I would like to simulate coupled PDEs with a Gaussian white noise field, and was unable to find any examples or documentation that suggests how it should be done. In particular, I am interested in Cahn-Hilliard-like systems with noise:
d/dt(phi) = div(grad(psi)) + div(noise)
psi = f(phi) + div(grad(phi))
Is there a way to implement this in Fipy?
You can add a GaussianNoiseVariable as a source to your equations.
For a non-conserved field, you would do, e.g.,
noise = fp.GaussianNoiseVariable(mesh=..., mean=..., variance=...)
:
:
eq = fp.TransientTerm(...) == fp.DiffusionTerm(...) + ... + noise
for step in steps:
noise.scramble()
:
:
eq.solve(...)
For a conserved field, you would use:
eq = fp.TransientTerm(...) == fp.DiffusionTerm(...) + ... + noise.faceGrad.divergence
Related
I am trying to solve the following coupled pde's in fipy. I tried the following
eq1 = (DiffusionTerm(coeff=1, var=f)-f*DiffusionTerm(coeff=1, var=phi)
+f-f**3 == 0)
eq2 = (2*DiffusionTerm(coeff=f, var=phi)+f*DiffusionTerm(coeff=1, var=phi)
== 0)
eq = eq1 & eq2
eq.solve()
but it does not like "f*DiffusionTerm(coeff=1, var=phi)" and I get the error.
"TermMultiplyError: Must multiply terms by int or float." Is there a way that I can implement a cell variable times a diffusion term?
Neither of the following will work in FiPy,
from fipy import CellVariable, DiffusionTerm, Grid1D
mesh = Grid1D(nx=10)
var = CellVariable(mesh=mesh)
# eqn = var * DiffusionTerm(coeff=1)
eqn = ImplicitSourceTerm(coeff=DiffusionTerm(coeff=1))
eqn.solve(var)
They both simply don't make sense with respect to the discretization in the finite volume method. Regardless, you can use the following identity to rewrite the term of interest
Basically, instead of using
var * DiffusionTerm(coeff=1)
you can use
DiffusionTerm(coeff=var) - var.grad.mag**2
Giving a regular diffusion term and an extra explicit source term.
I was trying to do out-of-sample prediction using the gof function from package btergm. When calculating the AUC value of a precision-recall curve from the testing set, I get the result of 1.012909, which seems to be theoretically impossible.
How can I interpret the result, or am I doing something wrong. Thank you! Here is my code:
network <- readRDS(url("https://www.dropbox.com/s/zxhqxa8h9awkzpd/network.rds?dl=1"))
model.3<- btergm(network[1:9]~edges+gwodegree(1, fixed=TRUE)+transitiveties + ctriple +
gwidegree(1, fixed=TRUE)+mutual+gwesp(1.5, fixed=TRUE)+ttriple+
memory(type="stability")+delrecip,R=1000)
gof.3 <-gof(model.3, nsim = 1000,target=network[[10]],formula = network[9:10]~edges+gwodegree(1,fixed=TRUE) +
transitiveties + ctriple +gwidegree(1,fixed=TRUE)+mutual+gwesp(1.5, fixed=TRUE)+ttriple+
memory(type="stability")+delrecip,coef = coef(model.3),
statistics = rocpr)
gof.3[[1]]$auc.pr
I am running a logistic generalized linear mixed model and would like to plot my effects together with confidence intervals.
I use the lme4 package to fit my model:
glmer (cbind(positive, negative) ~ F1 * F2 * F3 + V1 + F1 * I(V1^2) + V2 + F1 * I(V2^2) + V3 + I(V3^2) + V4 + I(V4^2) + F4 + (1|OLRE) + (1|ind), family = binomial, data = try, na.action = na.omit, control=glmerControl(optimizer = "optimx", calc.derivs = FALSE, optCtrl = list(method = "nlminb", starttests = FALSE, kkt = FALSE)))
OLRE means that I use an observational level random effect in order to overcome overdispersion.
If you wonder because of convergence warnings, I went through the lme4 troubleshooting protocol, and it should be fine.
In order to get effect plots with confidence intervals, I tried to use ggpredict:
sjPlot, e.g.:
plot_model(mod, type = "pred", terms = c("F1", "F2", "F3"), ci.lvl = 0.95)
I also tried to go through this protocol:
https://rpubs.com/hughes/84484
intdata <- expand.grid(
positive = c(0,1),
negative = c(0,1),
F1 = as.factor(c(0,1)),
F2 = as.factor(c(1,2,3)),
F3= as.factor(c(1,2,3,4)),
V1= as.numeric(median(try$V1)),
F4= as.factor(c(30, 31)),
ind= as.factor(c(68)),
OLRE = as.factor(c(2450)),
V2= as.numeric(median(try$V2)),
V3= as.numeric(median(try$V3)),
V4= as.numeric(median(try$V4))
)
#conditional variances
cV <- ranef(mod, condVar = TRUE)
ranvar <- attr(cV[[1]], "postVar")
sqrt(diag(ranvar[,,1]))
mm <- model.matrix(terms( mod), data=intdata,
contrasts.arg = lapply(intdata[,c(3:5, 7:9)], contrasts, contrasts=FALSE))
predFun<-function(.) mm%*%fixef(.)
bb<-bootMer(mod,FUN=predFun,nsim=3)
with some problems and warnings regarding contrasts and
Error in mm %*% fixef(.) : non-conformable arguments
Therefore, I am wondering if anyone might help me but so far I am not able to Nothing worked so far, so I would really appreciate some help.
Here the link to the data https://drive.google.com/file/d/1qZaJBbM1ggxwPnZ9bsTCXL7_BnXlvfkW/view?usp=sharing
I feel I must be missing something obvious, in struggling to get a positive control for logistic regression going in tensorflow probability.
I've modified the example for logistic regression here, and created a positive control features and labels data. I struggle to achieve accuracy over 60%, however this is an easy problem for a 'vanilla' Keras model (accuracy 100%). What am I missing? I tried different layers, activations, etc.. With this method of setting up the model, is posterior updating actually being performed? Do I need to specify an interceptor object? Many thanks..
### Added positive control
nSamples = 80
features1 = np.float32(np.hstack((np.reshape(np.ones(40), (40, 1)),
np.reshape(np.random.randn(nSamples), (40, 2)))))
features2 = np.float32(np.hstack((np.reshape(np.zeros(40), (40, 1)),
np.reshape(np.random.randn(nSamples), (40, 2)))))
features = np.vstack((features1, features2))
labels = np.concatenate((np.zeros(40), np.ones(40)))
featuresInt, labelsInt = build_input_pipeline(features, labels, 10)
###
#w_true, b_true, features, labels = toy_logistic_data(FLAGS.num_examples, 2)
#featuresInt, labelsInt = build_input_pipeline(features, labels, FLAGS.batch_size)
with tf.name_scope("logistic_regression", values=[featuresInt]):
layer = tfp.layers.DenseFlipout(
units=1,
activation=None,
kernel_posterior_fn=tfp.layers.default_mean_field_normal_fn(),
bias_posterior_fn=tfp.layers.default_mean_field_normal_fn())
logits = layer(featuresInt)
labels_distribution = tfd.Bernoulli(logits=logits)
neg_log_likelihood = -tf.reduce_mean(labels_distribution.log_prob(labelsInt))
kl = sum(layer.losses)
elbo_loss = neg_log_likelihood + kl
predictions = tf.cast(logits > 0, dtype=tf.int32)
accuracy, accuracy_update_op = tf.metrics.accuracy(
labels=labelsInt, predictions=predictions)
with tf.name_scope("train"):
optimizer = tf.train.AdamOptimizer(learning_rate=FLAGS.learning_rate)
train_op = optimizer.minimize(elbo_loss)
init_op = tf.group(tf.global_variables_initializer(),
tf.local_variables_initializer())
with tf.Session() as sess:
sess.run(init_op)
# Fit the model to data.
for step in range(FLAGS.max_steps):
_ = sess.run([train_op, accuracy_update_op])
if step % 100 == 0:
loss_value, accuracy_value = sess.run([elbo_loss, accuracy])
print("Step: {:>3d} Loss: {:.3f} Accuracy: {:.3f}".format(
step, loss_value, accuracy_value))
### Check with basic Keras
kerasModel = tf.keras.models.Sequential([
tf.keras.layers.Dense(1)])
optimizer = tf.train.AdamOptimizer(5e-2)
kerasModel.compile(optimizer = optimizer, loss = 'binary_crossentropy',
metrics = ['accuracy'])
kerasModel.fit(features, labels, epochs = 50) #100% accuracy
Compared to the github example, you forgot to divide by the number of examples when defining the KL divergence:
kl = sum(layer.losses) / FLAGS.num_examples
When I change this to your code, I quickly get to an accuracy of 99.9% on your toy data.
Additionaly, the output layer of your Keras model actually expects a sigmoid activation for this problem (binary classification):
kerasModel = tf.keras.models.Sequential([
tf.keras.layers.Dense(1, activation='sigmoid')])
It's a toy problem, but you will notice that the model gets to 100% accuracy faster with a sigmoid activation.
This is my source code and I want to reduce the possible errors. When running this code there is a lot of difference between trained output to target. I have tried different ways but didn't work so please help me reducing it.
a=[31 9333 2000;31 9500 1500;31 9700 2300;31 9700 2320;31 9120 2230;31 9830 2420;31 9300 2900;31 9400 2500]'
g=[35000;23000;3443;2343;1244;9483;4638;4739]'
h=[31 9333 2000]'
inputs =(a);
targets =[g];
% Create a Fitting Network
hiddenLayerSize = 1;
net = fitnet(hiddenLayerSize);
% Choose Input and Output Pre/Post-Processing Functions
% For a list of all processing functions type: help nnprocess
net.inputs{1}.processFcns = {'removeconstantrows','mapminmax'};
net.outputs{2}.processFcns = {'removeconstantrows','mapminmax'};
% Setup Division of Data for Training, Validation, Testing
% For a list of all data division functions type: help nndivide
net.divideFcn = 'dividerand'; % Divide data randomly
net.divideMode = 'sample'; % Divide up every sample
net.divideParam.trainRatio = 70/100;
net.divideParam.valRatio = 15/100;
net.divideParam.testRatio = 15/100;
% For help on training function 'trainlm' type: help trainlm
% For a list of all training functions type: help nntrain
net.trainFcn = 'trainlm'; % Levenberg-Marquardt
% Choose a Performance Function
% For a list of all performance functions type: help nnperformance
net.performFcn = 'mse'; % Mean squared error
% Choose Plot Functions
% For a list of all plot functions type: help nnplot
net.plotFcns = {'plotperform','plottrainstate','ploterrhist', ...
'plotregression','plotconfusion' 'plotfit','plotroc'};
% Train the Network
[net,tr] = train(net,inputs,targets);
plottrainstate(tr)
% Test the Network
outputs = net(inputs)
errors = gsubtract(targets,outputs)
fprintf('errors = %4.3f\t',errors);
performance = perform(net,targets,outputs);
% Recalculate Training, Validation and Test Performance
trainTargets = targets .* tr.trainMask{1};
valTargets = targets .* tr.valMask{1};
testTargets = targets .* tr.testMask{1};
trainPerformance = perform(net,trainTargets,outputs);
valPerformance = perform(net,valTargets,outputs);
testPerformance = perform(net,testTargets,outputs);
% View the Network
view(net);
sc=sim(net,h)
I think you need to be more specific.
What is the performance like on your training set and on your test set?
Have you tried doing any regularization?