-
Notifications
You must be signed in to change notification settings - Fork 0
/
PMTM.m
236 lines (193 loc) · 7.87 KB
/
PMTM.m
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
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
clear all;
close all;
N = 256; % number of spectrum samples in [0,fs/2) determining the
% frequency spacing fs/(2N) between samples
N_max = 256; % number of desired spectrum samples which tends to be much
% lower then N in neural signals because of oversampling
K = 512; % number of spiking samples per neuron
L = 40; % number of neurons
iter_Newton = 100; % number of newton iterations per EM iteration
iter_EM = 100; % number of EM iterations
tol_Newton = 1e-50; % Stopping criterion for Newton's method
%*************************Toy Example generation***************************
% AR process generation
B = 1;
multiplier = 0.025;
Coeff = conv( ...
conv([1 -0.965*exp(-2*1i*pi*0.10)], [1 -0.965*exp(2*1i*pi*0.10)]), ...
conv([1 -0.975*exp(-2*1i*pi*0.35)], [1 -0.975*exp(2*1i*pi*0.35)]) ...
); % AR coeffs
[H_AR,W] = freqz(multiplier, Coeff, 256); % Frequency response
% of the filter
b = randn(K+512, 1);
x_AR = filter(B, Coeff, multiplier * (b));
y2 = x_AR(end - K + 1:end);
CIF = (y2 + 0.12); % baseline spiking rate = 1.2
% Spiking data generation by Thinning
spikes = zeros(length(y2), L);
for i = 1:L
spikes(:, i) = (CIF > rand(length(y2), 1));
rng('shuffle');
end
% Plotting the data Set
figure,
subplot(2, 1, 1), plot(CIF);
xlabel('$$k$$', 'Interpreter', 'Latex');
ylabel('Amplitude','Interpreter','Latex');
xlim([200.5, 350.5]);
xlim([1,K]);
subplot(2, 1, 2), SpikeRasterPlot(spikes(:, 1:10)');
xlabel('$$k$$', 'Interpreter','Latex');
ylabel('Trials','Interpreter','Latex');
xlim([200.5, 350.5]);
%**************************************************************************
%*****************************PMTM ESTIMATION******************************
% Discrete Prolate Spheroidal Sequences
time_halfbandwidth = 5;
dps_seq= dpss(K, time_halfbandwidth);
% Initialization
S_mt = [];
S_pp = [];
S_ss = [];
S_pp_mt = [];
A=zeros(K, 2*N_max);
for i=1:K
for j=1:N_max
A(i,2*j-1) = cos(i*pi*(j-1)/N);
A(i,2*j) = -sin(i*pi*(j-1)/N);
end
end
A = 2*pi*A/N;
A(:, 2) = [];
PSTH = mean(spikes, 2); % PSTH calculation
mu_hat = mean(PSTH); % Estimate of true mu
gamma = 1e-0;
childPSTH = [];
for k = 1:size(dps_seq, 2) - 2
% ## Auxiliary spike generation ##
lamda0 = 1.5 * max(dps_seq(:, k)); % factor 1.5 is arbitrary, can
% use any factor > 1.
n_avg = zeros(K, 1);
for j = 1:L
n = spikes(:, j);
n_comp = 1 - spikes(:, j);
n = n .* dps_seq(:, k) / lamda0;
n_comp = - n_comp .* dps_seq(:, k) / lamda0;
n = n .* (n > 0);
n_comp = n_comp.*(n_comp > 0);
n_avg = n_avg + (n + n_comp);
end
n_avg = n_avg / L;
mu_child = mu_hat * (dps_seq(:, k) / lamda0) .* (dps_seq(:, k) >= 0) ...
+ (1 - mu_hat) * (-dps_seq(:, k) / lamda0) .* (dps_seq(:, k) <= 0);
% Plot Auxiliary Spike trains
figure, stem(0:K-1, n_avg);
str = sprintf('$$%d$$ th child process', k-1);
title(str, 'Interpreter', 'Latex');
xlabel('$$k$$', 'Interpreter', 'Latex');
ylabel('$$\sum_{l=1}^{L-1} n_k^{(l)}$$', 'Interpreter', 'Latex');
drawnow
% ## EM algorithm ##
childPSTH = [childPSTH, n_avg];
mu_v = zeros(2*N_max-1, 1);
theta = zeros(2*N_max-1, iter_EM);
PSD_est = zeros(N_max, iter_EM);
% initialize theta for EM
theta(:, 1) = 1e-1*ones(2*N_max-1, 1);
PSD_est(2:end, 1) = theta(2:2:end, 1) + theta(3:2:end, 1);
% initialize Newton
x = mu_child + 0.5 * zeros(K, 1);
% x = mu + SS_latent_1(n_avg);
g = L * A' * (childPSTH(:, k) ./ x - (1 - childPSTH(:, k)) ./ (1 - x));
H = -L * A' * diag(childPSTH(:,k)./(x.^2) ...
+ (1 - childPSTH(:, k)) ./ ((1 - x) .^ 2)) * A ...
- diag(1 ./ theta(:, 1));
d = H\g;
% EM iterations
for i=2:iter_EM
% Newton iterations
for j=1:iter_Newton
tau = 1;
while (sum(x - tau*A*d + 0.02 <= 0)~=0 || ...
sum(x - tau*A*d - 0.00 >= 1)~=0)
tau = tau / 2;
if tau < 1e-30
break
end
end
mu_v = mu_v - tau * d;
x = mu_child + A * mu_v;
g = L * A' *(childPSTH(:, k) ./ x - ...
(1 - childPSTH(:, k)) ./ (1 - x)) - mu_v ./ theta(:, i - 1);
H = - L * A' * diag(childPSTH(:, k) ./ (x .^ 2) ...
+ (1 - childPSTH(:, k)) ./ ((1 - x) .^ 2)) * A ...
- diag(1 ./ theta(:, i - 1));
d = H \ g;
if - (g' * d) * tau < tol_Newton
break
end
end
E = diag(-H \ eye(2 * N_max - 1)) + mu_v .^ 2;
% theta(:,i) = E; % Without regularization works!
theta(:, i) = (-1 + sqrt(1 + 8 * gamma * E)) / (4 * gamma); % But
% a little regularization helps fast convergence
PSD_est(2:end, i) = theta(2:2:end, i) + theta(3:2:end, i);
end
% % Estimation progress
figure, pcolor(PSD_est);
shading flat;
xlabel('Number of Iterations$$\rightarrow$$', 'Interpreter', 'Latex');
ylabel('Frequency$$\rightarrow$$', 'Interpreter', 'Latex');
str = [sprintf('$$%d$$th tapered auxiliary process:' , k-1) ...
'estimation progress'];
title(str, 'Interpreter', 'Latex');
% Eigen-Spectra
S_pp_mt = [S_pp_mt, PSD_est(:, iter_EM) * ((2* pi * lamda0) ^ 2)];
end
pp_mt_est = mean(S_pp_mt, 2); % PMTM PSD
%**************************************************************************
%%
%************************Oracle,PSTH,SS PSD********************************
% SS CIF estimation
y_ss = SS_latent_estimation(spikes);
for k = 1:size(dps_seq,2) - 2
% Oracle PSD
S_mt = [S_mt, abs(fft((y2 - mean(y2)) .* dps_seq(:, k))) .^ 2];
% PSTH PSD
S_pp = [S_pp, (abs(fft((PSTH - mean(PSTH)) .* dps_seq(:, k))) .^ 2)];
% SS PSD
S_ss = [S_ss, abs(fft((y_ss - mean(y_ss)) .* dps_seq(:, k))) .^ 2];
end
mt_est = mean(S_mt, 2); % Oracle PSD
pp_est = mean(S_pp, 2); % PSTH PSD
pp_ss_est = mean(S_ss, 2); % SS PSD
%**************************************************************************
%%
%********************************Overlay plot******************************
figure,
plot((1:N_max) / (2 * N), 10 * log10(pp_mt_est * K / N), ...
'k', 'LineWidth', 1.2);
hold on;
plot((0:(length(pp_est) - 1)) / length(pp_est), 10 * log10(pp_est), ...
'--g', 'LineWidth', 1.2);
plot((0:(length(mt_est) - 1)) / length(mt_est), 10 * log10(mt_est), ...
'--r', 'LineWidth', 1.2);
plot((0:(length(pp_ss_est) - 1)) / length(pp_ss_est), ...
10 * log10(pp_ss_est), '-.c', 'LineWidth', 1.2);
plot(W / (2 * pi), 20 * log10(abs(H_AR)), 'b', 'LineWidth', 1.2);
legend('Proposed Method','PSTH PSD','Oracle PSD','SS PSD','True PSD');
xlim([0 0.5]);
grid on
xlabel('Normalized Frequency', 'Interpreter', 'Latex');
ylabel('Power/Frequency $$(dB/rad/sample)$$', 'Interpreter', 'Latex');
%*********************************MSE calculation**************************
lin_MSE_pp = sum(((pp_est(2:256) - abs(H_AR(2:end)) .^ 2) .^2 ) ...
./ (abs(H_AR(2:end)) .^ 2));
lin_MSE_pp_mt = sum(((pp_mt_est(2:end) - abs(H_AR(2:end)) .^ 2) .^ 2) ...
./ (abs(H_AR(2:end)) .^ 2));
lin_MSE_pp_ss = sum(((pp_ss_est(2:256) - abs(H_AR(2:end)) .^ 2) .^ 2) ...
./ (abs(H_AR(2:end)) .^ 2));
fprintf('Method \t \t MSE\n');
fprintf('PSTH-PSD \t %f\n', lin_MSE_pp);
fprintf('PMTM-PSD \t %f\n', lin_MSE_pp_mt);
fprintf('SS-PSD \t \t %f\n', lin_MSE_pp_ss);