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Questions and Answers
What is the mean function of a stochastic process?
What is the mean function of a stochastic process?
- The autocovariance function of the process
- The distribution of the finite dimensional vector
- The variance of the process
- The expected value of the process at time t (correct)
Which function fully determines the probabilistic behavior of a stochastic process?
Which function fully determines the probabilistic behavior of a stochastic process?
- Autocovariance function
- Variance function
- Mean function
- Family of all finite dimensional distributions (correct)
What does the variance of a process calculate?
What does the variance of a process calculate?
- Distribution of the finite dimensional vector
- Difference between the value and the mean at time t (correct)
- Expected value of the process at time t
- Autocovariance function of the process
In a stochastic process, what is the autocovariance function?
In a stochastic process, what is the autocovariance function?
Which property helps define the probabilistic behavior of a stochastic process?
Which property helps define the probabilistic behavior of a stochastic process?
What aspect of a stochastic process does the mean function capture?
What aspect of a stochastic process does the mean function capture?
What is the most common method to eliminate seasonal effects in time series data?
What is the most common method to eliminate seasonal effects in time series data?
In which scenario is an additive model considered suitable for decomposition?
In which scenario is an additive model considered suitable for decomposition?
What determines whether an additive or multiplicative model is most suitable for time series decomposition?
What determines whether an additive or multiplicative model is most suitable for time series decomposition?
How is the error term transformed in Model (3) when dealing with seasonal effects?
How is the error term transformed in Model (3) when dealing with seasonal effects?
What transformation can change an error term into an additive model when seasonal effects are present?
What transformation can change an error term into an additive model when seasonal effects are present?
What approach is needed to estimate the components of a time series when seasonality is changing over time?
What approach is needed to estimate the components of a time series when seasonality is changing over time?
What condition must a stochastic process satisfy to be strictly stationary?
What condition must a stochastic process satisfy to be strictly stationary?
What is the autocorrelation function of the process defined as?
What is the autocorrelation function of the process defined as?
What characteristic defines a strictly stationary stochastic process?
What characteristic defines a strictly stationary stochastic process?
What is the relationship between the joint distribution of (X(t), X(s)) and the time difference t − s = τ in a strictly stationary process?
What is the relationship between the joint distribution of (X(t), X(s)) and the time difference t − s = τ in a strictly stationary process?
If Y has a normal distribution N(0,1) and X(t) = Y sin(ωt + θ), what is µ(t)?
If Y has a normal distribution N(0,1) and X(t) = Y sin(ωt + θ), what is µ(t)?
If ρ(t,s) is the cross-correlation function, what does a value of 0 imply?
If ρ(t,s) is the cross-correlation function, what does a value of 0 imply?
What does it mean for a process to be second-order stationary?
What does it mean for a process to be second-order stationary?
Is the process in Example 1 strongly stationary?
Is the process in Example 1 strongly stationary?
Can a weakly stationary process be strongly stationary?
Can a weakly stationary process be strongly stationary?
Are strong and weak stationarity equivalent for Gaussian processes?
Are strong and weak stationarity equivalent for Gaussian processes?
What characterizes a Gaussian process?
What characterizes a Gaussian process?
In a Gaussian process, what does the autocorrelation function depend on?
In a Gaussian process, what does the autocorrelation function depend on?
What is the purpose of fitting a seasonal model to a time series?
What is the purpose of fitting a seasonal model to a time series?
What does the process (1 − B)^(d) (1 − Bs)^(d) Xt represent?
What does the process (1 − B)^(d) (1 − Bs)^(d) Xt represent?
In the context of seasonal ARMA models, what do φp(B) and ΦP(B) represent?
In the context of seasonal ARMA models, what do φp(B) and ΦP(B) represent?
If p = q = P = Q = 1, d = D = 0, and s = 12, what model form is represented by (1, 0, 1) × (1, 0, 1)12?
If p = q = P = Q = 1, d = D = 0, and s = 12, what model form is represented by (1, 0, 1) × (1, 0, 1)12?
What is the purpose of fitting an (p, d, q) model to a time series without considering the seasonal effect?
What is the purpose of fitting an (p, d, q) model to a time series without considering the seasonal effect?
What does Φq(Bs) represent in the context of a seasonal ARMA model?
What does Φq(Bs) represent in the context of a seasonal ARMA model?
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Study Notes
Stochastic Processes
- A stochastic process is a collection of random variables indexed by a set t ∈ T, {X(t), t ∈ T}.
- The distribution of a finite dimensional vector (Xt1, Xt2, ..., Xtm) is a well-defined multivariate distribution function.
Characteristics of a Stochastic Process
- The mean function is the expected value of the process at time t, defined by µ(t) = E[X(t)].
- The variance of the process is defined for all t by σ²(t) = E[X(t) - µ(t)]².
- The autocovariance function of the process is defined for all t and s by γ(t, s) = E[(X(t) - µ(t))(X(s) - µ(s))].
- The autocorrelation function of the process is defined for all t and s by ρ(t, s) = γ(t, s) / σ(t) σ(s).
Seasonal Effect Analysis
- There are three models in common use: additive, multiplicative, and multiplicative with error term.
- Choosing a decomposition model depends on the time plot of the original series.
- If the magnitude of the seasonal component is relatively constant, an additive model is suitable.
- If the magnitude of the seasonal component varies with changes in the trend, a multiplicative model is suitable.
Removing Seasonal Effect
- The most common way to eliminate the seasonal effect is to calculate moving average filtering.
- If the seasonal effect is changing over time, an iterative approach is needed to estimate the components of a time series.
- The X11 method is a well-known method for adjusting series in this case.
Stationary Processes
- A stochastic process is strictly stationary if the joint distributions of (X(t1), ..., X(tm)) and (X(t1 + τ), ..., X(tm + τ)) are the same for all τ, m, and t1, ..., tm.
- A process is called second-order stationary if its mean is constant and its autocovariance function depends only on the lag.
- The autocorrelation function for a strictly stationary time series reduces to ρ(τ) = γ(τ) / γ(0).
Gaussian Processes
- A process is called a Gaussian process if all the finite-dimensional distributions are multivariate normal distributions.
- A Gaussian process is characterized by its mean and autocovariance functions.
- Strong and weak stationarity are equivalent for Gaussian processes.
Seasonal ARIMA Models
- Seasonal ARIMA models are extensions of the ARIMA model to account for the seasonal nonstationary behavior of some series.
- The process {X} is a (p, d, q) × (P, D, Q)s with period s, if the differenced series Yt is an ARMA process defined by a specific formula.
- Example: (1, 0, 1) × (1, 0, 1)12 has the form of an (13, 0, 13) with some coefficients equal to zero.
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