I am trying to convert equation 2 in this paper which is a quantum stochastic master equation to the form of equation 5 which is in Cartesian coordinate form.
This screenshotis taken from the following peer reviewed journal article https://doi.org/10.1016/j.jfranklin.2019.05.021]
What I did so far is I took the derivative of equation 4, then set it equal to equation 2.
Since there are commutators in equation 2, when I substitute rho_t into equation 2, I always
get really confused.
If anyone have any insight, please share. Thank you for your help.
Related
I'm investigating the task of training a neural network to predict one future value given a sinusoidal input. So for example, as seen in the Figure, the input signal is x and the expected output signal y. The model's output is y^. Doing the regression task is fairly straightforward, and there are a lot of choices for this problem. I'm using a simple recurrent neural network with mean-squared error (MSE) loss between y and y^.
Additionally, suppose I know that the sinusoid is made up of N modalities, e.g., at some points, the wave oscillates at 5 Hz, then 10 Hz, then back to 5 Hz, then up to 15 Hz maybe—i.e., N=3.
In this case, I have ground-truth class labels in a vector k and the model does both regression and classification, additionally outputting a vector k^. An example is shown in the Figure. As this is a multi-class problem with exclusivity, I figured binary cross entropy (BCE) loss should be relevant here.
I'm sure there is a lot of research about combining loss functions, but does just adding MSE and BCE make sense? Scaling one up or down by a factor of 10 doesn't seem to change the learning outcome too much. So I was wondering what is considered the standard approach to problems where there is a joint classification and regression objective.
Additionally, on top of just BCE, I want to penalize k^ for quickly jumping around between classes; for example, if the model guesses one class, I'd like it to stay in that one class and switch only when it's necessary. See how in the Figure, there are fast dark blue blips in k^. I would like the same solid bands as seen in k, and naive BCE loss doesn't account for that.
Appreciate any and all advice!
No idea if I am asking this question in the right place, but here goes...
I have a set of equations that were calculated based on numbers ranging from 4 to 8. So an equation for when this number is 5, one for when it is 6, one for when it is 7, etc. These equations were determined from graphing a best fit line to data points in a Google Sheet graph. Here is an example of a graph...
Example...
When the number is between 6 and 6.9, this equation is used: windGust6to7 = -29.2 + (17.7 * log(windSpeed))
When the number is between 7 and 7.9, this equation is used: windGust7to8 = -70.0 + (30.8 * log(windSpeed))
I am using these equations to create an image in python, but the image is too choppy since each equation covers a range from x to x.9. In order to smooth this image out and make it more accurate, I really would need an equation for every 0.1 change in number. So an equation for 6, a different equation for 6.1, one for 6.2, etc.
Here is an example output image that is created using the current equations:
So my question is: Is there a way to find the relationship between the two example equations I gave above in order to use that to create a smoother looking image?
This is not about logarithms; for the purposes of this derivation, log(windspeed) is a constant term. Rather, you're trying to find a fit for your mapping:
6 (-29.2, 17.7)
7 (-70.0, 30.8)
...
... and all of the other numbers you have already. You need to determine two basic search paramteres:
(1) Where in each range is your function an exact fit? For instance, for the first one, is it exactly correct at 6.0, 6.5, 7.0, or elsewhere? Change the left-hand column to reflect that point.
(2) What sort of fit do you want? You are basically fitting a pair of parameterized equations, one for each coefficient:
x y x y
6 -29.2 6 17.7
7 -70.0 7 30.8
For each of these, you want to find the coefficients of a good matching function. This is a large field of statistical and algebraic study. Since you have four ranges, you will have four points for each function. It is straightforward to fit a cubic equation to each set of points in Cartesian space. However, the resulting function may not be as smooth as you like; in such a case, you may well find that a 4th- or 5th- degree function fits better, or perhaps something exponential, depending on the actual distribution of your points.
You need to work with your own problem objectives and do a little more research into function fitting. Once you determine the desired characteristics, look into scikit for fitting functions to do the heavy computational work for you.
I am trying to implement a CNN in Tensorflow (quite similar architecture to VGG), which then splits into two branches after the first fully connected layer. It follows this paper: https://arxiv.org/abs/1612.01697
Each of the two branches of the network outputs a set of 32 numbers. I want to write a joint loss function, which will take 3 inputs:
The predictions of branch 1 (y)
The predictions of branch 2 (alpha)
The labels Y (ground truth) (q)
and calculate a weighted loss, as in the image below:
Loss function definition
q_hat = tf.divide(tf.reduce_sum(tf.multiply(alpha, y),0), tf.reduce_sum(alpha,0))
loss = tf.abs(tf.subtract(q_hat, q))
I understand the fact that I need to use the tf functions in order to implement this loss function. Having implemented the above function, the network is training, but once trained, it is not outputting the expected results.
Has anyone ever tried combining outputs of two branches of a network in one joint loss function? Is this something TensorFlow supports? Maybe I am making a mistake somewhere here? Any help whatsoever would be greatly appreciated. Let me know if you would like me to add any further details.
From TensorFlow perspective, there is absolutely no difference between a "regular" CNN graph and a "branched" graph. For TensorFlow, it is just a graph that needs to be executed. So, TensorFlow certainly supports this. "Combining two branches into joint loss" is also nothing special. In fact, it is "good" that loss depends on both branches. It means that when you ask TensorFlow to compute loss, it will have to do the forward pass through both branches, which is what you want.
One thing I noticed is that your code for loss is different than the image. Your code appears to do this https://ibb.co/kbEH95
lmerTest was designed as a wrapper to permit estimation of p-values from lmer mixed model analyses, using the Satterthwaite estimate of denominator degrees of freedom (ddf). But lmerTest now appears to be broken. It presently returns a message that there was an internal calculation error and returns only the lmer result (with no p-values). I have been able to calculate the p-values from the summary() function, using Dan Mirman's excellent code for calculating the Kenward-Rogers estimate of ddf. But I can't find equivalent code to calculate the p-values in an anova call on the lmer model. I suspect that one just needs to feed anova() a ddf, but I can't figure out how to do that.
Thanks in advance to anyone that can suggest solutions for this problem.
Larry Hunsicker
lmerTest returns the anova output of lme4 package whenever some computational error occurs in getting the Satterthwaite's approximation (such as e.g. in calculating the asymptotic variance covariance matrix). The lmerTest is not broken, it is just that there could be examples when the Satterthwaite's approximation cannot be calculated. In my experience this occurs not often.
I am using Pylearn2 OR Caffe to build a deep network. My target is ordered nominal. I am trying to find a proper loss function but cannot find any in Pylearn2 or Caffe.
I read a paper "Loss Functions for Preference Levels: Regression with Discrete Ordered Labels" . I get the general idea - but I am not sure I understand what will the thresholds be, if my final layer is a SoftMax over Logistic Regression (outputting probabilities).
Can some help me by pointing to any implementation of such a loss function ?
Thanks
Regards
For both pylearn2 and caffe, your labels will need to be 0-4 instead of 1-5...it's just the way they work. The output layer will be 5 units, each is a essentially a logistic unit...and the softmax can be thought of as an adaptor that normalizes the final outputs. But "softmax" is commonly used as an output type. When training, the value of any individual unit is rarely ever exactly 0.0 or 1.0...it's always a distribution across your units - which log-loss can be calculated on. This loss is used to compare against the "perfect" case and the error is back-propped to update your network weights. Note that a raw output from PL2 or Caffe is not a specific digit 0,1,2,3, or 5...it's 5 number, each associated to the likelihood of each of the 5 classes. When classifying, one just takes the class with the highest value as the 'winner'.
I'll try to give an example...
say I have a 3 class problem, I train a network with a 3 unit softmax.
the first unit represents the first class, second the second and third, third.
Say I feed a test case through and get...
0.25, 0.5, 0.25 ...0.5 is the highest, so a classifier would say "2". this is the softmax output...it makes sure the sum of the output units is one.
You should have a look at ordinal (logistic) regression. This is the formal solution to the problem setup you describe ( do not use plain regression as the distance measures of errors are wrong).
https://stats.stackexchange.com/questions/140061/how-to-set-up-neural-network-to-output-ordinal-data
In particular I recommend looking at Coral ordinal regression implementation at
https://github.com/ck37/coral-ordinal/issues.