Estimator Functions

Scikit-GStat implements various semi-variance estimators. These functions can be found in the skgstat.estimators submodule. Each of these functions can be used independently from Variogram class. In this case the estimator is expecting an array of pairwise differences to calculate the semi-variance. Not the values themselves.

Matheron

skgstat.estimators.matheron(x)

Matheron Semi-Variance

Calculates the Matheron Semi-Variance from an array of pairwise differences. Returns the semi-variance for the whole array. In case a semi-variance is needed for multiple groups, this function has to be mapped on each group. That is the typical use case in geostatistics.

Parameters:

x (numpy.ndarray) – Array of pairwise differences. These values should be the distances between pairwise observations in value space. If xi and x[i+h] fall into the h separating distance class, x should contain abs(xi - x[i+h]) as an element.

Return type:

numpy.float64

Notes

This implementation follows the original publication [1] and the notes on their application [2]. Following the 1962 publication [1], the semi-variance is calculated as:

\gamma (h) = \frac{1}{2N(h)} * \sum_{i=1}^{N(h)}(x)^2

with:

x = Z(x_i) - Z(x_{i+h})

where x is exactly the input array x.

References

[1] (1,2)

Matheron, G. (1962): Traité de Géostatistique Appliqué, Tonne 1. Memoires de Bureau de Recherches Géologiques et Miniéres, Paris.

[2]

Matheron, G. (1965): Les variables regionalisées et leur estimation. Editions Masson et Cie, 212 S., Paris.

Cressie

skgstat.estimators.cressie(x)

Cressie-Hawkins Semi-Variance

Calculates the Cressie-Hawkins Semi-Variance from an array of pairwise differences. Returns the semi-variance for the whole array. In case a semi-variance is needed for multiple groups, this function has to be mapped on each group. That is the typical use case in geostatistics.

Parameters:

x (numpy.ndarray) – Array of pairwise differences. These values should be the distances between pairwise observations in value space. If xi and x[i+h] fall into the h separating distance class, x should contain abs(xi - x[i+h]) as an element.

Return type:

numpy.float64

Notes

This implementation is done after the publication by Cressie and Hawkins from 1980 [3]:

2\gamma (h) = \frac{(\frac{1}{N(h)} \sum_{i=1}^{N(h)} |x|^{0.5})^4} {0.457 + \frac{0.494}{N(h)} + \frac{0.045}{N^2(h)}}

with:

x = Z(x_i) - Z(x_{i+h})

where x is exactly the input array x.

References

[3]

Cressie, N., and D. Hawkins (1980): Robust estimation of the variogram. Math. Geol., 12, 115-125.

Dowd

skgstat.estimators.dowd(x)

Dowd semi-variance

Calculates the Dowd semi-variance from an array of pairwise differences. Returns the semi-variance for the whole array. In case a semi-variance is needed for multiple groups, this function has to be mapped on each group. That is the typical use case in geostatistics.

Parameters:

x (numpy.ndarray) – Array of pairwise differences. These values should be the distances between pairwise observations in value space. If xi and x[i+h] fall into the h separating distance class, x should contain abs(xi - x[i+h]) as an element.

Return type:

numpy.float64

Notes

The Dowd estimator is based on the median of all pairwise differences in each lag class and is therefore robust to exteme values at the cost of variability. This implementation follows Dowd’s publication [4]:

2\gamma (h) = 2.198 * {median(x)}^2

with:

x = Z(x_i) - Z(x_{i+h})

where x is exactly the input array x.

References

[4]

Dowd, P. A., (1984): The variogram and kriging: Robust and resistant estimators, in Geostatistics for Natural Resources Characterization. Edited by G. Verly et al., pp. 91 - 106, D. Reidel, Dordrecht.

Genton

skgstat.estimators.genton(x)

Genton robust semi-variance estimator

Return the Genton semi-variance of the given sample x. Genton is a highly robust varigram estimator, that is designed to be location free and robust on extreme values in x. Genton is based on calculating kth order statistics and will for large data sets be close or equal to the 25% quartile of all ordered point pairs in X.

Parameters:

x (numpy.ndarray) – Array of pairwise differences. These values should be the distances between pairwise observations in value space. If xi and x[i+h] fall into the h separating distance class, x should contain abs(xi - x[i+h]) as an element.

Return type:

numpy.float64

Notes

The Genton estimator is described in great detail in the original publication [5] and is defined as:

Q_{N_h} = 2.2191\{|V_i(h) - V_j(h)|; i < j\}_{(k)}

and

k = \binom{[N_h / 2] + 1}{2}

and

q = \binom{N_h}{2}

where k is the kth quantile of all q point pairs. For large N (k/q) will be close to 0.25. For N >= 500, (k/q) is close to 0.25 by two decimals and will therefore be set to 0.5 and the two binomial coefficients k, q are not calculated.

References

[5]

Genton, M. G., (1998): Highly robust variogram estimation, Math. Geol., 30, 213 - 221.

Shannon Entropy

skgstat.estimators.entropy(x, bins=None)

Shannon Entropy estimator

Calculates the Shannon Entropy H as a variogram estimator. It is highly recommended to calculate the bins and explicitly set them as a list. In case this function is called for more than one lag class in a variogram, setting bins to None would result in different bin edges in each lag class. This would be very difficult to interpret.

Parameters:

• x (numpy.ndarray) – Array of pairwise differences. These values should be the distances between pairwise observations in value space. If xi and x[i+h] fall into the h separating distance class, x should contain abs(xi - x[i+h]) as an element.

• bins (int, list, str) – list of the bin edges used to calculate the empirical distribution of x. If bins is a list, these values are used directly. In case bins is a integer, as many even width bins will be calculated between the minimum and maximum value of x. In case bins is a string, it will be passed as bins argument to numpy.histograms function.

Returns:

entropy – Shannon entropy of the given pairwise differences.

Return type:

numpy.float64

Notes

MinMax

Warning

This is an experimental semi-variance estimator. It is heavily influenced by extreme values and outliers. That behaviour is usually not desired in geostatistics.

skgstat.estimators.minmax(x)

Minimum - Maximum Estimator

Returns a custom value. This estimator is the difference of maximum and minimum pairwise differences, normalized by the mean. MinMax will be very sensitive to extreme values.

Do only use this estimator, in case you know what you are doing. It is experimental and might change its behaviour in a future version.

Parameters:

x (numpy.ndarray) – Array of pairwise differences. These values should be the distances between pairwise observations in value space. If xi and x[i+h] fall into the h separating distance class, x should contain abs(xi - x[i+h]) as an element.

Return type:

numpy.float64

Percentile

Warning

This is an experimental semi-variance estimator. It uses just a percentile of the given pairwise differences and does not bear any information about their variance.

skgstat.estimators.percentile(x, p=50)

Percentile estimator

Returns a given percentile as semi-variance. Do only use this estimator, in case you know what you are doing.

Do only use this estimator, in case you know what you are doing. It is experimental and might change its behaviour in a future version.

Parameters:

• x (numpy.ndarray) – Array of pairwise differences. These values should be the distances between pairwise observations in value space. If xi and x[i+h] fall into the h separating distance class, x should contain abs(xi - x[i+h]) as an element.

• p (int) – Desired percentile. Should be given as whole numbers 0 < p < 100.

Return type:

np.float64