LSTM

• 记录下我学习的LSTM

• 学习的内容：

  • http://colah.github.io/posts/2015-08-Understanding-LSTMs/
  • https://blog.csdn.net/wangyangzhizhou/article/details/76651116

LSTM的概念

• LSTM（Long Short Memory Network）是基于RNN的一种神经网络

• RNN有个缺陷：有时需要很短的序列（如$X_1\ X_2\ X_3$）预测$X_4$，有时需要很长的序列（如$X_1 \cdots X_{100}$）预测

• LSTM的结构如下

  • 每条线都携带着整个向量，如$X_t$
  • 黄色是神经网络层
  • 粉色圆圈是逐点操作，如向量相加、矩阵对应点相乘

• LSTM的关键是cell state，细胞状态

LSTM的步骤

1. 决定哪些信息会进入cell state：遗忘门

$$f_{t}=\sigma\left(W_{f} \cdot\left[h_{t-1}, x_{t}\right]+b_{f}\right)$$

1. 决定哪些信息会储存在cell state：输入门
   • 这里有两个部分
   • 第一个部分是激活层
   • 第二个部分是$tanh$层

$$\begin{aligned} i_{t} &=\sigma\left(W_{i} \cdot\left[h_{t-1}, x_{t}\right]+b_{i}\right) \\ \tilde{C}_{t} &=\tanh \left(W_{C} \cdot\left[h_{t-1}, x_{t}\right]+b_{C}\right) \end{aligned}$$

1. 更新cell state

$$C_{t}=f_{t} * C_{t-1}+i_{t} * \tilde{C}_{t}$$

1. 决定输出，这个输出基于先现在的cell state：输出门

$$\begin{array}{l}{o_{t}=\sigma\left(W_{o}\left[h_{t-1}, x_{t}\right]+b_{o}\right)} \\ {h_{t}=o_{t} * \tanh \left(C_{t}\right)}\end{array}$$

代码

• 这里的LSTM输入为：(样本数量，步长，维度)
• 步长：如4个旧数据预测1个新数据
• 维度：新数据的维度

# -*- coding: utf-8 -*-
import warnings
from keras.layers.core import Dense, Activation, Dropout
from keras.layers.recurrent import LSTM
from keras.models import Sequential
warnings.filterwarnings("ignore")

def Model():
    model = Sequential()
    model.add(LSTM(64,return_sequences=True,input_shape=(None,1)))
    model.add(LSTM(32,return_sequences=Truem))
    model.add(LSTM(16))
    #model.add(Dropout(0.2))
    model.add(Dense(1, activation='linear'))
    model.compile(loss='mse',optimizer='adam')
    return model

keras的LSTM代码

• 从keras中找到的LSTM定义

class LSTMCell(Layer):
    """Cell class for the LSTM layer.

    # Arguments
        units: Positive integer, dimensionality of the output space.
        activation: Activation function to use
            (see [activations](../activations.md)).
            Default: hyperbolic tangent (`tanh`).
            If you pass `None`, no activation is applied
            (ie. "linear" activation: `a(x) = x`).
        recurrent_activation: Activation function to use
            for the recurrent step
            (see [activations](../activations.md)).
            Default: sigmoid (`sigmoid`).
            If you pass `None`, no activation is applied
            (ie. "linear" activation: `a(x) = x`).x
        use_bias: Boolean, whether the layer uses a bias vector.
        kernel_initializer: Initializer for the `kernel` weights matrix,
            used for the linear transformation of the inputs
            (see [initializers](../initializers.md)).
        recurrent_initializer: Initializer for the `recurrent_kernel`
            weights matrix,
            used for the linear transformation of the recurrent state
            (see [initializers](../initializers.md)).
        bias_initializer: Initializer for the bias vector
            (see [initializers](../initializers.md)).
        unit_forget_bias: Boolean.
            If True, add 1 to the bias of the forget gate at initialization.
            Setting it to true will also force `bias_initializer="zeros"`.
            This is recommended in [Jozefowicz et al. (2015)](
            http://www.jmlr.org/proceedings/papers/v37/jozefowicz15.pdf).
        kernel_regularizer: Regularizer function applied to
            the `kernel` weights matrix
            (see [regularizer](../regularizers.md)).
        recurrent_regularizer: Regularizer function applied to
            the `recurrent_kernel` weights matrix
            (see [regularizer](../regularizers.md)).
        bias_regularizer: Regularizer function applied to the bias vector
            (see [regularizer](../regularizers.md)).
        kernel_constraint: Constraint function applied to
            the `kernel` weights matrix
            (see [constraints](../constraints.md)).
        recurrent_constraint: Constraint function applied to
            the `recurrent_kernel` weights matrix
            (see [constraints](../constraints.md)).
        bias_constraint: Constraint function applied to the bias vector
            (see [constraints](../constraints.md)).
        dropout: Float between 0 and 1.
            Fraction of the units to drop for
            the linear transformation of the inputs.
        recurrent_dropout: Float between 0 and 1.
            Fraction of the units to drop for
            the linear transformation of the recurrent state.
        implementation: Implementation mode, either 1 or 2.
            Mode 1 will structure its operations as a larger number of
            smaller dot products and additions, whereas mode 2 will
            batch them into fewer, larger operations. These modes will
            have different performance profiles on different hardware and
            for different applications.
    """

    def __init__(self, units,
                 activation='tanh',
                 recurrent_activation='sigmoid',
                 use_bias=True,
                 kernel_initializer='glorot_uniform',
                 recurrent_initializer='orthogonal',
                 bias_initializer='zeros',
                 unit_forget_bias=True,
                 kernel_regularizer=None,
                 recurrent_regularizer=None,
                 bias_regularizer=None,
                 kernel_constraint=None,
                 recurrent_constraint=None,
                 bias_constraint=None,
                 dropout=0.,
                 recurrent_dropout=0.,
                 implementation=2,
                 **kwargs):
        super(LSTMCell, self).__init__(**kwargs)
        self.units = units
        self.activation = activations.get(activation)
        self.recurrent_activation = activations.get(recurrent_activation)
        self.use_bias = use_bias

        self.kernel_initializer = initializers.get(kernel_initializer)
        self.recurrent_initializer = initializers.get(recurrent_initializer)
        self.bias_initializer = initializers.get(bias_initializer)
        self.unit_forget_bias = unit_forget_bias

        self.kernel_regularizer = regularizers.get(kernel_regularizer)
        self.recurrent_regularizer = regularizers.get(recurrent_regularizer)
        self.bias_regularizer = regularizers.get(bias_regularizer)

        self.kernel_constraint = constraints.get(kernel_constraint)
        self.recurrent_constraint = constraints.get(recurrent_constraint)
        self.bias_constraint = constraints.get(bias_constraint)

        self.dropout = min(1., max(0., dropout))
        self.recurrent_dropout = min(1., max(0., recurrent_dropout))
        self.implementation = implementation
        self.state_size = (self.units, self.units)
        self.output_size = self.units
        self._dropout_mask = None
        self._recurrent_dropout_mask = None

    def build(self, input_shape):
        input_dim = input_shape[-1]

        if type(self.recurrent_initializer).__name__ == 'Identity':
            def recurrent_identity(shape, gain=1., dtype=None):
                del dtype
                return gain * np.concatenate(
                    [np.identity(shape[0])] * (shape[1] // shape[0]), axis=1)

            self.recurrent_initializer = recurrent_identity

        self.kernel = self.add_weight(shape=(input_dim, self.units * 4),
                                      name='kernel',
                                      initializer=self.kernel_initializer,
                                      regularizer=self.kernel_regularizer,
                                      constraint=self.kernel_constraint)
        self.recurrent_kernel = self.add_weight(
            shape=(self.units, self.units * 4),
            name='recurrent_kernel',
            initializer=self.recurrent_initializer,
            regularizer=self.recurrent_regularizer,
            constraint=self.recurrent_constraint)

        if self.use_bias:
            if self.unit_forget_bias:
                @K.eager
                def bias_initializer(_, *args, **kwargs):
                    return K.concatenate([
                        self.bias_initializer((self.units,), *args, **kwargs),
                        initializers.Ones()((self.units,), *args, **kwargs),
                        self.bias_initializer((self.units * 2,), *args, **kwargs),
                    ])
            else:
                bias_initializer = self.bias_initializer
            self.bias = self.add_weight(shape=(self.units * 4,),
                                        name='bias',
                                        initializer=bias_initializer,
                                        regularizer=self.bias_regularizer,
                                        constraint=self.bias_constraint)
        else:
            self.bias = None

        self.kernel_i = self.kernel[:, :self.units]
        self.kernel_f = self.kernel[:, self.units: self.units * 2]
        self.kernel_c = self.kernel[:, self.units * 2: self.units * 3]
        self.kernel_o = self.kernel[:, self.units * 3:]

        self.recurrent_kernel_i = self.recurrent_kernel[:, :self.units]
        self.recurrent_kernel_f = (
            self.recurrent_kernel[:, self.units: self.units * 2])
        self.recurrent_kernel_c = (
            self.recurrent_kernel[:, self.units * 2: self.units * 3])
        self.recurrent_kernel_o = self.recurrent_kernel[:, self.units * 3:]

        if self.use_bias:
            self.bias_i = self.bias[:self.units]
            self.bias_f = self.bias[self.units: self.units * 2]
            self.bias_c = self.bias[self.units * 2: self.units * 3]
            self.bias_o = self.bias[self.units * 3:]
        else:
            self.bias_i = None
            self.bias_f = None
            self.bias_c = None
            self.bias_o = None
        self.built = True

    def call(self, inputs, states, training=None):
        if 0 < self.dropout < 1 and self._dropout_mask is None:
            self._dropout_mask = _generate_dropout_mask(
                K.ones_like(inputs),
                self.dropout,
                training=training,
                count=4)
        if (0 < self.recurrent_dropout < 1 and
                self._recurrent_dropout_mask is None):
            self._recurrent_dropout_mask = _generate_dropout_mask(
                K.ones_like(states[0]),
                self.recurrent_dropout,
                training=training,
                count=4)

        # dropout matrices for input units
        dp_mask = self._dropout_mask
        # dropout matrices for recurrent units
        rec_dp_mask = self._recurrent_dropout_mask

        h_tm1 = states[0]  # previous memory state
        c_tm1 = states[1]  # previous carry state

        if self.implementation == 1:
            if 0 < self.dropout < 1.:
                inputs_i = inputs * dp_mask[0]
                inputs_f = inputs * dp_mask[1]
                inputs_c = inputs * dp_mask[2]
                inputs_o = inputs * dp_mask[3]
            else:
                inputs_i = inputs
                inputs_f = inputs
                inputs_c = inputs
                inputs_o = inputs
            x_i = K.dot(inputs_i, self.kernel_i)
            x_f = K.dot(inputs_f, self.kernel_f)
            x_c = K.dot(inputs_c, self.kernel_c)
            x_o = K.dot(inputs_o, self.kernel_o)
            if self.use_bias:
                x_i = K.bias_add(x_i, self.bias_i)
                x_f = K.bias_add(x_f, self.bias_f)
                x_c = K.bias_add(x_c, self.bias_c)
                x_o = K.bias_add(x_o, self.bias_o)

            if 0 < self.recurrent_dropout < 1.:
                h_tm1_i = h_tm1 * rec_dp_mask[0]
                h_tm1_f = h_tm1 * rec_dp_mask[1]
                h_tm1_c = h_tm1 * rec_dp_mask[2]
                h_tm1_o = h_tm1 * rec_dp_mask[3]
            else:
                h_tm1_i = h_tm1
                h_tm1_f = h_tm1
                h_tm1_c = h_tm1
                h_tm1_o = h_tm1
            i = self.recurrent_activation(x_i + K.dot(h_tm1_i,
                                                      self.recurrent_kernel_i))
            f = self.recurrent_activation(x_f + K.dot(h_tm1_f,
                                                      self.recurrent_kernel_f))
            c = f * c_tm1 + i * self.activation(x_c + K.dot(h_tm1_c,
                                                            self.recurrent_kernel_c))
            o = self.recurrent_activation(x_o + K.dot(h_tm1_o,
                                                      self.recurrent_kernel_o))
        else:
            if 0. < self.dropout < 1.:
                inputs *= dp_mask[0]
            z = K.dot(inputs, self.kernel)
            if 0. < self.recurrent_dropout < 1.:
                h_tm1 *= rec_dp_mask[0]
            z += K.dot(h_tm1, self.recurrent_kernel)
            if self.use_bias:
                z = K.bias_add(z, self.bias)

            z0 = z[:, :self.units]
            z1 = z[:, self.units: 2 * self.units]
            z2 = z[:, 2 * self.units: 3 * self.units]
            z3 = z[:, 3 * self.units:]

            i = self.recurrent_activation(z0)
            f = self.recurrent_activation(z1)
            c = f * c_tm1 + i * self.activation(z2)
            o = self.recurrent_activation(z3)

        h = o * self.activation(c)
        if 0 < self.dropout + self.recurrent_dropout:
            if training is None:
                h._uses_learning_phase = True
        return h, [h, c]

    def get_config(self):
        config = {'units': self.units,
                  'activation': activations.serialize(self.activation),
                  'recurrent_activation':
                      activations.serialize(self.recurrent_activation),
                  'use_bias': self.use_bias,
                  'kernel_initializer':
                      initializers.serialize(self.kernel_initializer),
                  'recurrent_initializer':
                      initializers.serialize(self.recurrent_initializer),
                  'bias_initializer': initializers.serialize(self.bias_initializer),
                  'unit_forget_bias': self.unit_forget_bias,
                  'kernel_regularizer':
                      regularizers.serialize(self.kernel_regularizer),
                  'recurrent_regularizer':
                      regularizers.serialize(self.recurrent_regularizer),
                  'bias_regularizer': regularizers.serialize(self.bias_regularizer),
                  'kernel_constraint': constraints.serialize(self.kernel_constraint),
                  'recurrent_constraint':
                      constraints.serialize(self.recurrent_constraint),
                  'bias_constraint': constraints.serialize(self.bias_constraint),
                  'dropout': self.dropout,
                  'recurrent_dropout': self.recurrent_dropout,
                  'implementation': self.implementation}
        base_config = super(LSTMCell, self).get_config()
        return dict(list(base_config.items()) + list(config.items()))
    
class LSTM(RNN):
    """Long Short-Term Memory layer - Hochreiter 1997.

    # Arguments
        units: Positive integer, dimensionality of the output space.
        activation: Activation function to use
            (see [activations](../activations.md)).
            Default: hyperbolic tangent (`tanh`).
            If you pass `None`, no activation is applied
            (ie. "linear" activation: `a(x) = x`).
        recurrent_activation: Activation function to use
            for the recurrent step
            (see [activations](../activations.md)).
            Default: sigmoid (`sigmoid`).
            If you pass `None`, no activation is applied
            (ie. "linear" activation: `a(x) = x`).
        use_bias: Boolean, whether the layer uses a bias vector.
        kernel_initializer: Initializer for the `kernel` weights matrix,
            used for the linear transformation of the inputs.
            (see [initializers](../initializers.md)).
        recurrent_initializer: Initializer for the `recurrent_kernel`
            weights matrix,
            used for the linear transformation of the recurrent state.
            (see [initializers](../initializers.md)).
        bias_initializer: Initializer for the bias vector
            (see [initializers](../initializers.md)).
        unit_forget_bias: Boolean.
            If True, add 1 to the bias of the forget gate at initialization.
            Setting it to true will also force `bias_initializer="zeros"`.
            This is recommended in [Jozefowicz et al. (2015)](
            http://www.jmlr.org/proceedings/papers/v37/jozefowicz15.pdf).
        kernel_regularizer: Regularizer function applied to
            the `kernel` weights matrix
            (see [regularizer](../regularizers.md)).
        recurrent_regularizer: Regularizer function applied to
            the `recurrent_kernel` weights matrix
            (see [regularizer](../regularizers.md)).
        bias_regularizer: Regularizer function applied to the bias vector
            (see [regularizer](../regularizers.md)).
        activity_regularizer: Regularizer function applied to
            the output of the layer (its "activation").
            (see [regularizer](../regularizers.md)).
        kernel_constraint: Constraint function applied to
            the `kernel` weights matrix
            (see [constraints](../constraints.md)).
        recurrent_constraint: Constraint function applied to
            the `recurrent_kernel` weights matrix
            (see [constraints](../constraints.md)).
        bias_constraint: Constraint function applied to the bias vector
            (see [constraints](../constraints.md)).
        dropout: Float between 0 and 1.
            Fraction of the units to drop for
            the linear transformation of the inputs.
        recurrent_dropout: Float between 0 and 1.
            Fraction of the units to drop for
            the linear transformation of the recurrent state.
        implementation: Implementation mode, either 1 or 2.
            Mode 1 will structure its operations as a larger number of
            smaller dot products and additions, whereas mode 2 will
            batch them into fewer, larger operations. These modes will
            have different performance profiles on different hardware and
            for different applications.
        return_sequences: Boolean. Whether to return the last output
            in the output sequence, or the full sequence.
        return_state: Boolean. Whether to return the last state
            in addition to the output. The returned elements of the
            states list are the hidden state and the cell state, respectively.
        go_backwards: Boolean (default False).
            If True, process the input sequence backwards and return the
            reversed sequence.
        stateful: Boolean (default False). If True, the last state
            for each sample at index i in a batch will be used as initial
            state for the sample of index i in the following batch.
        unroll: Boolean (default False).
            If True, the network will be unrolled,
            else a symbolic loop will be used.
            Unrolling can speed-up a RNN,
            although it tends to be more memory-intensive.
            Unrolling is only suitable for short sequences.

    # References
        - [Long short-term memory](
          http://www.bioinf.jku.at/publications/older/2604.pdf)
        - [Learning to forget: Continual prediction with LSTM](
          http://www.mitpressjournals.org/doi/pdf/10.1162/089976600300015015)
        - [Supervised sequence labeling with recurrent neural networks](
          http://www.cs.toronto.edu/~graves/preprint.pdf)
        - [A Theoretically Grounded Application of Dropout in
           Recurrent Neural Networks](https://arxiv.org/abs/1512.05287)
    """

    @interfaces.legacy_recurrent_support
    def __init__(self, units,
                 activation='tanh',
                 recurrent_activation='sigmoid',
                 use_bias=True,
                 kernel_initializer='glorot_uniform',
                 recurrent_initializer='orthogonal',
                 bias_initializer='zeros',
                 unit_forget_bias=True,
                 kernel_regularizer=None,
                 recurrent_regularizer=None,
                 bias_regularizer=None,
                 activity_regularizer=None,
                 kernel_constraint=None,
                 recurrent_constraint=None,
                 bias_constraint=None,
                 dropout=0.,
                 recurrent_dropout=0.,
                 implementation=2,
                 return_sequences=False,
                 return_state=False,
                 go_backwards=False,
                 stateful=False,
                 unroll=False,
                 **kwargs):
        if implementation == 0:
            warnings.warn('`implementation=0` has been deprecated, '
                          'and now defaults to `implementation=1`.'
                          'Please update your layer call.')
        if K.backend() == 'theano' and (dropout or recurrent_dropout):
            warnings.warn(
                'RNN dropout is no longer supported with the Theano backend '
                'due to technical limitations. '
                'You can either set `dropout` and `recurrent_dropout` to 0, '
                'or use the TensorFlow backend.')
            dropout = 0.
            recurrent_dropout = 0.

        cell = LSTMCell(units,
                        activation=activation,
                        recurrent_activation=recurrent_activation,
                        use_bias=use_bias,
                        kernel_initializer=kernel_initializer,
                        recurrent_initializer=recurrent_initializer,
                        unit_forget_bias=unit_forget_bias,
                        bias_initializer=bias_initializer,
                        kernel_regularizer=kernel_regularizer,
                        recurrent_regularizer=recurrent_regularizer,
                        bias_regularizer=bias_regularizer,
                        kernel_constraint=kernel_constraint,
                        recurrent_constraint=recurrent_constraint,
                        bias_constraint=bias_constraint,
                        dropout=dropout,
                        recurrent_dropout=recurrent_dropout,
                        implementation=implementation)
        super(LSTM, self).__init__(cell,
                                   return_sequences=return_sequences,
                                   return_state=return_state,
                                   go_backwards=go_backwards,
                                   stateful=stateful,
                                   unroll=unroll,
                                   **kwargs)
        self.activity_regularizer = regularizers.get(activity_regularizer)

    def call(self, inputs, mask=None, training=None, initial_state=None):
        self.cell._dropout_mask = None
        self.cell._recurrent_dropout_mask = None
        return super(LSTM, self).call(inputs,
                                      mask=mask,
                                      training=training,
                                      initial_state=initial_state)

    @property
    def units(self):
        return self.cell.units

    @property
    def activation(self):
        return self.cell.activation

    @property
    def recurrent_activation(self):
        return self.cell.recurrent_activation

    @property
    def use_bias(self):
        return self.cell.use_bias

    @property
    def kernel_initializer(self):
        return self.cell.kernel_initializer

    @property
    def recurrent_initializer(self):
        return self.cell.recurrent_initializer

    @property
    def bias_initializer(self):
        return self.cell.bias_initializer

    @property
    def unit_forget_bias(self):
        return self.cell.unit_forget_bias

    @property
    def kernel_regularizer(self):
        return self.cell.kernel_regularizer

    @property
    def recurrent_regularizer(self):
        return self.cell.recurrent_regularizer

    @property
    def bias_regularizer(self):
        return self.cell.bias_regularizer

    @property
    def kernel_constraint(self):
        return self.cell.kernel_constraint

    @property
    def recurrent_constraint(self):
        return self.cell.recurrent_constraint

    @property
    def bias_constraint(self):
        return self.cell.bias_constraint

    @property
    def dropout(self):
        return self.cell.dropout

    @property
    def recurrent_dropout(self):
        return self.cell.recurrent_dropout

    @property
    def implementation(self):
        return self.cell.implementation

    def get_config(self):
        config = {'units': self.units,
                  'activation': activations.serialize(self.activation),
                  'recurrent_activation':
                      activations.serialize(self.recurrent_activation),
                  'use_bias': self.use_bias,
                  'kernel_initializer':
                      initializers.serialize(self.kernel_initializer),
                  'recurrent_initializer':
                      initializers.serialize(self.recurrent_initializer),
                  'bias_initializer': initializers.serialize(self.bias_initializer),
                  'unit_forget_bias': self.unit_forget_bias,
                  'kernel_regularizer':
                      regularizers.serialize(self.kernel_regularizer),
                  'recurrent_regularizer':
                      regularizers.serialize(self.recurrent_regularizer),
                  'bias_regularizer': regularizers.serialize(self.bias_regularizer),
                  'activity_regularizer':
                      regularizers.serialize(self.activity_regularizer),
                  'kernel_constraint': constraints.serialize(self.kernel_constraint),
                  'recurrent_constraint':
                      constraints.serialize(self.recurrent_constraint),
                  'bias_constraint': constraints.serialize(self.bias_constraint),
                  'dropout': self.dropout,
                  'recurrent_dropout': self.recurrent_dropout,
                  'implementation': self.implementation}
        base_config = super(LSTM, self).get_config()
        del base_config['cell']
        return dict(list(base_config.items()) + list(config.items()))

    @classmethod
    def from_config(cls, config):
        if 'implementation' in config and config['implementation'] == 0:
            config['implementation'] = 1
        return cls(**config)