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Echoflow
1.0.0
A ROS2 Toolset for tracking marine radar targets
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Echoflow
1.0.0
A ROS2 Toolset for tracking marine radar targets
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Functions | |
| float | computeSequentialMean (float new_observation, float num_samples, float prior_mean) |
| Compute the arithmetic mean of a sample given a new observation. | |
| std::tuple< float, float > | computeSequentialVariance (float new_observation, float num_samples, float prior_mean, float new_mean, float prior_ssdm) |
| Compute the variance of a sample given a new observation. | |
| std::tuple< float, float > | computeSequentialStdDev (float new_observation, float num_samples, float prior_mean, float new_mean, float prior_ssdm) |
| Compute the standard deviation of a sample given a new observation. | |
| float | computeCircularMean (float sines_sum, float cosines_sum) |
| Compute the circular mean angle of a sample of angle data. | |
| float | computeCircularVariance (float sines_sum, float cosines_sum, float num_samples) |
| Compute the circular variance of a sample of angle data. | |
| float | computeCircularStdDev (float sines_sum, float cosines_sum, float num_samples) |
| Compute the circular standard deviation of a sample of angle data. | |
| float | computeMeanResultantLength (float sines_sum, float cosines_sum, float num_samples) |
| Compute the mean resultant length of a sample of angle data. | |
| float echoflow::statistics::computeCircularMean | ( | float | sines_sum, |
| float | cosines_sum ) |
Compute the circular mean angle of a sample of angle data.
Given \(n\) angles \(\alpha_1, ..., \alpha_n\) measured in radians, their circular mean is defined as (from [Mardia_1972], section 2.2.2):
\(\overline{\alpha} = \textrm{arg}\biggl(\sum_\limits{j=1}^{n} e^{i\cdot\alpha_j} \biggr)\)
In order to store the Cartesian coordinates of each angle as a real float without an imaginary component, we instead use the arctan2 formulation to compute the mean resultant angle back to polar coordinates for to obtain the mean angle, as follows:
\(\overline{\alpha} = \textrm{atan2}\biggl(\sum\limits_{j=1}^{n} \sin \alpha_j, \sum_\limits{j=1}^{n} \cos \alpha_j \biggr)\)
| sines_sum | Sum of the sines of a sample of angle data. |
| cosines_sum | Sum of the cosines of a sample of angle data. |
| float echoflow::statistics::computeCircularStdDev | ( | float | sines_sum, |
| float | cosines_sum, | ||
| float | num_samples ) |
Compute the circular standard deviation of a sample of angle data.
Given \(n\) angles \(\alpha_1, ..., \alpha_n\) measured in radians, their circular standard deviation is defined as (from [Mardia_1972], section 2.3.4, Eq. 2.3.14):
\( s_0 = \sqrt{-2.0 * \log(\overline{R})} \),
where \( \overline{R} \) is the mean resultant length of the data, as defined in computeMeanResultantLength().
| sines_sum | Sum of the sines of a sample of angle data. |
| cosines_sum | Sum of the cosines of a sample of angle data. |
| num_samples | Total number of samples in the set of angle data. |
| float echoflow::statistics::computeCircularVariance | ( | float | sines_sum, |
| float | cosines_sum, | ||
| float | num_samples ) |
Compute the circular variance of a sample of angle data.
Given \(n\) angles \(\alpha_1, ..., \alpha_n\) measured in radians, their circular variance \(S_0\) is defined as (from [Mardia_1972], section 2.3):
\( S_0 = 1 - \overline{R} \)
where \( \overline{R} \) is the mean resultant length of the data, as defined in computeMeanResultantLength().
| sines_sum | Sum of the sines of a sample of angle data. |
| cosines_sum | Sum of the cosines of a sample of angle data. |
| num_samples | Total number of samples in the set of angle data. |
| float echoflow::statistics::computeMeanResultantLength | ( | float | sines_sum, |
| float | cosines_sum, | ||
| float | num_samples ) |
Compute the mean resultant length of a sample of angle data.
Given \(n\) angles \(\alpha_1, ..., \alpha_n\) measured in radians, their mean resultant length is defined as (from [Mardia_1972]):
\(\overline{R} = \sqrt{\overline{C}^2 + \overline{S}^2} \),
where \(\overline{C} = \frac{1}{n} \sum_\limits{i=1}^{n} \cos \alpha_i \) and \(\overline{S} = \frac{1}{n} \sum_\limits{i=1}^{n} \sin \alpha_i \).
| sines_sum | Sum of the sines of a sample of angle data. |
| cosines_sum | Sum of the cosines of a sample of angle data. |
| num_samples | Total number of samples in the set of angle data. |
| float echoflow::statistics::computeSequentialMean | ( | float | new_observation, |
| float | num_samples, | ||
| float | prior_mean ) |
Compute the arithmetic mean of a sample given a new observation.
Given a new observation \(x_n\), the prior mean of the data \(\overline{x}_{n-1}\), and the total number of observations \(n\), the recurrence relation for computing the sequential mean \(\overline{x}_n\) is as follows:
\( \overline{x}_n = \overline{x}_{n-1} + \frac{x_n - \overline{x}_{n-1}}{n} \)
| new_observation | New value with which to update the mean. |
| num_samples | Total number of samples (including current new observation). |
| prior_mean | Prior mean of the sample data (without new observation). |
| std::tuple< float, float > echoflow::statistics::computeSequentialStdDev | ( | float | new_observation, |
| float | num_samples, | ||
| float | prior_mean, | ||
| float | new_mean, | ||
| float | prior_ssdm ) |
Compute the standard deviation of a sample given a new observation.
Computes the square root of the variance. See the documentation for the computeSequentialVariance() function for details on how the variance is computed.
Returns both standard deviation and the updated sum of squares of deviation from the mean \( M_{2,n} \).
| new_observation | New value with which to update the standard deviation. |
| num_samples | Total number of samples (including current new observation). |
| prior_mean | Prior mean of the sample data (without new observation). |
| new_mean | New mean of sample including current observation (computeSequentialMean() should be used to compute the mean of the sample with the current observation before computing the standard deviation). |
| prior_ssdm | Prior sum of squared deviations from the mean (without new observation) used to compute variance. |
| std::tuple< float, float > echoflow::statistics::computeSequentialVariance | ( | float | new_observation, |
| float | num_samples, | ||
| float | prior_mean, | ||
| float | new_mean, | ||
| float | prior_ssdm ) |
Compute the variance of a sample given a new observation.
This function uses Welford's algorithm [welford_1962] to compute the new variance of the sample given the new observation. The following recurrence relation computes the unbiased variance of the sample for \( n > 1\):
\( s^2_n = s^2_{n-1} + \frac{(x_n - \overline{x}_{n-1})^2}{n} - \frac{s^2_{n-1}}{n-1}\)
Directly using this formula can be numerically unstable, so following Welford's algorithm the sum of squares of deviation from current mean, \( M_{2,n} = \sum_\limits{i=1}^{n} (x_i - \overline{x}_n)^2 \) is used to update the variance:
\( M_{2,n} = M_{2,n-1} + (x_n - \overline{x}_{n-1})(x_n - \overline{x}_n) \)
The variance returned is: \( s^2_n = \frac{M_{2,n}}{n} \)
The function returns both the variance and the updated sum of squares of deviation from the mean \( M_{2,n} \).
| new_observation | New value with which to update the variance. |
| num_samples | Total number of samples (including current new observation). |
| prior_mean | Prior mean of the sample data (without new observation). |
| new_mean | New mean of sample including current observation (computeSequentialMean() should be used to compute the mean of the sample with the current observation before computing the variance). |
| prior_ssdm | Prior sum of squared deviations from the mean (without new observation). |
1.13.2