-
Notifications
You must be signed in to change notification settings - Fork 0
/
KF exact initialisation.jl
188 lines (134 loc) · 4.43 KB
/
KF exact initialisation.jl
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
using Base: thread_notifiers
# Kalman filter exact initialisation
using DataFrames
using Optim
using LinearAlgebra
import XLSX
# ---------------------------------------------------------
# Function 1 - initialisation
# Function 1 - Kalman filter recursions
# Function 2 - Log Likelihood estimation
# Function 3 - Kalman smoother recursions
# ---------------------------------------------------------
# ALT-J then ALT-O to resart REPL
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# Mod
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
mutable struct dlm
FF
H
Q
R
A
k
n
cur_ξ_hat
cur_Σ_hat
end
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# Initialisation
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# Function to initialse filter
# 1) Diffuse
# 2) Exact diffuse
# Sets up dlm, given system matrices
# ξ_t = FFξ_(t-1) + Qw_t
# y_t = A'x_t + H'ξ_t + Rv_t
# TODO: Expand to inlcude x and y?
function dlm(FF,H, Q, R, A)
k = size(H, 2) # Number of y variables (columns of H as it is transposed)
n = size(H, 1) # Number of states (row of H as it is transposed
ξ = n == 1 ? zero(eltype(FF)) : zeros(n) # IF n == 1 then 0, else zeros[n,1]
Σ = n == 1 ? zero(eltype(FF)) : zeros(n, n) # IF n == 1 then 0, else zeros[n,n]
return dlm(FF, H, Q, R, A, k, n, ξ, Σ)
end
# User defined initialisation
function initialise!(m::dlm, ξ, Σ)
m.cur_ξ_hat = ξ
m.cur_Σ_hat = Σ
Nothing
end
# Exact limit
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# Kalman filter
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# ξ_t = Fξ_(t-1) + w_t
# y_t = A'x_t + H'ξ_t + v_t
# Assumes states have been initialised
function KalmanFilter(m::dlm,y::AbstractArray, x = "NA")
#m =mod1
FF, A, H, Q, R = m.FF, m.A, m.H, m.Q, m.R
k, n, = m.k, m.n
# *********************************
# Initialisation of matrices
# *********************************
# Number of time periods
T = size(y,1)
# Filtered mean and variance
ξ_f = zeros(T,n)
Σ_f = zeros(n,n,T)
# Predicted mean and variance
ξ_p = zeros(T,n)
Σ_p = zeros(n,n,T)
if x == "NA"
x = zeros(T,1)
end
for t= 1:T
# ******************************
# Prediction step
# ******************************
#t = 1
#--------------------------------
# Conditional initialses the filter with priors if t = 1
#--------------------------------
if t ==1
ξ_p[t,:] = FF[n,n]*m.cur_ξ_hat
Σ_p[:,:,t] = FF[n,n]*m.cur_Σ_hat*transpose(FF[n,n]) .+ Q[n,n]
else
ξ_p[t,:] = FF[:,:]*ξ_f[t-1,:]
Σ_p[:,:,t] = FF[:,:]*Σ_f[:,:,t-1]*transpose(FF[:,:]) .+ Q[:,:]
end
# Ensure Y vector is y[vars,time]
if m.k > 1
reshape(y[t], m.k, 1)
end
# Matrix size error here!
# Prediction error
Hᵗ = transpose(H)
Aᵗ = transpose(A)
prediction_error = (y[t].-Aᵗ[:,:]*x[t].-Hᵗ[:,:]*ξ_p[t,:])
# Prediction variance
HᵗΣHR = Hᵗ[:,:]*Σ_p[:,:,t]*H[:,:] .+ R[:,:]
# ******************************
# Filtered step
# ******************************
# Kalman Gain
Gain = (Σ_p[:,:,t]*H[:,:])/(HᵗΣHR[:,:])
# Filtered mean
ξ_f[t,:] = ξ_p[t,:] .+ Gain[:,:]*prediction_error[:,:]
# Filtered variance
Σ_f[:,:,t] = Σ_p[:,:,t] .- Gain[:,:]*H[:,:]*Σ_p[:,:,t]
end
return ξ_p,Σ_p,ξ_f,Σ_f
end
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# Kalman smoother
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# Tests
#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# Nile river data for testing
dat = DataFrame(XLSX.readtable("C:/Users/aelde/OneDrive/Documents/GitHub/Tutorials/State space models/NileDat.xlsx","Sheet1")...)
# Sets up dlm, given system matrices
# ξ_t = FFξ_(t-1) + Qw_t
# y_t = A'x_t + H'ξ_t + Rv_t
A = [0.0]
H = [1.0; 0.0]
FF = [1.0 1.0;0.0 1.0]
Q = [1469.1 0.0; 0.0 100.0]
R = [15099.0]
mod1= dlm(FF,H,Q,R,A)
initialise!(mod1,[0.0 ;0.0],[100000 0; 0 1000])
a =KalmanFilter(mod1,dat[:,1])
a[3]
plot([a[3],y])