# Bayesian Statistics: Time Series Analysis Coursera Quiz Answers

## 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​

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

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

### Week 04 : Quiz Seasonal Models and Superposition

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

### Quiz : NDLM, Part II

Q1. Consider a full seasonal Fourier DLM with a 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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