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一些机器学习工具的简单实现

Softmax

  • 将数据通过Softmax映射，实际中会考虑排除数据中的最大值，这里不考虑 
    T Softmax({int dim = -1})
    $$ Softmax(X) = \frac{e^{x_i}}{\displaystyle \sum_{i \in N } e^{x_i}} $$

test

import 'package:flutter_matrix/matrix_type.dart';

main(){
  data_format = "%2.3f";
  var mt = Matrix.fromList([
    [1, 2, 3, 1],
    [2, 3, 4, 9],
    [3, 5, 7, 6]
  ]);
  print(mt.Softmax(dim: 0));
  print(mt.Softmax(dim: 1));
  print(mt.Softmax(dim: 2));
}

output

[
 [ 0.083  0.225  0.610  0.083]
 [ 0.001  0.002  0.007  0.990]
 [ 0.012  0.089  0.657  0.242]
]
[
 [ 0.090  0.042  0.017  0.000]
 [ 0.245  0.114  0.047  0.952]
 [ 0.665  0.844  0.936  0.047]
]
[
 [ 0.000  0.001  0.002  0.000]
 [ 0.001  0.002  0.006  0.820]
 [ 0.002  0.015  0.111  0.041]
]

LeakyReLU

  • LeakyReLU是一种改进的激活函数，旨在解决ReLU函数中负输入导致的零梯度问题 
    T LeakyReLU({double alpha = 0.01})

$$ LeakyReLU(X) = \begin{cases} x, & x 0 \ \alpha x, & x \le 0 \end{cases} $$

test

import 'package:flutter_matrix/matrix_type.dart';

main(){
  data_format = "%2.6f";
  var mt = Matrix.fromList([
    [1, 2, 3, 1],
    [2, 0, -4, -2],
    [0, 5, 7, -6]
  ]);
  print(mt.LeakyReLU());
}

output

[
 [ 1.000000  2.000000  3.000000  1.000000]
 [ 2.000000  0.000000 -0.040000 -0.020000]
 [ 0.000000  5.000000  7.000000 -0.060000]
]

ReLU

  • 激活函数ReLU 
    T ReLU()
    $$ ReLU(x) = max(x, 0) $$

test

import 'package:flutter_matrix/matrix_type.dart';

main(){
  data_format = "%2.0f";
  var mt = Matrix.fromList([
    [1, 2, 3, 1],
    [2, 0, -4, -2],
    [0, 5, 7, -6]
  ]);
  print(mt.ReLU());
}

output

[
 [  1   2   3   1]
 [  2   0  -0  -0]
 [  0   5   7  -0]
]

Sigmoid

  • 激活函数，也被称为S型生长曲线 
    T Sigmoid()
    $$ Sigmoid(x) = \frac{1}{1 + e^{-x}} $$

test

import 'package:flutter_matrix/matrix_type.dart';

main(){
  data_format = "%2.5f";
  var mt = Matrix.fromList([
    [1, 2, 3, 1],
    [2, 0, -4, -2],
    [0, 5, 7, -6]
  ]);
  print(mt.Sigmoid());
}

output

[
 [ 0.73106  0.88080  0.95257  0.73106]
 [ 0.88080  0.50000  0.01799  0.11920]
 [ 0.50000  0.99331  0.99909  0.00247]
]

ELU

  • 激活函数ELU 
    T ELU({required double alpha})
    $$ ELU(x) = \begin{cases} x, &x 0 \ \alpha (e^x - 1), & x \le 0\end{cases} $$

test

import 'package:flutter_matrix/matrix_type.dart';

main(){
  data_format = "%2.5f";
  var mt = Matrix.fromList([
    [1, 2, 3, 1],
    [2, 0, -4, -2],
    [0, 5, 7, -6]
  ]);
  print(mt.ELU(alpha: 0.1));
}

output

[
 [ 1.00000  2.00000  3.00000  1.00000]
 [ 2.00000  0.00000 -0.09817 -0.08647]
 [ 0.00000  5.00000  7.00000 -0.09975]
]

Swish

  • Swish是一种自我门控的激活函数 
    T Swish()
    $$ Swish(x) = x * Sigmoid(x) $$

test

import 'package:flutter_matrix/matrix_type.dart';

main(){
  data_format = "%2.5f";
  var mt = Matrix.fromList([
    [1, 2, 3, 1],
    [2, 0, -4, -2],
    [0, 5, 7, -6]
  ]);
  print(mt.Swish());
}

output

[
 [ 0.73106  1.76159  2.85772  0.73106]
 [ 1.76159  0.00000 -0.07194 -0.23841]
 [ 0.00000  4.96654  6.99362 -0.01484]
]

Softsign

  • Softsign函数是Tanh函数的另一个替代选择 
    T Softsign()
    $$ Softsign(x) = \frac{x}{1+ |x|} $$

test

import 'package:flutter_matrix/matrix_type.dart';

main(){
  data_format = "%2.5f";
  var mt = Matrix.fromList([
    [1, 2, 3, 1],
    [2, 0, -4, -2],
    [0, 5, 7, -6]
  ]);
  print(mt.Softsign());
}

output

[
 [ 0.50000  0.66667  0.75000  0.50000]
 [ 0.66667  0.00000 -0.80000 -0.66667]
 [ 0.00000  0.83333  0.87500 -0.85714]
]

Softplus

  • Softplus函数可以看作是ReLU函数的平滑 
    T Softplus()

$$ Softplus(x) = log(1 + e^x) $$

test

import 'package:flutter_matrix/matrix_type.dart';

main(){
  data_format = "%2.5f";
  var mt = Matrix.fromList([
    [1, 2, 3, 1],
    [2, 0, -4, -2],
    [0, 5, 7, -6]
  ]);
  print(mt.Softplus());
}

output

[
 [ 1.31326  2.12693  3.04859  1.31326]
 [ 2.12693  0.69315  0.01815  0.12693]
 [ 0.69315  5.00672  7.00091  0.00248]
]

MAE

  • 平均绝对值误差，它表示预测值和观测值之间绝对误差的平均值 
    Object MAE({required Matrix other, int dim = -1})
    $$ MAE(X, h) = \frac{1}{n} \displaystyle \sum_{i \in N} |y_i - \hat{y_i}| $$

test

import 'package:flutter_matrix/matrix_type.dart';

main(){
  var mt = Matrix.fromList([
    [1.2, 1.3, 4.2, 2.2, 1.4],
    [0, 0.3, 2.3, 1.3, 1.7],
    [1.9, 1.83, 1.2, 2, 2.1]
  ]);
  print(mt.MAE(other: Matrix.fill(number: 1, row: mt.shape[0], column: mt.shape[1]), dim: 0));
  print(mt.MAE(other: Matrix.fill(number: 1, row: mt.shape[0], column: mt.shape[1]), dim: 1));
  print(mt.MAE(other: Matrix.fill(number: 1, row: mt.shape[0], column: mt.shape[1]), dim: 2));
}

output

[1.06, 0.8, 0.8059999999999998]
[0.6999999999999998, 0.61, 1.5666666666666667, 0.8333333333333334, 0.7333333333333334]
0.8886666666666666

MSE

  • MSE通过计算预测值和真实值之间的误差平方的平均值，来衡量模型的预测性能 
    Object MSE({required Matrix other, int dim = -1})

$$ MSE = \frac{1}{n} \displaystyle \sum_{i \in N} (y_i - \hat{y}_i) ^2 $$

test

import 'package:flutter_matrix/matrix_type.dart';

main(){
  var mt = Matrix.fromList([
    [1.2, 1.3, 4.2, 2.2, 1.4],
    [0, 0.3, 2.3, 1.3, 1.7],
    [1.9, 1.83, 1.2, 2, 2.1]
  ]);
  print(mt.MSE(other: Matrix.fill(number: 1, row: mt.shape[0], column: mt.shape[1]), dim: 0));
  print(mt.MSE(other: Matrix.fill(number: 1, row: mt.shape[0], column: mt.shape[1]), dim: 1));
  print(mt.MSE(other: Matrix.fill(number: 1, row: mt.shape[0], column: mt.shape[1]), dim: 2));
}

output

[2.3940000000000006, 0.7519999999999999, 0.74978]
[0.6166666666666666, 0.42296666666666666, 3.99, 0.8433333333333334, 0.62]
1.2985933333333333

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