## 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=8*d*=8 from a time series?

- 1/8
*yt*−4+41(*yt*−3+*yt*−2+*yt*−1+*yt*+*yt*+1+*yt*+2+*yt*+3)+81*yt*+4 - 1/8∑
*j*=−88*yt*−*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=3*d*=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 = 0.8*ϕ*=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 0*ϕ*=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 0*ϕ*=0 are zero after lag 1 - The ACF of an AR(1) with coefficient \phi=0.5
*ϕ*=0.5 decays exponentially in an oscillatory manner - The ACF of an AR(1) with AR coefficient \phi=0.8
*ϕ*=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*=0.7*Yt*−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

*y**t*=−0.7*y**t*−1+*ϵ**t* with \epsilon_t \stackrel{iid}\sim N(0,1)*ϵ**t*∼*i**i**d**N*(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

*y**t*=0.6*y**t*−1+*ϵ**t* with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,v)*ϵ**t*∼*i*.*i*.*d*.*N*(0,*v*) ? with variance v=2*v*=2.

- 1.5625
- 1.875
- 1
- 0.6

Q7. Consider an AR(1) process y_t = -0.5 y_{t-1} + \epsilon_t,*yt*=−0.5*yt*−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 h
*h*and it is always negative - The autocovariance process of this function decays exponentially as a function of the lag h
*h*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 h
*h*oscillating between negative and positive values

**Week 02 : Properties of AR processes**

Q1. Consider the following AR(2)*A**R*(2) process,

Y_t = 0.5Y_{t-1} + 0.24Y_{t-2} + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v).*Y**t*=0.5*Y**t*−1+0.24*Y**t*−2+*ϵ**t*,*ϵ**t*∼N(0,*v*).

Give the value of one of the **reciprocal **roots of this process.

Q2. Assume the reciprocal roots of an AR(2)*A**R*(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*)0.3*h*,*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*)0.7*h*,*h*>0, where a*a*and b*b*are some constants.*ρ*(*h*)=(*a*+*bh*)(0.3*h*+0.7*h*),*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 = 0.95*r*=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)*h*cos(7.1*h*+*b*), where a*a*and b*b*are some constants.**ρ(h)=a(0.95)hcos(2πh/7.1+b), where aa and bb are some constants.***ρ*(*h*)=*a*0.95*h*,*h*>0, where a*a*and b*b*are some constants.*ρ*(*h*)=(*a*+*bh*)0.95*h*, where a*a*and b*b*are some constants.

Q4. Given the following AR(2)*A**R*(2) process,

Y_t = 0.5Y_{t-1} + 0.36Y_{t-2} + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v).*Y**t*=0.5*Y**t*−1+0.36*Y**t*−2+*ϵ**t*,*ϵ**t*∼N(0,*v*).

The h=3*h*=3 steps-ahead forecast function f_t(3)*ft*(3) has the following form:

*ft*(3)=*c*1*t*(1.1)3+*c*2*t*(−2.5)3 for c_{1t}*c*1*t* and c_{2t}*c*2*t* constants.*ft*(3)=(0.9)3(*c*1*t*+*c*2*t*3) for c_{1t}*c*1*t* and c_{2t}*c*2*t* constants**ft(3)=c1t(3)0.9+c2t(3)−0.4 for c_{1t}c1t and c_{2t}c2t constants.***ft*(3)=*c*1*t*(0.9)3+*c*2*t*(−0.4)3 for c_{1t}*c*1*t* and c_{2t}*c*2*t* 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*=*aθt*2+*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+*c*1+*θt*−12*θt*−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+*c*1+*θt*−12*θt*−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:{*F**t*,*G**t*,⋅,⋅}, for t = 1, \dots, T.*t*=1,…,*T*. Let’s assume \bm{F}_t*F**t* is K \times 1*K*×1 vector. What is the dimension of \bm{G}_t?*G**t*?

- T \times 1
*T*×1 - T \times T
*T*×*T* - K \times K
*K*×*K* - K \times 1
*K*×1

Q3. Consider the third order polynomial Normal Dynamic Linear Model \mathcal{M}: \{\bm{F}, \bm{G}, \cdot, \cdot\}, M:{** F**,

**,⋅,⋅}, where \bm{F} = (1 \quad 0 \quad 0)’**

*G***=(100)′ and \bm{G} = \bm{J}_3(1),**

*F***=**

*G***3(1), where \bm{J}**

*J***is Jordan block given by**

*J*J_3(1) = \left(

100110011

\right) *J*3(1)=⎝⎜⎛100110011⎠⎟⎞

Given the posterior mean E (\bm{\theta}_t | D_t) = (m_t, b_t, g_t)’,*E*(*θ**t*∣*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*+*h*2*gt*

Q4. Consider a forecast function of the following form:

*f**t*(*h*)=*a**t*,0+*a**t*,1*x**t*+*h*+*a**t*,3+*a**t*,4*h*, for h \geq 0.*h*≥0.

Which is the possible corresponding DLM \left\{ \bm{F}, \bm{G}, \cdot, \cdot \right\}?{** F**,

**,⋅,⋅}?**

*G*=(1*F**xt*10)′

\bm{G} = \bm{I}_4,** G**=

**4,**

*I* where \bm{I}_4** I**4 is the identity matrix of dimension 4 \times 44×4

=(1010)′*F*

\bm{G} = \textrm{block diag} [\bm{I}_2, \bm{J}_2(1)],** G**=block diag[

**2,**

*I***2(1)],**

*J* where \bm{I}_2** I**2 is the identity matrix of dimension 2 \times 22×2, and \bm{J}_2(1)

**2(1) is the Jordan block matrix given by**

*J*J_2(1) = \left(

1011 \right).*J*2(1)=(1011).

=(1*F**xt*10)′

\bm{G} = \textrm{block diag} [\bm{I}_2, \bm{J}_2(1)],** G**=block diag[

**2,**

*I***2(1)],**

*J* where \bm{I}_2** I**2 is the identity matrix, and \bm{J}_2(1)

**2(1) is the Jordan block matrix given by**

*J*J_2(1) = \left(

1011 \right).*J*2(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*(*θ**t*∣D*t*)=(1,0.5)′. What is the best description of the forecast function, f_t(h)*f**t*(*h*) for h \geq 0*h*≥0 of this model?

- .

- .

- .

- .

Q2. Which of the options below correspond the DLMs \{\bm{F}, \bm{G}, \cdot, \cdot \}{** F**,

**,⋅,⋅} with forecast function**

*G*f_t(h) = k_{t1}\lambda^h_1 – k_{t2}\lambda_2^h, *f**t*(*h*)=*k**t*1*λ*1*h*−*k**t*2*λ*2*h*,

where \lambda_1\lambda_2 \neq 0*λ*1*λ*2=0 and \lambda_1 \neq \lambda_2*λ*1=*λ*2?

=(1,0)′,*F*

\bm{G} =

(*λ*111*λ*2).** G**=(

*λ*111

*λ*2).

=(1,−1)′,*F*

\bm{G} =

(*λ*100*λ*2).** G**=(

*λ*100

*λ*2).

=(1,−1)′,*F*

\bm{G} =

(*λ*111*λ*2).** G**=(

*λ*111

*λ*2).

=(1,0)′,*F*

\bm{G} =

(*λ*100*λ*2).** G**=(

*λ*100

*λ*2).

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*,1*λh*+*kt*,2*xt*+*h*. Which of the following representations corresponds to this DLM?

- {
*F**t*,*G**t*,⋅,⋅} with

\bm{F}_t = (1,x_t)’*F**t*=(1,*x**t*)′ and \bm{G}_t= \left(

*λ*001 \right).*G**t*=(*λ*001).

- {
*F**t*,*G**t*,⋅,⋅} with

\bm{F}_t = (1,0)’*F**t*=(1,0)′ and \bm{G}_t= \left(

*λ*00*xt* \right).*G**t*=(*λ*00*xt*).

- {
*F**t*,*G**t*,⋅,⋅} with

\bm{F}_t = (1,x_t)’*F**t*=(1,*x**t*)′ and \bm{G}_t= \left(

1001 \right).*G**t*=(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\}{
*F**t*,*G**t*,*vt*,*W**t*} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t*F**t*,*G**t*,*vt*,*W**t* known for all t,*t*, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(*θt*∣D*t*) is normal if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(*θ*0∣D0) is normal - The filtering equations allow us to obtain the moments of the distribution of (\mathbf{\theta}_t| \mathcal{D}_T)(
*θt*∣D*T*) for t=1:T*t*=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*∣D*T*) for t=1:T*t*=1:*T*and T >t.*T*>*t*. - In a DLM of the form \{ \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t\}{
*F**t*,*G**t*,*vt*,*W**t*} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t*F**t*,*G**t*,*vt*,*W**t* known for all t,*t*, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(*θt*∣D*t*) is not normal even if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(*θ*0∣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}

**for such model?**

*G*=(10)′*F*

\bm{G} = \left(

−12−3√23√2−12 \right) ** G**=(−21−2323−21)

=(10)′*F*

\bm{G} = \left(

1011 \right) ** G**=(1011)

=(101)′*F*

\bm{G} = \left(

0−1010000−1

\right) ** G**=⎝⎜⎛0−1010000−1⎠⎟⎞

=(10)′*F*

\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*=2*π*4/12=2*π*/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*,1*h**ft*(*h*)=*at*,0+*at*,1*h*+*at*,3cos(32*πh*)+*at*,4sin(32*πh*)*ft*(*h*)=*at*,1cos(32*πh*)+*at*,2sin(32*πh*)*ft*(*h*)=*at*,0+*at*,1*h*+*at*,3cos(32*πh*)+*at*,4sin(32*πh*)+*at*,5(−1)*h*

Q4. A DLM \{ \bm{F}_t, \bm{G}_t, \cdot, \cdot \}{*F**t*,*G**t*,⋅,⋅} has the following forecast function:

f_t(h) = a_{t,1} x_{t+h} + (-1)^{h} a_{t,2}.*f**t*(*h*)=*a**t*,1*x**t*+*h*+(−1)*h**a**t*,2.

What are the corresponding \bm{F}_t*F**t* and \bm{G}_t*G**t* matrices?

*F**t*=(1,*xt*)′ and \bm{G}_t =\left(

1001 \right)*G**t*=(1001)

*F**t*=(1,*xt*)′ and \bm{G}_t =\left(

100−1 \right)*G**t*=(100−1)

*F**t*=(1,0)′ and \bm{G}_t =\left(

10*xt*−1 \right)*G**t*=(10*xt*−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\}{
1,*F*1,⋅,⋅} with \bm{F}_1 = (1,0,0)’,*G*1=(1,0,0)′, \bm{G}_1 = \bm{J}_3(1)*F*1=*G*3(1) and state vector \bm{\theta}_{1t} = (1, -0.5, 0.1)’.*J*1*θ**t*=(1,−0.5,0.1)′. - Seasonal component: \{\bm{F}_2, \bm{G}_2, \cdot, \cdot\}{
2,*F*2,⋅,⋅} with \bm{F}_2 = (1, 0)’*G*2=(1,0)′, \bm{G}_2 = \bm{J}_2(\lambda, \omega)*F*2=*G*2(*J**λ*,*ω*), where \omega = \frac{\pi}{2}*ω*=2*π* and \lambda = 0.9*λ*=0.9 and state vector \bm{\theta}_{2t} = (1, 0.45)’.2*θ**t*=(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**,

**,⋅,⋅} with period p = 6?**

*G**p*=6?

**1 point**

=(1,0,1,0,0)′*F*

\bm{G} =

⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜12−3√20003√21200000−12−3√20003√2−1200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟** G**=⎝⎜⎜⎜⎜⎜⎜⎛21−23000232100000−21−2300023−2100000−1⎠⎟⎟⎟⎟⎟⎟⎞

=(1,0,1,0,1)′*F*

\bm{G} =

⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜12−3√20003√21200000−12−3√20003√2−1200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟** G**=⎝⎜⎜⎜⎜⎜⎜⎛21−23000232100000−21−2300023−2100000−1⎠⎟⎟⎟⎟⎟⎟⎞

=(1,0,1,0,0)′*F*

\bm{G} =

⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜3√21200012−3√2000003√2−12000−12−3√200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟** G**=⎝⎜⎜⎜⎜⎜⎜⎛232100021−230000023−21000−21−2300000−1⎠⎟⎟⎟⎟⎟⎟⎞

=(1,0,1,0,1)′*F*

\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}_t*θ**t* at each time t*t*?

- None of the above
- 10
- 9
- 11

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