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- 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. 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
Conclusion
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