- About The Coursera
- About Bayesian Statistics: Time Series Analysis Course
- Bayesian Statistics: Time Series Analysis Quiz Answers
- Practice Quiz : Objectives of the course
- Quiz: Stationarity, the ACF and the PACF
- Quiz: The AR(1) definitions and properties
- Week 02 : Properties of AR processes
- Week 03: Practice Quiz The Normal Dynamic Linear Model
- Quiz : NDLM, Part I: Review
- Week 04 : Quiz Seasonal Models and Superposition
- Quiz : NDLM, Part II
- Conclusion

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**About Bayesian Statistics: Time Series Analysis** **Course**

This course is designed for working data scientists and aspiring statisticians alike.

This is the fourth and final course in a sequence of four that will introduce the fundamentals of Bayesian statistics.

It expands upon the topics covered in the course Bayesian Statistics: From Concept to Data Analysis, Techniques and Models, as well as Mixture models.

Analysis of time series focuses on modeling the dependencies that exist between the components of a sequence of variables that are time-related.

In order for you to do well in this class, you should have some background knowledge in calculus-based probability, the principles of maximum likelihood estimation, and Bayesian inference.

You will learn how to construct models that are able to describe temporal dependencies, as well as how to perform Bayesian inference and forecasting for the models that you have constructed.

Using the open-source and freely available software R, you will put what you’ve learned into practise using sample databases.

Your instructor, Raquel Prado, will guide you from the fundamental ideas behind modeling temporally dependent data to the implementation of particular classes of models.

**Course Apply Link – Bayesian Statistics: Time Series Analysis**

**Bayesian Statistics: Time Series Analysis** **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. YtโโYtโ1โ=etโโ0.8etโ1โ

How is this process written using **backshift operator** notation ({B}*B*) ?

- (1โB)Ytโ=(1โ0.8B)etโ
- None of the above
- BYtโ=(1โ0.8B)etโ
- B(YtโโYtโ1โ)=0.8Betโ

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*โ} is a strongly stationary time series process with finite first and second moments, the following statements are true:

- {
*Yt*โ} is also weakly or second order stationary - {
*Yt*โ} is a Gaussian process - The variance of
*Yt*โ, Var(Y_t),*Var*(*Yt*โ),changes over time - The expected value of
*Yt*โ, E(Y_t),*E*(*Yt*โ),does not depend on t.*t*.

Q4. If \{Y_t\}{*Yt*โ} is weakly or second order stationary with finite first and second moments, the following statements are true:

- If \{Y_t\}{
*Yt*โ} is also a Gaussian process then \{Y_t\}{*Yt*โ} is strongly stationary - {
*Yt*โ} is also strongly stationary - None of the above

Q5. Wโhich of the following moving averages can be used to remove a period d=8*d*=8 from a time series?

- 1โ/8
*yt*โ4โ+41โ(*yt*โ3โ+*yt*โ2โ+*yt*โ1โ+*yt*โ+*yt*+1โ+*yt*+2โ+*yt*+3โ)+81โ*yt*+4โ - 1/8โ
*j*=โ88โ*yt*โ*k* - 1/2โ(
*yt*โ4โ+*yt*โ3โ+*yt*โ2โ+*yt*โ1โ+*yt*โ+*yt*+1โ+*yt*+2โ+*yt*+3โ+*yt*+4โ) - 1/8โ(
*yt*โ4โ+*yt*โ3โ+*yt*โ2โ+*yt*โ1โ+*yt*โ+*yt*+1โ+*yt*+2โ+*yt*+3โ+*yt*+4โ)

Q6. Which of the following moving averages can be used to remove a period d=3*d*=3 from a time series?

- 1/2โ(
*yt*โ1โ+*yt*โ+*yt*+1โ) - 1/3โ(
*yt*โ1โ+*yt*โ+*yt*+1โ) - 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*ฯ*=0.8?

- None of above is correct
- .

- .

- .

Q2.

- Which of the following AR(1) processes are stable and therefore stationary?

- Ytโ=0.9Ytโ1โ+ฯตtโ,ฯตtโโผi.i.d.N(0,v)
- Ytโ=Ytโ1โ+ฯตtโ,ฯตtโโผi.i.d.N(0,v)
- Ytโ=โ2Ytโ1โ+ฯตtโ,ฯตtโโผi.i.d.N(0,v)
- Ytโ=โ0.8Ytโ1โ+ฯตtโ,ฯตtโโผi.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)
*ฯ*โ(โ1,1) and \phi \neq 0*ฯ*๎ โ=0 are zero after lag 1 - The PACF coefficients of an AR(1) with AR coefficient \phi \in (-1,1)
*ฯ*โ(โ1,1) and \phi \neq 0*ฯ*๎ โ=0 are zero after lag 1 - The ACF of an AR(1) with coefficient \phi=0.5
*ฯ*=0.5 decays exponentially in an oscillatory manner - The ACF of an AR(1) with AR coefficient \phi=0.8
*ฯ*=0.8 decays exponentially

Q4. Which of the following corresponds to the autocovariance function at lag h=2,*h*=2, \gamma(2)*ฮณ*(2), of the autoregressive process *Yt*โ=0.7*Yt*โ1โ+*ฯตt*โ,*ฯตt*โโผ*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*โ=โ0.7*y**t*โ1โ+*ฯต**t*โ with \epsilon_t \stackrel{iid}\sim N(0,1)*ฯต**t*โโผ*i**i**d**N*(0,1)?

- 0.70.7
- -0.7โ0.7
- \approx 1.96โ1.96
- 00

Q6. What is the autovariance function at lag 1, \gamma(1)*ฮณ*(1) of the AR(1) process

*y**t*โ=0.6*y**t*โ1โ+*ฯต**t*โ with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,v)*ฯต**t*โโผ*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*โ=โ0.5*yt*โ1โ+*ฯตt*โ, with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,1)*ฯตt*โโผ*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)
*ฯ*(1,1) is equal to -0.5โ0.5 - The PACF coefficients for lags greater than 1 are zero
- The PACF coefficient at lag 1 \phi(1,1)
*ฯ*(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*โ=0.5*Y**t*โ1โ+0.24*Y**t*โ2โ+*ฯต**t*โ,*ฯต**t*โโผN(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.โ0.2.

Which is the corresponding form of the autocorrelation function \rho(h)*ฯ*(*h*) of this process?

*ฯ*(*h*)=(*a*+*bh*)0.3*h*,*h*>0, where $a$ and $b$ are some constants.**ฯ(h)=a(0.7)h+b(โ0.3)h,h>0, where aa and bb are some constants.***ฯ*(*h*)=(*a*+*bh*)0.7*h*,*h*>0, where a*a*and b*b*are some constants.*ฯ*(*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.*ฮป*=7.1.

Which following options corresponds to the correct form of its autocorrelation function, \rho(h)*ฯ*(*h*) ?

*ฯ*(*h*)=*a*(0.95)*h*cos(7.1*h*+*b*), where a*a*and b*b*are some constants.**ฯ(h)=a(0.95)hcos(2ฯh/7.1+b), where aa and bb are some constants.***ฯ*(*h*)=*a*0.95*h*,*h*>0, where a*a*and b*b*are some constants.*ฯ*(*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*โ=0.5*Y**t*โ1โ+0.36*Y**t*โ2โ+*ฯต**t*โ,*ฯต**t*โโผN(0,*v*).

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

*ft*โ(3)=*c*1*t*โ(1.1)3+*c*2*t*โ(โ2.5)3 for c_{1t}*c*1*t*โ and c_{2t}*c*2*t*โ constants.*ft*โ(3)=(0.9)3(*c*1*t*โ+*c*2*t*โ3) for c_{1t}*c*1*t*โ and c_{2t}*c*2*t*โ constants**ftโ(3)=c1tโ(3)0.9+c2tโ(3)โ0.4 for c_{1t}c1tโ and c_{2t}c2tโ constants.***ft*โ(3)=*c*1*t*โ(0.9)3+*c*2*t*โ(โ0.4)3 for c_{1t}*c*1*t*โ and c_{2t}*c*2*t*โ 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*โ=*aฮธt*2โ+*ฯตt*โ,*ฯตt*โโผN(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).
*ฮธt*โ=*bฮธt*โ1โ+*c*1+*ฮธt*โ12โ*ฮธt*โ1โโ+*ฯt*โ,*ฯt*โโผN(0,*w*). - Observation equation: y_t = \mu_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v),
*yt*โ=*ฮผt*โ+*ฯตt*โ,*ฯตt*โโผN(0,*v*), - System equation: \mu_t = \mu_{t-1} + \omega_t, \quad \omega_t \sim \mathcal{N}(0, w).
*ฮผt*โ=*ฮผt*โ1โ+*ฯt*โ,*ฯt*โโผN(0,*w*). - Observation equation: y_t = \theta_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v),
*yt*โ=*ฮธt*โ+*ฯตt*โ,*ฯตt*โโผN(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).
*ฮธt*โ=*bฮธt*โ1โ+*c*1+*ฮธt*โ12โ*ฮธt*โ1โโ+*ฯt*โ,*ฯt*โโผN(0,*w*).

Q2. Consider the Normal Dynamic Linear Model \mathcal{M}: \left\{\bm{F}_t, \bm{G}_t, \cdot, \cdot\right\}, M:{*F**t*โ,*G**t*โ,โ
,โ
}, for t = 1, \dots, T.*t*=1,โฆ,*T*. Letโs assume \bm{F}_t*F**t*โ is K \times 1*K*ร1 vector. What is the dimension of \bm{G}_t?*G**t*โ?

- T \times 1
*T*ร1 - T \times T
*T*ร*T* - K \times K
*K*ร*K* - K \times 1
*K*ร1

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

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

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

*F***=**

*G***3โ(1), where \bm{J}**

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

*J*J_3(1) = \left(

100110011

\right) *J*3โ(1)=โโโโ100โ110โ011โโ โโโ

Given the posterior mean E (\bm{\theta}_t | D_t) = (m_t, b_t, g_t)โ,*E*(*ฮธ**t*โโฃ*Dt*โ)=(*mt*โ,*bt*โ,*gt*โ)โฒ, which of the following options is the one corresponding to the forecast function f_t(h) \quad (h \geq 0)*ft*โ(*h*)(*h*โฅ0) of the model?

*ft*โ(*h*)=*mt*โ+*hbt*โ+*h*(*h*โ1)*gt*โ/2- โ
*ft*โ(*h*)=*mt*โ+*hbt*โ *ft*โ(*h*)=*mt*โ+*hbt*โ+*h*(*h*+1)*gt*โ*ft*โ(*h*)=*mt*โ+*hbt*โ+*h*2*gt*โ

Q4. Consider a forecast function of the following form:

*f**t*โ(*h*)=*a**t*,0โ+*a**t*,1โ*x**t*+*h*โ+*a**t*,3โ+*a**t*,4โ*h*, for h \geq 0.*h*โฅ0.

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

**,โ ,โ }?**

*G*=(1*F**xt*โ10)โฒ

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

**4โ,**

*I*where \bm{I}_4** I**4โ is the identity matrix of dimension 4 \times 44ร4

=(1010)โฒ*F*

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

**2โ,**

*I***2โ(1)],**

*J*where \bm{I}_2** I**2โ is the identity matrix of dimension 2 \times 22ร2, and \bm{J}_2(1)

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

*J*J_2(1) = \left(

1011 \right).*J*2โ(1)=(10โ11โ).

=(1*F**xt*โ10)โฒ

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

**2โ,**

*I***2โ(1)],**

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

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

*J*J_2(1) = \left(

1011 \right).*J*2โ(1)=(10โ11โ).

- None is correct.

**Quiz : NDLM, Part I: Review**

Q1. Assume we have following DLM representation:

\mathcal{M}: \qquad \left\{

(10)

,

(โ0.7501โ0.75)

, \cdot, \cdot \right\} M:{(10โ),(โ0.750โ1โ0.75โ),โ ,โ } and

E(\bm{\theta}_t | \mathcal{D}_t) = (1, 0.5)โ*E*(*ฮธ**t*โโฃD*t*โ)=(1,0.5)โฒ. What is the best description of the forecast function, f_t(h)*f**t*โ(*h*) for h \geq 0*h*โฅ0 of this model?

- .

- .

- .

- .

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

**,โ ,โ } with forecast function**

*G*f_t(h) = k_{t1}\lambda^h_1 โ k_{t2}\lambda_2^h, *f**t*โ(*h*)=*k**t*1โ*ฮป*1*h*โโ*k**t*2โ*ฮป*2*h*โ,

where \lambda_1\lambda_2 \neq 0*ฮป*1โ*ฮป*2โ๎ โ=0 and \lambda_1 \neq \lambda_2*ฮป*1โ๎ โ=*ฮป*2โ?

=(1,0)โฒ,*F*

\bm{G} =

(*ฮป*111*ฮป*2).** G**=(

*ฮป*1โ1โ1

*ฮป*2โโ).

=(1,โ1)โฒ,*F*

\bm{G} =

(*ฮป*100*ฮป*2).** G**=(

*ฮป*1โ0โ0

*ฮป*2โโ).

=(1,โ1)โฒ,*F*

\bm{G} =

(*ฮป*111*ฮป*2).** G**=(

*ฮป*1โ1โ1

*ฮป*2โโ).

=(1,0)โฒ,*F*

\bm{G} =

(*ฮป*100*ฮป*2).** G**=(

*ฮป*1โ0โ0

*ฮป*2โโ).

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*โ(*h*)=*kt*,1โ*ฮปh*+*kt*,2โ*xt*+*h*โ. Which of the following representations corresponds to this DLM?

- {
*F**t*โ,*G**t*โ,โ ,โ } with

\bm{F}_t = (1,x_t)โ*F**t*โ=(1,*x**t*โ)โฒ and \bm{G}_t= \left(

*ฮป*001 \right).*G**t*โ=(*ฮป*0โ01โ).

- {
*F**t*โ,*G**t*โ,โ ,โ } with

\bm{F}_t = (1,0)โ*F**t*โ=(1,0)โฒ and \bm{G}_t= \left(

*ฮป*00*xt* \right).*G**t*โ=(*ฮป*0โ0*xt*โโ).

- {
*F**t*โ,*G**t*โ,โ ,โ } with

\bm{F}_t = (1,x_t)โ*F**t*โ=(1,*x**t*โ)โฒ and \bm{G}_t= \left(

1001 \right).*G**t*โ=(10โ01โ).

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*โ,*G**t*โ,*vt*โ,*W**t*โ} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t*F**t*โ,*G**t*โ,*vt*โ,*W**t*โ known for all t,*t*, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(*ฮธt*โโฃD*t*โ) is normal if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(*ฮธ*0โโฃD0โ) is normal - The filtering equations allow us to obtain the moments of the distribution of (\mathbf{\theta}_t| \mathcal{D}_T)(
*ฮธt*โโฃD*T*โ) 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)(
*ฮธt*โโฃD*T*โ) 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*โ,*G**t*โ,*vt*โ,*W**t*โ} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t*F**t*โ,*G**t*โ,*vt*โ,*W**t*โ known for all t,*t*, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(*ฮธt*โโฃD*t*โ) is not normal even if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(*ฮธ*0โโฃD0โ) 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)โฒ*F*

\bm{G} = \left(

โ12โ3โ23โ2โ12 \right) ** G**=(โ21โโ23โโโ23โโโ21โโ)

=(10)โฒ*F*

\bm{G} = \left(

1011 \right) ** G**=(10โ11โ)

=(101)โฒ*F*

\bm{G} = \left(

0โ1010000โ1

\right) ** G**=โโโโ0โ10โ100โ00โ1โโ โโโ

=(10)โฒ*F*

\bm{G} = \left(

12โ3โ23โ212 \right) ** G**=(21โโ23โโโ23โโ21โโ)

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.*ฯ*=2*ฯ*4/12=2*ฯ*/3. What is the forecast function f_t(h), h \geq 0,*ft*โ(*h*),*h*โฅ0, for a DLM with this linear trend and a seasonal component that considers only the fourth harmonic?

*ft*โ(*h*)=*at*,0โ+*at*,1โ*h**ft*โ(*h*)=*at*,0โ+*at*,1โ*h*+*at*,3โcos(32*ฯh*โ)+*at*,4โsin(32*ฯh*โ)*ft*โ(*h*)=*at*,1โcos(32*ฯh*โ)+*at*,2โsin(32*ฯh*โ)*ft*โ(*h*)=*at*,0โ+*at*,1โ*h*+*at*,3โcos(32*ฯh*โ)+*at*,4โsin(32*ฯh*โ)+*at*,5โ(โ1)*h*

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

f_t(h) = a_{t,1} x_{t+h} + (-1)^{h} a_{t,2}.*f**t*โ(*h*)=*a**t*,1โ*x**t*+*h*โ+(โ1)*h**a**t*,2โ.

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

*F**t*โ=(1,*xt*โ)โฒ and \bm{G}_t =\left(

1001 \right)*G**t*โ=(10โ01โ)

*F**t*โ=(1,*xt*โ)โฒ and \bm{G}_t =\left(

100โ1 \right)*G**t*โ=(10โ0โ1โ)

*F**t*โ=(1,0)โฒ and \bm{G}_t =\left(

10*xt*โ1 \right)*G**t*โ=(10โ*xt*โโ1โ)

**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\}{
1โ,*F*1โ,โ ,โ } with \bm{F}_1 = (1,0,0)โ,*G*1โ=(1,0,0)โฒ, \bm{G}_1 = \bm{J}_3(1)*F*1โ=*G*3โ(1) and state vector \bm{\theta}_{1t} = (1, -0.5, 0.1)โ.*J*1*ฮธ**t*โ=(1,โ0.5,0.1)โฒ. - Seasonal component: \{\bm{F}_2, \bm{G}_2, \cdot, \cdot\}{
2โ,*F*2โ,โ ,โ } with \bm{F}_2 = (1, 0)โ*G*2โ=(1,0)โฒ, \bm{G}_2 = \bm{J}_2(\lambda, \omega)*F*2โ=*G*2โ(*J**ฮป*,*ฯ*), where \omega = \frac{\pi}{2}*ฯ*=2*ฯ*โ and \lambda = 0.9*ฮป*=0.9 and state vector \bm{\theta}_{2t} = (1, 0.45)โ.2*ฮธ**t*โ=(1,0.45)โฒ.

Which graph is the description of forecast function f_t(h)*ft*โ(*h*) with h \geq 0?*h*โฅ0?

- .

- .

- .

- .

Q2. Which is the right Fourier DLM model \{\bm{F}, \bm{G}, \cdot, \cdot \}{** F**,

**,โ ,โ } with period p = 6?**

*G**p*=6?

**1 point**

=(1,0,1,0,0)โฒ*F*

\bm{G} =

โโโโโโโโโโโโโโโโ12โ3โ20003โ21200000โ12โ3โ20003โ2โ1200000โ1โโ โโโโโโโโโโโโโโ** G**=โโโโโโโโโ21โโ23โโ000โ23โโ21โ000โ00โ21โโ23โโ0โ0023โโโ21โ0โ0000โ1โโ โโโโโโโโ

=(1,0,1,0,1)โฒ*F*

\bm{G} =

โโโโโโโโโโโโโโโโ12โ3โ20003โ21200000โ12โ3โ20003โ2โ1200000โ1โโ โโโโโโโโโโโโโโ** G**=โโโโโโโโโ21โโ23โโ000โ23โโ21โ000โ00โ21โโ23โโ0โ0023โโโ21โ0โ0000โ1โโ โโโโโโโโ

=(1,0,1,0,0)โฒ*F*

\bm{G} =

โโโโโโโโโโโโโโโโ3โ21200012โ3โ2000003โ2โ12000โ12โ3โ200000โ1โโ โโโโโโโโโโโโโโ** G**=โโโโโโโโโ23โโ21โ000โ21โโ23โโ000โ0023โโโ21โ0โ00โ21โโ23โโ0โ0000โ1โโ โโโโโโโโ

=(1,0,1,0,1)โฒ*F*

\bm{G} =

โโโโโโโโโโโโโโโโ3โ21200012โ3โ2000003โ2โ12000โ12โ3โ200000โ1โโ โโโโโโโโโโโโโโ** G**=โโโโโโโโโ23โโ21โ000โ21โโ23โโ000โ0023โโโ21โ0โ00โ21โโ23โโ0โ0000โ1โโ โโโโโโโโ

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*ฮธ**t*โ at each time t*t*?

- None of the above
- 10
- 9
- 11

**Conclusion**

Hopefully, this article will be useful for you to find all theย **Week, final assessment, and Peer Graded Assessment Answers of Bayesian Statistics: Time Series Analysis** **Quiz of Coursera**ย and grab some premium knowledge with less effort. If this article really helped you in any way then make sure to share it with your friends on social media and let them also know about this amazing training. You can also check out our other courseย Answers.ย So, be with us guys we will share a lot more free courses and their exam/quiz solutions also, and follow ourย Techno-RJย **Blog**ย for more updates.

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์๋ฌ ๊ฐ๊ฒ์ ์พ์๋ง๋ ๋จ์ชฝ๊น์ง ๋ฌ๋ ค ์์๊ฐ์ ์๋์ ๋์ฐฉํ๋ค.

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Fang Jifan์ “๋ฌด์จ ์ผ์ด์ผ?”

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Hongzhi ํฉ์ ๋ “์ ๋ง ํ๊ฐํ์ง ์๋ ์ผ์ ๋๋ค. “๋ผ๊ณ ๋ฌด๊ด์ฌํ๊ฒ ๋งํ์ต๋๋ค.

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๋๋จธ์ง๋ ๊ฐ์์ ๊ด์ฌ์ฌ๊ฐ ์๊ณ Fang Jifan์ด ์ง์ ์ฒญ์ํ๋ ๊ฒ์ ์ ๋ง๋ก๋ณด๊ณ ์ถ์ดํฉ๋๋ค.

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Hongzhi ํฉ์ ๋ ๋์ ๋๋ฆฌ๊ณ Fang Jifan์ ๋ฐ๋ผ ๋ณด์์ต๋๋ค. “๋น์ ์ ์์๋ ๊ด์ฐฎ์ต๋๊น?”