Time Series Analysis: ACF and Stationarity
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Questions and Answers

What does the Autocorrelation Function (ACF) measure?

  • The variance of a time series.
  • The correlation between a time series and its lagged values. (correct)
  • The correlation between two different time series.
  • The mean of a time series.

    • The first value in an ACF plot, p0, is always 0.

    False (B)

    What is the primary characteristic of the ACF for a stationary process?

    The ACF decays quickly to zero as the lag increases.

    A stochastic process is considered weakly stationary if its mean, variance, and autocovariance do not change over ______.

    <p>time</p> Signup and view all the answers

    Match the following time series processes with their ACF behavior:

    <p>Random Walk = ACF declines slowly AR(p) Process = ACF decays geometrically MA(q) Process = ACF cuts off after lag q</p> Signup and view all the answers

    What is the defining characteristic of a random walk?

    <p>The value at each time step is the sum of the previous value and a random noise term. (A)</p> Signup and view all the answers

    An MA(q) process is not stationary.

    <p>False (B)</p> Signup and view all the answers

    Write the general equation for an AR(p) process

    <p>Yt = φ1Yt−1 + φ2Yt−2 + ··· + φpYt-p + εt</p> Signup and view all the answers

    For an AR(1) process to be stationary, the absolute value of the coefficient φ1 must be less than ______.

    <p>1</p> Signup and view all the answers

    What happens to the ACF of an MA(q) process after lag q?

    <p>It cuts off (becomes approximately zero) (A)</p> Signup and view all the answers

    Which function in R is commonly used to fit ARMA models?

    <p>auto.arima() (A), arma() (D)</p> Signup and view all the answers

    The ACF of a random walk process typically declines rapidly.

    <p>False (B)</p> Signup and view all the answers

    What does 'p' represent in an AR(p) model?

    <p>the number of lagged values used in the model</p> Signup and view all the answers

    A time series model assumes _ _ is a key assumption.

    <p>weak stationarity</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Transition Matrix = Describes the probabilities of moving from one state to another Stationary Distribution = A probability distribution that remains unchanged over time Irreducible Markov Chain = A chain with the possibility to get from any state to any other state Ergodic Markov Chain = A chain that is both irreducible and aperiodic</p> Signup and view all the answers

    What is a key property of the transition matrix P?

    <p>The rows of the matrix sum to 1. (B)</p> Signup and view all the answers

    The n-step transition probabilities can be obtained by subtracting the transition matrix from the identity matrix.

    <p>False (B)</p> Signup and view all the answers

    What does the equation π = πP represent in the context of Markov Chains?

    <p>it represents the stationary distribution</p> Signup and view all the answers

    A Markov chain is _ _ if it is possible to reach any state from any other state in a finite number of steps.

    <p>irreducible</p> Signup and view all the answers

    What does it mean for a Markov Chain to be ergodic?

    <p>It is both irreducible and aperiodic. (C)</p> Signup and view all the answers

    Flashcards

    Autocorrelation Function (ACF)

    A measure of the correlation between a time series and its past values at different lags.

    ACF Plot

    A plot that visually represents the autocorrelation coefficients of a time series for different lags.

    Weakly Stationary Process

    A stochastic process whose statistical properties like mean, variance, and autocovariance remain constant over time.

    Random Walk

    A type of stochastic process where each value is the sum of the previous value and a random noise term.

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    AR(p) Process

    A time series model where the current value depends linearly on its past p values and a random noise term.

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    Stationarity Condition for AR(p)

    A condition that must be met for an AR(p) process to be stationary. The roots of the characteristic equation must lie outside the unit circle.

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    ACF Behavior in AR(p)

    The ACF of an AR(p) process decays geometrically, with the speed of decay determined by the coefficients.

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    MA(q) Process

    A time series model where the current value depends linearly on the past q random noise terms.

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    Stationarity Condition for MA(q)

    MA(q) processes are always stationary as they are finite linear combinations of stationary (white noise) terms.

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    ACF Behavior in MA(q)

    The ACF of an MA(q) process stops after lag q, with correlation close to zero for lags greater than q.

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    Weak Stationarity

    A process where the correlation between observations decreases as the time lag increases.

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    AR(p) Model

    A statistical model for stationary time series that uses past values to predict future values.

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    MA(q) Model

    A statistical model for stationary time series that uses past noise terms to predict future values.

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    Transition Matrix

    A square matrix describing the probabilities of transitioning between states in a Markov Chain over one step.

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    Stationary Distribution

    A probability distribution that remains unchanged as the Markov Chain evolves over time.

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    Ergodic Markov Chain

    A Markov chain that satisfies the properties of irreducibility and aperiodicity.

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    ACF (Autocorrelation Function)

    A correlation function that measures the correlation between values in a time series at different time lags.

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    Model Diagnostics

    A statistical approach for checking the adequacy of a fitted model by analyzing the residuals.

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    ARMA Model Fitting

    A process of estimating the parameters of an ARMA model using functions like arma() or auto.arima() in R.

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    Study Notes

    Autocorrelation Function (ACF) and Plot

    • ACF measures correlation between time series and lagged values.
    • Formula: pk = Cor(Yt, Yt-k)
    • ACF plot visualises autocorrelation coefficients for different lags (k).
    • First value (p0) is always 1 (correlation with itself).
    • High pk (close to 1 or -1) indicates strong positive/negative correlation at lag k.
    • Stationary processes show ACF decaying quickly to zero as lag increases.
    • Non-stationary processes (e.g., random walk) show ACF declining slowly.

    Weakly Stationary Process

    • A stochastic process (Yt) is weakly stationary if:
      • Mean (μ(t)) is constant over time.
      • Variance (σ²(t)) is constant over time.
      • Autocovariance (γ(t, s)) depends only on the lag (t-s), not specific time points (t and s).
    • Implications: Statistical properties (mean, variance, autocovariance) remain unchanged over time.
    • Many time series models (e.g., ARMA) assume weak stationarity.

    Random Walk

    • A stochastic process where current value is previous value plus random noise.

    AR(p) Process (Autoregressive)

    • AR(p) model: Current value (Yt) depends linearly on past p values and random noise.
    • Formula: Yt = φ₁Yt₋₁ + φ₂Yt₋₂ + ... + φpYt₋p + εt
    • εt is white noise.
    • Stationarity condition: Roots of characteristic equation must lie outside the unit circle. Simplified for AR(1): |φ₁| < 1.
    • ACF decays geometrically, rate depends on coefficients (φ₁...φp).

    MA(q) Process (Moving Average)

    • MA(q) model: Current value (Yt) depends linearly on past q random noise terms.
    • Formula: Yt = εt + θ₁εt₋₁ + θ₂εt₋₂ + ... + θqεt₋q
    • εt is white noise.
    • Always stationary because finite linear combination of white noise.
    • ACF cuts off after lag q (pk = 0 for k > q).

    Interpretation of Time Series Output in R

    • Model fitting: R functions (e.g., arma(), auto.arima()) estimate ARMA model parameters.
    • Diagnostics: Check residuals for patterns and autocorrelations to verify white noise characteristics. This involves examining residual plots and ACF of residuals.
    • Forecasting: Use fitted model to predict future values using R's forecast() function.

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    Description

    This quiz explores the concepts of Autocorrelation Function (ACF), weakly stationary processes, and random walks. Understand how ACF measures correlations in time series and the implications of weak stationarity in statistical models. Test your knowledge on these fundamental concepts in time series analysis.

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