Bayesian Statistics: Time Series Analysis Coursera Quiz Answers

All Weeks Bayesian Statistics: Time Series Analysis Coursera Quiz Answers

This course for practicing and aspiring data scientists and statisticians. It is the fourth of a four-course sequence introducing the fundamentals of Bayesian statistics. It builds on the course Bayesian Statistics: From Concept to Data Analysis, Techniques and Models, and Mixture models.

Time series analysis is concerned with modeling the dependency among elements of a sequence of temporally related variables. To succeed in this course, you should be familiar with calculus-based probability, the principles of maximum likelihood estimation, and Bayesian inference. You will learn how to build models that can describe temporal dependencies and how to perform Bayesian inference and forecasting for the models.

You will apply what you’ve learned with the open-source, freely available software R with sample databases. Your instructor Raquel Prado will take you from basic concepts for modeling temporally dependent data to implementation of specific classes of models

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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. W​hich 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​)+81​yt+4​
  • 1/8∑j=−88​ytk
  • 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.

  1. 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​+c2t​3) 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−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​+c1+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:{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)=⎝⎜⎛​100​110​011​⎠⎟⎞​

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,1​xt+h​+at,3​+at,4​h, for h \geq 0.h≥0.

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

  • F=(1xt​10)′

\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)=(10​11​).

  • F=(1xt​10)′

\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)=(10​11​).

  • 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​),(−0.750​1−0.75​),⋅,⋅} 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)=kt1​λ1h​−kt2​λ2h​,

where \lambda_1\lambda_2 \neq 0λ1​λ2​​=0 and \lambda_1 \neq \lambda_2λ1​​=λ2​?

  • F=(1,0)′,

\bm{G} =

(λ111λ2).G=(λ1​1​1λ2​​).

  • F=(1,−1)′,

\bm{G} =

(λ100λ2).G=(λ1​0​0λ2​​).

  • F=(1,−1)′,

\bm{G} =

(λ111λ2).G=(λ1​1​1λ2​​).

  • F=(1,0)′,

\bm{G} =

(λ100λ2).G=(λ1​0​0λ2​​).

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?

  • {Ft​,Gt​,⋅,⋅} with

\bm{F}_t = (1,x_t)’Ft​=(1,xt​)′ and \bm{G}_t= \left(

λ001 \right).Gt​=(λ0​01​).

  • {Ft​,Gt​,⋅,⋅} with

\bm{F}_t = (1,0)’Ft​=(1,0)′ and \bm{G}_t= \left(

λ00xt \right).Gt​=(λ0​0xt​​).

  • {Ft​,Gt​,⋅,⋅} with

\bm{F}_t = (1,x_t)’Ft​=(1,xt​)′ and \bm{G}_t= \left(

1001 \right).Gt​=(10​01​).

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​−23​​​23​​−21​​)

  • F=(10)′

\bm{G} = \left(

1011 \right) G=(10​11​)

  • F=(101)′

\bm{G} = \left(

0−1010000−1

\right) G=⎝⎜⎛​0−10​100​00−1​⎠⎟⎞​

  • F=(10)′

\bm{G} = \left(

12−3√23√212 \right) G=(21​−23​​​23​​21​​)

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,1​h
  • ft​(h)=at,0​+at,1​h+at,3​cos(32πh​)+at,4​sin(32πh​)
  • ft​(h)=at,1​cos(32πh​)+at,2​sin(32πh​)
  • ft​(h)=at,0​+at,1​h+at,3​cos(32πh​)+at,4​sin(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,1​xt+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​=(10​01​)

  • Ft​=(1,xt​)′ and \bm{G}_t =\left(

100−1 \right)Gt​=(10​0−1​)

  • Ft​=(1,0)′ and \bm{G}_t =\left(

10xt−1 \right)Gt​=(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\}{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​−23​​000​23​​21​000​00−21​−23​​0​0023​​−21​0​0000−1​⎠⎟⎟⎟⎟⎟⎟⎞​

  • F=(1,0,1,0,1)′

\bm{G} =

⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜12−3√20003√21200000−12−3√20003√2−1200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=⎝⎜⎜⎜⎜⎜⎜⎛​21​−23​​000​23​​21​000​00−21​−23​​0​0023​​−21​0​0000−1​⎠⎟⎟⎟⎟⎟⎟⎞​

  • F=(1,0,1,0,0)′

\bm{G} =

⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜3√21200012−3√2000003√2−12000−12−3√200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=⎝⎜⎜⎜⎜⎜⎜⎛​23​​21​000​21​−23​​000​0023​​−21​0​00−21​−23​​0​0000−1​⎠⎟⎟⎟⎟⎟⎟⎞​

  • F=(1,0,1,0,1)′

\bm{G} =

⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜3√21200012−3√2000003√2−12000−12−3√200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=⎝⎜⎜⎜⎜⎜⎜⎛​23​​21​000​21​−23​​000​0023​​−21​0​00−21​−23​​0​0000−1​⎠⎟⎟⎟⎟⎟⎟⎞​

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
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Bayesian Statistics: Time Series Analysis Quiz Answer

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