Why extreme large value to 0 frequency fft (numpy.fft.fft method) - numpy

I have a signal ts which has rougly mean 40 and applied fft on that with code
ts = array([25, 40, 30, 40, 29, 48, 36, 32, 34, 38, 15, 33, 40, 32, 41, 25, 37,49, 41, 35, 23, 22, 36, 44, 28, 36, 32, 37, 39, 51])
index = fftshift(fftfreq(len(ts)))
ft_ts =fftshift(fft(ts))
output
ft_ts = array([ -76.00000000 +8.34887715e-14j, -57.72501110 +1.17054586e+01j,
7.69492662 +9.79582336e+00j, -29.11145618 -7.22493645e+00j,
14.92140414 +4.58471353e+01j, -26.00000000 -4.67653718e+01j,
-39.61803399 -2.83601821e+01j, -11.34044003 +8.66215368e+00j,
23.68703939 +1.57391882e+01j, -64.88854382 -2.44499549e+01j,
50.00000000 -3.98371686e+01j, 4.09382150 -6.27663403e+00j,
-37.38196601 -3.06708342e+01j, 35.97162964 +1.31929223e+01j,
18.69662985 -2.20453671e+00j, 1048.00000000 +0.00000000e+00j,
18.69662985 +2.20453671e+00j, 35.97162964 -1.31929223e+01j,
-37.38196601 +3.06708342e+01j, 4.09382150 +6.27663403e+00j,
50.00000000 +3.98371686e+01j, -64.88854382 +2.44499549e+01j,
23.68703939 -1.57391882e+01j, -11.34044003 -8.66215368e+00j,
-39.61803399 +2.83601821e+01j, -26.00000000 +4.67653718e+01j,
14.92140414 -4.58471353e+01j, -29.11145618 +7.22493645e+00j,
7.69492662 -9.79582336e+00j, -57.72501110 -1.17054586e+01j])
at 0 frequency ft_ts has value of 1048. Shouldn't that be the mean of my original signal ts which is 40 ? What happened here ?
Many thanks

The FFT is not normalized, so the first term should be the sum, not the mean.
For example, see the definition here
and you can see, that when k=0, the exponential term is 1, and you'll just get the sum of x_n.
This is why the first item in fft(np.ones(10)) is 10, not 1. 1 is the mean (since it's an array of ones), and 10 is the sum.

Related

Feeding Word Embedding Matrix into a Pytorch LSTM Model

I have a LSTM model I am using to predict the unemployment rate from federal reserve filings. It uses glove vectors and vocab2index embedding and the training went as planned. However, upon attempting to feed a word embedding into the model for prediction testing it keeps throwing various errors.
Here is the model:
def load_glove_vectors(glove_file= glove_embedding_vectors_text_file):
"""Load the glove word vectors"""
word_vectors = {}
with open(glove_file) as f:
for line in f:
split = line.split()
word_vectors[split[0]] = np.array([float(x) for x in split[1:]])
return word_vectors
def get_emb_matrix(pretrained, word_counts, emb_size = 300):
""" Creates embedding matrix from word vectors"""
vocab_size = len(word_counts) + 2
vocab_to_idx = {}
vocab = ["", "UNK"]
W = np.zeros((vocab_size, emb_size), dtype="float32")
W[0] = np.zeros(emb_size, dtype='float32') # adding a vector for padding
W[1] = np.random.uniform(-0.25, 0.25, emb_size) # adding a vector for unknown words
vocab_to_idx["UNK"] = 1
i = 2
for word in word_counts:
if word in word_vecs:
W[i] = word_vecs[word]
else:
W[i] = np.random.uniform(-0.25,0.25, emb_size)
vocab_to_idx[word] = i
vocab.append(word)
i += 1
return W, np.array(vocab), vocab_to_idx
word_vecs = load_glove_vectors()
pretrained_weights, vocab, vocab2index = get_emb_matrix(word_vecs, counts)
Unfortunately when I feed this array
[array([ 3, 10, 6287, 6, 113, 271, 3, 6639, 104, 5105, 7525,
104, 7526, 9, 23, 9, 10, 11, 24, 7527, 7528, 104,
11, 24, 7529, 7530, 104, 11, 24, 7531, 7530, 104, 11,
24, 7532, 7530, 104, 11, 24, 7533, 7534, 24, 7535, 7536,
104, 7537, 104, 7538, 7539, 7540, 6643, 7541, 7354, 7542, 7543,
7544, 9, 23, 9, 10, 11, 24, 25, 8, 10, 11,
24, 3, 10, 663, 168, 9, 10, 290, 291, 3, 4909,
198, 10, 1478, 169, 15, 4621, 3, 3244, 3, 59, 1967,
113, 59, 520, 198, 25, 5105, 7545, 7546, 7547, 7546, 7548,
7549, 7550, 1874, 10, 7551, 9, 10, 11, 24, 7552, 6287,
7553, 7554, 7555, 24, 7556, 24, 7557, 7558, 7559, 6, 7560,
323, 169, 10, 7561, 1432, 6, 3134, 3, 7562, 6, 7563,
1862, 7144, 741, 3, 3961, 7564, 7565, 520, 7566, 4833, 7567,
7568, 4901, 7569, 7570, 4901, 7571, 1874, 7572, 12, 13, 7573,
10, 7574, 7575, 59, 7576, 59, 638, 1620, 7577, 271, 6488,
59, 7578, 7579, 7580, 7581, 271, 7582, 7583, 24, 669, 5932,
7584, 9, 113, 271, 3764, 3, 5930, 3, 59, 4901, 7585,
793, 7586, 7587, 6, 1482, 520, 7588, 520, 7589, 3246, 7590,
13, 7591])
into torch.LongTensor() I keep getting the following error:
TypeError: can't convert np.ndarray of type numpy.object_. The only supported types are: float64, float32, float16, complex64, complex128, int64, int32, int16, int8, uint8, and bool.
Any ideas on how to remedy? I am fairly new to AI in general, and I am an economist by trade so I am almost certain I have made a boneheaded error.

How to merge classes in multiclass image segmentation

I am performing an image segmentation with a u-net model.
My mask has classes from 0-50.
I also have a text file dictionary with codes representing each class.
For example -
{1: '1234', 2:'5678', 3:'1245'} etc.
How do I combine when the 2 first string characters are the same so for example above key 1 and 3 are the same because they both start with "12".
How can I do this for all classes?
firstTwoCharDict = {}
for key, value in dictionary.items():
if key == 0:
value == value
firstTwoCharDict[key] = value
else:
value = value[:2]
firstTwoCharDict[key] = value
newDict = {}
for key, value in firstTwoCharDict.items():
if value not in newDict:
newDict[value] = [key]
else:
newDict[value].append(key)
This provides this
{'62': [1, 39],
'90': [2, 5, 9, 20, 32, 42, 47, 72, 88, 91, 95],
'97': [3, 49, 55],
'98': [4, 24, 34, 40, 53, 76, 81, 90, 96],
'31': [6, 17, 30, 48, 83],
'69': [7, 13, 15, 16, 27, 44, 51, 54, 56, 75],
'79': [8, 50],
'71': [10, 19, 22, 35, 61, 63, 65],
'99': [11, 12, 21, 46, 52, 69, 78, 84, 89],
'48': [14, 36, 74],
'60': [18],
'64': [23, 38, 66, 97]
```
Now i have an 2d array with integers, how do I replace them with they keys if the array values are equal to the values in the dict?

How to increase 1d-data resolution though interpolating in between measurements?

I'm developing a pythong script where I receive angular measurements from a motor which has a low resolution encoder attached to it. The data I get from the motor has a very low resolution (about 5 degrees division in between measurments). This is an example of the sensor output whilst it is rotating with a constant speed (in degrees):
sensor output = ([5, 5, 5, 5, 5, 10, 10, 10, 10 ,10, 15, 15, 20, 20, 20, 20, 25, 25, 30, 30, 30, 30, 30, 35, 35....])
As you can see, some of these measurements are repeating themselves.
From these measurements, I would like to interpolate in order to get the measurements in between the 1D data-points. For instance, if I at time k receive the angular measurement theta=5 and in the next instance at t=k+1 also receive a measurement of theta=5, I would like to compute an estimate that would be something like theta = 5+(1/5).
I have also been considering using some sort of predictive filtering, but I'm not sure which one to apply if that is even applicable in this case (e.g. Kalman filtering). The estimated output should be in a linear form since the motor is rotating with a constast angular velocity.
I have tried using numpy.linspace in order to acheive what I want, but cannot seem to get it to work the way I want:
# Interpolate for every 'theta_div' values in angle received through
# modbus
for k in range(np.size(rx)):
y = T.readSensorData() # take measurement (call read sensor function)
fp = np.linspace(y, y+1, num=theta_div)
for n in range(theta_div):
if k % 6 == 0:
if not y == fp[n]:
z = fp[n]
else:
z = y
print(z)
So for the sensor readings: ([5, 5, 5, 5, 5, 10, 10, 10, 10 ,10, 15, 15, 20, 20, 20, 20, 25, 25, 30, 30, 30, 30, 30, 35, 35....]) # each element at time=k0...kn
I would like the output to be something similar to:
theta = ([5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17.5, 20...])
So in short, I need some sort of prediction and then update the value with the actual reading from the sensor, similar to the procedure in a Kalman filter.
why dont just make a linear fit?
import numpy as np
import matplotlib.pyplot as plt
messurements = np.array([5, 5, 5, 5, 5, 10, 10, 10, 10 ,10, 15, 15, 20, 20, 20, 20, 25, 25, 30, 30, 30, 30, 30, 35, 35])
time_array = np.arange(messurements.shape[0])
fitparms = np.polyfit(time_array,messurements,1)
def line(x,a,b):
return a*x +b
better_time_array = np.linspace(0,np.max(time_array))
plt.plot(time_array,messurements)
plt.plot(better_time_array,line(better_time_array,fitparms[0],fitparms[1]))

Numpy array changes shape when accessing with indices

I have a small matrix A with dimensions MxNxO
I have a large matrix B with dimensions KxMxNxP, with P>O
I have a vector ind of indices of dimension Ox1
I want to do:
B[1,:,:,ind] = A
But, the lefthand of my equation
B[1,:,:,ind].shape
is of dimension Ox1xMxN and therefore I can not broadcast A (MxNxO) into it.
Why does accessing B in this way change the dimensions of the left side?
How can I easily achieve my goal?
Thanks
There's a feature, if not a bug, that when slices are mixed in the middle of advanced indexing, the sliced dimensions are put at the end.
Thus for example:
In [204]: B = np.zeros((2,3,4,5),int)
In [205]: ind=[0,1,2,3,4]
In [206]: B[1,:,:,ind].shape
Out[206]: (5, 3, 4)
The 3,4 dimensions have been placed after the ind, 5.
We can get around that by indexing first with 1, and then the rest:
In [207]: B[1][:,:,ind].shape
Out[207]: (3, 4, 5)
In [208]: B[1][:,:,ind] = np.arange(3*4*5).reshape(3,4,5)
In [209]: B[1]
Out[209]:
array([[[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14],
[15, 16, 17, 18, 19]],
[[20, 21, 22, 23, 24],
[25, 26, 27, 28, 29],
[30, 31, 32, 33, 34],
[35, 36, 37, 38, 39]],
[[40, 41, 42, 43, 44],
[45, 46, 47, 48, 49],
[50, 51, 52, 53, 54],
[55, 56, 57, 58, 59]]])
This only works when that first index is a scalar. If it too were a list (or array), we'd get an intermediate copy, and couldn't set the value like this.
https://docs.scipy.org/doc/numpy-1.15.0/reference/arrays.indexing.html#combining-advanced-and-basic-indexing
It's come up in other SO questions, though not recently.
weird result when using both slice indexing and boolean indexing on a 3d array

Pymc size / indexing issue

I am trying to model Kruschke's "filtration-condensation experiment" with pymc 2.3.5. (numpy 1.10.1)
Basicaly there are:
4 groups
each group has 40 individuals
each individual has 64 Bernoulli trials (correct/incorrect)
What I am modeling:
each individual's results are Binomial distribution (e.g. 45 correct out of 64).
my belief about each individual's performance is Beta distribution.
this Beta distribution is influenced by group to which individual belongs (through parameters A=mu*kappa and B=(1-mu)*kappa)
my belief about how strong each group's influence is Gamma distribution (kappa variable)
my belief about each group's average is Beta distribution (mu variable)
The problem:
when I do modeling with "size=" parameters, pymc get's lost
when I seperate each distribution manually (no size=) the pymc does good job
I include the code below:
Data
import numpy as np
import seaborn as sns
import pymc as pm
from pymc.Matplot import plot as mcplot
%matplotlib inline
# Data
ncond = 4
nSubj = 40
trials = 64
N = np.repeat([trials], (ncond * nSubj))
z = np.array([45, 63, 58, 64, 58, 63, 51, 60, 59, 47, 63, 61, 60, 51, 59, 45,
61, 59, 60, 58, 63, 56, 63, 64, 64, 60, 64, 62, 49, 64, 64, 58, 64, 52, 64, 64,
64, 62, 64, 61, 59, 59, 55, 62, 51, 58, 55, 54, 59, 57, 58, 60, 54, 42, 59, 57,
59, 53, 53, 42, 59, 57, 29, 36, 51, 64, 60, 54, 54, 38, 61, 60, 61, 60, 62, 55,
38, 43, 58, 60, 44, 44, 32, 56, 43, 36, 38, 48, 32, 40, 40, 34, 45, 42, 41, 32,
48, 36, 29, 37, 53, 55, 50, 47, 46, 44, 50, 56, 58, 42, 58, 54, 57, 54, 51, 49,
52, 51, 49, 51, 46, 46, 42, 49, 46, 56, 42, 53, 55, 51, 55, 49, 53, 55, 40, 46,
56, 47, 54, 54, 42, 34, 35, 41, 48, 46, 39, 55, 30, 49, 27, 51, 41, 36, 45, 41,
53, 32, 43, 33])
condition = np.repeat([0,1,2,3], nSubj)
Does not work
# modeling
mu = pm.Beta('mu', 1, 1, size=ncond)
kappa = pm.Gamma('gamma', 1, 0.1, size=ncond)
# Prior
theta = pm.Beta('theta', mu[condition] * kappa[condition], (1 - mu[condition]) * kappa[condition], size=len(z))
# likelihood
y = pm.Binomial('y', p=theta, n=N, value=z, observed=True)
# model
model = pm.Model([mu, kappa, theta, y])
mcmc = pm.MCMC(model)
#mcmc.use_step_method(pm.Metropolis, mu)
#mcmc.use_step_method(pm.Metropolis, theta)
#mcmc.assign_step_methods()
mcmc.sample(100000, burn=20000, thin=3)
# outputs never converge and does vary in new simulations
mcplot(mcmc.trace('mu'), common_scale=False)
Works
z1 = z[:40]
z2 = z[40:80]
z3 = z[80:120]
z4 = z[120:]
Nv = N[:40]
mu1 = pm.Beta('mu1', 1, 1)
mu2 = pm.Beta('mu2', 1, 1)
mu3 = pm.Beta('mu3', 1, 1)
mu4 = pm.Beta('mu4', 1, 1)
kappa1 = pm.Gamma('gamma1', 1, 0.1)
kappa2 = pm.Gamma('gamma2', 1, 0.1)
kappa3 = pm.Gamma('gamma3', 1, 0.1)
kappa4 = pm.Gamma('gamma4', 1, 0.1)
# Prior
theta1 = pm.Beta('theta1', mu1 * kappa1, (1 - mu1) * kappa1, size=len(Nv))
theta2 = pm.Beta('theta2', mu2 * kappa2, (1 - mu2) * kappa2, size=len(Nv))
theta3 = pm.Beta('theta3', mu3 * kappa3, (1 - mu3) * kappa3, size=len(Nv))
theta4 = pm.Beta('theta4', mu4 * kappa4, (1 - mu4) * kappa4, size=len(Nv))
# likelihood
y1 = pm.Binomial('y1', p=theta1, n=Nv, value=z1, observed=True)
y2 = pm.Binomial('y2', p=theta2, n=Nv, value=z2, observed=True)
y3 = pm.Binomial('y3', p=theta3, n=Nv, value=z3, observed=True)
y4 = pm.Binomial('y4', p=theta4, n=Nv, value=z4, observed=True)
# model
model = pm.Model([mu1, kappa1, theta1, y1, mu2, kappa2, theta2, y2,
mu3, kappa3, theta3, y3, mu4, kappa4, theta4, y4])
mcmc = pm.MCMC(model)
#mcmc.use_step_method(pm.Metropolis, mu)
#mcmc.use_step_method(pm.Metropolis, theta)
#mcmc.assign_step_methods()
mcmc.sample(100000, burn=20000, thin=3)
# outputs converge and are not too much different in every simulation
mcplot(mcmc.trace('mu1'), common_scale=False)
mcplot(mcmc.trace('mu2'), common_scale=False)
mcplot(mcmc.trace('mu3'), common_scale=False)
mcplot(mcmc.trace('mu4'), common_scale=False)
mcmc.summary()
Can someone please explain it to me why mu[condition] and gamma[condition] does not work? :)
I guess that not splitting thetas into different variables is the problem but cannot understand why and maybe there is a way to pass a shape parameter to size= on theta?
First of all, I can confirm that the first version doesn't lead to stable results. What I can't confirm is that the second one is much better; I have seen very different results also with the second code, with values for the first mu parameter varying between 0.17 and 0.9 for different runs.
The convergence problems can be cured by using good starting values for the Markov chain. This can be done by first doing a maximum a posteriori (MAP) estimate, and then starting the Markov chain from there. The MAP step is computationally inexpensive and leads to a converging Markov chain with reproducible results for both variants of your code. For reference and comparison: The values I see for the four mu parameters are around 0.94 / 0.86 / 0.72 and 0.71.
You can do the MAP estimation by inserting the following two lines of code right after the line in which you define your model with "model=pm.Model(...":
map_ = pm.MAP(model)
map_.fit()
This technique is covered in more detail in Cameron Davidson-Pilon's Bayesian Methods for Hackers, together with other helpful topics around PyMC.