fem.discrete package¶
Submodules¶
fem.discrete.combinatorics module¶
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fem.discrete.combinatorics.binomial_coefficients(n, k)[source]¶ Return a list of binomial coefficients.
Parameters: - n (int) – the first argument to the choose operation
- k (int) – the maximum second argument to the choose operation
Returns: a list of binomial coefficients
Return type: ndarray
Examples
Compute the binomial coefficients \({4\choose 0}\), \({4\choose 1}\), and \({4\choose 2}\)
>>> from fem.discrete.combinatorics import binomial_coefficients >>> binomial_coefficients(4, 2) array([1, 4, 6])
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fem.discrete.combinatorics.mixed_radix_to_base_10(x, b)[source]¶ Convert the mixed radix integer with digits x and bases b to base 10.
Parameters: - x (list) – a list of digits ordered by increasing place values
- b (list) – a list of bases corresponding to the digits
Examples
Generally, the base 10 representation of the mixed radix number \(x_n\ldots x_1\) where \(x_i\) is a digit in place value \(i\) with base \(b_i\) is
\[\sum_{i=1}^nx_i\prod_{j=i+1}^nb_j = x_n + b_nx_{n-1} + b_nb_{n-1}x_{n-2} + \cdots + b_n\cdots b_2x_1\]Convert 111 with bases \((b_1,b_2,b_3)=(2,3,4)\) to base 10:
>>> from fem.discrete.combinatorics import mixed_radix_to_base_10 >>> mixed_radix_to_base_10([1,1,1], [2,3,4]) 17
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fem.discrete.combinatorics.multiindices(n, k)[source]¶ Return an ordered list of distinct tuples of the first \(n\) nonnegative integers.
Parameters: - n (int) – The least integer not included in the list of distinct tuples
- k (int) – The length of each tuple
Returns: a list of distinct tupes of shape \({n\choose k}\) by \(k\)
Return type: ndarray
Examples
>>> from fem.discrete.combinatorics import multiindices >>> multiindices(4, 2) array([[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]])
fem.discrete.fit module¶
This module implements a class that can fit the Potts model to data and make predictions with the model. FEM for discrete data
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fem.discrete.fit.categorize(x)[source]¶ Convert symbolic x data to integer data. x contains a list of lists that are samples of different variables. Each list of samples is mapped to the first number of unique values detected of nonnegative integers.
Parameters: x (list) – A list where each element is a list of samples of a different variable Returns: The integer data and a list of dictionaries that map symbol data to integer data for each row of x Return type: (list, list) Examples
>>> from fem.discrete import categorize >>> x = [['a', 'b', 'a', 'a'], [10, 11, 12, 10, 11]] >>> x_int, cat_x = categorize(x) >>> x_int [array([0, 1, 0, 0]), array([0, 1, 2, 0, 1])] >>> cat_x [{'a': 0, 'b': 1}, {10: 0, 11: 1, 12: 2}]
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class
fem.discrete.fit.model(degs=[1])[source]¶ Bases:
objectThis class implements a Potts model that can be fit to data and used to make predictions.
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degs¶ list – A list of model degrees
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n_x¶ int – The number of input variables
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n_y¶ int – The number of output variables
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m_x¶ ndarray – An array of length n_x listing the number of values that each input variable takes
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m_y¶ ndarray
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m_x_cumsum¶ ndarray
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m_y_cumsum¶ ndarray
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x_int¶ ndarray – The input variable data x mapped to nonnegative integers
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y_int¶ ndarray – The output variable data y mapped to nonnegative integers
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cat_x¶ list – A list of per-row dicts that map the input x symbol data to the integer data in x_int
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cat_y¶ list
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cat_x_inv¶ list – A list of dicts that are the inverses of the list of dicts in cat_x
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cat_y_inv¶ list
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w¶ dict – A dictionary of length len(degs) keyed by degs. w[deg] are the model parameters for degree deg
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disc¶ list – A list of length n_x of the running discrepancies per variable computed during the model fit
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impute¶ bool – If true, the model will map each combination of hold-one-out input variables to the held out variable
Examples
>>> import fem >>> n, m = 10, 3 >>> fem.discrete.time_series(fem.discrete.model_parameters(n, m), n, m) >>> model = fem.discrete.model() >>> model.fit(x[:, :-1], x[:, 1:]) >>> model.predict(x[:, -1])
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fem.discrete.fit.one_hot(x, m, degs=[1])[source]¶ Compute the one-hot encoding of x
Parameters: - x (ndarray) – n by l array of nonnegative integers to be converted to one-hot encoding
- m (ndarray) – length n array storing the number of possible integers in each row of x
- degs (list) – list of degrees of one-hot-encodings to compute
Returns: sparse matrix that is the one-hot encoding of x. The degrees of one-hot encodings are vertically stacked.
Return type: csc_matrix
Examples
The one-hot encoding of a discrete variable records the state of the variable by a boolean vector with a 1 at the index which is the state of the variable. The one-hot encoding of a system of discrete variables is the concatenation of the one-hot encodings of each variable. For example if the system of discrete variables \((x_1, x_2, x_3)\) has state \((x_1,x_2,x_3)=(1,1,2)\) and the variables may take values from the first \(m=(3,3,4)\) nonnegative integers, i.e. \(x_1,x_2\in\{0,1,2\}\) and \(x_3\in\{0,1,2,3\}\), then the one-hot encoding of \((x_1, x_2, x_3)^T\) is \(\sigma=(0,1,0,0,1,0,0,0,1,0)^T\).
>>> from fem.discrete import one_hot >>> x = [1,1,2] >>> m = [3,3,4] >>> x_oh = one_hot(x, m) >>> print x_oh.todense() matrix([[0.], [1.], [0.], [0.], [1.], [0.], [0.], [0.], [1.], [0.]])