All Weeks Bayesian Statistics: Time Series Analysis Coursera Quiz Answers
Bayesian Statistics: Time Series Analysis Coursera Quiz Answers
Practice Quiz : Objectives of the course
Q1. In this course will focus on models that assume that (mark all the options that apply):
- The observations are realizations from spatial processes, where the random variables are spatially related
- The observations are realizations from time series processes, where the random variables are temporally related
- The observations are realizations from independent random variables
Q2. In this course we will focus on the following topics
- Some classes of models for non-stationary time series
- Models for univariate time series
- Models for multivariate time series
- Some classes of models for stationary time series
Q3. Some of the goals of time series analysis that we will illustrate in this course include:
- Online monitoring
- Analysis and inference
- Forecasting
- Clustering
Q4. In this course we will study models and methods for
- Equally spaced time series processes
- Discrete time processes
- Unequally spaced time series processes
- Continuous time processes
Q5. In this course you will learn about
- Nonparametric methods of estimation for time series analysis
- Normal dynamic linear models for non-stationary univariate time series
- Bayesian inference and forecasting for some classes of time series models
- Spatio-temporal models
- Non-linear dynamic models for non-stationary time series
- Autoregressive processes
Quiz: Stationarity, the ACF and the PACF
Q1. Yt−Yt−1=et−0.8et−1
How is this process written using backshift operator notation ({B}B) ?
- (1−B)Yt=(1−0.8B)et
- None of the above
- BYt=(1−0.8B)et
- B(Yt−Yt−1)=0.8Bet
Q2. Which of the following plots is the most likely to correspond to a realization of a stationary time series process?
- .
- .
- .
Q3. If \{Y_t\}{Yt} is a strongly stationary time series process with finite first and second moments, the following statements are true:
- {Yt} is also weakly or second order stationary
- {Yt} is a Gaussian process
- The variance of Yt, Var(Y_t),Var(Yt),changes over time
- The expected value of Yt, E(Y_t),E(Yt),does not depend on t.t.
Q4. If \{Y_t\}{Yt} is weakly or second order stationary with finite first and second moments, the following statements are true:
- If \{Y_t\}{Yt} is also a Gaussian process then \{Y_t\}{Yt} is strongly stationary
- {Yt} is also strongly stationary
- None of the above
Q5. Which of the following moving averages can be used to remove a period d=8d=8 from a time series?
- 1/8yt−4+41(yt−3+yt−2+yt−1+yt+yt+1+yt+2+yt+3)+81yt+4
- 1/8∑j=−88yt−k
- 1/2(yt−4+yt−3+yt−2+yt−1+yt+yt+1+yt+2+yt+3+yt+4)
- 1/8(yt−4+yt−3+yt−2+yt−1+yt+yt+1+yt+2+yt+3+yt+4)
Q6. Which of the following moving averages can be used to remove a period d=3d=3 from a time series?
- 1/2(yt−1+yt+yt+1)
- 1/3(yt−1+yt+yt+1)
- None of the above
Quiz: The AR(1) definitions and properties
Q1. Which of the following plots corresponds to the PACF for an AR(1) with \phi = 0.8ϕ=0.8?
- None of above is correct
- .
- .
- .
Q2.
- Which of the following AR(1) processes are stable and therefore stationary?
- Yt=0.9Yt−1+ϵt,ϵt∼i.i.d.N(0,v)
- Yt=Yt−1+ϵt,ϵt∼i.i.d.N(0,v)
- Yt=−2Yt−1+ϵt,ϵt∼i.i.d.N(0,v)
- Yt=−0.8Yt−1+ϵt,ϵt∼i.i.d.N(0,v)
Q3. Which of the statements below are true?
- The ACF coefficients of an AR(1) with AR coefficient \phi \in (-1,1)ϕ∈(−1,1) and \phi \neq 0ϕ=0 are zero after lag 1
- The PACF coefficients of an AR(1) with AR coefficient \phi \in (-1,1)ϕ∈(−1,1) and \phi \neq 0ϕ=0 are zero after lag 1
- The ACF of an AR(1) with coefficient \phi=0.5ϕ=0.5 decays exponentially in an oscillatory manner
- The ACF of an AR(1) with AR coefficient \phi=0.8ϕ=0.8 decays exponentially
Q4. Which of the following corresponds to the autocovariance function at lag h=2,h=2, \gamma(2)γ(2), of the autoregressive process Yt=0.7Yt−1+ϵt,ϵt∼i.i.d.N(0,v), with v=2.v=2.
- 3.9216
- 0.490.49
- 1.9216
Q5. What is the PACF coefficient at lag 1 for the AR(1) process
yt=−0.7yt−1+ϵt with \epsilon_t \stackrel{iid}\sim N(0,1)ϵt∼iidN(0,1)?
- 0.70.7
- -0.7−0.7
- \approx 1.96≈1.96
- 00
Q6. What is the autovariance function at lag 1, \gamma(1)γ(1) of the AR(1) process
yt=0.6yt−1+ϵt with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,v)ϵt∼i.i.d.N(0,v) ? with variance v=2v=2.
- 1.5625
- 1.875
- 1
- 0.6
Q7. Consider an AR(1) process y_t = -0.5 y_{t-1} + \epsilon_t,yt=−0.5yt−1+ϵt, with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,1)ϵt∼i.i.d.N(0,1). Which of the following statements are true?
- The autocovariance process of this function decays exponentially as a function of the lag hh and it is always negative
- The autocovariance process of this function decays exponentially as a function of the lag hh and it is always positive
- The PACF coefficient at lag 1 \phi(1,1)ϕ(1,1) is equal to -0.5−0.5
- The PACF coefficients for lags greater than 1 are zero
- The PACF coefficient at lag 1 \phi(1,1)ϕ(1,1) is equal to 0.50.5
- The autocovariance process of this function decays exponentially as a function of the lag hh oscillating between negative and positive values
Week 02 : Properties of AR processes
Q1. Consider the following AR(2)AR(2) process,
Y_t = 0.5Y_{t-1} + 0.24Y_{t-2} + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v).Yt=0.5Yt−1+0.24Yt−2+ϵt,ϵt∼N(0,v).
Give the value of one of the reciprocal roots of this process.
Q2. Assume the reciprocal roots of an AR(2)AR(2) characteristic polynomial are 0.70.7 and -0.2.−0.2.
Which is the corresponding form of the autocorrelation function \rho(h)ρ(h) of this process?
- ρ(h)=(a+bh)0.3h,h>0, where $a$ and $b$ are some constants.
- ρ(h)=a(0.7)h+b(−0.3)h,h>0, where aa and bb are some constants.
- ρ(h)=(a+bh)0.7h,h>0, where aa and bb are some constants.
- ρ(h)=(a+bh)(0.3h+0.7h),h>0, where $a$ and $b$ are some constants.
Q3. Assume that an AR(2) process has a pair of complex reciprocal roots with modulus r = 0.95r=0.95 and period \lambda = 7.1.λ=7.1.
Which following options corresponds to the correct form of its autocorrelation function, \rho(h)ρ(h) ?
- ρ(h)=a(0.95)hcos(7.1h+b), where aa and bb are some constants.
- ρ(h)=a(0.95)hcos(2πh/7.1+b), where aa and bb are some constants.
- ρ(h)=a0.95h,h>0, where aa and bb are some constants.
- ρ(h)=(a+bh)0.95h, where aa and bb are some constants.
Q4. Given the following AR(2)AR(2) process,
Y_t = 0.5Y_{t-1} + 0.36Y_{t-2} + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v).Yt=0.5Yt−1+0.36Yt−2+ϵt,ϵt∼N(0,v).
The h=3h=3 steps-ahead forecast function f_t(3)ft(3) has the following form:
- ft(3)=c1t(1.1)3+c2t(−2.5)3 for c_{1t}c1t and c_{2t}c2t constants.
- ft(3)=(0.9)3(c1t+c2t3) for c_{1t}c1t and c_{2t}c2t constants
- ft(3)=c1t(3)0.9+c2t(3)−0.4 for c_{1t}c1t and c_{2t}c2t constants.
- ft(3)=c1t(0.9)3+c2t(−0.4)3 for c_{1t}c1t and c_{2t}c2t constants.
Week 03: Practice Quiz The Normal Dynamic Linear Model
Q1. Which of the models below is a Dynamic Normal Linear Model?
- Observation equation: y_t = a\theta^2_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v), yt=aθt2+ϵt,ϵt∼N(0,v),
- System equation: \theta_t = b\theta_{t-1} + c \frac{\theta_{t-1}}{1+ \theta^2_{t-1}} + \omega_t, \quad \omega_t \sim \mathcal{N}(0, w). θt=bθt−1+c1+θt−12θt−1+ωt,ωt∼N(0,w).
- Observation equation: y_t = \mu_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v),yt=μt+ϵt,ϵt∼N(0,v),
- System equation: \mu_t = \mu_{t-1} + \omega_t, \quad \omega_t \sim \mathcal{N}(0, w).μt=μt−1+ωt,ωt∼N(0,w).
- Observation equation: y_t = \theta_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v), yt=θt+ϵt,ϵt∼N(0,v),
- System equation: \theta_t = b\theta_{t-1} + c \frac{\theta_{t-1}}{1+ \theta^2_{t-1}} + \omega_t, \quad \omega_t \sim \mathcal{N}(0, w). θt=bθt−1+c1+θt−12θt−1+ωt,ωt∼N(0,w).
Q2. Consider the Normal Dynamic Linear Model \mathcal{M}: \left\{\bm{F}_t, \bm{G}_t, \cdot, \cdot\right\}, M:{Ft,Gt,⋅,⋅}, for t = 1, \dots, T.t=1,…,T. Let’s assume \bm{F}_tFt is K \times 1K×1 vector. What is the dimension of \bm{G}_t?Gt?
- T \times 1T×1
- T \times TT×T
- K \times KK×K
- K \times 1K×1
Q3. Consider the third order polynomial Normal Dynamic Linear Model \mathcal{M}: \{\bm{F}, \bm{G}, \cdot, \cdot\}, M:{F,G,⋅,⋅}, where \bm{F} = (1 \quad 0 \quad 0)’F=(100)′ and \bm{G} = \bm{J}_3(1),G=J3(1), where \bm{J}J is Jordan block given by
J_3(1) = \left(
100110011
\right) J3(1)=⎝⎜⎛100110011⎠⎟⎞
Given the posterior mean E (\bm{\theta}_t | D_t) = (m_t, b_t, g_t)’,E(θt∣Dt)=(mt,bt,gt)′, which of the following options is the one corresponding to the forecast function f_t(h) \quad (h \geq 0)ft(h)(h≥0) of the model?
- ft(h)=mt+hbt+h(h−1)gt/2
- ft(h)=mt+hbt
- ft(h)=mt+hbt+h(h+1)gt
- ft(h)=mt+hbt+h2gt
Q4. Consider a forecast function of the following form:
ft(h)=at,0+at,1xt+h+at,3+at,4h, for h \geq 0.h≥0.
Which is the possible corresponding DLM \left\{ \bm{F}, \bm{G}, \cdot, \cdot \right\}?{F,G,⋅,⋅}?
- F=(1xt10)′
\bm{G} = \bm{I}_4,G=I4,
where \bm{I}_4I4 is the identity matrix of dimension 4 \times 44×4
- F=(1010)′
\bm{G} = \textrm{block diag} [\bm{I}_2, \bm{J}_2(1)],G=block diag[I2,J2(1)],
where \bm{I}_2I2 is the identity matrix of dimension 2 \times 22×2, and \bm{J}_2(1)J2(1) is the Jordan block matrix given by
J_2(1) = \left(
1011 \right).J2(1)=(1011).
- F=(1xt10)′
\bm{G} = \textrm{block diag} [\bm{I}_2, \bm{J}_2(1)],G=block diag[I2,J2(1)],
where \bm{I}_2I2 is the identity matrix, and \bm{J}_2(1)J2(1) is the Jordan block matrix given by
J_2(1) = \left(
1011 \right).J2(1)=(1011).
- None is correct.
Quiz : NDLM, Part I: Review
Q1. Assume we have following DLM representation:
\mathcal{M}: \qquad \left\{
(10)
,
(−0.7501−0.75)
, \cdot, \cdot \right\} M:{(10),(−0.7501−0.75),⋅,⋅} and
E(\bm{\theta}_t | \mathcal{D}_t) = (1, 0.5)’E(θt∣Dt)=(1,0.5)′. What is the best description of the forecast function, f_t(h)ft(h) for h \geq 0h≥0 of this model?
- .
- .
- .
- .
Q2. Which of the options below correspond the DLMs \{\bm{F}, \bm{G}, \cdot, \cdot \}{F,G,⋅,⋅} with forecast function
f_t(h) = k_{t1}\lambda^h_1 – k_{t2}\lambda_2^h, ft(h)=kt1λ1h−kt2λ2h,
where \lambda_1\lambda_2 \neq 0λ1λ2=0 and \lambda_1 \neq \lambda_2λ1=λ2?
- F=(1,0)′,
\bm{G} =
(λ111λ2).G=(λ111λ2).
- F=(1,−1)′,
\bm{G} =
(λ100λ2).G=(λ100λ2).
- F=(1,−1)′,
\bm{G} =
(λ111λ2).G=(λ111λ2).
- F=(1,0)′,
\bm{G} =
(λ100λ2).G=(λ100λ2).
Q3. Consider a DLM with a forecast function of the form f_t(h) = k_{t,1} \lambda^h + k_{t,2} x_{t+h}.ft(h)=kt,1λh+kt,2xt+h. Which of the following representations corresponds to this DLM?
- {Ft,Gt,⋅,⋅} with
\bm{F}_t = (1,x_t)’Ft=(1,xt)′ and \bm{G}_t= \left(
λ001 \right).Gt=(λ001).
- {Ft,Gt,⋅,⋅} with
\bm{F}_t = (1,0)’Ft=(1,0)′ and \bm{G}_t= \left(
λ00xt \right).Gt=(λ00xt).
- {Ft,Gt,⋅,⋅} with
\bm{F}_t = (1,x_t)’Ft=(1,xt)′ and \bm{G}_t= \left(
1001 \right).Gt=(1001).
Q4. Which of the following statements are true?
- In order to obtain the filtering equations one must first obtain the smoothing equations
- In a DLM of the form \{ \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t\}{Ft,Gt,vt,Wt} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_tFt,Gt,vt,Wt known for all t,t, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(θt∣Dt) is normal if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(θ0∣D0) is normal
- The filtering equations allow us to obtain the moments of the distribution of (\mathbf{\theta}_t| \mathcal{D}_T)(θt∣DT) for t=1:Tt=1:T and T >t.T>t.
- The smoothing equations allow us to obtain the moments of the distribution of (\mathbf{\theta}_t| \mathcal{D}_T)(θt∣DT) for t=1:Tt=1:T and T >t.T>t.
- In a DLM of the form \{ \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t\}{Ft,Gt,vt,Wt} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_tFt,Gt,vt,Wt known for all t,t, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(θt∣Dt) is not normal even if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(θ0∣D0) is normal
- In order to obtain the DLM smoothing equations one must first obtain the filtering equations
Week 04 : Quiz Seasonal Models and Superposition
Q1. Consider a full seasonal Fourier DLM with fundamental period p=3.p=3. Which of the choices below corresponds to the specification of \bm{F}F and \bm{G}G for such model?
- F=(10)′
\bm{G} = \left(
−12−3√23√2−12 \right) G=(−21−2323−21)
- F=(10)′
\bm{G} = \left(
1011 \right) G=(1011)
- F=(101)′
\bm{G} = \left(
0−1010000−1
\right) G=⎝⎜⎛0−1010000−1⎠⎟⎞
- F=(10)′
\bm{G} = \left(
12−3√23√212 \right) G=(21−232321)
Q2. Assume monthly data have an annual cycle and so the fundamental period is p=12.p=12. Further assume that we want to fit a model with a linear trend and seasonal component to this dataset. For the seasonal component, assume we only consider the fourth harmonic, i.e., we only consider the Fourier component for the frequency \omega= 2\pi 4/12= 2 \pi/3.ω=2π4/12=2π/3. What is the forecast function f_t(h), h \geq 0,ft(h),h≥0, for a DLM with this linear trend and a seasonal component that considers only the fourth harmonic?
- ft(h)=at,0+at,1h
- ft(h)=at,0+at,1h+at,3cos(32πh)+at,4sin(32πh)
- ft(h)=at,1cos(32πh)+at,2sin(32πh)
- ft(h)=at,0+at,1h+at,3cos(32πh)+at,4sin(32πh)+at,5(−1)h
Q4. A DLM \{ \bm{F}_t, \bm{G}_t, \cdot, \cdot \}{Ft,Gt,⋅,⋅} has the following forecast function:
f_t(h) = a_{t,1} x_{t+h} + (-1)^{h} a_{t,2}.ft(h)=at,1xt+h+(−1)hat,2.
What are the corresponding \bm{F}_tFt and \bm{G}_tGt matrices?
- Ft=(1,xt)′ and \bm{G}_t =\left(
1001 \right)Gt=(1001)
- Ft=(1,xt)′ and \bm{G}_t =\left(
100−1 \right)Gt=(100−1)
- Ft=(1,0)′ and \bm{G}_t =\left(
10xt−1 \right)Gt=(10xt−1)
Quiz : NDLM, Part II
Q1. Assume that we want a model with the following $2$ components:
- Linear trend: \{\bm{F}_1, \bm{G}_1, \cdot, \cdot\}{F1,G1,⋅,⋅} with \bm{F}_1 = (1,0,0)’,F1=(1,0,0)′, \bm{G}_1 = \bm{J}_3(1) G1=J3(1) and state vector \bm{\theta}_{1t} = (1, -0.5, 0.1)’.θ1t=(1,−0.5,0.1)′.
- Seasonal component: \{\bm{F}_2, \bm{G}_2, \cdot, \cdot\}{F2,G2,⋅,⋅} with \bm{F}_2 = (1, 0)’F2=(1,0)′, \bm{G}_2 = \bm{J}_2(\lambda, \omega)G2=J2(λ,ω), where \omega = \frac{\pi}{2}ω=2π and \lambda = 0.9λ=0.9 and state vector \bm{\theta}_{2t} = (1, 0.45)’.θ2t=(1,0.45)′.
Which graph is the description of forecast function f_t(h)ft(h) with h \geq 0?h≥0?
- .
- .
- .
- .
Q2. Which is the right Fourier DLM model \{\bm{F}, \bm{G}, \cdot, \cdot \}{F,G,⋅,⋅} with period p = 6?p=6?
1 point
- F=(1,0,1,0,0)′
\bm{G} =
⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜12−3√20003√21200000−12−3√20003√2−1200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=⎝⎜⎜⎜⎜⎜⎜⎛21−23000232100000−21−2300023−2100000−1⎠⎟⎟⎟⎟⎟⎟⎞
- F=(1,0,1,0,1)′
\bm{G} =
⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜12−3√20003√21200000−12−3√20003√2−1200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=⎝⎜⎜⎜⎜⎜⎜⎛21−23000232100000−21−2300023−2100000−1⎠⎟⎟⎟⎟⎟⎟⎞
- F=(1,0,1,0,0)′
\bm{G} =
⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜3√21200012−3√2000003√2−12000−12−3√200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=⎝⎜⎜⎜⎜⎜⎜⎛232100021−230000023−21000−21−2300000−1⎠⎟⎟⎟⎟⎟⎟⎞
- F=(1,0,1,0,1)′
\bm{G} =
⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜3√21200012−3√2000003√2−12000−12−3√200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=⎝⎜⎜⎜⎜⎜⎜⎛232100021−230000023−21000−21−2300000−1⎠⎟⎟⎟⎟⎟⎟⎞
Q3. Consider a full seasonal Fourier DLM with fundamental period p=10.p=10. What is the dimension of the state vector \bm{\theta}_tθt at each time tt?
- None of the above
- 10
- 9
- 11
Get All Course Quiz Answers of Bayesian Statistics Specialization
Bayesian Statistics: From Concept to Data Analysis Quiz Answers
Bayesian Statistics: Techniques and Models Quiz Answers
Bayesian Statistics: Mixture Models Coursera Quiz Answers
Bayesian Statistics: Time Series Analysis Quiz Answer
