309 lines
10 KiB
Python
309 lines
10 KiB
Python
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import itertools
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from sympy.core.add import Add
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from sympy.core.expr import Expr
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from sympy.core.function import expand as _expand
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from sympy.core.mul import Mul
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from sympy.core.singleton import S
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from sympy.matrices.common import ShapeError
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from sympy.matrices.expressions.matexpr import MatrixExpr
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from sympy.matrices.expressions.matmul import MatMul
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from sympy.matrices.expressions.special import ZeroMatrix
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from sympy.stats.rv import RandomSymbol, is_random
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from sympy.core.sympify import _sympify
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from sympy.stats.symbolic_probability import Variance, Covariance, Expectation
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class ExpectationMatrix(Expectation, MatrixExpr):
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"""
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Expectation of a random matrix expression.
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Examples
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========
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>>> from sympy.stats import ExpectationMatrix, Normal
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>>> from sympy.stats.rv import RandomMatrixSymbol
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>>> from sympy import symbols, MatrixSymbol, Matrix
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>>> k = symbols("k")
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>>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k)
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>>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1)
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>>> ExpectationMatrix(X)
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ExpectationMatrix(X)
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>>> ExpectationMatrix(A*X).shape
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(k, 1)
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To expand the expectation in its expression, use ``expand()``:
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>>> ExpectationMatrix(A*X + B*Y).expand()
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A*ExpectationMatrix(X) + B*ExpectationMatrix(Y)
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>>> ExpectationMatrix((X + Y)*(X - Y).T).expand()
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ExpectationMatrix(X*X.T) - ExpectationMatrix(X*Y.T) + ExpectationMatrix(Y*X.T) - ExpectationMatrix(Y*Y.T)
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To evaluate the ``ExpectationMatrix``, use ``doit()``:
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>>> N11, N12 = Normal('N11', 11, 1), Normal('N12', 12, 1)
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>>> N21, N22 = Normal('N21', 21, 1), Normal('N22', 22, 1)
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>>> M11, M12 = Normal('M11', 1, 1), Normal('M12', 2, 1)
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>>> M21, M22 = Normal('M21', 3, 1), Normal('M22', 4, 1)
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>>> x1 = Matrix([[N11, N12], [N21, N22]])
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>>> x2 = Matrix([[M11, M12], [M21, M22]])
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>>> ExpectationMatrix(x1 + x2).doit()
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Matrix([
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[12, 14],
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[24, 26]])
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"""
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def __new__(cls, expr, condition=None):
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expr = _sympify(expr)
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if condition is None:
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if not is_random(expr):
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return expr
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obj = Expr.__new__(cls, expr)
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else:
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condition = _sympify(condition)
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obj = Expr.__new__(cls, expr, condition)
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obj._shape = expr.shape
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obj._condition = condition
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return obj
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@property
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def shape(self):
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return self._shape
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def expand(self, **hints):
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expr = self.args[0]
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condition = self._condition
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if not is_random(expr):
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return expr
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if isinstance(expr, Add):
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return Add.fromiter(Expectation(a, condition=condition).expand()
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for a in expr.args)
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expand_expr = _expand(expr)
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if isinstance(expand_expr, Add):
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return Add.fromiter(Expectation(a, condition=condition).expand()
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for a in expand_expr.args)
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elif isinstance(expr, (Mul, MatMul)):
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rv = []
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nonrv = []
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postnon = []
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for a in expr.args:
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if is_random(a):
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if rv:
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rv.extend(postnon)
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else:
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nonrv.extend(postnon)
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postnon = []
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rv.append(a)
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elif a.is_Matrix:
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postnon.append(a)
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else:
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nonrv.append(a)
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# In order to avoid infinite-looping (MatMul may call .doit() again),
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# do not rebuild
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if len(nonrv) == 0:
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return self
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return Mul.fromiter(nonrv)*Expectation(Mul.fromiter(rv),
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condition=condition)*Mul.fromiter(postnon)
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return self
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class VarianceMatrix(Variance, MatrixExpr):
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"""
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Variance of a random matrix probability expression. Also known as
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Covariance matrix, auto-covariance matrix, dispersion matrix,
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or variance-covariance matrix.
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Examples
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========
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>>> from sympy.stats import VarianceMatrix
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>>> from sympy.stats.rv import RandomMatrixSymbol
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>>> from sympy import symbols, MatrixSymbol
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>>> k = symbols("k")
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>>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k)
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>>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1)
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>>> VarianceMatrix(X)
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VarianceMatrix(X)
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>>> VarianceMatrix(X).shape
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(k, k)
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To expand the variance in its expression, use ``expand()``:
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>>> VarianceMatrix(A*X).expand()
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A*VarianceMatrix(X)*A.T
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>>> VarianceMatrix(A*X + B*Y).expand()
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2*A*CrossCovarianceMatrix(X, Y)*B.T + A*VarianceMatrix(X)*A.T + B*VarianceMatrix(Y)*B.T
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"""
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def __new__(cls, arg, condition=None):
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arg = _sympify(arg)
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if 1 not in arg.shape:
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raise ShapeError("Expression is not a vector")
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shape = (arg.shape[0], arg.shape[0]) if arg.shape[1] == 1 else (arg.shape[1], arg.shape[1])
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if condition:
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obj = Expr.__new__(cls, arg, condition)
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else:
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obj = Expr.__new__(cls, arg)
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obj._shape = shape
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obj._condition = condition
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return obj
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@property
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def shape(self):
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return self._shape
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def expand(self, **hints):
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arg = self.args[0]
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condition = self._condition
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if not is_random(arg):
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return ZeroMatrix(*self.shape)
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if isinstance(arg, RandomSymbol):
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return self
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elif isinstance(arg, Add):
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rv = []
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for a in arg.args:
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if is_random(a):
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rv.append(a)
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variances = Add(*(Variance(xv, condition).expand() for xv in rv))
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map_to_covar = lambda x: 2*Covariance(*x, condition=condition).expand()
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covariances = Add(*map(map_to_covar, itertools.combinations(rv, 2)))
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return variances + covariances
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elif isinstance(arg, (Mul, MatMul)):
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nonrv = []
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rv = []
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for a in arg.args:
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if is_random(a):
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rv.append(a)
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else:
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nonrv.append(a)
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if len(rv) == 0:
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return ZeroMatrix(*self.shape)
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# Avoid possible infinite loops with MatMul:
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if len(nonrv) == 0:
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return self
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# Variance of many multiple matrix products is not implemented:
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if len(rv) > 1:
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return self
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return Mul.fromiter(nonrv)*Variance(Mul.fromiter(rv),
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condition)*(Mul.fromiter(nonrv)).transpose()
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# this expression contains a RandomSymbol somehow:
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return self
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class CrossCovarianceMatrix(Covariance, MatrixExpr):
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"""
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Covariance of a random matrix probability expression.
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Examples
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========
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>>> from sympy.stats import CrossCovarianceMatrix
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>>> from sympy.stats.rv import RandomMatrixSymbol
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>>> from sympy import symbols, MatrixSymbol
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>>> k = symbols("k")
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>>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k)
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>>> C, D = MatrixSymbol("C", k, k), MatrixSymbol("D", k, k)
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>>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1)
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>>> Z, W = RandomMatrixSymbol("Z", k, 1), RandomMatrixSymbol("W", k, 1)
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>>> CrossCovarianceMatrix(X, Y)
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CrossCovarianceMatrix(X, Y)
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>>> CrossCovarianceMatrix(X, Y).shape
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(k, k)
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To expand the covariance in its expression, use ``expand()``:
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>>> CrossCovarianceMatrix(X + Y, Z).expand()
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CrossCovarianceMatrix(X, Z) + CrossCovarianceMatrix(Y, Z)
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>>> CrossCovarianceMatrix(A*X, Y).expand()
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A*CrossCovarianceMatrix(X, Y)
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>>> CrossCovarianceMatrix(A*X, B.T*Y).expand()
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A*CrossCovarianceMatrix(X, Y)*B
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>>> CrossCovarianceMatrix(A*X + B*Y, C.T*Z + D.T*W).expand()
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A*CrossCovarianceMatrix(X, W)*D + A*CrossCovarianceMatrix(X, Z)*C + B*CrossCovarianceMatrix(Y, W)*D + B*CrossCovarianceMatrix(Y, Z)*C
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"""
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def __new__(cls, arg1, arg2, condition=None):
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arg1 = _sympify(arg1)
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arg2 = _sympify(arg2)
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if (1 not in arg1.shape) or (1 not in arg2.shape) or (arg1.shape[1] != arg2.shape[1]):
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raise ShapeError("Expression is not a vector")
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shape = (arg1.shape[0], arg2.shape[0]) if arg1.shape[1] == 1 and arg2.shape[1] == 1 \
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else (1, 1)
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if condition:
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obj = Expr.__new__(cls, arg1, arg2, condition)
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else:
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obj = Expr.__new__(cls, arg1, arg2)
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obj._shape = shape
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obj._condition = condition
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return obj
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@property
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def shape(self):
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return self._shape
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def expand(self, **hints):
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arg1 = self.args[0]
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arg2 = self.args[1]
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condition = self._condition
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if arg1 == arg2:
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return VarianceMatrix(arg1, condition).expand()
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if not is_random(arg1) or not is_random(arg2):
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return ZeroMatrix(*self.shape)
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if isinstance(arg1, RandomSymbol) and isinstance(arg2, RandomSymbol):
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return CrossCovarianceMatrix(arg1, arg2, condition)
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coeff_rv_list1 = self._expand_single_argument(arg1.expand())
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coeff_rv_list2 = self._expand_single_argument(arg2.expand())
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addends = [a*CrossCovarianceMatrix(r1, r2, condition=condition)*b.transpose()
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for (a, r1) in coeff_rv_list1 for (b, r2) in coeff_rv_list2]
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return Add.fromiter(addends)
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@classmethod
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def _expand_single_argument(cls, expr):
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# return (coefficient, random_symbol) pairs:
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if isinstance(expr, RandomSymbol):
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return [(S.One, expr)]
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elif isinstance(expr, Add):
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outval = []
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for a in expr.args:
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if isinstance(a, (Mul, MatMul)):
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outval.append(cls._get_mul_nonrv_rv_tuple(a))
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elif is_random(a):
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outval.append((S.One, a))
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return outval
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elif isinstance(expr, (Mul, MatMul)):
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return [cls._get_mul_nonrv_rv_tuple(expr)]
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elif is_random(expr):
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return [(S.One, expr)]
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@classmethod
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def _get_mul_nonrv_rv_tuple(cls, m):
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rv = []
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nonrv = []
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for a in m.args:
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if is_random(a):
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rv.append(a)
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else:
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nonrv.append(a)
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return (Mul.fromiter(nonrv), Mul.fromiter(rv))
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