Modern Signal Processing Review

Fuzzy Control Neural Network

1. BP neural network computing questions:
First, make sure that the input is a column vector $ x = \ begin {bmatrix}
X_1 \\
X_2
\ End {bmatrix}, and then the weight vector matrix is W = \ begin {bmatrix}
W _ {11} & W _ {12 }\\
W _ {21} & W _ {22}
\ End {bmatrix} The number of W _ {IJ} $ here is a bit messy, so I don't feel like remembering the order, the principle is that each row of $ W $ corresponds to a fixed output neuron. That is, $ net_y = wx $. This sequence also applies to subsequent neurons.
When calculating the output of each layer of neurons, You need to calculate three quantities at the same time: $ net_y, y = f (net_y), f' (net_y) $
Back Propagation of BP neural networks: $ \ Delta _ {output} =-(Z _ {t}-Z ). * f' (net_z) $. For other Reverse propagation parameters $ \ Delta = w ^ t \ Delta _ {last_level }. * f' (net_y) $. For a real gradient, you can use $ \ Delta W = \ ETA \ Delta * y ^ t $, where $ y $ is the input of the current layer.
This calculates the real gradient and uses the formula below to update the weight. $ W = W-\ Delta W $
TIPS: When reversely propagating $ \ Delta $, you must note that the top layer of the weight vector $ W $ is not used, because $ W $ is used for reverse propagation $ \ Delta $, compared with $ W $ at the top layer, no other $ \ Delta $ above needs to be calculated, therefore, the $ W $ at the top layer is not required for reverse propagation $ \ Delta $. For a weight of $ W $ in a neural network diagram, updating this value is calculated using the data on the right of the Weight Value in addition to the input variable.

2. Kohonen self-organizing network clustering algorithm

If $ \ ETA = 0.5 $, it is calculated based on the angle relationship. When the key point is closest to the point, it is used to update the cluster center until it finally becomes stable.

3. Questions about Fuzzy Control

This part of the question is mainly to calculate $ A \ circ B $, using the "maximum-Minimum synthesis" method. The technique is to calculate by the order of Matrix Multiplication. The calculation must be performed step by step. The elements in the matrix are first minimized in autumn, and then the maximum value is obtained by element. One step at a time, multiple checks are performed, which is prone to errors, but this question is not difficult in general.

Fuzzy Logic
Representation of Fuzzy Sets: Zadeh notation, ordinal notation, and vector notation
Fuzzy set terms: Platform set, $ \ Alpha $ intercept, Regular Fuzzy Set, convex fuzzy set, demarcation point, and single point Fuzzy Set
Calculation of Fuzzy Sets: the Union of fuzzy sets, the intersection of fuzzy sets, and the complement of fuzzy sets.
The synthesis of fuzzy relationships (the maximum and minimum synthesis, here we must write the calculation formula and solve the problem step by step, so that it is not prone to errors .)
Fuzzy Relationship: If $ x $ and $ y $ are two non-empty sets, the values of $ x \ times Y =\{ (x, y) | x \ In X, Y \ In Y \} $ A Fuzzy Set R becomes a fuzzy relationship from X to Y, which is recorded as $ R _ {XY} $

 

Wavelet Analysis
1. Proof $ \ {w _ {j, k}: K =, \ cdots, 2 ^ J-1 \} $ constitute an orthogonal basis for $ w_j $
2. proof $ Haar $ Wavelet Transform Formula: $ A _ {n-1, k} = (A _ {n, 2 k} + A _ {n, 2 k + 1 }) /SQRT (2), k = 0, 1, \ cdots, 2 ^ {n-1}-1 $