Develop Reference

2d laplacian computation in tensorflow

I want to do a toy code computing the laplacian of the function f(x,y) = sin(pi (x+1)/2)*sin(pi (y+1)/2) for (x,y) in [-1,1]^2.
I have tried multiple methods:
The first one works
the second one has a bug when calculating u_xx
the third one gives wrong results. I dont even know what is in the hess variable
the fourth one is a try of the tf.hessians function that doesnt work.
If someone can shed some light, I would be indebted.
import tensorflow as tf
import tensorflow as tf
from tensorflow import keras
import matplotlib.pyplot as plt
pi = np.pi
# function that will be passed to autmoatic differentiation
#tf.function
def sol(X):
x,y = X[:,0] , X[:,1]
return tf.sin(pi*(x+1)/2)*tf.sin(pi*(y+1)/2)
#tf.function
def sol2(X,Y):
return tf.sin(pi*(X+1)/2)*tf.sin(pi*(Y+1)/2)
# true laplacian for comparison
def f(X):
x , y = X[:,0] , X[:,1]
return -pi**2/2*tf.sin(pi*(x+1)/2)*tf.sin(pi*(y+1)/2)
def fgrid(X,Y):
return -pi**2/2*tf.sin(pi*(X+1)/2)*tf.sin(pi*(Y+1)/2)
#######"MESH OVER THE SQUARE"
n = 500
x1 , x2 = -1 , 1
vec = tf.linspace(x1,x2,n)
xgrid,ygrid = tf.meshgrid(vec,vec)
xrow,yrow = tf.reshape(xgrid,(-1,1)),tf.reshape(ygrid,(-1,1))
Xdata = tf.Variable(tf.concat((xrow,yrow),axis=1))
#######COMPUTING LAPLACIAN
# FIRST WORKING METHOD
"""
with tf.GradientTape(persistent=True) as tape:
xx = tf.reshape(Xdata[:,0],(-1,1))
yy = tf.reshape(Xdata[:,1],(-1,1))
tape.watch(xx)
tape.watch(yy)
u = sol(tf.concat([xx,yy],axis=1))
u_x = tape.gradient(u,xx)
u_xx = tape.gradient(u_x,xx)
u_y = tape.gradient(u,yy)
u_yy = tape.gradient(u_y,yy)
lapl = (u_xx+u_yy)
del(tape)
#displaying results
plt.contourf(xgrid,ygrid,lapl.numpy().reshape(n,n))
#"""
# SECOND METHOD NOT WORKING : error in u_xx computation
"""
with tf.GradientTape(persistent=True) as tape:
xx = tf.Variable(Xdata[:,0])
yy = tf.Variable(Xdata[:,1])
u = sol(Xdata)
u_x = tape.gradient(u,xx)
u_xx = tape.gradient(u_x,xx)
u_y = tape.gradient(u,yy)
u_yy = tape.gradient(u_y,yy)
lapl = u_xx + u_yy
del(tape)
#"""
#plt.contourf(xgrid,ygrid,lapl.numpy().reshape(n,n))
# THIRD METHOD NOT WORKING : wrong results
"""
with tf.GradientTape(persistent=True) as tape:
u = sol(Xdata)
grads = tape.gradient(u,Xdata)
hess = tape.gradient(grads,Xdata) # shape (-1,2)
lapl = hess[:,0] + hess[:,1]
del(tape)
plt.contourf(xgrid,ygrid,lapl.numpy().reshape(n,n))
#"""
# FOURTH METHOD NOT WORKING : tf.hessians (use Gradient.tape instead) or 'GradientTape' object has no attribute 'hessians'
"""
with tf.GradientTape() as tape:
u = sol(Xdata)
hess = tape.hessians(u,Xdata)
lapl = hess[:,0] + hess[:,1]
plt.contourf(xgrid,ygrid,lapl.numpy().reshape(n,n))
#"""
# second try with tf.hessians
#"""
hess = tf.hessians(sol(Xdata),Xdata) #tf.gradients is not supported when eager execution is enabled. Use tf.GradientTape instead.
#""" ```

SKlearn Gaussian Process with constant, manually set correlation

I want to use the Gaussian Process approximation for a simple 1D test function to illustrate a few things. I want to iterate over a few different values for the correlation matrix (since this is 1D it is just a single value) and show what effect different values have on the approximation. My understanding is, that "theta" is the parameter for this. Therefore I want to set the theta value manually and don't want any optimization/changes to it. I thought the constant kernel and the clone_with_theta function might get me what I want but I didn't get it to work. Here is what I have so far:
import numpy as np
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF, ConstantKernel as ConstantKernel
def f(x):
"""The function to predict."""
return x/2 + ((1/10 + x) * np.sin(5*x - 1))/(1 + x**2 * (np.sin(x - (1/2))**2))
# ----------------------------------------------------------------------
# Data Points
X = np.atleast_2d(np.delete(np.linspace(-1,1, 7),4)).T
y = f(X).ravel()
# Instantiate a Gaussian Process model
kernel = ConstantKernel(constant_value=1, constant_value_bounds='fixed')
theta = np.array([0.5,0.5])
kernel = kernel.clone_with_theta(theta)
gp = GaussianProcessRegressor(kernel=kernel, optimizer=None)
# Fit to data using Maximum Likelihood Estimation of the parameters
gp.fit(X, y)
# Make the prediction on the meshed x-axis (ask for MSE as well)
y_pred, sigma = gp.predict(x, return_std=True)
# Plot
# ...

I programmed a simple implementation myself now, which allows to set correlation (here 'b') manually:
import numpy as np
from numpy.linalg import inv
def f(x):
"""The function to predict."""
return x/2 + ((1/10 + x) * np.sin(5*x - 1))/(1 + x**2 * (np.sin(x - (1/2))**2))
def kriging_approx(x,xt,yt,b,mu,R_inv):
N = yt.size
one = np.matrix(np.ones((yt.size))).T
r = np.zeros((N))
for i in range(0,N):
r[i]= np.exp(-b * (xt[i]-x)**2)
y = mu + np.matmul(np.matmul(r.T,R_inv),yt - mu*one)
y = y[0,0]
return y
def calc_R (x,b):
N = x.size
# setup R
R = np.zeros((N,N))
for i in range(0,N):
for j in range(0,N):
R[i][j] = np.exp(-b * (x[i]-x[j])**2)
R_inv = inv(R)
return R, R_inv
def calc_mu_sig (yt, R_inv):
N = yt.size
one = np.matrix(np.ones((N))).T
mu = np.matmul(np.matmul(one.T,R_inv),yt) / np.matmul(np.matmul(one.T,R_inv),one)
mu = mu[0,0]
sig2 = (np.matmul(np.matmul((yt - mu*one).T,R_inv),yt - mu*one))/(N)
sig2 = sig2[0,0]
return mu, sig2
# ----------------------------------------------------------------------
# Data Points
xt = np.linspace(-1,1, 7)
yt = np.matrix((f(xt))).T
# Calc R
R, R_inv = calc_R(xt, b)
# Calc mu and sigma
mu_dach, sig_dach2 = calc_mu_sig(yt, R_inv)
# Point to get approximation for
x = 1
y_approx = kriging_approx(x, xt, yt, b, mu_dach, R_inv)

Incremental Bayesian updates with multi-dimensional parameters

I am trying to use PYMC3 for a Bayesian model where I would like to repeatedly train my model on new unseen data. I am thinking I would need to update the priors with the posterior of the previously trained model every time I see the data, similar to how is achieved here https://docs.pymc.io/notebooks/updating_priors.html. They use the following function that finds the KDE from the samples and replacing each of the original definitions of the parameters in the model with a call to from_posterior.
def from_posterior(param, samples):
smin, smax = np.min(samples), np.max(samples)
width = smax - smin
x = np.linspace(smin, smax, 100)
y = stats.gaussian_kde(samples)(x)
# what was never sampled should have a small probability but not 0,
# so we'll extend the domain and use linear approximation of density on it
x = np.concatenate([[x[0] - 3 * width], x, [x[-1] + 3 * width]])
y = np.concatenate([[0], y, [0]])
return Interpolated(param, x, y)
And here is my original model.
def create_model(batsmen, bowlers, id1, id2, X):
testval = [[-5,0,1,2,3.5,5] for i in range(0, 9)]
l = [i for i in range(9)]
model = pm.Model()
with model:
delta_1 = pm.Uniform("delta_1", lower=0, upper=1)
delta_2 = pm.Uniform("delta_2", lower=0, upper=1)
inv_sigma_sqr = pm.Gamma("sigma^-2", alpha=1.0, beta=1.0)
inv_tau_sqr = pm.Gamma("tau^-2", alpha=1.0, beta=1.0)
mu_1 = pm.Normal("mu_1", mu=0, sigma=1/pm.math.sqrt(inv_tau_sqr), shape=len(batsmen))
mu_2 = pm.Normal("mu_2", mu=0, sigma=1/pm.math.sqrt(inv_tau_sqr), shape=len(bowlers))
delta = pm.math.ge(l, 3) * delta_1 + pm.math.ge(l, 6) * delta_2
eta = [pm.Deterministic("eta_" + str(i), delta[i] + mu_1[id1[i]] - mu_2[id2[i]]) for i in range(9)]
cutpoints = pm.Normal("cutpoints", mu=0, sigma=1/pm.math.sqrt(inv_sigma_sqr), transform=pm.distributions.transforms.ordered, shape=(9,6), testval=testval)
X_ = [pm.OrderedLogistic("X_" + str(i), cutpoints=cutpoints[i], eta=eta[i], observed=X[i]-1) for i in range(9)]
return model
Here, the problem is that some of my parameters such as mu_1, are multidimensional. This is why I get the following error:
ValueError: points have dimension 1, dataset has dimension 1500
because of the line y = stats.gaussian_kde(samples)(x).
Can someone please help me make this work for multi-dimensional parameters? I don't properly understand what KDE is and how the code computes it.
Thank you in advance!!

Implementing simple probabilistic model with negative log likelihood loss

First a quick disclaimer would be that I posted this question on Reddit, in the Deep Learning and Learning Machine Learning first, but I thought I might also request your expertise here too. Without further ado:
I am currently challenging myself on this year Deep Unsupervised Learning Course of Berkeley University and although I just started the warmup exercise of week 1, I am already having 'technical' difficulties.
The exercise in question is the "1. Warmup" in the following document: Week 1 Exercises. (My apologies as I am not familiar enough with Reddit formating to seemlessly include images.
In my understanding, we have a variable x which can take values from 1..100 which a specific probability of being sampled ( defined in sample_data() function).
The task is therefore to fit a vector of parameters theta which is passed to a softmax function, and is supposed to give the likelihood of a specific element x_i to be sampled. Namely, theta_1 should the parameter which "bumps up" the soft-max value corresponding to the variable x = 1 and so on.
Using Tensorflow, I think I was able to create such a model, but when it comes to training, I believe I am missing a crucial point as the program cannot compute gradients with respect to the theta parameters.
I would like to know if am not misunderstanding the task, and if there is any better method to achieve the result of the exercise.
Here is the code, where the failing par is located from the # Computing gradients.
import numpy as np
import tensorflow as tf
import tensorflow_probability as tfp
if __name__ == "__main__":
# Sampling function of the x variable provided in the exercise
def sample_data():
count = 10000
rand = np.random.RandomState(0)
a = 0.3 + 0.1 * rand.randn(count)
b = 0.8 + 0.05 * rand.randn(count)
mask = rand.rand(count) < 0.5
samples = np.clip(a * mask + b * (1 - mask), 0.0, 1.0)
return np.digitize(samples, np.linspace(0.0, 1.0, 100))
full_data = sample_data()
train_ds = full_data[:int(.8*len( full_data))]
val_ds = full_data[int(.8*len( full_data)):]
# Declaring parameters theta
w_init = tf.zeros_initializer()
params = tf.Variable(
initial_value=w_init(shape=(1, 100),
dtype='float32'), trainable=True, name='params')
softmax = tf.squeeze( tf.nn.softmax( params, axis=1))
#Should materialize the loss of the model
def get_neg_log_likelihood( inputs):
return - tf.math.log( softmax)
neg_log_likelihoods = get_neg_log_likelihood( softmax)
dist = tfp.distributions.Categorical( probs=softmax, dtype=tf.int32)
optimizer = tf.keras.optimizers.Adam()
for epoch in range( 100):
minibatch_size = 200
n_minibatches = len( train_ds) // minibatch_size
# Running over minibatches of the data
for minibatch in range( n_minibatches):
# Minibatching
start_index = (minibatch*minibatch_size)
end_index = (minibatch_size*minibatch + minibatch_size)
x = train_ds[start_index:end_index]
with tf.GradientTape() as tape:
tape.watch( params)
loss = tf.reduce_mean( - dist.log_prob( x))
# Computing gradients
grads = tape.gradient( loss, params)
print( grads) # Result: None
# input()
optimizer.apply_gradients( zip( grads, params))
Thank you in advance for your time.
PS: I mainly have a background in Deep Reinforcement Learning, therefore I can understand the various models used there ( policy, value functions ...), but I am trying to refine my grasp over the internals of the models themselves, namely in generative probabilistic models (GAN, VAE) and other unsupervised learning models in general ( RealNVP, Norm Flows, ...)

Pretty sure nobody is gonna see this, but I thought I might as well bring some closure to this.
First of all, I calculated the gradients by directly deriving its expression from the negative log likelihood of the soft-max value, thus dropping the Tensorflow framework by the same occasion.
Although the results are a little bit under my expectations, the program was able to fit the model to a distribution somewhat similar to the empirical distribution of the sampled data. I guess this is due to the fact that just a 1 dimensional theta parameter vector is not enough to fully model the real data distribution, as well as the finite amount of sampled data.
An updated version of the code:
import numpy as np
from matplotlib import pyplot as plt
np.random.seed( 42)
def softmax(X, theta = 1.0, axis = None):
# Shamefull copy paste from SO
y = np.atleast_2d(X)
if axis is None:
axis = next(j[0] for j in enumerate(y.shape) if j[1] > 1)
y = y * float(theta)
y = y - np.expand_dims(np.max(y, axis = axis), axis)
y = np.exp(y)
ax_sum = np.expand_dims(np.sum(y, axis = axis), axis)
p = y / ax_sum
if len(X.shape) == 1: p = p.flatten()
return p
if __name__ == "__main__":
def sample_data():
count = 10000
rand = np.random.RandomState(0)
a = 0.3 + 0.1 * rand.randn(count)
b = 0.8 + 0.05 * rand.randn(count)
mask = rand.rand(count) < 0.5
samples = np.clip(a * mask + b * (1 - mask), 0.0, 1.0)
return np.digitize(samples, np.linspace(0.0, 1.0, 100))
full_data = sample_data()
train_ds = full_data[:int(.8*len( full_data))]
val_ds = full_data[int(.8*len( full_data)):]
# Declaring parameters
params = np.zeros(100)
# Use for loss computation
def get_neg_log_likelihood( softmax):
return - np.log( softmax)
def get_loss( params, x):
return np.mean( [get_neg_log_likelihood( softmax( params))[i-1] for i in x])
lr = .0005
for epoch in range( 1000):
# Shuffling training data
np.random.shuffle( train_ds)
minibatch_size = 100
n_minibatches = len( train_ds) // minibatch_size
# Running over minibatches of the data
for minibatch in range( n_minibatches):
smax = softmax( params)
# Jacobian of neg log likelishood
jacobian = [[ smax[j] - 1 if i == j else
smax[j] for j in range(100)] for i in range(100)]
# Minibatching
start_index = (minibatch*minibatch_size)
end_index = (minibatch_size*minibatch + minibatch_size)
x = train_ds[start_index:end_index]
# Compute the gradient matrix for each sample data and mean over it
grad_matrix = np.vstack( [jacobian[i] for i in x])
grads = np.sum( grad_matrix, axis=0)
params -= lr * grads
print( "Epoch %d -- Train loss: %.4f , Val loss: %.4f" %(epoch, get_loss( params, train_ds), get_loss( params, val_ds)))
# Plotting each ~100 epochs
if epoch % 100 == 0:
counters = { i+1: 0 for i in range(100)}
for x in full_data:
counters[x]+= 1
histogram = np.array( [ counters[i+1] / len( full_data) for i in range( 100)])
fsmax = softmax( params)
fig, ax = plt.subplots()
ax.set_title('Dist. Comp. after %d epochs of training (from scratch)' % epoch)
x = np.arange( 1,101)
width = 0.35
rects1 = ax.bar(x - width/2, fsmax, width, label='Model')
rects2 = ax.bar(x + width/2, histogram, width, label='Empirical')
ax.set_ylabel('Likelihood')
ax.set_xlabel('Variable x\s values')
ax.legend()
def autolabel(rects):
for rect in rects:
height = rect.get_height()
autolabel(rects1)
autolabel(rects2)
fig.tight_layout()
plt.savefig( 'plots/results_after_%d_epochs.png' % epoch)
Picture of the final model distribution included for completeness. Modeled vs Empirical Distribution

How to get accurate predictions from Neural Network?

I'm doing a project on water quality prediction using Artificial Neural Network. I implemented this using python. I have completed my prediction model but the generated predictions are not much accurate.
What I'm doing is I have collected data from a river for past 4 and half years on daily basis and I'm predicting a pattern for a specific parameter by inputting data from past records. Simply what I need to do is to predict "Turbidity level" of water on 2015 by feeding data on turbidity from 2012-2014.
From the model which I have created it is not much accurate when I compare to the real data I have gathered for 2015. Please help me to solve this. I tried this by changing hidden layer sizes and the Lambda value.
//This is my code
import xlrd
import numpy as np
from numpy import zeros
from scipy.optimize import minimize
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from scipy import optimize
#Neural Network
class Neural_Network(object):
def __init__(self,Lambda):
#Define Hyperparameters
self.inputLayerSize = 2
self.outputLayerSize = 1
self.hiddenLayerSize = 10
#Weights (parameters)
self.W1 = np.random.randn(self.inputLayerSize,self.hiddenLayerSize)
self.W2 = np.random.randn(self.hiddenLayerSize,self.outputLayerSize)
#Regularization Parameter:
self.Lambda = Lambda
def forward(self, arrayInput):
#Propogate inputs though network
self.z2 = np.dot(arrayInput, self.W1)
self.a2 = self.sigmoid(self.z2)
self.z3 = np.dot(self.a2, self.W2)
yHat = self.sigmoid(self.z3)
return yHat
def sigmoid(self, z):
#Apply sigmoid activation function to scalar, vector, or matrix
return 1/(1+np.exp(-z))
def sigmoidPrime(self,z):
#Gradient of sigmoid
return np.exp(-z)/((1+np.exp(-z))**2)
def costFunction(self, arrayInput, arrayOutput):
#Compute cost for given input,output use weights already stored in class.
self.yHat = self.forward(arrayInput)
#J = 0.5*sum((arrayOutput-self.yHat)**2)
#J = 0.5*sum((arrayOutput-self.yHat)**2)/arrayInput.shape[0] + (self.Lambda/2)
J = 0.5*sum((arrayOutput-self.yHat)**2)/arrayInput.shape[0] + (self.Lambda/2)*sum(sum(self.W1**2),sum(self.W2**2))
#J = 0.5*sum((arrayOutput-self.yHat)**2)/arrayInput.shape[0] + (self.Lambda/2)*(sum(self.W1**2)+sum(self.W2**2))
return J
def costFunctionPrime(self, arrayInput, arrayOutput):
#Compute derivative with respect to W and W2 for a given X and y:
self.yHat = self.forward(arrayInput)
delta3 = np.multiply(-(arrayOutput-self.yHat), self.sigmoidPrime(self.z3))
#Add gradient of regularization term:
#dJdW2 = np.dot(self.a2.T, delta3) + self.Lambda*self.W2
dJdW2 = np.dot(self.a2.T, delta3)
delta2 = np.dot(delta3, self.W2.T)*self.sigmoidPrime(self.z2)
#Add gradient of regularization term:
#dJdW1 = np.dot(arrayInput.T, delta2)+ self.Lambda*self.W1
dJdW1 = np.dot(arrayInput.T, delta2)
return dJdW1, dJdW2
#Helper Functions for interacting with other classes:
def getParams(self):
#Get W1 and W2 unrolled into vector:
params = np.concatenate((self.W1.ravel(), self.W2.ravel()))
return params
def setParams(self, params):
#Set W1 and W2 using single paramater vector.
W1_start = 0
W1_end = self.hiddenLayerSize * self.inputLayerSize
self.W1 = np.reshape(params[W1_start:W1_end], (self.inputLayerSize , self.hiddenLayerSize))
W2_end = W1_end + self.hiddenLayerSize*self.outputLayerSize
self.W2 = np.reshape(params[W1_end:W2_end], (self.hiddenLayerSize, self.outputLayerSize))
def computeGradients(self, arrayInput, arrayOutput):
dJdW1, dJdW2 = self.costFunctionPrime(arrayInput, arrayOutput)
return np.concatenate((dJdW1.ravel(), dJdW2.ravel()))
def computeNumericalGradient(self,N, X, y):
paramsInitial = N.getParams()
numgrad = np.zeros(paramsInitial.shape)
perturb = np.zeros(paramsInitial.shape)
e = 1e-4
for p in range(len(paramsInitial)):
#Set perturbation vector
perturb[p] = e
N.setParams(paramsInitial + perturb)
loss2 = N.costFunction(X, y)
N.setParams(paramsInitial - perturb)
loss1 = N.costFunction(X, y)
#Compute Numerical Gradient
numgrad[p] = (loss2 - loss1) / (2*e)
#Return the value we changed to zero:
perturb[p] = 0
#Return Params to original value:
N.setParams(paramsInitial)
return numgrad
#Trainer class
class trainer(object):
def __init__(self, N):
self.N = N
def costFunctionWrapper(self, params, arrayInput, arrayOutput):
self.N.setParams(params)
cost = self.N.costFunction(arrayInput, arrayOutput)
#grad = self.N.computeGradients(arrayInput, arrayOutput)
grad = self.N.computeNumericalGradient(self.N,arrayInput, arrayOutput)
return cost, grad
def callbackF(self, params):
self.N.setParams(params)
self.J.append(self.N.costFunction(self.arrayInput, self.arrayOutput))
self.testJ.append(self.N.costFunction(self.TestInput, self.TestOutput))
def train(self, arrayInput, arrayOutput,TestInput,TestOutput):
#Make an internal variable for the callback function:
self.arrayInput = arrayInput
self.arrayOutput = arrayOutput
self.TestInput = TestInput
self.TestOutput = TestOutput
#Make empty list to store costs:
self.J = []
self.testJ= []
params0 = self.N.getParams()
options = {'maxiter': 200, 'disp' : True}
_res = optimize.minimize(self.costFunctionWrapper, params0, jac=True, method='BFGS', \
args=(arrayInput, arrayOutput), options=options, callback=self.callbackF)
self.N.setParams(_res.x)
self.optimizationResults = _res
#Main Program
path = "F:\prototype\\newdata\\tody\\turbidity\\c.xlsx"
book = xlrd.open_workbook(path)
input1=[]
output=[]
testinput=[]
testoutput=[]
#training data set
first_sheet = book.sheet_by_index(1)
for row in range(first_sheet.ncols-1):
input1.append(first_sheet.col_values(row))
for row in range((first_sheet.ncols-1),first_sheet.ncols ):
output.append(first_sheet.col_values(row))
arrayInput = np.asarray(input1)
arrayInput = arrayInput.T
arrayOutput = np.asarray(output)
arrayOutput = arrayOutput.T
#testing data set
first_sheet1 = book.sheet_by_index(0)
for row in range(first_sheet1.ncols-1):
testinput.append(first_sheet1.col_values(row))
for row in range((first_sheet1.ncols-1),first_sheet1.ncols ):
testoutput.append(first_sheet1.col_values(row))
TestInput = np.asarray(testinput)
TestInput = TestInput.T
TestOutput = np.asarray(testoutput)
TestOutput = TestOutput.T
#2016
input2016=[]
first_sheet2 = book.sheet_by_index(2)
for row in range(first_sheet2.ncols):
input2016.append(first_sheet2.col_values(row))
Input = np.asarray(input2016)
Input = Input.T
# Scaling
arrayInput = arrayInput / np.amax(arrayInput, axis=0)
arrayOutput = arrayOutput / np.amax(arrayOutput, axis=0)
TestInput = TestInput / np.amax(TestInput, axis=0)
Input = Input / np.amax(Input, axis=0)
TestOutput = TestOutput / np.amax(TestOutput, axis=0)
NN=Neural_Network(Lambda=0.00000000000001)
T = trainer(NN)
T.train(arrayInput,arrayOutput,TestInput,TestOutput)
print NN.costFunctionPrime(arrayInput,arrayOutput)
Output = NN.forward(Input)
print Output
print '----------'
#print TestOutput
#plt.plot(T.J)
plt.plot(Output)
plt.grid(1)
plt.xlabel('Iterations')
plt.ylabel('cost')
plt.show()
//Turbidity means 2015 real data and prediction means data predicted using this code

Some of the comments suggest scaling the output sigmoidal layer to match the correct data. If you look at your predictions, you will see that with some scaling they are pretty accurate. I advise against scaling a sigmoidal function, however.
A sigmoidal output is meant to be interpreted as a probability (given certain constraints are followed), so scaling it would be breaking that contract and could give undefined results. What happens if you scale from 0-100, but then start receiving training targets larger than 100? (assuming you are training an online system, otherwise perhaps that example is not relevant)
I would change your code to use a linear output layer. This would not require any manipulation of the data after training the network. Also given that your cost function is least squares, the linear output layer will be convex (which reduces the number of local optima that your algorithm can get stuck in).