Week 45, Recurrent Neural Networks¶

Morten Hjorth-Jensen, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

Date: November 6-10

Plan for week 45¶

Material for the active learning sessions on Tuesday and Wednesday.

Discussion of project 2
Video of lab session from week 43
Video of lab session from week 44
Video of lab session from week 45
See also whiteboard notes from lab session week 44

Material for the lecture on Thursday November 9, 2023.

Short repetition on Convolutional Neural Networks
Recurrent Neural Networks (RNNs)
Readings and Videos:
- These lecture notes
- Video of lecture
- Whiteboard notes
- For a more in depth discussion on neural networks we recommend Goodfellow et al chapter 10. See also chapter 11 and 12 on practicalities and applications
- Reading suggestions for implementation of RNNs: Aurelien Geron’s chapter 14.
- Video on Recurrent Neural Networks from MIT
- Video on Deep Learning

Material for the lab sessions, additional ways to present classification results and other practicalities¶

Searching for Optimal Regularization Parameters \(\lambda\)¶

In project 1, when using Ridge and Lasso regression, we end up searching for the optimal parameter \(\lambda\) which minimizes our selected scores (MSE or \(R2\) values for example). The brute force approach, as discussed in the code here for Ridge regression, consists in evaluating the MSE as function of different \(\lambda\) values. Based on these calculations, one tries then to determine the value of the hyperparameter \(\lambda\) which results in optimal scores (for example the smallest MSE or an \(R2=1\)).

%matplotlib inline

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn import linear_model

def MSE(y_data,y_model):
    n = np.size(y_model)
    return np.sum((y_data-y_model)**2)/n
# A seed just to ensure that the random numbers are the same for every run.
# Useful for eventual debugging.
np.random.seed(2021)

n = 100
x = np.random.rand(n)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.randn(n)

Maxpolydegree = 5
X = np.zeros((n,Maxpolydegree-1))

for degree in range(1,Maxpolydegree): #No intercept column
    X[:,degree-1] = x**(degree)

# We split the data in test and training data
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)

# Decide which values of lambda to use
nlambdas = 500
MSERidgePredict = np.zeros(nlambdas)
lambdas = np.logspace(-4, 2, nlambdas)
for i in range(nlambdas):
    lmb = lambdas[i]
    RegRidge = linear_model.Ridge(lmb)
    RegRidge.fit(X_train,y_train)
    ypredictRidge = RegRidge.predict(X_test)
    MSERidgePredict[i] = MSE(y_test,ypredictRidge)

# Now plot the results
plt.figure()
plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')
plt.xlabel('log10(lambda)')
plt.ylabel('MSE')
plt.legend()
plt.show()

Here we have performed a rather data greedy calculation as function of the regularization parameter \(\lambda\). There is no resampling here. The latter can easily be added by employing the function RidgeCV instead of just calling the Ridge function. For RidgeCV we need to pass the array of \(\lambda\) values. By inspecting the figure we can in turn determine which is the optimal regularization parameter. This becomes however less functional in the long run.

Grid Search¶

An alternative is to use the so-called grid search functionality included with the library Scikit-Learn, as demonstrated for the same example here.

import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.linear_model import Ridge
from sklearn.model_selection import GridSearchCV

def R2(y_data, y_model):
    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)

def MSE(y_data,y_model):
    n = np.size(y_model)
    return np.sum((y_data-y_model)**2)/n

# A seed just to ensure that the random numbers are the same for every run.
# Useful for eventual debugging.
np.random.seed(2021)

n = 100
x = np.random.rand(n)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.randn(n)

Maxpolydegree = 5
X = np.zeros((n,Maxpolydegree-1))

for degree in range(1,Maxpolydegree): #No intercept column
    X[:,degree-1] = x**(degree)

# We split the data in test and training data
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)

# Decide which values of lambda to use
nlambdas = 10
lambdas = np.logspace(-4, 2, nlambdas)
# create and fit a ridge regression model, testing each alpha
model = Ridge()
gridsearch = GridSearchCV(estimator=model, param_grid=dict(alpha=lambdas))
gridsearch.fit(X_train, y_train)
print(gridsearch)
ypredictRidge = gridsearch.predict(X_test)
# summarize the results of the grid search
print(f"Best estimated lambda-value: {gridsearch.best_estimator_.alpha}")
print(f"MSE score: {MSE(y_test,ypredictRidge)}")
print(f"R2 score: {R2(y_test,ypredictRidge)}")

GridSearchCV(estimator=Ridge(),
             param_grid={'alpha': array([1.00000000e-04, 4.64158883e-04, 2.15443469e-03, 1.00000000e-02,
       4.64158883e-02, 2.15443469e-01, 1.00000000e+00, 4.64158883e+00,
       2.15443469e+01, 1.00000000e+02])})
Best estimated lambda-value: 100.0
MSE score: 1.0892144853354966
R2 score: -0.0038332550504751595

By default the grid search function includes cross validation with five folds. The Scikit-Learn documentation contains more information on how to set the different parameters.

If we take out the random noise, running the above codes results in \(\lambda=0\) yielding the best fit.

Randomized Grid Search¶

An alternative to the above manual grid set up, is to use a random search where the parameters are tuned from a random distribution (uniform below) for a fixed number of iterations. A model is constructed and evaluated for each combination of chosen parameters. We repeat the previous example but now with a random search. Note that values of \(\lambda\) are now limited to be within \(x\in [0,1]\). This domain may not be the most relevant one for the specific case under study.

import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.linear_model import Ridge
from sklearn.model_selection import GridSearchCV
from scipy.stats import uniform as randuniform
from sklearn.model_selection import RandomizedSearchCV


def R2(y_data, y_model):
    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)

def MSE(y_data,y_model):
    n = np.size(y_model)
    return np.sum((y_data-y_model)**2)/n

# A seed just to ensure that the random numbers are the same for every run.
# Useful for eventual debugging.
np.random.seed(2021)

n = 100
x = np.random.rand(n)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.randn(n)

Maxpolydegree = 5
X = np.zeros((n,Maxpolydegree-1))

for degree in range(1,Maxpolydegree): #No intercept column
    X[:,degree-1] = x**(degree)

# We split the data in test and training data
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)

param_grid = {'alpha': randuniform()}
# create and fit a ridge regression model, testing each alpha
model = Ridge()
gridsearch = RandomizedSearchCV(estimator=model, param_distributions=param_grid, n_iter=100)
gridsearch.fit(X_train, y_train)
print(gridsearch)
ypredictRidge = gridsearch.predict(X_test)
# summarize the results of the grid search
print(f"Best estimated lambda-value: {gridsearch.best_estimator_.alpha}")
print(f"MSE score: {MSE(y_test,ypredictRidge)}")
print(f"R2 score: {R2(y_test,ypredictRidge)}")

RandomizedSearchCV(estimator=Ridge(), n_iter=100,
                   param_distributions={'alpha': <scipy.stats._distn_infrastructure.rv_frozen object at 0x1646da640>})
Best estimated lambda-value: 0.9849967686928113
MSE score: 1.0853136633465326
R2 score: -0.0002382102844775691

Wisconsin Cancer Data¶

We show here how we can use a simple regression case on the breast cancer data using Logistic regression as our algorithm for classification.

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import  train_test_split 
from sklearn.datasets import load_breast_cancer
from sklearn.linear_model import LogisticRegression

# Load the data
cancer = load_breast_cancer()

X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
print(X_train.shape)
print(X_test.shape)
# Logistic Regression
logreg = LogisticRegression(solver='lbfgs')
logreg.fit(X_train, y_train)
print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))

(426, 30)
(143, 30)
Test set accuracy with Logistic Regression: 0.94

/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(

Using the correlation matrix¶

In addition to the above scores, we could also study the covariance (and the correlation matrix). We use Pandas to compute the correlation matrix.

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import  train_test_split 
from sklearn.datasets import load_breast_cancer
from sklearn.linear_model import LogisticRegression
cancer = load_breast_cancer()
import pandas as pd
# Making a data frame
cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)

fig, axes = plt.subplots(15,2,figsize=(10,20))
malignant = cancer.data[cancer.target == 0]
benign = cancer.data[cancer.target == 1]
ax = axes.ravel()

for i in range(30):
    _, bins = np.histogram(cancer.data[:,i], bins =50)
    ax[i].hist(malignant[:,i], bins = bins, alpha = 0.5)
    ax[i].hist(benign[:,i], bins = bins, alpha = 0.5)
    ax[i].set_title(cancer.feature_names[i])
    ax[i].set_yticks(())
ax[0].set_xlabel("Feature magnitude")
ax[0].set_ylabel("Frequency")
ax[0].legend(["Malignant", "Benign"], loc ="best")
fig.tight_layout()
plt.show()

import seaborn as sns
correlation_matrix = cancerpd.corr().round(1)
# use the heatmap function from seaborn to plot the correlation matrix
# annot = True to print the values inside the square
plt.figure(figsize=(15,8))
sns.heatmap(data=correlation_matrix, annot=True)
plt.show()

Discussing the correlation data¶

In the above example we note two things. In the first plot we display the overlap of benign and malignant tumors as functions of the various features in the Wisconsing breast cancer data set. We see that for some of the features we can distinguish clearly the benign and malignant cases while for other features we cannot. This can point to us which features may be of greater interest when we wish to classify a benign or not benign tumour.

In the second figure we have computed the so-called correlation matrix, which in our case with thirty features becomes a \(30\times 30\) matrix.

We constructed this matrix using pandas via the statements

cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)

and then

correlation_matrix = cancerpd.corr().round(1)

Diagonalizing this matrix we can in turn say something about which features are of relevance and which are not. This leads us to the classical Principal Component Analysis (PCA) theorem with applications. This will be discussed later this semester (week 43).

Combinations of classification results¶

It is common in the literature to define various combinations the above numbers. The most commonly used are

Sensitivity, recall, hit rate, or true positive rate \(TPR\). It is the probability of a positive test result, conditioned on the individual truly being positive.

\[ {\displaystyle \mathrm {TPR} ={\frac {\mathrm {TP} }{\mathrm {P} }}={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FN} }}=1-\mathrm {FNR} } \]

The \(TPR\) defines how many correct positive results occur among all positive samples available during the test

Miss rate or false negative rate \(FNR\).

\[ {\displaystyle \mathrm {FNR} ={\frac {\mathrm {FN} }{\mathrm {P} }}={\frac {\mathrm {FN} }{\mathrm {FN} +\mathrm {TP} }} } \]

Specificity, selectivity or true negative rate \(TNR\). It is the probability of a negative test result, conditioned on the individual truly being negative.

\[ {\displaystyle \mathrm {TNR} ={\frac {\mathrm {TN} }{\mathrm {N} }}={\frac {\mathrm {TN} }{\mathrm {TN} +\mathrm {FP} }}=1-\mathrm {FPR} } \]

with the fall-out false positive rate

\[ {\displaystyle \mathrm {FPR} ={\frac {\mathrm {FP} }{\mathrm {N} }}={\frac {\mathrm {FP} }{\mathrm {FP} +\mathrm {TN} }}=1-\mathrm {TNR} } \]

The \(FPR\) defines how many incorrect positive results occur among all negative samples available during the test.

Positive and negative prediction values¶

The positive and negative predictive values are the proportions of positive and negative results in statistics and diagnostic tests that are true positive and true negative results, respectively.[1] The PPV and NPV describe the performance of a diagnostic test or other statistical measure. A high result can be interpreted as indicating the accuracy of such a statistic.

Precision or positive predictive value \(PPV\).

\[ {\displaystyle \mathrm {PPV} ={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FP} }}=1-\mathrm {FDR} } \]

Negative predictive value \(NPV\).

\[ {\displaystyle \mathrm {NPV} ={\frac {\mathrm {TN} }{\mathrm {TN} +\mathrm {FN} }}=1-\mathrm {FOR} } \]

Other quantities¶

False discovery rate \(FDR\).

\[ {\displaystyle \mathrm {FDR} ={\frac {\mathrm {FP} }{\mathrm {FP} +\mathrm {TP} }}=1-\mathrm {PPV} } \]

False omission rate \(FOR\).

\[ {\displaystyle \mathrm {FOR} ={\frac {\mathrm {FN} }{\mathrm {FN} +\mathrm {TN} }}=1-\mathrm {NPV} } \]

\(F_1\) score¶

In statistical analysis of binary classification, the F-score or F-measure is a measure of a test’s accuracy. It is calculated from the precision and recall of the test, where the precision is the number of true positive results divided by the number of all positive results, including those not identified correctly, and the recall is the number of true positive results divided by the number of all samples that should have been identified as positive. Precision is also known as positive predictive value, and recall is also known as sensitivity in diagnostic binary classification.

The F1 score is the harmonic mean of the precision and recall. It thus symmetrically represents both precision and recall in one metric. The highest possible value of an F-score is 1.0, indicating perfect precision and recall, and the lowest possible value is 0, if either precision or recall are zero.

It is defined as

\[ {\displaystyle \mathrm {F} _{1}=2\times {\frac {\mathrm {PPV} \times \mathrm {TPR} }{\mathrm {PPV} +\mathrm {TPR} }}={\frac {2\mathrm {TP} }{2\mathrm {TP} +\mathrm {FP} +\mathrm {FN} }}} \]

ROC curve¶

A receiver operating characteristic curve, or ROC curve, is a graphical plot that illustrates the performance of a binary classifier model at varying threshold values.

The ROC curve is the plot of the true positive rate (TPR) against the false positive rate (FPR) at each threshold setting.

To draw a ROC curve, only the true positive rate (TPR) and false positive rate (FPR) are needed (as functions of some classifier parameter). The TPR defines how many correct positive results occur among all positive samples available during the test. FPR, on the other hand, defines how many incorrect positive results occur among all negative samples available during the test.

See https://en.wikipedia.org/wiki/Receiver_operating_characteristic for more discussions.

Cumulative gain curve¶

The cumulative gain curve is a performance evaluation used typically for binary classification problems. It plots the \(TPR\) True Positive Rate or Sensitivity (which represents the fraction of examples correctly classified against Predictive Positive Rate, which represents the fraction of positively predicted examples.

The examples below show the confusion matrix, the ROC curve and the cumulative gain for the Wisconsin cancer data.

Other measures in classification studies: Cancer Data again¶

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import  train_test_split 
from sklearn.datasets import load_breast_cancer
from sklearn.linear_model import LogisticRegression

# Load the data
cancer = load_breast_cancer()

X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
print(X_train.shape)
print(X_test.shape)
# Logistic Regression
logreg = LogisticRegression(solver='lbfgs')
logreg.fit(X_train, y_train)

from sklearn.preprocessing import LabelEncoder
from sklearn.model_selection import cross_validate
#Cross validation
accuracy = cross_validate(logreg,X_test,y_test,cv=10)['test_score']
print(accuracy)
print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))

import scikitplot as skplt
y_pred = logreg.predict(X_test)
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = logreg.predict_proba(X_test)
skplt.metrics.plot_roc(y_test, y_probas)
plt.show()
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
plt.show()

(426, 30)
(143, 30)
[1.         0.86666667 1.         0.92857143 1.         0.85714286
 1.         0.92857143 0.92857143 1.        ]
Test set accuracy with Logistic Regression: 0.94

/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:814: ConvergenceWarning: lbfgs failed to converge (status=1):
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
Please also refer to the documentation for alternative solver options:
    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
  n_iter_i = _check_optimize_result(

Material for Lecture Thursday November 9¶

Recurrent neural networks (RNNs): Overarching view¶

Till now our focus has been, including convolutional neural networks as well, on feedforward neural networks. The output or the activations flow only in one direction, from the input layer to the output layer.

A recurrent neural network (RNN) looks very much like a feedforward neural network, except that it also has connections pointing backward.

RNNs are used to analyze time series data such as stock prices, and tell you when to buy or sell. In autonomous driving systems, they can anticipate car trajectories and help avoid accidents. More generally, they can work on sequences of arbitrary lengths, rather than on fixed-sized inputs like all the nets we have discussed so far. For example, they can take sentences, documents, or audio samples as input, making them extremely useful for natural language processing systems such as automatic translation and speech-to-text.

A simple example¶

# Start importing packages
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
from tensorflow.keras import datasets, layers, models
from tensorflow.keras.layers import Input
from tensorflow.keras.models import Model, Sequential 
from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU
from tensorflow.keras import optimizers     
from tensorflow.keras import regularizers           
from tensorflow.keras.utils import to_categorical 



# convert into dataset matrix
def convertToMatrix(data, step):
 X, Y =[], []
 for i in range(len(data)-step):
  d=i+step  
  X.append(data[i:d,])
  Y.append(data[d,])
 return np.array(X), np.array(Y)

step = 4
N = 1000    
Tp = 800    

t=np.arange(0,N)
x=np.sin(0.02*t)+2*np.random.rand(N)
df = pd.DataFrame(x)
df.head()

values=df.values
train,test = values[0:Tp,:], values[Tp:N,:]

# add step elements into train and test
test = np.append(test,np.repeat(test[-1,],step))
train = np.append(train,np.repeat(train[-1,],step))
 
trainX,trainY =convertToMatrix(train,step)
testX,testY =convertToMatrix(test,step)
trainX = np.reshape(trainX, (trainX.shape[0], 1, trainX.shape[1]))
testX = np.reshape(testX, (testX.shape[0], 1, testX.shape[1]))

model = Sequential()
model.add(SimpleRNN(units=32, input_shape=(1,step), activation="relu"))
model.add(Dense(8, activation="relu")) 
model.add(Dense(1))
model.compile(loss='mean_squared_error', optimizer='rmsprop')
model.summary()

model.fit(trainX,trainY, epochs=100, batch_size=16, verbose=2)
trainPredict = model.predict(trainX)
testPredict= model.predict(testX)
predicted=np.concatenate((trainPredict,testPredict),axis=0)

trainScore = model.evaluate(trainX, trainY, verbose=0)
print(trainScore)
plt.plot(df)
plt.plot(predicted)
plt.show()

Metal device set to: Apple M1
Model: "sequential"

_________________________________________________________________

 Layer (type)                Output Shape              Param #

=================================================================

 simple_rnn (SimpleRNN)      (None, 32)                1184

 dense (Dense)               (None, 8)                 264

 dense_1 (Dense)             (None, 1)                 9

=================================================================

Total params: 1,457

Trainable params: 1,457

Non-trainable params: 0

_________________________________________________________________

Epoch 1/100

2023-11-21 06:15:36.594141: W tensorflow/core/platform/profile_utils/cpu_utils.cc:128] Failed to get CPU frequency: 0 Hz

50/50 - 1s - loss: 0.6155 - 1s/epoch - 20ms/step

Epoch 2/100

50/50 - 0s - loss: 0.4372 - 444ms/epoch - 9ms/step

Epoch 3/100

50/50 - 0s - loss: 0.4090 - 443ms/epoch - 9ms/step

Epoch 4/100

50/50 - 0s - loss: 0.4011 - 441ms/epoch - 9ms/step

Epoch 5/100

50/50 - 0s - loss: 0.3967 - 440ms/epoch - 9ms/step

Epoch 6/100

50/50 - 0s - loss: 0.3961 - 439ms/epoch - 9ms/step

Epoch 7/100

50/50 - 0s - loss: 0.3966 - 443ms/epoch - 9ms/step

Epoch 8/100

50/50 - 0s - loss: 0.3936 - 442ms/epoch - 9ms/step

Epoch 9/100

50/50 - 0s - loss: 0.3915 - 444ms/epoch - 9ms/step

Epoch 10/100

50/50 - 0s - loss: 0.3908 - 446ms/epoch - 9ms/step

Epoch 11/100

50/50 - 0s - loss: 0.3897 - 446ms/epoch - 9ms/step

Epoch 12/100

50/50 - 0s - loss: 0.3897 - 446ms/epoch - 9ms/step

Epoch 13/100

50/50 - 0s - loss: 0.3896 - 446ms/epoch - 9ms/step

Epoch 14/100

50/50 - 0s - loss: 0.3881 - 445ms/epoch - 9ms/step

Epoch 15/100

50/50 - 0s - loss: 0.3876 - 445ms/epoch - 9ms/step

Epoch 16/100

50/50 - 0s - loss: 0.3861 - 444ms/epoch - 9ms/step

Epoch 17/100

50/50 - 0s - loss: 0.3860 - 446ms/epoch - 9ms/step

Epoch 18/100

50/50 - 0s - loss: 0.3831 - 448ms/epoch - 9ms/step

Epoch 19/100

50/50 - 0s - loss: 0.3857 - 446ms/epoch - 9ms/step

Epoch 20/100

50/50 - 0s - loss: 0.3854 - 446ms/epoch - 9ms/step

Epoch 21/100

50/50 - 0s - loss: 0.3856 - 446ms/epoch - 9ms/step

Epoch 22/100

50/50 - 0s - loss: 0.3845 - 448ms/epoch - 9ms/step

Epoch 23/100

50/50 - 0s - loss: 0.3831 - 446ms/epoch - 9ms/step

Epoch 24/100

50/50 - 0s - loss: 0.3839 - 446ms/epoch - 9ms/step

Epoch 25/100

50/50 - 0s - loss: 0.3827 - 446ms/epoch - 9ms/step

Epoch 26/100

50/50 - 0s - loss: 0.3814 - 448ms/epoch - 9ms/step

Epoch 27/100

50/50 - 0s - loss: 0.3829 - 445ms/epoch - 9ms/step

Epoch 28/100

50/50 - 0s - loss: 0.3814 - 444ms/epoch - 9ms/step

Epoch 29/100

50/50 - 0s - loss: 0.3810 - 446ms/epoch - 9ms/step

Epoch 30/100

50/50 - 0s - loss: 0.3803 - 446ms/epoch - 9ms/step

Epoch 31/100

50/50 - 0s - loss: 0.3805 - 446ms/epoch - 9ms/step

Epoch 32/100

50/50 - 0s - loss: 0.3812 - 446ms/epoch - 9ms/step

Epoch 33/100

50/50 - 0s - loss: 0.3812 - 447ms/epoch - 9ms/step

Epoch 34/100

50/50 - 0s - loss: 0.3794 - 444ms/epoch - 9ms/step

Epoch 35/100

50/50 - 0s - loss: 0.3778 - 447ms/epoch - 9ms/step

Epoch 36/100

50/50 - 0s - loss: 0.3787 - 446ms/epoch - 9ms/step

Epoch 37/100

50/50 - 0s - loss: 0.3802 - 445ms/epoch - 9ms/step

Epoch 38/100

50/50 - 0s - loss: 0.3758 - 447ms/epoch - 9ms/step

Epoch 39/100

50/50 - 0s - loss: 0.3795 - 445ms/epoch - 9ms/step

Epoch 40/100

50/50 - 0s - loss: 0.3788 - 444ms/epoch - 9ms/step

Epoch 41/100

50/50 - 0s - loss: 0.3701 - 447ms/epoch - 9ms/step

Epoch 42/100

50/50 - 0s - loss: 0.3802 - 443ms/epoch - 9ms/step

Epoch 43/100

50/50 - 0s - loss: 0.3770 - 444ms/epoch - 9ms/step

Epoch 44/100

50/50 - 0s - loss: 0.3758 - 446ms/epoch - 9ms/step

Epoch 45/100

50/50 - 0s - loss: 0.3763 - 446ms/epoch - 9ms/step

Epoch 46/100

50/50 - 0s - loss: 0.3768 - 446ms/epoch - 9ms/step

Epoch 47/100

50/50 - 0s - loss: 0.3745 - 445ms/epoch - 9ms/step

Epoch 48/100

50/50 - 0s - loss: 0.3773 - 443ms/epoch - 9ms/step

Epoch 49/100

50/50 - 0s - loss: 0.3753 - 447ms/epoch - 9ms/step

Epoch 50/100

50/50 - 0s - loss: 0.3747 - 446ms/epoch - 9ms/step

Epoch 51/100

50/50 - 0s - loss: 0.3750 - 440ms/epoch - 9ms/step

Epoch 52/100

50/50 - 0s - loss: 0.3740 - 448ms/epoch - 9ms/step

Epoch 53/100

50/50 - 0s - loss: 0.3767 - 448ms/epoch - 9ms/step

Epoch 54/100

50/50 - 0s - loss: 0.3745 - 455ms/epoch - 9ms/step

Epoch 55/100

50/50 - 0s - loss: 0.3750 - 452ms/epoch - 9ms/step

Epoch 56/100

50/50 - 0s - loss: 0.3730 - 452ms/epoch - 9ms/step

Epoch 57/100

50/50 - 0s - loss: 0.3728 - 452ms/epoch - 9ms/step

Epoch 58/100

50/50 - 0s - loss: 0.3736 - 451ms/epoch - 9ms/step

Epoch 59/100

50/50 - 0s - loss: 0.3727 - 450ms/epoch - 9ms/step

Epoch 60/100

50/50 - 0s - loss: 0.3732 - 453ms/epoch - 9ms/step

Epoch 61/100

50/50 - 0s - loss: 0.3705 - 451ms/epoch - 9ms/step

Epoch 62/100

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
Input In [9], in <cell line: 53>()
model.compile(loss='mean_squared_error', optimizer='rmsprop')
model.summary()
---> 53 model.fit(trainX,trainY, epochs=100, batch_size=16, verbose=2)
trainPredict = model.predict(trainX)
testPredict= model.predict(testX)

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/keras/utils/traceback_utils.py:64, in filter_traceback.<locals>.error_handler(*args, **kwargs)
filtered_tb = None
try:
---> 64   return fn(*args, **kwargs)
except Exception as e:  # pylint: disable=broad-except
 filtered_tb = _process_traceback_frames(e.__traceback__)

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/keras/engine/training.py:1384, in Model.fit(self, x, y, batch_size, epochs, verbose, callbacks, validation_split, validation_data, shuffle, class_weight, sample_weight, initial_epoch, steps_per_epoch, validation_steps, validation_batch_size, validation_freq, max_queue_size, workers, use_multiprocessing)
with tf.profiler.experimental.Trace(
   'train',
   epoch_num=epoch,
   step_num=step,
   batch_size=batch_size,
   _r=1):
 callbacks.on_train_batch_begin(step)
-> 1384   tmp_logs = self.train_function(iterator)
 if data_handler.should_sync:
   context.async_wait()

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/util/traceback_utils.py:150, in filter_traceback.<locals>.error_handler(*args, **kwargs)
filtered_tb = None
try:
--> 150   return fn(*args, **kwargs)
except Exception as e:
 filtered_tb = _process_traceback_frames(e.__traceback__)

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/def_function.py:915, in Function.__call__(self, *args, **kwds)
compiler = "xla" if self._jit_compile else "nonXla"
with OptionalXlaContext(self._jit_compile):
--> 915   result = self._call(*args, **kwds)
new_tracing_count = self.experimental_get_tracing_count()
without_tracing = (tracing_count == new_tracing_count)

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/def_function.py:947, in Function._call(self, *args, **kwds)
 self._lock.release()
 # In this case we have created variables on the first call, so we run the
 # defunned version which is guaranteed to never create variables.
--> 947   return self._stateless_fn(*args, **kwds)  # pylint: disable=not-callable
elif self._stateful_fn is not None:
 # Release the lock early so that multiple threads can perform the call
 # in parallel.
 self._lock.release()

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/function.py:2956, in Function.__call__(self, *args, **kwargs)
with self._lock:
 (graph_function,
  filtered_flat_args) = self._maybe_define_function(args, kwargs)
-> 2956 return graph_function._call_flat(
   filtered_flat_args, captured_inputs=graph_function.captured_inputs)

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/function.py:1853, in ConcreteFunction._call_flat(self, args, captured_inputs, cancellation_manager)
possible_gradient_type = gradients_util.PossibleTapeGradientTypes(args)
if (possible_gradient_type == gradients_util.POSSIBLE_GRADIENT_TYPES_NONE
   and executing_eagerly):
 # No tape is watching; skip to running the function.
-> 1853   return self._build_call_outputs(self._inference_function.call(
     ctx, args, cancellation_manager=cancellation_manager))
forward_backward = self._select_forward_and_backward_functions(
   args,
   possible_gradient_type,
   executing_eagerly)
forward_function, args_with_tangents = forward_backward.forward()

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/function.py:499, in _EagerDefinedFunction.call(self, ctx, args, cancellation_manager)
with _InterpolateFunctionError(self):
 if cancellation_manager is None:
--> 499     outputs = execute.execute(
       str(self.signature.name),
       num_outputs=self._num_outputs,
       inputs=args,
       attrs=attrs,
       ctx=ctx)
 else:
   outputs = execute.execute_with_cancellation(
       str(self.signature.name),
       num_outputs=self._num_outputs,
   (...)
       ctx=ctx,
       cancellation_manager=cancellation_manager)

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/execute.py:54, in quick_execute(op_name, num_outputs, inputs, attrs, ctx, name)
try:
 ctx.ensure_initialized()
---> 54   tensors = pywrap_tfe.TFE_Py_Execute(ctx._handle, device_name, op_name,
                                     inputs, attrs, num_outputs)
except core._NotOkStatusException as e:
 if name is not None:

KeyboardInterrupt: 

RNNs¶

RNNs are very powerful, because they combine two properties:

Distributed hidden state that allows them to store a lot of information about the past efficiently.
Non-linear dynamics that allows them to update their hidden state in complicated ways.

With enough neurons and time, RNNs can compute anything that can be computed by your computer!

Basic layout¶

Figure 1:

We need to specify the initial activity state of all the hidden and output units¶

We could just fix these initial states to have some default value like 0.5.
But it is better to treat the initial states as learned parameters.
We learn them in the same way as we learn the weights.

Start off with an initial random guess for the initial states.

a. At the end of each training sequence, backpropagate through time all the way to the initial states to get the gradient of the error function with respect to each initial state.

b. Adjust the initial states by following the negative gradient.

We can specify inputs in several ways¶

Specify the initial states of all the units.
Specify the initial states of a subset of the units.
Specify the states of the same subset of the units at every time step.

This is the natural way to model most sequential data.

We can specify targets in several ways¶

Specify desired final activities of all the units
Specify desired activities of all units for the last few steps

Good for learning attractors
It is easy to add in extra error derivatives as we backpropagate.
- Specify the desired activity of a subset of the units.
The other units are input or hidden units.

Figure 1:

Backpropagation through time¶

We can think of the recurrent net as a layered, feed-forward net with shared weights and then train the feed-forward net with weight constraints.

We can also think of this training algorithm in the time domain:

The forward pass builds up a stack of the activities of all the units at each time step.
The backward pass peels activities off the stack to compute the error derivatives at each time step.
After the backward pass we add together the derivatives at all the different times for each weight.

The backward pass is linear¶

There is a big difference between the forward and backward passes.
In the forward pass we use squashing functions (like the logistic) to prevent the activity vectors from exploding.
The backward pass, is completely linear. If you double the error derivatives at the final layer, all the error derivatives will double.

The forward pass determines the slope of the linear function used for backpropagating through each neuron

Figure 1:

The problem of exploding or vanishing gradients¶

What happens to the magnitude of the gradients as we backpropagate through many layers?

a. If the weights are small, the gradients shrink exponentially.

b. If the weights are big the gradients grow exponentially.

Typical feed-forward neural nets can cope with these exponential effects because they only have a few hidden layers.
In an RNN trained on long sequences (e.g. 100 time steps) the gradients can easily explode or vanish.

a. We can avoid this by initializing the weights very carefully.

Even with good initial weights, its very hard to detect that the current target output depends on an input from many time-steps ago.

RNNs have difficulty dealing with long-range dependencies.

Four effective ways to learn an RNN¶

Long Short Term Memory Make the RNN out of little modules that are designed to remember values for a long time.
Hessian Free Optimization: Deal with the vanishing gradients problem by using a fancy optimizer that can detect directions with a tiny gradient but even smaller curvature.
Echo State Networks: Initialize the input a hidden and hidden-hidden and output-hidden connections very carefully so that the hidden state has a huge reservoir of weakly coupled oscillators which can be selectively driven by the input.

ESNs only need to learn the hidden-output connections.

Good initialization with momentum Initialize like in Echo State Networks, but then learn all of the connections using momentum

Long Short Term Memory (LSTM)¶

LSTM uses a memory cell for modeling long-range dependencies and avoid vanishing gradient problems.

Introduced by Hochreiter and Schmidhuber (1997) who solved the problem of getting an RNN to remember things for a long time (like hundreds of time steps).
They designed a memory cell using logistic and linear units with multiplicative interactions.
Information gets into the cell whenever its “write” gate is on.
The information stays in the cell so long as its keep gate is on.
Information can be read from the cell by turning on its read gate.

Implementing a memory cell in a neural network¶

To preserve information for a long time in the activities of an RNN, we use a circuit that implements an analog memory cell.

A linear unit that has a self-link with a weight of 1 will maintain its state.
Information is stored in the cell by activating its write gate.
Information is retrieved by activating the read gate.
We can backpropagate through this circuit because logistics are have nice derivatives.

Figure 1:

An extrapolation example¶

The following code provides an example of how recurrent neural networks can be used to extrapolate to unknown values of physics data sets. Specifically, the data sets used in this program come from a quantum mechanical many-body calculation of energies as functions of the number of particles.

# For matrices and calculations
import numpy as np
# For machine learning (backend for keras)
import tensorflow as tf
# User-friendly machine learning library
# Front end for TensorFlow
import tensorflow.keras
# Different methods from Keras needed to create an RNN
# This is not necessary but it shortened function calls 
# that need to be used in the code.
from tensorflow.keras import datasets, layers, models
from tensorflow.keras.layers import Input
from tensorflow.keras import regularizers
from tensorflow.keras.models import Model, Sequential
from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU
# For timing the code
from timeit import default_timer as timer
# For plotting
import matplotlib.pyplot as plt


# The data set
datatype='VaryDimension'
X_tot = np.arange(2, 42, 2)
y_tot = np.array([-0.03077640549, -0.08336233266, -0.1446729567, -0.2116753732, -0.2830637392, -0.3581341341, -0.436462435, -0.5177783846,
	-0.6019067271, -0.6887363571, -0.7782028952, -0.8702784034, -0.9649652536, -1.062292565, -1.16231451, 
	-1.265109911, -1.370782966, -1.479465113, -1.591317992, -1.70653767])

Formatting the Data¶

The way the recurrent neural networks are trained in this program differs from how machine learning algorithms are usually trained. Typically a machine learning algorithm is trained by learning the relationship between the x data and the y data. In this program, the recurrent neural network will be trained to recognize the relationship in a sequence of y values. This is type of data formatting is typically used time series forcasting, but it can also be used in any extrapolation (time series forecasting is just a specific type of extrapolation along the time axis). This method of data formatting does not use the x data and assumes that the y data are evenly spaced.

For a standard machine learning algorithm, the training data has the form of (x,y) so the machine learning algorithm learns to assiciate a y value with a given x value. This is useful when the test data has x values within the same range as the training data. However, for this application, the x values of the test data are outside of the x values of the training data and the traditional method of training a machine learning algorithm does not work as well. For this reason, the recurrent neural network is trained on sequences of y values of the form ((y1, y2), y3), so that the network is concerned with learning the pattern of the y data and not the relation between the x and y data. As long as the pattern of y data outside of the training region stays relatively stable compared to what was inside the training region, this method of training can produce accurate extrapolations to y values far removed from the training data set.

# FORMAT_DATA
def format_data(data, length_of_sequence = 2):  
    """
        Inputs:
            data(a numpy array): the data that will be the inputs to the recurrent neural
                network
            length_of_sequence (an int): the number of elements in one iteration of the
                sequence patter.  For a function approximator use length_of_sequence = 2.
        Returns:
            rnn_input (a 3D numpy array): the input data for the recurrent neural network.  Its
                dimensions are length of data - length of sequence, length of sequence, 
                dimnsion of data
            rnn_output (a numpy array): the training data for the neural network
        Formats data to be used in a recurrent neural network.
    """

    X, Y = [], []
    for i in range(len(data)-length_of_sequence):
        # Get the next length_of_sequence elements
        a = data[i:i+length_of_sequence]
        # Get the element that immediately follows that
        b = data[i+length_of_sequence]
        # Reshape so that each data point is contained in its own array
        a = np.reshape (a, (len(a), 1))
        X.append(a)
        Y.append(b)
    rnn_input = np.array(X)
    rnn_output = np.array(Y)

    return rnn_input, rnn_output


# ## Defining the Recurrent Neural Network Using Keras
# 
# The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer.

def rnn(length_of_sequences, batch_size = None, stateful = False):
    """
        Inputs:
            length_of_sequences (an int): the number of y values in "x data".  This is determined
                when the data is formatted
            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
        Returns:
            model (a Keras model): The recurrent neural network that is built and compiled by this
                method
        Builds and compiles a recurrent neural network with one hidden layer and returns the model.
    """
    # Number of neurons in the input and output layers
    in_out_neurons = 1
    # Number of neurons in the hidden layer
    hidden_neurons = 200
    # Define the input layer
    inp = Input(batch_shape=(batch_size, 
                length_of_sequences, 
                in_out_neurons))  
    # Define the hidden layer as a simple RNN layer with a set number of neurons and add it to 
    # the network immediately after the input layer
    rnn = SimpleRNN(hidden_neurons, 
                    return_sequences=False,
                    stateful = stateful,
                    name="RNN")(inp)
    # Define the output layer as a dense neural network layer (standard neural network layer)
    #and add it to the network immediately after the hidden layer.
    dens = Dense(in_out_neurons,name="dense")(rnn)
    # Create the machine learning model starting with the input layer and ending with the 
    # output layer
    model = Model(inputs=[inp],outputs=[dens])
    # Compile the machine learning model using the mean squared error function as the loss 
    # function and an Adams optimizer.
    model.compile(loss="mean_squared_error", optimizer="adam")  
    return model

Predicting New Points With A Trained Recurrent Neural Network¶

def test_rnn (x1, y_test, plot_min, plot_max):
    """
        Inputs:
            x1 (a list or numpy array): The complete x component of the data set
            y_test (a list or numpy array): The complete y component of the data set
            plot_min (an int or float): the smallest x value used in the training data
            plot_max (an int or float): the largest x valye used in the training data
        Returns:
            None.
        Uses a trained recurrent neural network model to predict future points in the 
        series.  Computes the MSE of the predicted data set from the true data set, saves
        the predicted data set to a csv file, and plots the predicted and true data sets w
        while also displaying the data range used for training.
    """
    # Add the training data as the first dim points in the predicted data array as these
    # are known values.
    y_pred = y_test[:dim].tolist()
    # Generate the first input to the trained recurrent neural network using the last two 
    # points of the training data.  Based on how the network was trained this means that it
    # will predict the first point in the data set after the training data.  All of the 
    # brackets are necessary for Tensorflow.
    next_input = np.array([[[y_test[dim-2]], [y_test[dim-1]]]])
    # Save the very last point in the training data set.  This will be used later.
    last = [y_test[dim-1]]

    # Iterate until the complete data set is created.
    for i in range (dim, len(y_test)):
        # Predict the next point in the data set using the previous two points.
        next = model.predict(next_input)
        # Append just the number of the predicted data set
        y_pred.append(next[0][0])
        # Create the input that will be used to predict the next data point in the data set.
        next_input = np.array([[last, next[0]]], dtype=np.float64)
        last = next

    # Print the mean squared error between the known data set and the predicted data set.
    print('MSE: ', np.square(np.subtract(y_test, y_pred)).mean())
    # Save the predicted data set as a csv file for later use
    name = datatype + 'Predicted'+str(dim)+'.csv'
    np.savetxt(name, y_pred, delimiter=',')
    # Plot the known data set and the predicted data set.  The red box represents the region that was used
    # for the training data.
    fig, ax = plt.subplots()
    ax.plot(x1, y_test, label="true", linewidth=3)
    ax.plot(x1, y_pred, 'g-.',label="predicted", linewidth=4)
    ax.legend()
    # Created a red region to represent the points used in the training data.
    ax.axvspan(plot_min, plot_max, alpha=0.25, color='red')
    plt.show()

# Check to make sure the data set is complete
assert len(X_tot) == len(y_tot)

# This is the number of points that will be used in as the training data
dim=12

# Separate the training data from the whole data set
X_train = X_tot[:dim]
y_train = y_tot[:dim]


# Generate the training data for the RNN, using a sequence of 2
rnn_input, rnn_training = format_data(y_train, 2)


# Create a recurrent neural network in Keras and produce a summary of the 
# machine learning model
model = rnn(length_of_sequences = rnn_input.shape[1])
model.summary()

# Start the timer.  Want to time training+testing
start = timer()
# Fit the model using the training data genenerated above using 150 training iterations and a 5%
# validation split.  Setting verbose to True prints information about each training iteration.
hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
                 verbose=True,validation_split=0.05)

for label in ["loss","val_loss"]:
    plt.plot(hist.history[label],label=label)

plt.ylabel("loss")
plt.xlabel("epoch")
plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
plt.legend()
plt.show()

# Use the trained neural network to predict more points of the data set
test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
# Stop the timer and calculate the total time needed.
end = timer()
print('Time: ', end-start)

Other Things to Try¶

Changing the size of the recurrent neural network and its parameters can drastically change the results you get from the model. The below code takes the simple recurrent neural network from above and adds a second hidden layer, changes the number of neurons in the hidden layer, and explicitly declares the activation function of the hidden layers to be a sigmoid function. The loss function and optimizer can also be changed but are kept the same as the above network. These parameters can be tuned to provide the optimal result from the network. For some ideas on how to improve the performance of a recurrent neural network.

def rnn_2layers(length_of_sequences, batch_size = None, stateful = False):
    """
        Inputs:
            length_of_sequences (an int): the number of y values in "x data".  This is determined
                when the data is formatted
            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
        Returns:
            model (a Keras model): The recurrent neural network that is built and compiled by this
                method
        Builds and compiles a recurrent neural network with two hidden layers and returns the model.
    """
    # Number of neurons in the input and output layers
    in_out_neurons = 1
    # Number of neurons in the hidden layer, increased from the first network
    hidden_neurons = 500
    # Define the input layer
    inp = Input(batch_shape=(batch_size, 
                length_of_sequences, 
                in_out_neurons))  
    # Create two hidden layers instead of one hidden layer.  Explicitly set the activation
    # function to be the sigmoid function (the default value is hyperbolic tangent)
    rnn1 = SimpleRNN(hidden_neurons, 
                    return_sequences=True,  # This needs to be True if another hidden layer is to follow
                    stateful = stateful, activation = 'sigmoid',
                    name="RNN1")(inp)
    rnn2 = SimpleRNN(hidden_neurons, 
                    return_sequences=False, activation = 'sigmoid',
                    stateful = stateful,
                    name="RNN2")(rnn1)
    # Define the output layer as a dense neural network layer (standard neural network layer)
    #and add it to the network immediately after the hidden layer.
    dens = Dense(in_out_neurons,name="dense")(rnn2)
    # Create the machine learning model starting with the input layer and ending with the 
    # output layer
    model = Model(inputs=[inp],outputs=[dens])
    # Compile the machine learning model using the mean squared error function as the loss 
    # function and an Adams optimizer.
    model.compile(loss="mean_squared_error", optimizer="adam")  
    return model

# Check to make sure the data set is complete
assert len(X_tot) == len(y_tot)

# This is the number of points that will be used in as the training data
dim=12

# Separate the training data from the whole data set
X_train = X_tot[:dim]
y_train = y_tot[:dim]


# Generate the training data for the RNN, using a sequence of 2
rnn_input, rnn_training = format_data(y_train, 2)


# Create a recurrent neural network in Keras and produce a summary of the 
# machine learning model
model = rnn_2layers(length_of_sequences = 2)
model.summary()

# Start the timer.  Want to time training+testing
start = timer()
# Fit the model using the training data genenerated above using 150 training iterations and a 5%
# validation split.  Setting verbose to True prints information about each training iteration.
hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
                 verbose=True,validation_split=0.05)


# This section plots the training loss and the validation loss as a function of training iteration.
# This is not required for analyzing the couple cluster data but can help determine if the network is
# being overtrained.
for label in ["loss","val_loss"]:
    plt.plot(hist.history[label],label=label)

plt.ylabel("loss")
plt.xlabel("epoch")
plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
plt.legend()
plt.show()

# Use the trained neural network to predict more points of the data set
test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
# Stop the timer and calculate the total time needed.
end = timer()
print('Time: ', end-start)

Other Types of Recurrent Neural Networks¶

Besides a simple recurrent neural network layer, there are two other commonly used types of recurrent neural network layers: Long Short Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short introduction to these layers see https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b and https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b.

The first network created below is similar to the previous network, but it replaces the SimpleRNN layers with LSTM layers. The second network below has two hidden layers made up of GRUs, which are preceeded by two dense (feeddorward) neural network layers. These dense layers “preprocess” the data before it reaches the recurrent layers. This architecture has been shown to improve the performance of recurrent neural networks (see the link above and also https://arxiv.org/pdf/1807.02857.pdf.

def lstm_2layers(length_of_sequences, batch_size = None, stateful = False):
    """
        Inputs:
            length_of_sequences (an int): the number of y values in "x data".  This is determined
                when the data is formatted
            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
        Returns:
            model (a Keras model): The recurrent neural network that is built and compiled by this
                method
        Builds and compiles a recurrent neural network with two LSTM hidden layers and returns the model.
    """
    # Number of neurons on the input/output layer and the number of neurons in the hidden layer
    in_out_neurons = 1
    hidden_neurons = 250
    # Input Layer
    inp = Input(batch_shape=(batch_size, 
                length_of_sequences, 
                in_out_neurons)) 
    # Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers)
    rnn= LSTM(hidden_neurons, 
                    return_sequences=True,
                    stateful = stateful,
                    name="RNN", use_bias=True, activation='tanh')(inp)
    rnn1 = LSTM(hidden_neurons, 
                    return_sequences=False,
                    stateful = stateful,
                    name="RNN1", use_bias=True, activation='tanh')(rnn)
    # Output layer
    dens = Dense(in_out_neurons,name="dense")(rnn1)
    # Define the midel
    model = Model(inputs=[inp],outputs=[dens])
    # Compile the model
    model.compile(loss='mean_squared_error', optimizer='adam')  
    # Return the model
    return model

def dnn2_gru2(length_of_sequences, batch_size = None, stateful = False):
    """
        Inputs:
            length_of_sequences (an int): the number of y values in "x data".  This is determined
                when the data is formatted
            batch_size (an int): Default value is None.  See Keras documentation of SimpleRNN.
            stateful (a boolean): Default value is False.  See Keras documentation of SimpleRNN.
        Returns:
            model (a Keras model): The recurrent neural network that is built and compiled by this
                method
        Builds and compiles a recurrent neural network with four hidden layers (two dense followed by
        two GRU layers) and returns the model.
    """    
    # Number of neurons on the input/output layers and hidden layers
    in_out_neurons = 1
    hidden_neurons = 250
    # Input layer
    inp = Input(batch_shape=(batch_size, 
                length_of_sequences, 
                in_out_neurons)) 
    # Hidden Dense (feedforward) layers
    dnn = Dense(hidden_neurons/2, activation='relu', name='dnn')(inp)
    dnn1 = Dense(hidden_neurons/2, activation='relu', name='dnn1')(dnn)
    # Hidden GRU layers
    rnn1 = GRU(hidden_neurons, 
                    return_sequences=True,
                    stateful = stateful,
                    name="RNN1", use_bias=True)(dnn1)
    rnn = GRU(hidden_neurons, 
                    return_sequences=False,
                    stateful = stateful,
                    name="RNN", use_bias=True)(rnn1)
    # Output layer
    dens = Dense(in_out_neurons,name="dense")(rnn)
    # Define the model
    model = Model(inputs=[inp],outputs=[dens])
    # Compile the mdoel
    model.compile(loss='mean_squared_error', optimizer='adam')  
    # Return the model
    return model

# Check to make sure the data set is complete
assert len(X_tot) == len(y_tot)

# This is the number of points that will be used in as the training data
dim=12

# Separate the training data from the whole data set
X_train = X_tot[:dim]
y_train = y_tot[:dim]


# Generate the training data for the RNN, using a sequence of 2
rnn_input, rnn_training = format_data(y_train, 2)


# Create a recurrent neural network in Keras and produce a summary of the 
# machine learning model
# Change the method name to reflect which network you want to use
model = dnn2_gru2(length_of_sequences = 2)
model.summary()

# Start the timer.  Want to time training+testing
start = timer()
# Fit the model using the training data genenerated above using 150 training iterations and a 5%
# validation split.  Setting verbose to True prints information about each training iteration.
hist = model.fit(rnn_input, rnn_training, batch_size=None, epochs=150, 
                 verbose=True,validation_split=0.05)


# This section plots the training loss and the validation loss as a function of training iteration.
# This is not required for analyzing the couple cluster data but can help determine if the network is
# being overtrained.
for label in ["loss","val_loss"]:
    plt.plot(hist.history[label],label=label)

plt.ylabel("loss")
plt.xlabel("epoch")
plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
plt.legend()
plt.show()

# Use the trained neural network to predict more points of the data set
test_rnn(X_tot, y_tot, X_tot[0], X_tot[dim-1])
# Stop the timer and calculate the total time needed.
end = timer()
print('Time: ', end-start)


# ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data)
# 
# Finally, comparing the performace of a recurrent neural network using the standard data formatting to the performance of the network with time sequence data formatting shows the benefit of this type of data formatting with extrapolation.

# Check to make sure the data set is complete
assert len(X_tot) == len(y_tot)

# This is the number of points that will be used in as the training data
dim=12

# Separate the training data from the whole data set
X_train = X_tot[:dim]
y_train = y_tot[:dim]

# Reshape the data for Keras specifications
X_train = X_train.reshape((dim, 1))
y_train = y_train.reshape((dim, 1))


# Create a recurrent neural network in Keras and produce a summary of the 
# machine learning model
# Set the sequence length to 1 for regular data formatting 
model = rnn(length_of_sequences = 1)
model.summary()

# Start the timer.  Want to time training+testing
start = timer()
# Fit the model using the training data genenerated above using 150 training iterations and a 5%
# validation split.  Setting verbose to True prints information about each training iteration.
hist = model.fit(X_train, y_train, batch_size=None, epochs=150, 
                 verbose=True,validation_split=0.05)


# This section plots the training loss and the validation loss as a function of training iteration.
# This is not required for analyzing the couple cluster data but can help determine if the network is
# being overtrained.
for label in ["loss","val_loss"]:
    plt.plot(hist.history[label],label=label)

plt.ylabel("loss")
plt.xlabel("epoch")
plt.title("The final validation loss: {}".format(hist.history["val_loss"][-1]))
plt.legend()
plt.show()

# Use the trained neural network to predict the remaining data points
X_pred = X_tot[dim:]
X_pred = X_pred.reshape((len(X_pred), 1))
y_model = model.predict(X_pred)
y_pred = np.concatenate((y_tot[:dim], y_model.flatten()))

# Plot the known data set and the predicted data set.  The red box represents the region that was used
# for the training data.
fig, ax = plt.subplots()
ax.plot(X_tot, y_tot, label="true", linewidth=3)
ax.plot(X_tot, y_pred, 'g-.',label="predicted", linewidth=4)
ax.legend()
# Created a red region to represent the points used in the training data.
ax.axvspan(X_tot[0], X_tot[dim], alpha=0.25, color='red')
plt.show()

# Stop the timer and calculate the total time needed.
end = timer()
print('Time: ', end-start)

Applied Data Analysis and Machine Learning

Week 45, Recurrent Neural Networks

Contents

Week 45, Recurrent Neural Networks¶

Plan for week 45¶

Material for the lab sessions, additional ways to present classification results and other practicalities¶

Searching for Optimal Regularization Parameters \(\lambda\)¶

Grid Search¶

Randomized Grid Search¶

Wisconsin Cancer Data¶

Using the correlation matrix¶

Discussing the correlation data¶

Other ways of presenting a classification problem¶

Combinations of classification results¶

Positive and negative prediction values¶

Other quantities¶

\(F_1\) score¶

ROC curve¶

Cumulative gain curve¶

Other measures in classification studies: Cancer Data again¶

Material for Lecture Thursday November 9¶

Recurrent neural networks (RNNs): Overarching view¶

A simple example¶

RNNs¶

Basic layout¶

We need to specify the initial activity state of all the hidden and output units¶

We can specify inputs in several ways¶

We can specify targets in several ways¶

Backpropagation through time¶

The backward pass is linear¶

The problem of exploding or vanishing gradients¶

Four effective ways to learn an RNN¶

Long Short Term Memory (LSTM)¶

Implementing a memory cell in a neural network¶

An extrapolation example¶

Formatting the Data¶

Predicting New Points With A Trained Recurrent Neural Network¶

Other Things to Try¶

Other Types of Recurrent Neural Networks¶