realize these are actually AR(3) models, but…
Stats newb
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Under what circumstances is an MA process or AR process appropriate?

I understand that if a process depends on previous values of itself, then it is an AR process. If it depends on previous errors, then it is an MA process. When would one of either of these two situations occur? Does anyone have a solid example that…
tumultous_rooster
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How does ACF & PACF identify the order of MA and AR terms?

It's been more than 2 years that I am working on different time series. I have read on many articles that ACF is used to identify order of MA term, and PACF for AR. There is a thumb rule that for MA, the lag where ACF shuts off suddenly is the order…
Arpit Sisodia
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Extract data points from moving average?

Is it possible to extract data points from moving average data? In other words, if a set of data only has simple moving averages of the previous 30 points, is it possible to extract the original data points? If so, how?
user16679
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7 answers

Correcting for outliers in a running average

We have a daemon that reads in data from some sensors, and among the things it calculates (besides simply just reporting the state) is the average time it takes for the sensors to change from one value to another. It keeps a running average of 64…
Edward Z. Yang
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Why do we care if an MA process is invertible?

I am having trouble understanding why we care if an MA process is invertible or not. Please correct me if I'm wrong, but I can understand why we care whether or not an AR process is causal, ie if we can "re-write it," so to speak, as the sum of some…
agra94
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What guarantees the existence of a finite representation of the Wold decomposition? Mechanics and Intuition

Every covariance stationary process can be written as a linear, infinite distributed lag of white noise. In other words, every covariance stationary process has a Wold representation. Then we go on to say that this infinite distributed lag of white…
ColorStatistics
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Writing AR(1) as a MA($\infty$) process

The AR(1) process is $$ X_t = \phi X_{t-1} + \varepsilon_t $$ if we use this formula recursively, we get $$ X_t = \phi(\phi X_{t-2} + \varepsilon_{t-1}) + \varepsilon_t = \phi^2X_{t-2} + \phi\varepsilon_{t-1} + \varepsilon_t = \cdots = \phi^k…
Frank Vel
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Help in Expectation Maximization from paper :how to include prior distribution?

The Question is based on the paper titled : Image reconstruction in diffuse optical tomography using the coupled radiative transport–diffusion model Download link The Authors apply EM algorithm with $l_1$ sparsity regularization of an unknown vector…
SKM
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Tuning an exponential moving average to a moving window mean?

The alpha parameter of an exponential moving average defines the smoothing that the average applies to a time series. In a similar way, the window size of a moving window mean also defines the smoothing. Is there some way to tune the alpha parameter…
naught101
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What is the implication of unit root of MA?

A ARMA(p,q) process is weakly stationary, iff the root of its AR part is not on the unit circle. So its weak stationarity doesn't depend on its MA part. But what can the positions of the roots of its MA part imply? In the unit root tests for ARIMA,…
Tim
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Gauss-Newton method for MA parameter estimation

Please check my solution below for estimating Moving Average parameter using the Gauss-Newton (Linearization) method. I consider MA(1). MA(1) model: $$z_t=a_t-\theta_1a_{t-1}.$$ Solution: The residual of this model…
Al-Ahmadgaid Asaad
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Do non-invertible MA models imply that the effect of past observations increases with the distance?

Update (2019-06-25): changing title from "Do non-invertible MA models make sense?" to distinguish it from Question 333802. While reviewing MA($q$) models, I came across these slides (Alonso and Garcia-Martos, 2012). The authors state that, while all…
Ben Ogorek
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Intuition behind the characteristic equation of an AR or MA process

Ok, so I've just started learning Time Series Analysis. We can write an $MA(q)$ process as $Y_t = \theta(L) \varepsilon_t$ and an $AR(p)$ process as $\varepsilon_t = \phi(L) Y_t$ in terms of the lag operator. Then, with no explanation (from my…
mynameisthis
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