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The Lee–Carter model is a numerical algorithm used in mortality forecasting and life expectancy forecasting. [1] The input to the model is a matrix of age specific mortality rates ordered monotonically by time, usually with ages in columns and years in rows. The output is a forecasted matrix of mortality rates in the same format as the input.
Models used in multivariate regression are parameterized by a matrix of coefficients. In the Frobenius inner product above, each matrix X {\displaystyle X} is X i t = e t ⊗ x i {\displaystyle X_{i}^{t}=e_{t}\otimes x_{i}} such that the output of the inner product is the dot product of one row of the input with one column of the coefficient ...
Equivalent to the definition, a 0-1 matrix is balanced if and only if it does not contain a submatrix that is the incidence matrix of any odd cycle (a cycle graph of odd order). [2] Therefore, the only three by three 0-1 matrix that is not balanced is (up to permutation of the rows and columns) the following incidence matrix of a cycle graph of ...
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector , v → {\displaystyle {\vec {v}}\!} , adding two matrices would have the geometric effect of applying each matrix transformation separately onto v → {\displaystyle {\vec {v}}\!} , then adding the transformed vectors.
Extensions of this result can be made for more than two random variables, using the covariance matrix. Note that the condition that X and Y are known to be jointly normally distributed is necessary for the conclusion that their sum is normally distributed to apply.
This model can be generalized to any number of species competing against each other. One can think of the populations and growth rates as vectors, α 's as a matrix.Then the equation for any species i becomes = (=) or, if the carrying capacity is pulled into the interaction matrix (this doesn't actually change the equations, only how the interaction matrix is defined), = (=) where N is the ...
The key observation is that multiplying two 2 × 2 matrices can be done with only 7 multiplications, instead of the usual 8 (at the expense of 11 additional addition and subtraction operations). This means that, treating the input n × n matrices as block 2 × 2 matrices, the task of multiplying n × n matrices can be reduced to 7 subproblems ...
Detailed balance, which is a requirement of reversibility, states that if you observe the system for a long enough time, the system goes from state to with the same frequency as going from to . In equilibrium, the probability of observing the system at state A is given by the Boltzmann weight , e − E A / T {\displaystyle e^{-E_{A}/T}} .