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Contents

# Classical Filtering With Linear Algebra¶

## Overview¶

This is a sequel to the earlier lecture Classical Control with Linear Algebra

That lecture used linear algebra – in particular, the LU decomposition – to formulate and solve a class of linear-quadratic optimal control problems

In this lecture, we’ll be using a closely related decomposition, the Cholesky decomposition , to solve linear prediction and filtering problems

We exploit the useful fact that there is an intimate connection between two superficially different classes of problems:

- deterministic linear-quadratic (LQ) optimal control problems
- linear least squares prediction and filtering problems

The first class of problems involves no randomness, while the second is all about randomness

Nevertheless, essentially the same mathematics solves both type of problem

This connection, which is often termed “duality,” is present whether one uses “classical” or “recursive” solution procedures

In fact we saw duality at work earlier when we formulated control and prediction problems recursively in lectures LQ dynamic programming problems, A first look at the Kalman filter, and The permanent income model

A useful consequence of duality is that

- With every LQ control problem there is implicitly affiliated a linear least squares prediction or filtering problem
- With every linear least squares prediction or filtering problem there is implicitly affiliated a LQ control problem

An understanding of these connections has repeatedly proved useful in cracking interesting applied problems

For example, Sargent [Sar87] [chs. IX, XIV] and Hansen and Sargent [HS80] formulated and solved control and filtering problems using \(z\)-transform methods

In this lecture we investigate these ideas using mostly elementary linear algebra

## Infinite Horizon Prediction and Filtering Problems¶

We pose two related prediction and filtering problems

We let \(Y_t\) be a univariate \(m^{\rm th}\) order moving average, covariance stationary stochastic process,

where \(d(L) = \sum^m_{j=0} d_j L^j\), and \(u_t\) is a serially uncorrelated stationary random process satisfying

We impose no conditions on the zeros of \(d(z)\)

A second covariance stationary process is \(X_t\) given by

where \(\varepsilon_t\) is a serially uncorrelated stationary random process with \(\mathbb{E} \varepsilon_t = 0\) and \(\mathbb{E} \varepsilon_t \varepsilon_s\) = \(0\) for all distinct \(t\) and \(s\)

We also assume that \(\mathbb{E} \varepsilon_t u_s = 0\) for all \(t\) and \(s\)

The **linear least squares prediction problem** is to find the \(L_2\)
random variable \(\hat X_{t+j}\) among linear combinations of
\(\{ X_t,\ X_{t-1},
\ldots \}\) that minimizes \(\mathbb{E}(\hat X_{t+j} - X_{t+j})^2\)

That is, the problem is to find a \(\gamma_j (L) = \sum^\infty_{k=0} \gamma_{jk}\, L^k\) such that \(\sum^\infty_{k=0} \vert \gamma_{jk} \vert^2 < \infty\) and \(\mathbb{E} [\gamma_j \, (L) X_t -X_{t+j}]^2\) is minimized

The **linear least squares filtering problem** is to find a \(b\,(L) = \sum^\infty_{j=0} b_j\, L^j\) such that \(\sum^\infty_{j=0}\vert b_j \vert^2 < \infty\) and \(\mathbb{E} [b\, (L) X_t -Y_t ]^2\) is minimized

Interesting versions of these problems related to the permanent income theory were studied by [Mut60]

### Problem formulation¶

These problems are solved as follows

The covariograms of \(Y\) and \(X\) and their cross covariogram are, respectively,

The covariance and cross covariance generating functions are defined as

The generating functions can be computed by using the following facts

Let \(v_{1t}\) and \(v_{2t}\) be two mutually and serially uncorrelated white noises with unit variances

That is, \(\mathbb{E}v^2_{1t} = \mathbb{E}v^2_{2t} = 1, \mathbb{E}v_{1t} = \mathbb{E}v_{2t} = 0, \mathbb{E}v_{1t} v_{2s} = 0\) for all \(t\) and \(s\), \(\mathbb{E}v_{1t} v_{1t-j} = \mathbb{E}v_{2t} v_{2t-j} = 0\) for all \(j \not = 0\)

Let \(x_t\) and \(y_t\) be two random process given by

Then, as shown for example in [Sar87] [ch. XI], it is true that

Applying these formulas to (1) – (4), we have

The key step in obtaining solutions to our problems is to factor the covariance generating function \(g_X(z)\) of \(X\)

The solutions of our problems are given by formulas due to Wiener and Kolmogorov

These formulas utilize the Wold moving average representation of the \(X_t\) process,

where \(c(L) = \sum^m_{j=0} c_j\, L^j\), with

Here \(\mathbb{\hat E}\) is the linear least squares projection operator

Equation (9) is the condition that \(c_0 \eta_t\) can be the one-step ahead error in predicting \(X_t\) from its own past values

Condition (9) requires that \(\eta_t\) lie in the closed linear space spanned by \([X_t,\ X_{t-1}, \ldots]\)

This will be true if and only if the zeros of \(c(z)\) do not lie inside the unit circle

It is an implication of (9) that \(\eta_t\) is a serially uncorrelated random process, and that a normalization can be imposed so that \(\mathbb{E}\eta_t^2 = 1\)

Consequently, an implication of (8) is that the covariance generating function of \(X_t\) can be expressed as

It remains to discuss how \(c(L)\) is to be computed

Therefore, we have already showed constructively how to factor the covariance generating function \(g_X(z) = d(z)\,d\,(z^{-1}) + h\)

We now introduce the **annihilation operator**:

In words, \([\phantom{00}]_+\) means “ignore negative powers of \(L\)“

We have defined the solution of the prediction problem as \(\mathbb{\hat E} [X_{t+j} \vert X_t,\, X_{t-1}, \ldots] = \gamma_j\, (L) X_t\)

Assuming that the roots of \(c(z) = 0\) all lie outside the unit circle, the Wiener-Kolmogorov formula for \(\gamma_j (L)\) holds:

We have defined the solution of the filtering problem as \(\mathbb{\hat E}[Y_t \mid X_t, X_{t-1}, \ldots] = b (L)X_t\)

The Wiener-Kolomogorov formula for \(b(L)\) is

or

Formulas (13) and (14) are discussed in detail in [Whi83] and [Sar87]

The interested reader can there find several examples of the use of these formulas in economics Some classic examples using these formulas are due to [Mut60]

As an example of the usefulness of formula (14), we let \(X_t\) be a stochastic process with Wold moving average representation

where \(\mathbb{E}\eta^2_t = 1, \hbox { and } c_0 \eta_t = X_t - \mathbb{\hat E} [X_t \vert X_{t-1}, \ldots], c (L) = \sum^m_{j=0} c_j L\)

Suppose that at time \(t\), we wish to predict a geometric sum of future \(X\)‘s, namely

given knowledge of \(X_t, X_{t-1}, \ldots\)

We shall use (14) to obtain the answer

Using the standard formulas (6), we have that

Then (14) becomes

In order to evaluate the term in the annihilation operator, we use the following result from [HS80]

**Proposition** Let

- \(g(z) = \sum^\infty_{j=0} g_j \, z^j\) where \(\sum^\infty_{j=0} \vert g_j \vert^2 < + \infty\)
- \(h\,(z^{-1}) =\) \((1- \delta_1 z^{-1}) \ldots (1-\delta_n z^{-1})\), where \(\vert \delta_j \vert < 1\), for \(j = 1, \ldots, n\)

Then

and, alternatively,

where \(B_j = 1 / \prod^n_{k=1\atop k+j} (1 - \delta_k / \delta_j)\)

Applying formula (17) of the proposition to evaluating (15) with \(g(z) = c(z)\) and \(h(z^{-1}) = 1 - \delta z^{-1}\) gives

or

Thus, we have

This formula is useful in solving stochastic versions of problem 1 of lecture Classical Control with Linear Algebra in which the randomness emerges because \(\{a_t\}\) is a stochastic process

The problem is to maximize

where \(\mathbb{E}_t\) is mathematical expectation conditioned on information known at \(t\), and where \(\{ a_t\}\) is a covariance stationary stochastic process with Wold moving average representation

where

and

The problem is to maximize (19) with respect to a contingency plan expressing \(y_t\) as a function of information known at \(t\), which is assumed to be \((y_{t-1},\ y_{t-2}, \ldots, a_t, \ a_{t-1}, \ldots)\)

The solution of this problem can be achieved in two steps

First, ignoring the uncertainty, we can solve the problem assuming that \(\{a_t\}\) is a known sequence

The solution is, from above,

or

Second, the solution of the problem under uncertainty is obtained by replacing the terms on the right-hand side of the above expressions with their linear least squares predictors.

Using (18) and (20), we have the following solution

## Finite Dimensional Prediction¶

Let \((x_1, x_2, \ldots, x_T)^\prime = x\) be a \(T \times 1\) vector of random variables with mean \(\mathbb{E} x = 0\) and covariance matrix \(\mathbb{E} xx^\prime = V\)

Here \(V\) is a \(T \times T\) positive definite matrix

We shall regard the random variables as being ordered in time, so that \(x_t\) is thought of as the value of some economic variable at time \(t\)

For example, \(x_t\) could be generated by the random process described by the Wold representation presented in equation (8)

In this case, \(V_{ij}\) is given by the coefficient on \(z^{\mid i-j \mid}\) in the expansion of \(g_x (z) = d(z) \, d(z^{-1}) + h\), which equals \(h+\sum^\infty_{k=0} d_k d_{k+\mid i-j \mid}\)

We shall be interested in constructing \(j\) step ahead linear least squares predictors of the form

where \(\mathbb{\hat E}\) is the linear least squares projection operator

The solution of this problem can be exhibited by first constructing an orthonormal basis of random variables \(\varepsilon\) for \(x\)

Since \(V\) is a positive definite and symmetric, we know that there exists a (Cholesky) decomposition of \(V\) such that

or

where \(L\) is lower-trangular, and therefore so is \(L^{-1}\)

Form the random variable \(Lx = \varepsilon\)

Then \(\varepsilon\) is an orthonormal basis for \(x\), since \(L\) is nonsingular, and \(\mathbb{E} \, \varepsilon \, \varepsilon^\prime = L \mathbb{E} xx^\prime L^\prime = I\)

It is convenient to write out the equations \(Lx = \varepsilon\) and \(L^{-1} \varepsilon = x\)

or

We also have

Notice from (23) that \(x_t\) is in the space spanned by \(\varepsilon_t, \, \varepsilon_{t-1}, \ldots, \varepsilon_1\), and from (22) that \(\varepsilon_t\) is in the space spanned by \(x_t,\, x_{t-1}, \ldots,\, x_1\)

Therefore, we have that for \(t-1\geq m \geq 1\)

For \(t-1 \geq m \geq 1\) rewrite (23) as

Representation (25) is an orthogonal decomposition of \(x_t\) into a part \(\sum^{t-1}_{j=m} L_{t, t-j}^{-1}\, \varepsilon_{t-j}\) that lies in the space spanned by \([x_{t-m},\, x_{t-m+1},\, \ldots, x_1]\), and an orthogonal component not in this space

### Implementation¶

Code that computes solutions to LQ control and filtering problems using the methods described here and in Classical Control with Linear Algebra can be found in the file control_and_filter.jl

Here’s how it looks

```
#=
Author: Shunsuke Hori
=#
using Polynomials
struct LQFilter{TR<:Real, TI<:Integer, TF<:AbstractFloat}
d::Vector{TF}
h::TR
y_m::Vector{TF}
m::TI
ϕ::Vector{TF}
β::TR
ϕ_r::Union{Vector{TF},Void}
k::Union{TI,Void}
end
"""
Parameters
----------
d : Vector
The order of the coefficients: [d_0, d_1, ..., d_m]
h : Real
Parameter of the objective function (corresponding to the
quadratic term)
y_m : Vector
Initial conditions for y
r : Vector
The order of the coefficients: [r_0, r_1, ..., r_k]
(optional, if not defined -> deterministic problem)
β : Real or nothing
Discount factor (optional, default value is one)
h_eps :
"""
function LQFilter{TR<:Real}(d::Vector{TR},
h::TR,
y_m::Vector{TR};
r::Union{Vector{TR},Void}=nothing,
β::Union{TR,Void}=nothing,
h_eps::Union{TR,Void}=nothing,
)
m = length(d) - 1
m == length(y_m) ||
throw(ArgumentError("y_m and d must be of same length = $m"))
#---------------------------------------------
# Define the coefficients of ϕ up front
#---------------------------------------------
ϕ = Vector{TR}(2m + 1)
for i in -m:m
ϕ[m-i+1] = sum(diag(d*d', -i))
end
ϕ[m+1] = ϕ[m+1] + h
#-----------------------------------------------------
# If r is given calculate the vector ϕ_r
#-----------------------------------------------------
if r == nothing
k=nothing
ϕ_r = nothing
else
k = size(r, 1) - 1
ϕ_r = Vector{TR}(2k + 1)
for i = -k:k
ϕ_r[k-i+1] = sum(diag(r*r', -i))
end
if h_eps != nothing
ϕ_r[k+1] = ϕ_r[k+1] + h_eps
end
end
#-----------------------------------------------------
# If β is given, define the transformed variables
#-----------------------------------------------------
if β == nothing
β = 1.0
else
d = β.^(collect(0:m)/2) * d
y_m = y_m * β.^(- collect(1:m)/2)
end
return LQFilter(d, h, y_m, m, ϕ, β, ϕ_r, k)
end
"""
This constructs the matrices W and W_m for a given number of periods N
"""
function construct_W_and_Wm(lqf::LQFilter, N::Integer)
d, m = lqf.d, lqf.m
W = zeros(N + 1, N + 1)
W_m = zeros(N + 1, m)
#---------------------------------------
# Terminal conditions
#---------------------------------------
D_m1 = zeros(m + 1, m + 1)
M = zeros(m + 1, m)
# (1) Constuct the D_{m+1} matrix using the formula
for j in 1:(m+1)
for k in j:(m+1)
D_m1[j, k] = dot(d[1:j, 1], d[k-j+1:k, 1])
end
end
# Make the matrix symmetric
D_m1 = D_m1 + D_m1' - diagm(diag(D_m1))
# (2) Construct the M matrix using the entries of D_m1
for j in 1:m
for i in (j + 1):(m + 1)
M[i, j] = D_m1[i-j, m+1]
end
end
M
#----------------------------------------------
# Euler equations for t = 0, 1, ..., N-(m+1)
#----------------------------------------------
ϕ, h = lqf.ϕ, lqf.h
W[1:(m + 1), 1:(m + 1)] = D_m1 + h * eye(m + 1)
W[1:(m + 1), (m + 2):(2m + 1)] = M
for (i, row) in enumerate((m + 2):(N + 1 - m))
W[row, (i + 1):(2m + 1 + i)] = ϕ'
end
for i in 1:m
W[N - m + i + 1 , end-(2m + 1 - i)+1:end] = ϕ[1:end-i]
end
for i in 1:m
W_m[N - i + 2, 1:(m - i)+1] = ϕ[(m + 1 + i):end]
end
return W, W_m
end
"""
This function calculates z_0 and the 2m roots of the characteristic equation
associated with the Euler equation (1.7)
Note:
------
`poly(roots)` from Polynomial.jll defines a polynomial using its roots that can be
evaluated at any point by `polyval(Poly,x)`. If x_1, x_2, ... , x_m are the roots then
polyval(poly(roots),x) = (x - x_1)(x - x_2)...(x - x_m)
"""
function roots_of_characteristic(lqf::LQFilter)
m, ϕ = lqf.m, lqf.ϕ
# Calculate the roots of the 2m-polynomial
ϕ_poly=Poly(ϕ[end:-1:1])
proots = roots(ϕ_poly)
# sort the roots according to their length (in descending order)
roots_sorted = sort(proots, by=abs)[end:-1:1]
z_0 = sum(ϕ) / polyval(poly(proots), 1.0)
z_1_to_m = roots_sorted[1:m] # we need only those outside the unit circle
λ = 1 ./ z_1_to_m
return z_1_to_m, z_0, λ
end
"""
This function computes the coefficients {c_j, j = 0, 1, ..., m} for
c(z) = sum_{j = 0}^{m} c_j z^j
Based on the expression (1.9). The order is
c_coeffs = [c_0, c_1, ..., c_{m-1}, c_m]
"""
function coeffs_of_c(lqf::LQFilter)
m = lqf.m
z_1_to_m, z_0, λ = roots_of_characteristic(lqf)
c_0 = (z_0 * prod(z_1_to_m) * (-1.0)^m)^(0.5)
c_coeffs = coeffs(poly(z_1_to_m)) * z_0 / c_0
return c_coeffs
end
"""
This function calculates {λ_j, j=1,...,m} and {A_j, j=1,...,m}
of the expression (1.15)
"""
function solution(lqf::LQFilter)
z_1_to_m, z_0, λ = roots_of_characteristic(lqf)
c_0 = coeffs_of_c(lqf)[end]
A = zeros(lqf.m)
for j in 1:m
denom = 1 - λ/λ[j]
A[j] = c_0^(-2) / prod(denom[1:m .!= j])
end
return λ, A
end
"""
This function constructs the covariance matrix for x^N (see section 6)
for a given period N
"""
function construct_V(lqf::LQFilter; N::Integer=nothing)
if N == nothing
error("N must be provided!!")
end
if !(typeof(N) <: Integer)
throw(ArgumentError("N must be Integer!"))
end
ϕ_r, k = lqf.ϕ_r, lqf.k
V = zeros(N, N)
for i in 1:N
for j in 1:N
if abs(i-j) <= k
V[i, j] = ϕ_r[k + abs(i-j)+1]
end
end
end
return V
end
"""
Assuming that the u's are normal, this method draws a random path
for x^N
"""
function simulate_a(lqf::LQFilter, N::Integer)
V = construct_V(N + 1)
d = MVNSampler(zeros(N + 1), V)
return rand(d)
end
"""
This function implements the prediction formula discussed is section 6 (1.59)
It takes a realization for a^N, and the period in which the prediciton is formed
Output: E[abar | a_t, a_{t-1}, ..., a_1, a_0]
"""
function predict(lqf::LQFilter, a_hist::Vector, t::Integer)
N = length(a_hist) - 1
V = construct_V(N + 1)
aux_matrix = zeros(N + 1, N + 1)
aux_matrix[1:t+1 , 1:t+1 ] = eye(t + 1)
L = chol(V)'
Ea_hist = inv(L) * aux_matrix * L * a_hist
return Ea_hist
end
"""
- if t is NOT given it takes a_hist (Vector or Array) as a deterministic a_t
- if t is given, it solves the combined control prediction problem (section 7)
(by default, t == nothing -> deterministic)
for a given sequence of a_t (either determinstic or a particular realization),
it calculates the optimal y_t sequence using the method of the lecture
Note:
------
lufact normalizes L, U so that L has unit diagonal elements
To make things cosistent with the lecture, we need an auxiliary diagonal
matrix D which renormalizes L and U
"""
function optimal_y(lqf::LQFilter, a_hist::Vector, t = nothing)
β, y_m, m = lqf.β, lqf.y_m, lqf.m
N = length(a_hist) - 1
W, W_m = construct_W_and_Wm(lqf, N)
F = lufact(W, Val{true})
L, U = F[:L], F[:U]
D = diagm(1.0./diag(U))
U = D * U
L = L * diagm(1.0./diag(D))
J = flipdim(eye(N + 1), 2)
if t == nothing # if the problem is deterministic
a_hist = J * a_hist
#--------------------------------------------
# Transform the a sequence if β is given
#--------------------------------------------
if β != 1
a_hist = reshape(a_hist * (β^(collect(N:0)/ 2)), N + 1, 1)
end
a_bar = a_hist - W_m * y_m # a_bar from the lecutre
Uy = \(L, a_bar) # U @ y_bar = L^{-1}a_bar from the lecture
y_bar = \(U, Uy) # y_bar = U^{-1}L^{-1}a_bar
# Reverse the order of y_bar with the matrix J
J = flipdim(eye(N + m + 1), 2)
y_hist = J * vcat(y_bar, y_m) # y_hist : concatenated y_m and y_bar
#--------------------------------------------
# Transform the optimal sequence back if β is given
#--------------------------------------------
if β != 1
y_hist = y_hist .* β.^(- collect(-m:N)/2)
end
else # if the problem is stochastic and we look at it
Ea_hist = reshape(predict(a_hist, t), N + 1, 1)
Ea_hist = J * Ea_hist
a_bar = Ea_hist - W_m * y_m # a_bar from the lecutre
Uy = \(L, a_bar) # U @ y_bar = L^{-1}a_bar from the lecture
y_bar = \(U, Uy) # y_bar = U^{-1}L^{-1}a_bar
# Reverse the order of y_bar with the matrix J
J = flipdim(eye(N + m + 1), 2)
y_hist = J * vcat(y_bar, y_m) # y_hist : concatenated y_m and y_bar
end
return y_hist, L, U, y_bar
end
```

Let’s use this code to tackle two interesting examples

### Example 1¶

Consider a stochastic process with moving average representation

where \(\varepsilon_t\) is a serially uncorrelated random process with mean zero and variance unity

We want to use the Wiener-Kolmogorov formula (13) to compute the linear least squares forecasts \(\mathbb{E} [x_{t+j} \mid x_t, x_{t-1}, \ldots]\), for \(j = 1,\, 2\)

We can do everything we want by setting \(d = r\), generating an instance of LQFilter, then invoking pertinent methods of LQFilter

```
m = 1
y_m = zeros(m)
d = [1.0, -2.0]
r = [1.0, -2.0]
h = 0.0
example = LQFilter(d, h, y_m, r=d)
```

The Wold representation is computed by example.coefficients_of_c()

Let’s check that it “flips roots” as required

```
coeffs_of_c(example)
```

```
2-element Array{Float64,1}:
2.0
-1.0
```

```
roots_of_characteristic(example)
```

```
([2.0],-2.0,[0.5])
```

Now let’s form the covariance matrix of a time series vector of length \(N\) and put it in \(V\)

Then we’ll take a Cholesky decomposition of \(V = L^{-1} L^{-1} = Li Li'\) and use it to form the vector of “moving average representations” \(x = Li \varepsilon\) and the vector of “autoregressive representations” \(L x = \varepsilon\)

```
V = construct_V(example,N=5)
```

```
5×5 Array{Float64,2}:
5.0 -2.0 0.0 0.0 0.0
-2.0 5.0 -2.0 0.0 0.0
0.0 -2.0 5.0 -2.0 0.0
0.0 0.0 -2.0 5.0 -2.0
0.0 0.0 0.0 -2.0 5.0
```

Notice how the lower rows of the “moving average representations” are converging to the appropriate infinite history Wold representation

```
F = cholfact(V)
Li = F[:L]
```

```
5×5 LowerTriangular{Float64,Array{Float64,2}}:
2.23607 ⋅ ⋅ ⋅ ⋅
-0.894427 2.04939 ⋅ ⋅ ⋅
0.0 -0.9759 2.01187 ⋅ ⋅
0.0 0.0 -0.9941 2.00294 ⋅
0.0 0.0 0.0 -0.998533 2.00073
```

Notice how the lower rows of the “autoregressive representations” are converging to the appropriate infinite history autoregressive representation

```
L = inv(Li)
```

```
5×5 LowerTriangular{Float64,Array{Float64,2}}:
0.447214 ⋅ ⋅ ⋅ ⋅
0.19518 0.48795 ⋅ ⋅ ⋅
0.0946762 0.236691 0.49705 ⋅ ⋅
0.0469898 0.117474 0.246696 0.499266 ⋅
0.0234518 0.0586295 0.123122 0.249176 0.499817
```

**Remark** Let \(\pi (z) = \sum^m_{j=0} \pi_j z^j\) and let \(z_1, \ldots,
z_k\) be the zeros of \(\pi (z)\) that are inside the unit circle, \(k < m\)

Then define

The term multiplying \(\pi (z)\) is termed a “Blaschke factor”

Then it can be proved directly that

and that the zeros of \(\theta (z)\) are not inside the unit circle

### Example 2¶

Consider a stochastic process \(X_t\) with moving average representation

where \(\varepsilon_t\) is a serially uncorrelated random process with mean zero and variance unity

Let’s find a Wold moving average representation for \(x_t\)

Let’s use the Wiener-Kolomogorov formula (13) to compute the linear least squares forecasts \(\mathbb{\hat E}\left[X_{t+j} \mid X_{t-1}, \ldots\right] \hbox { for } j = 1,\, 2,\, 3\)

We proceed in the same way as example 1

```
m = 2
y_m = [0.0, 0.0]
d = [1, 0, -sqrt(2)]
r = [1, 0, -sqrt(2)]
h = 0.0
example = LQFilter(d, h, y_m, r = d)
```

```
coeffs_of_c(example)
```

```
3-element Array{Float64,1}:
1.41421
-0.0
-1.0
```

```
roots_of_characteristic(example)
```

```
([1.18921,-1.18921],-1.4142135623731122,[0.840896,-0.840896])
```

```
V=construct_V(example, N = 8)
```

```
8×8 Array{Float64,2}:
3.0 0.0 -1.41421 0.0 … 0.0 0.0 0.0
0.0 3.0 0.0 -1.41421 0.0 0.0 0.0
-1.41421 0.0 3.0 0.0 0.0 0.0 0.0
0.0 -1.41421 0.0 3.0 -1.41421 0.0 0.0
0.0 0.0 -1.41421 0.0 0.0 -1.41421 0.0
0.0 0.0 0.0 -1.41421 … 3.0 0.0 -1.41421
0.0 0.0 0.0 0.0 0.0 3.0 0.0
0.0 0.0 0.0 0.0 -1.41421 0.0 3.0
```

```
F = cholfact(V)
Li = F[:L]
Li[end-2:end, :]
```

```
3×8 Array{Float64,2}:
0.0 0.0 0.0 -0.92582 0.0 1.46385 0.0 0.0
0.0 0.0 0.0 0.0 -0.966092 0.0 1.43759 0.0
0.0 0.0 0.0 0.0 0.0 -0.966092 0.0 1.43759
```

```
L = inv(Li)
```

```
8×8 LowerTriangular{Float64,Array{Float64,2}}:
0.57735 ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅
0.0 0.57735 ⋅ ⋅ ⋅ ⋅ ⋅
0.308607 0.0 0.654654 ⋅ ⋅ ⋅ ⋅
0.0 0.308607 0.0 0.654654 ⋅ ⋅ ⋅
0.19518 0.0 0.414039 0.0 ⋅ ⋅ ⋅
0.0 0.19518 0.0 0.414039 … 0.68313 ⋅ ⋅
0.131165 0.0 0.278243 0.0 0.0 0.695608 ⋅
0.0 0.131165 0.0 0.278243 0.459078 0.0 0.695608
```

### Prediction¶

It immediately follows from the “orthogonality principle” of least squares (see [AP91] or [Sar87] [ch. X]) that

This can be interpreted as a finite-dimensional version of the Wiener-Kolmogorov \(m\)-step ahead prediction formula

We can use (26) to represent the linear least squares projection of the vector \(x\) conditioned on the first \(s\) observations \([x_s, x_{s-1} \ldots, x_1]\)

We have

This formula will be convenient in representing the solution of control problems under uncertainty

Equation (23) can be recognized as a finite dimensional version of a moving average representation

Equation (22) can be viewed as a finite dimension version of an autoregressive representation

Notice that even if the \(x_t\) process is covariance stationary, so that \(V\) is such that \(V_{ij}\) depends only on \(\vert i-j\vert\), the coefficients in the moving average representation are time-dependent, there being a different moving average for each \(t\)

If \(x_t\) is a covariance stationary process, the last row of \(L^{-1}\) converges to the coefficients in the Wold moving average representation for \(\{ x_t\}\) as \(T \rightarrow \infty\)

Further, if \(x_t\) is covariance stationary, for fixed \(k\) and \(j > 0, \, L^{-1}_{T,T-j}\) converges to \(L^{-1}_{T-k, T-k-j}\) as \(T \rightarrow \infty\)

That is, the “bottom” rows of \(L^{-1}\) converge to each other and to the Wold moving average coefficients as \(T \rightarrow \infty\)

This last observation gives one simple and widely-used practical way of forming a finite \(T\) approximation to a Wold moving average representation

First, form the covariance matrix \(\mathbb{E}xx^\prime = V\), then obtain the Cholesky decomposition \(L^{-1} L^{-1^\prime}\) of \(V\), which can be accomplished quickly on a computer

The last row of \(L^{-1}\) gives the approximate Wold moving average coefficients

This method can readily be generalized to multivariate systems.

## Combined Finite Dimensional Control and Prediction¶

Consider the finite-dimensional control problem, maximize

where \(d(L) = d_0 + d_1 L+ \ldots + d_m L^m\), \(L\) is the lag operator, \(\bar a = [ a_N, a_{N-1} \ldots, a_1, a_0]^\prime\) a random vector with mean zero and \(\mathbb{E}\,\bar a \bar a^\prime = V\)

The variables \(y_{-1}, \ldots, y_{-m}\) are given

Maximization is over choices of \(y_0, y_1 \ldots, y_N\), where \(y_t\) is required to be a linear function of \(\{y_{t-s-1}, t+m-1\geq 0;\ a_{t-s}, t\geq s\geq 0\}\)

We saw in the lecture Classical Control with Linear Algebra that the solution of this problem under certainty could be represented in feedback-feedforward form

for some \((N+1)\times m\) matrix \(K\)

Using a version of formula (26), we can express \(\mathbb{\hat E}[\bar a \mid a_s,\, a_{s-1}, \ldots, a_0 ]\) as

where \(I_{(s + 1)}\) is the \((s+1) \times (s+1)\) identity
matrix, and \(V = \tilde U^{-1} \tilde U^{-1^{\prime}}\), where
\(\tilde U\) is the *upper* trangular Cholesky factor of the
covariance matrix \(V\)

(We have reversed the time axis in dating the \(a\)‘s relative to earlier)

The time axis can be reversed in representation (27) by replacing \(L\) with \(L^T\)

The optimal decision rule to use at time \(0 \leq t \leq N\) is then given by the \((N-t +1)^{\rm th}\) row of

## Exercises¶

### Exercise 1¶

Let \(Y_t = (1 - 2 L ) u_t\) where \(u_t\) is a mean zero white noise with \(\mathbb{E} u^2_t = 1\). Let

where \(\varepsilon_t\) is a serially uncorrelated white noise with \(\mathbb{E} \varepsilon^2_t = 9\), and \(\mathbb{E} \varepsilon_t u_s = 0\) for all \(t\) and \(s\)

Find the Wold moving average representation for \(X_t\)

Find a formula for the \(A_{1j}\)‘s in

Find a formula for the \(A_{2j}\)‘s in

### Exercise 2¶

(Multivariable Prediction) Let \(Y_t\) be an \((n\times 1)\) vector stochastic process with moving average representation

where \(D(L) = \sum^m_{j=0} D_j L^J, D_j\) an \(n \times n\) matrix, \(U_t\) an \((n \times 1)\) vector white noise with :math: mathbb{E} U_t =0 for all \(t\), \(\mathbb{E} U_t U_s' = 0\) for all \(s \neq t\), and \(\mathbb{E} U_t U_t' = I\) for all \(t\)

Let \(\varepsilon_t\) be an \(n \times 1\) vector white noise with mean \(0\) and contemporaneous covariance matrix \(H\), where \(H\) is a positive definite matrix

Let \(X_t = Y_t +\varepsilon_t\)

Define the covariograms as \(C_X (\tau) = \mathbb{E} X_t X^\prime_{t-\tau}, C_Y (\tau) = \mathbb{E} Y_t Y^\prime_{t-\tau}, C_{YX} (\tau) = \mathbb{E} Y_t X^\prime_{t-\tau}\)

Then define the matrix covariance generating function, as in (21), only interpret all the objects in (21) as matrices

Show that the covariance generating functions are given by

A factorization of \(g_X (z)\) can be found (see [Roz67] or [Whi83]) of the form

where the zeros of \(\vert C(z)\vert\) do not lie inside the unit circle

A vector Wold moving average representation of \(X_t\) is then

where \(\eta_t\) is an \((n\times 1)\) vector white noise that is “fundamental” for \(X_t\)

That is, \(X_t - \mathbb{\hat E}\left[X_t \mid X_{t-1}, X_{t-2} \ldots\right] = C_0 \, \eta_t\)

The optimum predictor of \(X_{t+j}\) is

If \(C(L)\) is invertible, i.e., if the zeros of \(\det\) \(C(z)\) lie strictly outside the unit circle, then this formula can be written