Skip to content

covar

covar_sep(col, X1, n1, X2, n2, d, g)

Calculate the correlation (K) between X1 and X2 with a separable power exponential correlation function with range d and nugget g.

Parameters:

Name Type Description Default
col int

Number of columns in the input matrices X1 and X2.

 required
X1 ndarray

First input matrix of shape (n1, col).

 required
n1 int

Number of rows in the first input matrix X1.

 required
X2 ndarray

Second input matrix of shape (n2, col).

 required
n2 int

Number of rows in the second input matrix X2.

 required
d ndarray

Array of length col representing the range parameters.

 required
g float

Nugget parameter.

 required

Returns:

Name Type Description
ndarray ndarray

The calculated covariance matrix K of shape (n1, n2).

Examples:

>>> import numpy as np
>>> from spotpython.gp.covar import covar_sep
>>> col = 2
>>> X1 = np.array([[1, 2], [3, 4], [5, 6]])
>>> n1 = 3
>>> X2 = np.array([[7, 8], [9, 10]])
>>> n2 = 2
>>> d = np.array([1.0, 1.0])
>>> g = 0.1
>>> K = covar_sep(col, X1, n1, X2, n2, d, g)
>>> print(K)
[[1.12535175e-07 3.72007598e-44]
 [3.72007598e-44 1.38389653e-87]
 [1.38389653e-87 5.14820022e-131]]
Source code in spotpython/gp/covar.py 
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
def covar_sep(col, X1, n1, X2, n2, d, g) -> np.ndarray:
    """
    Calculate the correlation (K) between X1 and X2 with
    a separable power exponential correlation function
    with range d and nugget g.

    Args:
        col (int): Number of columns in the input matrices X1 and X2.
        X1 (ndarray): First input matrix of shape (n1, col).
        n1 (int): Number of rows in the first input matrix X1.
        X2 (ndarray): Second input matrix of shape (n2, col).
        n2 (int): Number of rows in the second input matrix X2.
        d (ndarray): Array of length col representing the range parameters.
        g (float): Nugget parameter.

    Returns:
        ndarray: The calculated covariance matrix K of shape (n1, n2).

    Examples:
        >>> import numpy as np
        >>> from spotpython.gp.covar import covar_sep
        >>> col = 2
        >>> X1 = np.array([[1, 2], [3, 4], [5, 6]])
        >>> n1 = 3
        >>> X2 = np.array([[7, 8], [9, 10]])
        >>> n2 = 2
        >>> d = np.array([1.0, 1.0])
        >>> g = 0.1
        >>> K = covar_sep(col, X1, n1, X2, n2, d, g)
        >>> print(K)
        [[1.12535175e-07 3.72007598e-44]
         [3.72007598e-44 1.38389653e-87]
         [1.38389653e-87 5.14820022e-131]]
    """
    K = np.zeros((n1, n2))
    X1 = prepare_X(X1)
    X2 = prepare_X(X2)

    for i in range(n1):
        for j in range(n2):
            K[i, j] = 0.0
            for k in range(col):
                K[i, j] += (X1[i, k] - X2[j, k]) ** 2 / d[k]
            if i == j and K[i, j] == 0.0:
                K[i, j] = 1.0 + g
            else:
                K[i, j] = np.exp(0.0 - K[i, j])

    return K

covar_sep_symm(col, X, n, d, g)

Calculate the correlation (K) between X1 and X2 with a separable power exponential correlation function with range d and nugget g.

Parameters:

Name Type Description Default
col int

Number of columns in the input matrix X (features).

 required
X ndarray

Input matrix of shape (n, col).

 required
n int

Number of rows in the input matrix X.

 required
d ndarray

Array of length col representing the range parameters, shape (col,).

 required
g float

Nugget parameter.

 required

Returns:

Name Type Description
ndarray ndarray

The calculated covariance matrix K of shape (n, n).

Examples:

>>> from spotpython.gp.covar import covar_sep_symm
>>> import numpy as np
>>> col = 2
>>> X = np.array([[1, 2], [3, 4], [5, 6]])
>>> n = 3
>>> d = np.array([1.0, 1.0])
>>> g = 0.1
>>> K = covar_sep_symm(col, X, n, d, g)
>>> print(K)
[[1.1        0.01831564 0.00012341]
 [0.01831564 1.1        0.01831564]
 [0.00012341 0.01831564 1.1       ]]
Source code in spotpython/gp/covar.py 
 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
def covar_sep_symm(col, X, n, d, g) -> np.ndarray:
    """
    Calculate the correlation (K) between X1 and X2 with a separable power exponential correlation function with range d and nugget g.

    Args:
        col (int): Number of columns in the input matrix X (features).
        X (ndarray): Input matrix of shape (n, col).
        n (int): Number of rows in the input matrix X.
        d (ndarray): Array of length col representing the range parameters, shape (col,).
        g (float): Nugget parameter.

    Returns:
        ndarray: The calculated covariance matrix K of shape (n, n).

    Examples:
        >>> from spotpython.gp.covar import covar_sep_symm
        >>> import numpy as np
        >>> col = 2
        >>> X = np.array([[1, 2], [3, 4], [5, 6]])
        >>> n = 3
        >>> d = np.array([1.0, 1.0])
        >>> g = 0.1
        >>> K = covar_sep_symm(col, X, n, d, g)
        >>> print(K)
        [[1.1        0.01831564 0.00012341]
         [0.01831564 1.1        0.01831564]
         [0.00012341 0.01831564 1.1       ]]
    """
    K = np.zeros((n, n))
    X = prepare_X(X)

    # calculate the covariance matrix K
    for i in range(n):
        K[i, i] = 1.0 + g
        for j in range(i + 1, n):
            K[i, j] = 0.0
            for k in range(col):
                K[i, j] += (X[i, k] - X[j, k]) ** 2 / d[k]
            K[i, j] = np.exp(-K[i, j])
            K[j, i] = K[i, j]

    return K

diff_covar_sep(col, X1, n1, X2, n2, d, K)

Calculate the first and second derivative (wrt d) of the correlation (K) between X1 and X2 with a separable power exponential correlation function with range d and nugget g (though g not needed).

Parameters:

Name Type Description Default
col int

Number of columns in the input matrices X1 and X2.

 required
X1 ndarray

First input matrix of shape (n1, col).

 required
n1 int

Number of rows in the first input matrix X1.

 required
X2 ndarray

Second input matrix of shape (n2, col).

 required
n2 int

Number of rows in the second input matrix X2.

 required
d ndarray

Array of length col representing the range parameters.

 required
K ndarray

Covariance matrix of shape (n1, n2).

 required

Returns:

Name Type Description
ndarray ndarray

The calculated derivative covariance matrix dK of shape (col, n1, n2).

Examples:

>>> col = 2
>>> X1 = np.array([[1, 2], [3, 4], [5, 6]])
>>> n1 = 3
>>> X2 = np.array([[7, 8], [9, 10]])
>>> n2 = 2
>>> d = np.array([1.0, 1.0])
>>> K = np.exp(-np.array([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]))
>>> dK = diff_covar_sep(col, X1, n1, X2, n2, d, K)
>>> print(dK)
[[[1.12535175e-07 3.72007598e-44]
  [3.72007598e-44 1.38389653e-87]
  [1.38389653e-87 5.14820022e-131]]
 [[1.12535175e-07 3.72007598e-44]
  [3.72007598e-44 1.38389653e-87]
  [1.38389653e-87 5.14820022e-131]]]
Source code in spotpython/gp/covar.py 
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
def diff_covar_sep(col, X1, n1, X2, n2, d, K) -> np.ndarray:
    """
    Calculate the first and second derivative (wrt d) of the correlation (K)
    between X1 and X2 with a separable power exponential correlation function
    with range d and nugget g (though g not needed).

    Args:
        col (int): Number of columns in the input matrices X1 and X2.
        X1 (ndarray): First input matrix of shape (n1, col).
        n1 (int): Number of rows in the first input matrix X1.
        X2 (ndarray): Second input matrix of shape (n2, col).
        n2 (int): Number of rows in the second input matrix X2.
        d (ndarray): Array of length col representing the range parameters.
        K (ndarray): Covariance matrix of shape (n1, n2).

    Returns:
        ndarray: The calculated derivative covariance matrix dK of shape (col, n1, n2).

    Examples:
        >>> col = 2
        >>> X1 = np.array([[1, 2], [3, 4], [5, 6]])
        >>> n1 = 3
        >>> X2 = np.array([[7, 8], [9, 10]])
        >>> n2 = 2
        >>> d = np.array([1.0, 1.0])
        >>> K = np.exp(-np.array([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]))
        >>> dK = diff_covar_sep(col, X1, n1, X2, n2, d, K)
        >>> print(dK)
        [[[1.12535175e-07 3.72007598e-44]
          [3.72007598e-44 1.38389653e-87]
          [1.38389653e-87 5.14820022e-131]]
         [[1.12535175e-07 3.72007598e-44]
          [3.72007598e-44 1.38389653e-87]
          [1.38389653e-87 5.14820022e-131]]]
    """
    X1 = prepare_X(X1)
    X2 = prepare_X(X2)
    dK = np.zeros((col, n1, n2))

    for k in range(col):
        d2k = d[k] ** 2
        for i in range(n1):
            for j in range(n2):
                dK[k, i, j] = K[i, j] * ((X1[i, k] - X2[j, k]) ** 2) / d2k

    return dK

diff_covar_sep_symm(col, X, n, d, K)

Calculate the first and second derivative (wrt d) of the correlation (K) between X1 and X2 with a separable power exponential correlation function with range d and nugget g (though g not needed) – assumes symmetric matrix.

Parameters:

Name Type Description Default
col int

Number of columns in the input matrix X.

 required
X ndarray

Input matrix of shape (n, col).

 required
n int

Number of rows in the input matrix X.

 required
d ndarray

Array of length col representing the range parameters.

 required
K ndarray

Covariance matrix of shape (n, n).

 required

Returns:

Name Type Description
ndarray ndarray

The calculated derivative covariance matrix dK of shape (col, n, n).

Examples:

>>> col = 2
>>> X = np.array([[1, 2], [3, 4], [5, 6]])
>>> n = 3
>>> d = np.array([1.0, 1.0])
>>> K = np.exp(-np.array([[0.0, 1.0, 2.0], [1.0, 0.0, 1.0], [2.0, 1.0, 0.0]]))
>>> dK = diff_covar_sep_symm(col, X, n, d, K)
>>> print(dK)
[[[0.         0.36787944 0.01831564]
  [0.36787944 0.         0.36787944]
  [0.01831564 0.36787944 0.        ]]
 [[0.         0.36787944 0.01831564]
  [0.36787944 0.         0.36787944]
  [0.01831564 0.36787944 0.        ]]]
Source code in spotpython/gp/covar.py 
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
def diff_covar_sep_symm(col, X, n, d, K) -> np.ndarray:
    """
    Calculate the first and second derivative (wrt d) of the correlation (K)
    between X1 and X2 with a separable power exponential correlation function
    with range d and nugget g (though g not needed) -- assumes symmetric matrix.

    Args:
        col (int): Number of columns in the input matrix X.
        X (ndarray): Input matrix of shape (n, col).
        n (int): Number of rows in the input matrix X.
        d (ndarray): Array of length col representing the range parameters.
        K (ndarray): Covariance matrix of shape (n, n).

    Returns:
        ndarray: The calculated derivative covariance matrix dK of shape (col, n, n).

    Examples:
        >>> col = 2
        >>> X = np.array([[1, 2], [3, 4], [5, 6]])
        >>> n = 3
        >>> d = np.array([1.0, 1.0])
        >>> K = np.exp(-np.array([[0.0, 1.0, 2.0], [1.0, 0.0, 1.0], [2.0, 1.0, 0.0]]))
        >>> dK = diff_covar_sep_symm(col, X, n, d, K)
        >>> print(dK)
        [[[0.         0.36787944 0.01831564]
          [0.36787944 0.         0.36787944]
          [0.01831564 0.36787944 0.        ]]
         [[0.         0.36787944 0.01831564]
          [0.36787944 0.         0.36787944]
          [0.01831564 0.36787944 0.        ]]]
    """
    X = prepare_X(X)
    dK = np.zeros((col, n, n))

    for k in range(col):
        d2k = d[k] ** 2
        for i in range(n):
            for j in range(i + 1, n):
                dK[k, i, j] = dK[k, j, i] = K[i, j] * ((X[i, k] - X[j, k]) ** 2) / d2k
            dK[k, i, i] = 0.0

    return dK