HMS_map/docs/Example_Simulation.html at main · DylanEsguerra/HMS_map · GitHub

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  </style></head><body><div class="content"><h1></h1><!--introduction--><!--/introduction--><h2>Contents</h2><div><ul><li><a href="#1">Simulate Discrete Logistic Map</a></li><li><a href="#2">Apply HMSmap to Time Series</a></li><li><a href="#3">Figures</a></li><li><a href="#4">Lyapunov Exponent Calculation</a></li><li><a href="#5">Error Calculation</a></li></ul></div><h2 id="1">Simulate Discrete Logistic Map</h2><p><i>T</i>: length of series</p><p><i>r</i>: set to 4 for chaotic dynamics</p><p><i>obs</i>: noise added as a percent of the variation in x.</p><p><i>Y</i>: stores observed data, <i>XP</i>: smoothed time series, <i>X_pred</i> : Forecast</p><p>Add observation noise to the deterministic system to create <i>Y</i> the observed data</p><pre class="codeinput">addpath <span class="string">'/Users/dylanesguerra/Desktop/HMS_map/main'</span>

T = 100;
r = 4;
x = rand(T, 1);
<span class="keyword">for</span> t = 1:(T-1)
    x(t+1) = r * x(t) * (1 - x(t));
<span class="keyword">end</span>

obs = 0.1;

Y = x + obs * std(x) * randn(length(x), 1);
</pre><h2 id="2">Apply HMSmap to Time Series</h2><p>step: sets how many steps ahead to forecast.</p><p><b>f_nn</b> uses the false nearest neighbors algorithm to estimate the Embedding Dimension of each series from the observed data, (Kennel et al., 1992)</p><p>The optimal <img src="Example_Simulation_eq15003285036546140546.png" alt="$$ \theta $$" style="width:5px;height:8px;"> is determind based on minimizing the observation error of the furthest step ahead forecast with fminbnd.</p><pre class="codeinput">[FNN] = f_fnn(Y, 1, 10, 15, 2);
[~, E] = min(FNN);


noise = (obs .* std(x)).^2;
step = 3;

fun = @(z) HMSmap_lags(Y, <span class="string">'gaussian'</span>, z, noise, E-1, 1, 0, step,[]).oe(step);
z = fminbnd(fun, 0, 50);

out = HMSmap_lags(Y, <span class="string">'gaussian'</span>, z, noise, E-1, 1, 1, step,[]);

XP = out.states;
Theta = z;
Coefs = out.coef;
X_pred = out.pred;
</pre><img vspace="5" hspace="5" src="Example_Simulation_01.png" alt=""> <h2 id="3">Figures</h2><p>By setting the figs parameter of <b>HMSmao_lags.m</b> to 1 rather than zero some diagnostic plots are automatically created.</p><p>HMSmap_lags(x,model,kernel,theta,vobs,E,tau,figs,stepsahead,inits)</p><p>Top row plots observed data in red and smoothed data in blue. The data is plotted against a time lag x(E:T-k),x(E+k:T) to show an emndedding of the data in 2d space k steps ahead. Bottom row plots observed data vs smoothed data in black and the observed data vs predictions in green at each step ahead.</p><h2 id="4">Lyapunov Exponent Calculation</h2><p>The <b>Lyapunov Exponent</b> is estimated with the <b>lyapunov_QR_lags</b> function taking in the coeficients of HMSmap as well as the embedding dimension. This is the Jacobian method of <b>LE</b> estimation using the coefficients of local linear regression in place of partial derivatives,(Deyle et al., 2016b).</p><pre class="codeinput">Lyp = lyapunov_QR_lags(Coefs, T-(E-1), E-1)
</pre><pre class="codeoutput">
Lyp =

    0.4930

</pre><h2 id="5">Error Calculation</h2><p>Compare the smoothed time series to the noise free data to get the filtered error <i>Filter_err</i>.</p><p>This should be significantly lower than the added observation error <i>obs</i></p><pre class="codeinput">Filter_err = sqrt(mean((XP(2:T) - x(2:T)).^2) / var(x(2:T))) <span class="comment">% Calculate filter error</span>
</pre><pre class="codeoutput">
Filter_err =

    0.1133

</pre><p class="footer"><br><a href="https://www.mathworks.com/products/matlab/">Published with MATLAB&reg; R2021b</a><br></p></div><!--
##### SOURCE BEGIN #####

%% Simulate Discrete Logistic Map
% _T_: length of series
%
% _r_: set to 4 for chaotic dynamics
%
% _obs_: noise added as a percent of the variation in x.
%
% _Y_: stores observed data, _XP_: smoothed time series, _X_pred_ : Forecast
%
% Add observation noise to the deterministic system to create _Y_ the observed data


addpath '/Users/dylanesguerra/Desktop/HMS_map/main'

T = 100;
r = 4;
x = rand(T, 1);
for t = 1:(T-1)
    x(t+1) = r * x(t) * (1 - x(t));
end

obs = 0.1;

Y = x + obs * std(x) * randn(length(x), 1);


%% Apply HMSmap to Time Series
% step: sets how many steps ahead to forecast.
%
% *f_nn* uses the false nearest neighbors algorithm to estimate the Embedding
% Dimension of each series from the observed data, (Kennel et al., 1992)
%
% The optimal $$ \theta $$ is determind based on minimizing the observation error of the furthest step ahead forecast with fminbnd.

[FNN] = f_fnn(Y, 1, 10, 15, 2);
[~, E] = min(FNN);


noise = (obs .* std(x)).^2;
step = 3;

fun = @(z) HMSmap_lags(Y, 'gaussian', z, noise, E-1, 1, 0, step,[]).oe(step);
z = fminbnd(fun, 0, 50);

out = HMSmap_lags(Y, 'gaussian', z, noise, E-1, 1, 1, step,[]);

XP = out.states;
Theta = z;
Coefs = out.coef;
X_pred = out.pred;


%% Figures
% By setting the figs parameter of *HMSmao_lags.m* to 1 rather than zero
% some diagnostic plots are automatically created.
%
% HMSmap_lags(x,model,kernel,theta,vobs,E,tau,figs,stepsahead,inits)
%
% Top row plots observed data in red and smoothed data in blue. The data is plotted against a time lag x(E:T-k),x(E+k:T) to show an emndedding of the data in 2d space k steps ahead.
% Bottom row plots observed data vs smoothed data in black and the observed
% data vs predictions in green at each step ahead.


%% Lyapunov Exponent Calculation
% The *Lyapunov Exponent* is estimated with the *lyapunov_QR_lags* function taking in the coeficients of HMSmap as well as the embedding dimension.
% This is the Jacobian method of *LE* estimation using the coefficients of
% local linear regression in place of partial derivatives,(Deyle et al.,
% 2016b).

Lyp = lyapunov_QR_lags(Coefs, T-(E-1), E-1)

%% Error Calculation
% Compare the smoothed time series to the noise free data to get the
% filtered error _Filter_err_.
%
% This should be significantly lower than the added observation error _obs_

Filter_err = sqrt(mean((XP(2:T) - x(2:T)).^2) / var(x(2:T))) % Calculate filter error


##### SOURCE END #####
--></body></html>