Below is this second code update.

```
% select rolling window length to use - an optimisable parameter via pso?
rolling_window_length = 50 ;
batchsize = 5 ;
% how-many timesteps do we look back for directed connections - this is what we call the "order" of the model
n1 = 3 ; % first "gaussian" layer order, a best guess just for batchdata creation purposes
n2 = 3 ; % second "binary" layer order, a best guess just for batchdata creation purposes
% taking into account rolling_window_length, n1, n2 and batchsize, get total lookback length
remainder = rem( ( rolling_window_length + n1 + n2 ) , batchsize ) ;
if ( remainder > 0 ) % number of training examples with lookback and orders n1 and n2 not exactly divisable by batchsize
lookback_length = ( rolling_window_length + n1 + n2 + ( batchsize - remainder ) ) ; % increase the lookback_length
else % number of training examples with lookback and orders n1 and n2 exactly divisable by batchsize
lookback_length = ( rolling_window_length + n1 + n2 ) ;
end
% create batchdataindex using lookback_length to index bars in the features matrix
batchdataindex = ( ( training_point_index - ( lookback_length - 1 ) ) : 1 : training_point_index )' ;
batchdata = features( batchdataindex , : ) ;
% now that the batchdata has been created, check it for autocorrelation in the features
all_ar_coeff = zeros( size( batchdata , 2 ) , 1 ) ;
for ii = 1 : size( batchdata , 2 )
ar_coeffs = arburg( batchdata( : , ii ) , 10 , 'FPE' ) ;
all_ar_coeff( ii ) = length( ar_coeffs ) - 1 ;
end
% set order of gaussian_crbm, n1, to be equal to the average length of any autocorrelation in the data
n1 = round( mean( all_ar_coeff ) ) ;
% z-normalise the batchdata matrix with the mean and std of columns
data_mean = mean( batchdata , 1 ) ;
data_std = std( batchdata , 1 ) ;
batchdata = ( batchdata .- repmat( data_mean , size( batchdata , 1 ) , 1 ) ) ./ repmat( data_std , size( batchdata , 1 ) , 1 ) ; % batchdata is now z-normalised by data_mean & data_std
% create the minibatch index matrix for gaussian rbm pre-training of directed weights w
minibatch = ( 1 : 1 : size( batchdata , 1 ) ) ; minibatch( 1 : ( size( batchdata , 1 ) - rolling_window_length ) ) = [] ;
minibatch = minibatch( randperm( size( minibatch , 2 ) ) ) ; minibatch = reshape( minibatch , batchsize , size( minibatch , 2 ) / batchsize ) ;
% PRE-TRAINING FOR THE VISABLE TO HIDDEN AND THE VISIBLE TO VISIBLE WEIGHTS %%%%
% First create a training set and target set for the pre-training of gaussian layer
dAuto_Encode_targets = batchdata ; dAuto_Encode_training_data = [] ;
% dAuto_Encode_targets = batchdata( : , 2 : end ) ; dAuto_Encode_training_data = [] ; % if bias added to raw data
% loop to create the dAuto_Encode_training_data ( n1 == "order" of the gaussian layer of crbm )
for ii = 1 : n1
dAuto_Encode_training_data = [ dAuto_Encode_training_data shift( batchdata , ii ) ] ;
end
% now delete the first n1 rows due to circular shift induced mismatch of data and targets
dAuto_Encode_targets( 1 : n1 , : ) = [] ; dAuto_Encode_training_data( 1 : n1 , : ) = [] ;
% DO RBM PRE-TRAINING FOR THE BOTTOM UP DIRECTED WEIGHTS W %%%%%%%%%%%%%%%%%%%%%
% use rbm trained initial weights instead of using random initialisation for weights
% Doing this because we are not using regularisation in the autoencoder pre-training
epochs = 10000 ;
hidden_layer_size = 4 * size( dAuto_Encode_targets , 2 ) ;
[ w_weights , w_weights_hid_bias , w_weights_vis_bias ] = cc_gaussian_rbm( dAuto_Encode_targets , minibatch , epochs , hidden_layer_size , 0.05 ) ;
% keep a copy of these original w_weights
w1 = w_weights ;
[ A_weights , A_weights_hid_bias , A_weights_vis_bias ] = cc_gaussian_rbm( dAuto_Encode_training_data , minibatch , epochs , size( dAuto_Encode_targets , 2 ) , 0.05 ) ;
[ B_weights , B_weights_hid_bias , B_weights_vis_bias ] = cc_gaussian_rbm( dAuto_Encode_training_data , minibatch , epochs , hidden_layer_size , 0.05 ) ;
% END OF RBM PRE-TRAINING OF AUTOENCODER WEIGHTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(1) ; surf( A_weights ) ; title( 'A Weights after RBM training' ) ;
figure(2) ; surf( B_weights ) ; title( 'B Weights after RBM training' ) ;
figure(3) ; surf( w_weights ) ; title( 'w Weights after RBM training' ) ;
figure(4) ; plot( A_weights_hid_bias , 'b' , B_weights_hid_bias , 'r' , w_weights_vis_bias , 'g' ) ; title( 'Biases after RBM training' ) ; legend( 'A' , 'B' , 'w' ) ;
% DO THE AUTOENCODER TRAINING %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% create weight update matrices
A_weights_update = zeros( size( A_weights ) ) ;
A_weights_hid_bias_update = zeros( size( A_weights_hid_bias ) ) ;
B_weights_update = zeros( size( B_weights ) ) ;
B_weights_hid_bias_update = zeros( size( B_weights_hid_bias ) ) ;
w_weights_update = zeros( size( w_weights ) ) ;
w_weights_vis_bias_update = zeros( size( w_weights_vis_bias ) ) ;
% for adagrad
historical_A = zeros( size( A_weights ) ) ;
historical_A_hid_bias = zeros( size( A_weights_hid_bias ) ) ;
historical_B = zeros( size( B_weights ) ) ;
historical_B_hid_bias = zeros( size( B_weights_hid_bias ) ) ;
historical_w = zeros( size( w_weights ) ) ;
historical_w_vis_bias = zeros( size( w_weights_vis_bias ) ) ;
% set some training parameters
n = size( dAuto_Encode_training_data , 1 ) ; % number of training examples in dAuto_Encode_training_data
input_layer_size = size( dAuto_Encode_training_data , 2 ) ;
fudge_factor = 1e-6 ; % for numerical stability for adagrad
learning_rate = 0.01 ; % will be changed to 0.001 after 50 iters through epoch loop
mom = 0 ; % will be changed to 0.9 after 50 iters through epoch loop
noise = 0.5 ;
epochs = 1000 ;
cost = zeros( epochs , 1 ) ;
lowest_cost = inf ;
% Stochastic Gradient Descent training over dAuto_Encode_training_data
for iter = 1 : epochs
% change momentum and learning_rate after 50 iters
if iter == 50
mom = 0.9 ;
learning_rate = 0.001 ;
end
index = randperm( n ) ; % randomise the order of training examples
for training_example = 1 : n
% Select data for this training batch
tmp_X = dAuto_Encode_training_data( index( training_example ) , : ) ;
tmp_T = dAuto_Encode_targets( index( training_example ) , : ) ;
% Randomly black out some of the input training data
tmp_X( rand( size( tmp_X ) ) < noise ) = 0 ;
% feedforward tmp_X through B_weights and get sigmoid e.g ret = 1.0 ./ ( 1.0 + exp(-input) )
tmp_X_through_sigmoid = 1.0 ./ ( 1.0 .+ exp( - ( tmp_X * B_weights .+ B_weights_hid_bias ) ) ) ;
% Randomly black out some of tmp_X_through_sigmoid for dropout training
tmp_X_through_sigmoid( rand( size( tmp_X_through_sigmoid ) ) < noise ) = 0 ;
% feedforward tmp_X through A_weights and add to tmp_X_through_sigmoid * w_weights for linear output layer
final_output_layer = ( tmp_X * A_weights .+ A_weights_hid_bias ) .+ ( tmp_X_through_sigmoid * w_weights' .+ w_weights_vis_bias ) ;
% now do backpropagation
% this is the derivative of weights for the linear final_output_layer
delta_out = ( tmp_T - final_output_layer ) ;
% NOTE! gradient of sigmoid function g = sigmoid(z) .* ( 1.0 .- sigmoid(z) )
sig_grad = tmp_X_through_sigmoid .* ( 1 .- tmp_X_through_sigmoid ) ;
% backpropagation only through the w_weights that are connected to tmp_X_through_sigmoid
delta_hidden = ( delta_out * w_weights ) .* sig_grad ;
% apply deltas from backpropagation with adagrad for the weight updates
historical_A = historical_A .+ ( tmp_X' * delta_out ).^2 ;
A_weights_update = mom .* A_weights_update .+ ( learning_rate .* ( tmp_X' * delta_out ) ) ./ ( fudge_factor .+ sqrt( historical_A ) ) ;
historical_A_hid_bias = historical_A_hid_bias .+ delta_out.^2 ;
A_weights_hid_bias_update = mom .* A_weights_hid_bias_update .+ ( learning_rate .* delta_out ) ./ ( fudge_factor .+ sqrt( historical_A_hid_bias ) ) ;
historical_w = historical_w .+ ( delta_out' * tmp_X_through_sigmoid ).^2 ;
w_weights_update = mom .* w_weights_update .+ ( learning_rate .* ( delta_out' * tmp_X_through_sigmoid ) ) ./ ( fudge_factor .+ sqrt( historical_w ) ) ;
historical_w_vis_bias = historical_w_vis_bias .+ delta_out.^2 ;
w_weights_vis_bias_update = mom .* w_weights_vis_bias_update .+ ( learning_rate .* delta_out ) ./ ( fudge_factor .+ sqrt( historical_w_vis_bias ) ) ;
historical_B = historical_B .+ ( tmp_X' * delta_hidden ).^2 ;
B_weights_update = mom .* B_weights_update .+ ( learning_rate .* ( tmp_X' * delta_hidden ) ) ./ ( fudge_factor .+ sqrt( historical_B ) ) ;
historical_B_hid_bias = historical_B_hid_bias .+ delta_hidden.^2 ;
B_weights_hid_bias_update = mom .* B_weights_hid_bias_update .+ ( learning_rate .* delta_hidden ) ./ ( fudge_factor .+ sqrt( historical_B_hid_bias ) ) ;
% update the weight matrices with weight_updates
A_weights = A_weights + A_weights_update ;
A_weights_hid_bias = A_weights_hid_bias + A_weights_hid_bias_update ;
B_weights = B_weights + B_weights_update ;
B_weights_hid_bias = B_weights_hid_bias + B_weights_hid_bias_update ;
w_weights = w_weights + w_weights_update ;
w_weights_vis_bias = w_weights_vis_bias + w_weights_vis_bias_update ;
end % end of training_example loop
% feedforward with this epoch's updated weights
epoch_trained_tmp_X_through_sigmoid = 1.0 ./ ( 1.0 .+ exp( -( dAuto_Encode_training_data * B_weights .+ repmat( B_weights_hid_bias , size( dAuto_Encode_training_data , 1 ) , 1 ) ) ) ) ;
epoch_trained_output = ( dAuto_Encode_training_data * A_weights .+ repmat( A_weights_hid_bias , size( dAuto_Encode_training_data , 1 ) , 1 ) )...
.+ ( epoch_trained_tmp_X_through_sigmoid * w_weights' .+ repmat( w_weights_vis_bias , size( epoch_trained_tmp_X_through_sigmoid , 1 ) , 1 ) ) ;
% get sum squared error cost
cost( iter , 1 ) = sum( sum( ( dAuto_Encode_targets .- epoch_trained_output ) .^ 2 ) ) ;
% record best so far
if cost( iter , 1 ) <= lowest_cost
lowest_cost = cost( iter , 1 ) ;
iter_min = iter ;
best_A = A_weights ;
best_B = B_weights ;
best_w = w_weights ;
end
end % end of backpropagation epoch loop
% plot weights
figure(5) ; surf( best_A ) ; title( 'Best A Weights' ) ;
figure(6) ; surf( best_B ) ; title( 'Best B Weights' ) ;
figure(7) ; surf( best_w ) ; title( 'Best w Weights' ) ;
figure(8) ; plot( A_weights_hid_bias , 'b' , B_weights_hid_bias , 'r' , w_weights_vis_bias , 'g' ) ; title( 'Biases after Autoencoder training' ) ; legend( 'A' , 'B' , 'w' ) ;
figure(9) ; plot( cost ) ; title( 'Evolution of Autoencoder cost' ) ;
% END OF CRBM WEIGHT PRE-TRAINING %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```

The changes from the previous code update are a slightly different way to handle the bias units, the introduction of hidden and visible bias units from Restricted Boltzmann machine (RBM) pre-training and the introduction of an automated way to select the “order” of the Conditional Restricted Boltzmann machine (CRBM).

The order of a CRBM is how many time steps we look back in order to model the autoregressive components. This could be decided heuristically or through cross validation but I have decided to use the Octave “arburg” function to “auto-magically” select this look back length, the idea being that the data itself informs this decision and makes the whole CRBM training algorithm adaptive to current conditions. Since the ultimate point of the CRBM will be to make predictions of future OHLC values I have chosen to use the final prediction error model selection criteria for the arburg function.

Now that the bulk of this coding has been completed I think it would be useful to describe the proposed work flow of the various components.

- the data and its derived inputs, such as indicators etc, are input to a Gaussian RBM as a weight initialisation step for the denoising autoencoder training. A Gaussian RBM is used because the data are real valued and not binary. This step is typical of what happens in deep learning and helps to extract meaningful features from the raw data in an unsupervised manner
- the data and RBM initialised weights are then input to the denoising autoencoder to further model the weights and to take into account the autoregressive components of the data
- these twice modelled weights are then used as the initial weights for the CRBM training of a Gaussian-Binary CRBM layer
- the hidden layer of the above Gaussian-Binary CRBM is then used as data for a second Binary-Binary CRBM layer which will be stacked. The training for this second layer will follow the format above, i.e. RBM and denoising autoencoder pre-training of weights

The next step will be for me to compile the denoising autoencoder code into an Octave C++ .oct function for speed optimisation purposes.