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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. 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\}{Ytโ} is a strongly stationary time series process with finite first and second moments, the following statements are true:
- {Ytโ} is also weakly or second order stationary
- {Ytโ} is a Gaussian process
- The variance of Ytโ, Var(Y_t),Var(Ytโ),changes over time
- The expected value of Ytโ, E(Y_t),E(Ytโ),does not depend on t.t.
Q4. If \{Y_t\}{Ytโ} is weakly or second order stationary with finite first and second moments, the following statements are true:
- If \{Y_t\}{Ytโ} is also a Gaussian process then \{Y_t\}{Ytโ} is strongly stationary
- {Ytโ} is also strongly stationary
- None of the above
Q5. Wโhich of the following moving averages can be used to remove a period d=8d=8 from a time series?
- 1โ/8ytโ4โ+41โ(ytโ3โ+ytโ2โ+ytโ1โ+ytโ+yt+1โ+yt+2โ+yt+3โ)+81โyt+4โ
- 1/8โj=โ88โytโk
- 1/2โ(ytโ4โ+ytโ3โ+ytโ2โ+ytโ1โ+ytโ+yt+1โ+yt+2โ+yt+3โ+yt+4โ)
- 1/8โ(ytโ4โ+ytโ3โ+ytโ2โ+ytโ1โ+ytโ+yt+1โ+yt+2โ+yt+3โ+yt+4โ)
Q6. Which of the following moving averages can be used to remove a period d=3d=3 from a time series?
- 1/2โ(ytโ1โ+ytโ+yt+1โ)
- 1/3โ(ytโ1โ+ytโ+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 = 0.8ฯ=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)ฯโ(โ1,1) and \phi \neq 0ฯ๎ โ=0 are zero after lag 1
- The PACF coefficients of an AR(1) with AR coefficient \phi \in (-1,1)ฯโ(โ1,1) and \phi \neq 0ฯ๎ โ=0 are zero after lag 1
- The ACF of an AR(1) with coefficient \phi=0.5ฯ=0.5 decays exponentially in an oscillatory manner
- The ACF of an AR(1) with AR coefficient \phi=0.8ฯ=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 Ytโ=0.7Ytโ1โ+ฯตtโ,ฯตtโโผi.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
ytโ=โ0.7ytโ1โ+ฯตtโ with \epsilon_t \stackrel{iid}\sim N(0,1)ฯตtโโผiidN(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
ytโ=0.6ytโ1โ+ฯตtโ with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,v)ฯตtโโผi.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,ytโ=โ0.5ytโ1โ+ฯตtโ, with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,1)ฯตtโโผi.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)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 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).Ytโ=0.5Ytโ1โ+0.24Ytโ2โ+ฯตtโ,ฯตtโโผN(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.โ0.2.
Which is the corresponding form of the autocorrelation function \rho(h)ฯ(h) of this process?
- ฯ(h)=(a+bh)0.3h,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)0.7h,h>0, where aa and bb are some constants.
- ฯ(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.ฮป=7.1.
Which following options corresponds to the correct form of its autocorrelation function, \rho(h)ฯ(h) ?
- ฯ(h)=a(0.95)hcos(7.1h+b), where aa and bb are some constants.
- ฯ(h)=a(0.95)hcos(2ฯh/7.1+b), where aa and bb are some constants.
- ฯ(h)=a0.95h,h>0, where aa and bb are some constants.
- ฯ(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).Ytโ=0.5Ytโ1โ+0.36Ytโ2โ+ฯตtโ,ฯตtโโผN(0,v).
The h=3h=3 steps-ahead forecast function f_t(3)ftโ(3) has the following form:
- ftโ(3)=c1tโ(1.1)3+c2tโ(โ2.5)3 for c_{1t}c1tโ and c_{2t}c2tโ constants.
- ftโ(3)=(0.9)3(c1tโ+c2tโ3) for c_{1t}c1tโ and c_{2t}c2tโ constants
- ftโ(3)=c1tโ(3)0.9+c2tโ(3)โ0.4 for c_{1t}c1tโ and c_{2t}c2tโ constants.
- ftโ(3)=c1tโ(0.9)3+c2tโ(โ0.4)3 for c_{1t}c1tโ and c_{2t}c2tโ 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), ytโ=aฮธt2โ+ฯตtโ,ฯตtโโผN(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). ฮธtโ=bฮธtโ1โ+c1+ฮธtโ12โฮธtโ1โโ+ฯtโ,ฯtโโผN(0,w).
- Observation equation: y_t = \mu_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v),ytโ=ฮผtโ+ฯตtโ,ฯตtโโผN(0,v),
- System equation: \mu_t = \mu_{t-1} + \omega_t, \quad \omega_t \sim \mathcal{N}(0, w).ฮผtโ=ฮผtโ1โ+ฯtโ,ฯtโโผN(0,w).
- Observation equation: y_t = \theta_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v), ytโ=ฮธtโ+ฯตtโ,ฯตtโโผN(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). ฮธtโ=bฮธtโ1โ+c1+ฮธtโ12โฮธtโ1โโ+ฯtโ,ฯtโโผN(0,w).
Q2. Consider the Normal Dynamic Linear Model \mathcal{M}: \left\{\bm{F}_t, \bm{G}_t, \cdot, \cdot\right\}, M:{Ftโ,Gtโ,โ ,โ }, for t = 1, \dots, T.t=1,โฆ,T. Letโs assume \bm{F}_tFtโ is K \times 1Kร1 vector. What is the dimension of \bm{G}_t?Gtโ?
- T \times 1Tร1
- T \times TTรT
- K \times KKรK
- K \times 1Kร1
Q3. Consider the third order polynomial Normal Dynamic Linear Model \mathcal{M}: \{\bm{F}, \bm{G}, \cdot, \cdot\}, M:{F,G,โ ,โ }, where \bm{F} = (1 \quad 0 \quad 0)โF=(100)โฒ and \bm{G} = \bm{J}_3(1),G=J3โ(1), where \bm{J}J is Jordan block given by
J_3(1) = \left(
100110011
\right) J3โ(1)=โโโโ100โ110โ011โโ โโโ
Given the posterior mean E (\bm{\theta}_t | D_t) = (m_t, b_t, g_t)โ,E(ฮธtโโฃDtโ)=(mtโ,btโ,gtโ)โฒ, which of the following options is the one corresponding to the forecast function f_t(h) \quad (h \geq 0)ftโ(h)(hโฅ0) of the model?
- ftโ(h)=mtโ+hbtโ+h(hโ1)gtโ/2
- โftโ(h)=mtโ+hbtโ
- ftโ(h)=mtโ+hbtโ+h(h+1)gtโ
- ftโ(h)=mtโ+hbtโ+h2gtโ
Q4. Consider a forecast function of the following form:
ftโ(h)=at,0โ+at,1โxt+hโ+at,3โ+at,4โh, for h \geq 0.hโฅ0.
Which is the possible corresponding DLM \left\{ \bm{F}, \bm{G}, \cdot, \cdot \right\}?{F,G,โ ,โ }?
- F=(1xtโ10)โฒ
\bm{G} = \bm{I}_4,G=I4โ,
where \bm{I}_4I4โ is the identity matrix of dimension 4 \times 44ร4
- F=(1010)โฒ
\bm{G} = \textrm{block diag} [\bm{I}_2, \bm{J}_2(1)],G=block diag[I2โ,J2โ(1)],
where \bm{I}_2I2โ is the identity matrix of dimension 2 \times 22ร2, and \bm{J}_2(1)J2โ(1) is the Jordan block matrix given by
J_2(1) = \left(
1011 \right).J2โ(1)=(10โ11โ).
- F=(1xtโ10)โฒ
\bm{G} = \textrm{block diag} [\bm{I}_2, \bm{J}_2(1)],G=block diag[I2โ,J2โ(1)],
where \bm{I}_2I2โ is the identity matrix, and \bm{J}_2(1)J2โ(1) is the Jordan block matrix given by
J_2(1) = \left(
1011 \right).J2โ(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)โE(ฮธtโโฃDtโ)=(1,0.5)โฒ. What is the best description of the forecast function, f_t(h)ftโ(h) for h \geq 0hโฅ0 of this model?
- .
- .
- .
- .
Q2. Which of the options below correspond the DLMs \{\bm{F}, \bm{G}, \cdot, \cdot \}{F,G,โ ,โ } with forecast function
f_t(h) = k_{t1}\lambda^h_1 โ k_{t2}\lambda_2^h, ftโ(h)=kt1โฮป1hโโkt2โฮป2hโ,
where \lambda_1\lambda_2 \neq 0ฮป1โฮป2โ๎ โ=0 and \lambda_1 \neq \lambda_2ฮป1โ๎ โ=ฮป2โ?
- F=(1,0)โฒ,
\bm{G} =
(ฮป111ฮป2).G=(ฮป1โ1โ1ฮป2โโ).
- F=(1,โ1)โฒ,
\bm{G} =
(ฮป100ฮป2).G=(ฮป1โ0โ0ฮป2โโ).
- F=(1,โ1)โฒ,
\bm{G} =
(ฮป111ฮป2).G=(ฮป1โ1โ1ฮป2โโ).
- F=(1,0)โฒ,
\bm{G} =
(ฮป100ฮป2).G=(ฮป1โ0โ0ฮป2โโ).
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}.ftโ(h)=kt,1โฮปh+kt,2โxt+hโ. Which of the following representations corresponds to this DLM?
- {Ftโ,Gtโ,โ ,โ } with
\bm{F}_t = (1,x_t)โFtโ=(1,xtโ)โฒ and \bm{G}_t= \left(
ฮป001 \right).Gtโ=(ฮป0โ01โ).
- {Ftโ,Gtโ,โ ,โ } with
\bm{F}_t = (1,0)โFtโ=(1,0)โฒ and \bm{G}_t= \left(
ฮป00xt \right).Gtโ=(ฮป0โ0xtโโ).
- {Ftโ,Gtโ,โ ,โ } with
\bm{F}_t = (1,x_t)โFtโ=(1,xtโ)โฒ and \bm{G}_t= \left(
1001 \right).Gtโ=(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\}{Ftโ,Gtโ,vtโ,Wtโ} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_tFtโ,Gtโ,vtโ,Wtโ known for all t,t, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(ฮธtโโฃDtโ) is normal if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(ฮธ0โโฃD0โ) is normal
- The filtering equations allow us to obtain the moments of the distribution of (\mathbf{\theta}_t| \mathcal{D}_T)(ฮธtโโฃDTโ) 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)(ฮธtโโฃDTโ) 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\}{Ftโ,Gtโ,vtโ,Wtโ} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_tFtโ,Gtโ,vtโ,Wtโ known for all t,t, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(ฮธtโโฃDtโ) is not normal even if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(ฮธ0โโฃD0โ) 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)โฒ
\bm{G} = \left(
โ12โ3โ23โ2โ12 \right) G=(โ21โโ23โโโ23โโโ21โโ)
- F=(10)โฒ
\bm{G} = \left(
1011 \right) G=(10โ11โ)
- F=(101)โฒ
\bm{G} = \left(
0โ1010000โ1
\right) G=โโโโ0โ10โ100โ00โ1โโ โโโ
- F=(10)โฒ
\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=2ฯ4/12=2ฯ/3. What is the forecast function f_t(h), h \geq 0,ftโ(h),hโฅ0, for a DLM with this linear trend and a seasonal component that considers only the fourth harmonic?
- ftโ(h)=at,0โ+at,1โh
- ftโ(h)=at,0โ+at,1โh+at,3โcos(32ฯhโ)+at,4โsin(32ฯhโ)
- ftโ(h)=at,1โcos(32ฯhโ)+at,2โsin(32ฯhโ)
- ftโ(h)=at,0โ+at,1โh+at,3โcos(32ฯhโ)+at,4โsin(32ฯhโ)+at,5โ(โ1)h
Q4. A DLM \{ \bm{F}_t, \bm{G}_t, \cdot, \cdot \}{Ftโ,Gtโ,โ ,โ } has the following forecast function:
f_t(h) = a_{t,1} x_{t+h} + (-1)^{h} a_{t,2}.ftโ(h)=at,1โxt+hโ+(โ1)hat,2โ.
What are the corresponding \bm{F}_tFtโ and \bm{G}_tGtโ matrices?
- Ftโ=(1,xtโ)โฒ and \bm{G}_t =\left(
1001 \right)Gtโ=(10โ01โ)
- Ftโ=(1,xtโ)โฒ and \bm{G}_t =\left(
100โ1 \right)Gtโ=(10โ0โ1โ)
- Ftโ=(1,0)โฒ and \bm{G}_t =\left(
10xtโ1 \right)Gtโ=(10โxtโโ1โ)
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\}{F1โ,G1โ,โ ,โ } with \bm{F}_1 = (1,0,0)โ,F1โ=(1,0,0)โฒ, \bm{G}_1 = \bm{J}_3(1) G1โ=J3โ(1) and state vector \bm{\theta}_{1t} = (1, -0.5, 0.1)โ.ฮธ1tโ=(1,โ0.5,0.1)โฒ.
- Seasonal component: \{\bm{F}_2, \bm{G}_2, \cdot, \cdot\}{F2โ,G2โ,โ ,โ } with \bm{F}_2 = (1, 0)โF2โ=(1,0)โฒ, \bm{G}_2 = \bm{J}_2(\lambda, \omega)G2โ=J2โ(ฮป,ฯ), where \omega = \frac{\pi}{2}ฯ=2ฯโ and \lambda = 0.9ฮป=0.9 and state vector \bm{\theta}_{2t} = (1, 0.45)โ.ฮธ2tโ=(1,0.45)โฒ.
Which graph is the description of forecast function f_t(h)ftโ(h) with h \geq 0?hโฅ0?
- .
- .
- .
- .
Q2. Which is the right Fourier DLM model \{\bm{F}, \bm{G}, \cdot, \cdot \}{F,G,โ ,โ } with period p = 6?p=6?
1 point
- F=(1,0,1,0,0)โฒ
\bm{G} =
โโโโโโโโโโโโโโโโ12โ3โ20003โ21200000โ12โ3โ20003โ2โ1200000โ1โโ โโโโโโโโโโโโโโG=โโโโโโโโโ21โโ23โโ000โ23โโ21โ000โ00โ21โโ23โโ0โ0023โโโ21โ0โ0000โ1โโ โโโโโโโโ
- F=(1,0,1,0,1)โฒ
\bm{G} =
โโโโโโโโโโโโโโโโ12โ3โ20003โ21200000โ12โ3โ20003โ2โ1200000โ1โโ โโโโโโโโโโโโโโG=โโโโโโโโโ21โโ23โโ000โ23โโ21โ000โ00โ21โโ23โโ0โ0023โโโ21โ0โ0000โ1โโ โโโโโโโโ
- F=(1,0,1,0,0)โฒ
\bm{G} =
โโโโโโโโโโโโโโโโ3โ21200012โ3โ2000003โ2โ12000โ12โ3โ200000โ1โโ โโโโโโโโโโโโโโG=โโโโโโโโโ23โโ21โ000โ21โโ23โโ000โ0023โโโ21โ0โ00โ21โโ23โโ0โ0000โ1โโ โโโโโโโโ
- F=(1,0,1,0,1)โฒ
\bm{G} =
โโโโโโโโโโโโโโโโ3โ21200012โ3โ2000003โ2โ12000โ12โ3โ200000โ1โโ โโโโโโโโโโโโโโG=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}_tฮธtโ at each time tt?
- None of the above
- 10
- 9
- 11
Conclusion
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๊ฐ๋์ ์ธ ๊ฒ์ ๋ค, Xiao Zhu๋ ์ฌ์ ํ ํ๋ฅญํ๊ณ ์ค์ ๋ก ์์ ์ ๋๋ณด๋ ๋ฐฉ๋ฒ์ ์๊ณ ์๋ค๋ ๊ฒ์ ๋๋ค.
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๊ทธ๋ฌ๋ ํฅํ ํ๋ ฅ์ ๊ฒฐ๊ตญ ํ ๋ฌ ํ์ ๋๋ค.
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๋ง์ ์ฌ๋๋ค์ด ํฌ๋ฏธํ๊ฒ๋๋ง ํฅ๋ถํ๊ธฐ ์์ํ์ต๋๋ค.
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“์ฆ์ ๋๊ตฐ๊ฐ์๊ฒ ๋ชจ๋ ํต์ง๋ฅผ ์ฒ ํํ๋๋ก ์์ฒญํ์ญ์์ค! ํํ ์ด๋ ๊ณ์ญ๋๊น?”
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๋ ๋๊ทธ ํ์ฐ์ค ๋ฉ๊ฐ์จ์ด์ฆ
Fang Jifan์ “์กฐ์นด Chaowen์ด ๋ฐฐ๋ฌํ๊ฒํ์ญ์์ค. “๋ผ๊ณ ํํํ์ต๋๋ค.
๋ ํ๋ฆฌ์นธ ๋ฆฌ์น์ค
Fang Jifan์ ์์ผ๋ก ๋์๊ฐ์ต๋๋ค. “๊ทธ์ ์ง์ ์ด๋์ ๋๊น? ์ฌ๊ธฐ๋ก ๊ตด๋ฌ ๊ฐ๋ผ๊ณ ์ ํ์ญ์์ค.”
์๊ทธ๋ฒณ ๊ณ์ด
Xiao Jing์ ์์ ์์๊ณ ์๋๋ฌ ๋ฏธ์๋ฅผ ์ง์ผ๋ฉฐ “์ถํํฉ๋๋ค, ํํ, ์ถํํฉ๋๋ค, ํํ.”
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์ด๊ฒ์ ๋๊ตฐ๊ฐ์ ์๋ฒ์ง๋ฅผ ๋๋ฉดํ์ฌ ๊พธ์ง๊ณ ์๋ฌด๊ฒ๋ํ์ง ๋ชปํ๊ฒํ๋ ๊ฒ๊ณผ ๊ฐ์ต๋๋ค.
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ํ๋ผ์ค ์ฌ๋กฏ
“๋ถ๋ ์์ฌํ์ญ์์ค, ํํ. ์ ๋ ์ฝ๊ฐ์ ๊ฒฝํ์ด ์์ผ๋ฉฐ ์ ๋ ์ค์ํ์ง ์์ ๊ฒ์ ๋๋ค.”
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Wang Shouren์ ์ ์ ์ ์ฐจ๋ฆฌ๊ณ Wang Hua๋ฅผ ํ๋ ๋ณด์์ต๋๋ค. “์๋ฒ์ง …”
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“…” Fang Jifan์ “์ด๋ป๊ฒ ์์ ํ์ต๋๊น? “๋ผ๊ณ ๋๋ผ์ ๋ฌผ์์ต๋๋ค.
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Zhu Houzhao๋ ๊ทธ์ ์ฆ๊ธฐ ์ฐ๊ตฌ์์์ ์ฌ๋๋ค์ ์ด๋๊ณ ํฐ ํ๋ ฅ์ ๊ฐ์ง๊ณ ๋ํ๋ฌ์ต๋๋ค.
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๋ง์ข ์จ์ด์ฆ 2
“๋๋ ๋น์ ์ด ๋ฐ์ญ์๋ผ๊ณ ๋งํ์ง ์์๋๋ฐ ์ ๋ถ๋นํ ๋์ฐ๋ฅผ ๋ฐ์์ต๋๊น?”
ะฟัะธัะฝะธะปะฐัั ะพะฑะตะทััะฝะฐ ััะพ
ะทะฝะฐัะธั ะบ ัะตะผั ัะฝะธััั ััะถะฐั ะบะพัะฐ, ะบ ัะตะผั ัะฝะธััั ะบะพัะฐ
ะดะปั ััะฐะฒั ัะฐะผะฑั ะธะฝัะพ ะณะพัะพัะบะพะฟ, ัะพัะฝัะน ะฐัััะพะปะพะณะธัะตัะบะธะน ะณะพัะพัะบะพะฟ ะฝะฐ ัะตะณะพะดะฝั ะปัะบะฐ: ะทะฝะฐัะตะฝะธะต
ะธะผะตะฝะธ ั ะฐัะฐะบัะตั ะธ ััะดัะฑะฐ,
ะธะผั ะปัะบะฐ ะฒ ัะพััะธะธ ัะพะฝะฝะธะบ ะฒะธะดะตัั ัะพะถะดะตะฝะธะต ะดะพัะตัะธ,
ัะพะดะธัั ะดะตะฒะพัะบั ะฒะพ ัะฝะต ะฝะตะทะฐะผัะถะฝะตะน
๋ฌธ ํ๋ฆฐ์ธ์ค
Fang Jifan์ “ํํ, ์๋ฆฌํ ๋ง์์ ํญ์ ์๋ก ์ผ์นํฉ๋๋ค. “๋ผ๊ณ ๋ฐํฅํ์ต๋๋ค.
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79 ์ฌ๋กฏ
Fang Jifan์ “๊ฑฑ์ ํ์ง ๋ง๊ณ ๊ธฐ๋ค๋ฆฌ์ญ์์ค. “๋ผ๊ณ ์นจ์ฐฉํ๊ฒ ๋งํ์ต๋๋ค.
Heya fantastic blog! Does running a blog like this take a lot of work? I have very little knowledge of computer programming but I was hoping to start my own blog soon. Anyways, if you have any suggestions or tips for new blog owners please share. I know this is off topic nevertheless I just needed to ask. Thank you!
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์ฌ๋กฏ ๋จธ์ ์ฌ์ดํธ
Shen Wen์ Shen Ao์๊ฒ ํฌํจํ์ต๋๋ค. “๋๊ตฐ๊ฐ๋ฅผ ๊ตฌํ์ธ์. ๋๊ตฌ๋ฅผ ๊ตฌํ ๊ฑด๊ฐ์?”
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๊ฒ์ด์ธ ์ค๋ธ ์ฌ๋ฆผํธ์ค
Liu Wenshan์ ์์ ์ ๋ํ ์ ์๋์ ํ๊ฐ๋ฅผ ๋ค์์ ๋ ๋ฐ๋ปํจ์ ๋๊ผ์ต๋๋ค.
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ํ ํ ๋ณด์ฆ
์ด๋ ์ธ์๊ถ์์ ์์ฃผํ ์ ์ฐ์น์ด ๋๋ฌผ์ ํ๋ฆฌ๊ณ ์์๋ค.
ะะฒัะพะผะพะนะบะฐ ัะฐะผะพะพะฑัะปัะถะธะฒะฐะฝะธั ะฟะพะด ะบะปัั โ ััะพ ัะพะฒัะตะผะตะฝะฝะพะต ัะตัะตะฝะธะต ะดะปั ะผะฐะบัะธะผะฐะปัะฝะพ ะฒัะณะพะดะฝะพะณะพ ะฑะธะทะฝะตัะฐ. ะะตะท ะฟะตัะตััะฒะพะฒ ะธ ะฒัั ะพะดะฝัั , 24/7 ะดะปั ะบะปะธะตะฝัะพะฒ.
๋ฉ์ด์ ์ฌ๋กฏ ์ฌ์ดํธ
“๋นจ๋ฆฌ, ๋นจ๋ฆฌ ์๋ฆฌ๋ฅผ ๋น์ฐ์ธ์, ์ฅ ํ์ ๋, ์์์ ์๊ธฐํ์ธ์.”
ํ ํ ์์ด์ฆ
๊ทธ๋ ์ฃผ์ธ์ด ์๊ธฐ๋งํผ ๋์๋ค๋ ์ฌ์ค์ ๋๋๋ค.
๋ฌธ ํ๋ฆฐ์ธ์ค 100
์๋ง์ ์ฌ๋๋ค์ด ์์ ๋ค๊ณ ์๋ ๋๋ช ๊ถ์ด ํ์ง๊ฐ ๋ ๊น ๋ด ๊ฑฑ์ ํ๊ณ ์์ต๋๋ค.
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๋จนํ ์๋ ํ ํ ์ฌ์ดํธ
Hongzhi ํฉ์ ๋ ์ด๊ฒ์ ๋ํด ์ ์ ๋ ๊ฑฑ์ ํ์ต๋๋ค.
์๋ ์ฌ๋กฏ
์๋ง์ ์ค์ด ์ ํ์ ์ฐ๊ฒฐ๋์ด ๋น ๋ฅด๊ฒ ๊ตด๋ฌ๊ฐ๋๋ค.
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ํ๋ผ๊ทธ๋งํฑ ์ฌ๋กฏ ๋ฌด๋ฃ
Zhu Houzhao๋ ๊ฐ์๊ธฐ ๊ธด์ฅํ๊ณ ๋์น์ ์น์ผ ์ฌ๋ฆด ์ ์์์ต๋๋ค. “์ด ๊ถ์ ์์ ๋ฌด์์ ์ฐพ๊ณ ์์ต๋๊น?”
79 ์ฌ๋กฏ
์ฌ์ง์ด ๊ทธ๋ค์ ์๋ฏผ ์์์ ๋ณํ๋ฅผ ์๋๊น์ง ์ฌํํ ์ ์์ต๋๋ค.
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์ฌ๋กฏ ๊ฒ์
Fang Jifan์ “ํํ, ์ธ์ ๋ ๋์ผํ ์ง ๋ชจ๋ฅด๊ฒ ์ต๋๋ค. “๋ผ๊ณ ํธ๋ฆฌํ๊ฒ ๋งํ์ต๋๋ค.
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ํ ํ ๊ฒ์ฆ ์ฌ์ดํธ
Fang Jifan์ ์๋๋ฌ ๊ณ ๊ฐ๋ฅผ ๋๋์์ต๋๋ค. “์, ์ต์ ์ ๋คํ ๊ฒ์์ ์ดํดํฉ๋๋ค.”
์ผ์ด ์ฌ๋กฏ
์ฐ์ด์ ๊ฐ์ฅ ๋จผ์ ํ ์ผ์ ๊ณ์ฐ ๋ฐฉ๋ฒ์ ๋ํด ๋ฌป๋ ๊ฒ์ธ๋ฐ, ์ด๋ ์ด ๋ฌธ์ ์ ๋ํ ํํ์ ๊ด์ฌ์ ๋ณด์ฌ์ค๋๋ค.
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