Time Series Midterm Review
1
Preparing for the test
2
Expected Values, Variance, etc.
2.1
Notation and Some Definitions
2.1.1
Realizations and Ensembles
2.2
Population Means and Variances
2.2.1
The Mean of
\(X_t\)
2.2.2
Variance of
\(X_t\)
2.3
Estimating Mean and Variance for Multiple Realizations
2.4
Expected Value
2.4.1
Discrete RVs
2.4.2
Continuous RVs
2.4.3
Some Rules
2.4.4
A challenge in Expected Values
2.4.5
A solution
2.5
Variance and Covariance
2.5.1
Variance
2.5.2
Covariance
2.5.3
Some generalizations on covariance:
2.5.4
A challenge
2.5.5
A solution
2.5.6
A note on covariance and correlation
3
A Brief Discussion of Stationarity
3.1
Condition One: Constant Mean
3.2
Condition Two: Constant Variance
3.3
Condition 3: Constant autocorrelation
4
Autocorrelation
4.1
Independence
4.2
Serial Dependence / Autocorrelation
4.2.1
A definition
4.2.2
Autocorrelation Plots
4.2.3
Dependent(ish) data
4.2.4
Independent(ish) data!
5
Autocorrelation Concepts and Notation
5.1
Theoretical concept of
\(\rho\)
5.2
autocovariance
5.3
autocorrelation
5.4
Stationary Covariance
6
Practical Autocorrelation
6.1
Estimation!
6.1.1
\(\rho_k \rightarrow 0\)
7
The Frequency Domain
7.1
Review of sine and cosine
7.1.1
Phase shifts
7.2
Frequency and Period
7.3
psueudo-periodicity
7.4
Aperiodicity
7.5
Frequency
7.6
Frequency Terminology
7.7
Composite sine function
7.8
Fundamental idea
7.8.1
A Brief Look at Fourier Series
8
Spectral Density
8.1
Derivation of spectral density of white noise
8.2
Estimating Spectral Density from Real Data
8.3
Spectral Density in the Wild
8.4
Why do we do spectral density from -0.5 to 0.5?
9
AR(1) Models and Filtering
9.1
Algebra review
9.2
Linear filters
9.2.1
Example
9.2.2
5 point moving average
9.3
Types of filters:
9.3.1
An example in R
9.3.2
An another example
10
GLPs
10.1
AR(1) Intro
10.1.1
AR(1) math zone
10.1.2
the zero mean form of AR1
10.1.3
AR1 with positive phi
10.1.4
AR1 with negative phi
10.1.5
Nonstationary
11
Characteristic Equations for AR(1) Models
12
AR(2) models
12.1
Notes
12.2
Stationarity in AR(2)
12.3
AR(2) zero mean form
12.4
More on backshift operator notation
12.5
Characteristic equation
12.6
Key result
12.6.1
Two real roots
12.6.2
Complex conjugate roots
13
AR(p)
14
MA(q) Models
14.1
Properties and Characteristics
14.1.1
A quote for your thoughts:
14.2
MA(q)
14.3
Operator Zero-Mean Form
14.3.1
Characteristic Equation
14.3.2
Some definitions:
15
Some examples of MA(1)
15.0.1
Positive theta1
15.0.2
Negative theta
16
MA(2)
16.1
Invertibility
16.1.1
Criteria for invertibility
17
ARMA(p,q)
18
Blending AR and MA components
19
Psi weights with AR models:
19.1
Psi weights for ARMA
20
ARIMA and ARUMA
20.1
ARIMA
20.2
ARUMA
20.3
But what do they do????
20.3.1
ARIMA Transform
20.3.2
ARUMA Transform
21
Forecasting
21.1
Forecasts from Signal-Plus-Noise
22
Forecasting setting
22.1
Box-Jenkins approach
23
Strategy + Notation
23.1
Notation
23.2
some math
24
Discussion of Ar1 forcasts
25
AR(p) forecast math
25.1
AR(p) forecasts:
25.1.1
With tswge
26
Eventual Forecast Function:
27
ARMA(p,q)
27.1
Some math
28
Example
29
psi weights
30
Probability limits
30.1
An alternatve way to calculate psi weights
30.2
Calculating them
31
ASE
32
ARIMA forecasts
32.1
An example:
33
Seasonal Forecasts
33.1
Airline models
34
signal plus noise forecasts
34.1
Linear Signal
34.2
Cyclic signal
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Time Series Midterm Review
Time Series Midterm Review
David Josephs
2019-07-02
Unit 1
Preparing for the test