一些机器学习工具的简单实现
Softmax
Matrix Softmax({int dim = -1})
将数据映射为概率分布，定义如下（实际中会考虑排除数据中的最大值，这里不考虑）：
$$Softmax(X) = \frac{e^{x_i}}{\displaystyle \sum_{i \in N } e^{x_i}} $$
test
import '../lib/flutter_matrix.dart';
main(){
data_format = "%2.6f";
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
[
[ 0.082595 0.224515 0.610296 0.082595]
[ 0.000903 0.002454 0.006670 0.989973]
[ 0.012038 0.088947 0.657233 0.241783]
]
[
[ 0.090031 0.042010 0.017148 0.000319]
[ 0.244728 0.114195 0.046613 0.952270]
[ 0.665241 0.843795 0.936240 0.047411]
]
[
[ 0.000275 0.000747 0.002032 0.000275]
[ 0.000747 0.002032 0.005522 0.819600]
[ 0.002032 0.015012 0.110921 0.040805]
]
LeakyReLU
Matrix LeakyReLU({double alpha = 0.01})
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。
$$LeakyReLU(X) = \begin{cases} x, & x > 0 \ \alpha x, & x \le 0 \end{cases}$$
test
import '../lib/flutter_matrix.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
Matrix ReLU()
激活函数ReLU。
$$ReLU(x) = max(x, 0)$$
test
import '../lib/flutter_matrix.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
Matrix Sigmoid()
激活函数，也被称为S型生长曲线。
$$Sigmoid(x) = \frac{1}{1 + e^{-x}}$$
test
import '../lib/flutter_matrix.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
Matrix ELU({required double alpha})
激活函数ELU。
$$ELU(x) = \begin{cases} x, &x > 0 \ \alpha (e^x - 1), & x \le 0\end{cases}$$
test
import '../lib/flutter_matrix.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
Matrix Swish()
Swish是一种自我门控的激活函数。
$$Swish(x) = x * Sigmoid(x)$$
test
import '../lib/flutter_matrix.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
Matrix Softsign()
Softsign函数是Tanh函数的另一个替代选择。
$$Softsign(x) = \frac{x}{1+ |x|}$$
test
import '../lib/flutter_matrix.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
Matrix Softplus()
Softplus函数可以看作是ReLU函数的平滑。
$$Softplus(x) = log(1 + e^x)$$
test
import '../lib/flutter_matrix.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
Object MAE({required Matrix other, int dim = -1})
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。
$$MAE(X, h) = \frac{1}{n} \displaystyle \sum_{i \in N} |y_i - \hat{y_i}|$$
test
import '../lib/flutter_matrix.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
Object MSE({required Matrix other, int dim = -1})
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。
$$MSE = \frac{1}{n} \displaystyle \sum_{i \in N} (y_i - \hat{y}_i) ^2$$
test
import '../lib/flutter_matrix.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