Analyzing noisy data

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I recently launched a rocket with a barometric altimeter that is accurate to roughly 10 ft (calculated via data acquired during flight). The recorded data is in time increments of 0.05 sec per sample and a graph of altitude vs. time looks pretty much like it should when zoomed out over the entire flight.

The problem is when I try to calculate other values such as velocity or acceleration from the data, the accuracy of the measurements makes the calculated values pretty much worthless. What techniques can I use to smooth out the data so that I can calculate (or approximate) reasonable values for the velocity and acceleration? It is important that major events remain in place in time, most notably the 0 for for the first entry and the highest point during flight (2707).

The altitude data follows and is measured in ft above ground level. The first time would be 0.00 and each sample is 0.05 seconds after the previous sample. The spike at the beginning of the flight is due to a technical problem that occurred during liftoff and removing the spike is optimal.

I originally tried using linear interpolation, averaging nearby data points, but it took many iterations to smooth the data enough for integration and the flattening of the curve removed the important apogee and ground level events.

All help is greatly appreciated. Please note this is not the complete data set and I am looking for suggestions on better ways to analyze the data, not for someone to reply with a transformed data set. It would be nice to use an algorithm on board future rockets which can predict current altitude/velocity/acceleration without knowing the full flight data, though that is not required.

562

asked Dec 24 '09 05:12

Nick Larsen

3 Answers

Here is my solution, using a Kalman filter. You will need to tune the parameters (even +- orders of magnitude) if you want to smooth more or less.

#!/usr/bin/env octave

% Kalman filter to smooth measures of altitude and estimate
% speed and acceleration. The continuous time model is more or less as follows:
% derivative of altitude := speed
% derivative of speed := acceleration
% acceleration is a Wiener process

%------------------------------------------------------------
% Discretization of the continuous-time linear system
% 
%   d  |x|   | 0 1 0 | |x|
%  --- |v| = | 0 0 1 | |v|   + "noise"
%   dt |a|   | 0 0 0 | |a|
%
%   y = [1 0 0] |x|     + "measurement noise"
%               |v|
%               |a|
%
st = 0.05;    % Sampling time
A = [1  st st^2/2;
     0  1  st    ;
     0  0  1];
C = [1 0 0];

%------------------------------------------------------------
% Fine-tune these parameters! (in particular qa and R)
% The acceleration follows a "random walk". The greater is the variance qa,
% the more "reactive" the system is expected to be, i.e.
% the more the acceleration is expected to vary
% The greater is R, the more noisy is your measurement instrument
% (less "accuracy" of the barometric altimeter);
% if you increase R, you will smooth the estimate more
qx = 1.0;                      % Variance of model noise for position
qv = 1.0;                      % Variance of model noise for speed
qa = 50.0;                     % Variance of model noise for acceleration
Q  = diag([qx, qv, qa]);
R  = 100.0;                    % Variance of measurement noise
                               % (10^2, if 10ft is the standard deviation)

load data.txt  % Put your measures in this file

est_position     = zeros(length(data), 1);
est_speed        = zeros(length(data), 1);
est_acceleration = zeros(length(data), 1);

%------------------------------------------------------------
% Kalman filter
xhat = [0;0;0];     % Initial estimate
P    = zeros(3,3);  % Initial error variance
for i=1:length(data),
   y = data(i);
   xpred = A*xhat;                                    % Prediction
   Ppred = A*P*A' + Q;                                % Prediction error variance
   Lambdainv = 1/(C*Ppred*C' + R);
   xhat  = xpred + Ppred*C'*Lambdainv*(y - C*xpred);  % Update estimation
   P = Ppred - Ppred*C'*Lambdainv*C*Ppred;            % Update estimation error variance
   est_position(i)     = xhat(1);
   est_speed(i)        = xhat(2);
   est_acceleration(i) = xhat(3);
end

%------------------------------------------------------------
% Plot
figure(1);
hold on;
plot(data, 'k');               % Black: real data
plot(est_position, 'b');       % Blue:  estimated position
plot(est_speed, 'g');          % Green: estimated speed
plot(est_acceleration, 'r');   % Red:   estimated acceleration
pause

answered Oct 28 '22 22:10

Federico A. Ramponi

You could try running the data through a low-pass filter. This will smooth out high frequency noise. Maybe a simple FIR.

Also, you could pull your major events from the raw data, but use a polynomial fit for velocity and acceleration data.

answered Oct 28 '22 21:10

Rob Curtis

have you tried performing a scrolling window average of your values ? Basically you perform a window of, say 10 values (from 0 to 9), and calculate its average. then you scroll the window one point (from 1 to 10) and recalculate. This will smooth the values while keeping the number of points relatively unchanged. Larger windows give smoother data at the price of loosing more high-frequency information.

You can use the median instead of the average if your data happen to present outlier spikes.

You can also try with Autocorrelation.

answered Oct 28 '22 22:10

Stefano Borini

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Analyzing noisy data

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Analyzing noisy data

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numerical-analysis

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