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Exponential fit in pandas

I have this data:
puf = pd.DataFrame({'id':[1,2,3,4,5,6,7,8],
'val':[850,1889,3289,6083,10349,17860,28180,41236]})
The data seems to follow an exponential curve. Let's see the plot:
puf.plot('id','val')
I want to fit an exponential curve ($$ y = Ae^{Bx} $$, A times e to the B*X)and add it as a column in Pandas. Firstly I tried to log the values:
puf['log_val'] = np.log(puf['val'])
And then to use Numpy to fit the equation:
puf['fit'] = np.polyfit(puf['id'],puf['log_val'],1)
But I get an error:
ValueError: Length of values (2) does not match length of index (8)
My expected result is the fitted values as a new column in Pandas. I attach an image with the column fitted values I want (in orange):
I'm stuck in this code. I'm not sure what I am doing wrong. How can I create a new column with my fitted values?

Note that you asked for an exponential model yet you have the results for log-linear model.
Check out the work below:
For log-linear, we are fitting E(log(Y))ie log(y) - (log(b[0]) +b[1]*x):
from scipy.optimize import least_squares
least_squares(lambda b: np.log(puf['val']) -(np.log(b[0]) + b[1] * puf['id']),
[1,1])['x']
array([5.99531305e+02, 5.51106793e-01])
These are the values that excel gives.
On the other hand to fit an exponential curve, the randomness is on Y and not on its logarithm, E(Y)=b[0]*exp(b[1] *x) Hence we have:
least_squares(lambda b: puf['val'] - b[0]*exp(b[1] * puf['id']), [0,1])['x']
array([1.08047304e+03, 4.58116127e-01]) # correct results for exponential fit
Depending on your model choice, the values are alittle different.
Better Model? Since you have same number of parameters, consider the one that gives you lower deviance or better out of sample prediction
Note that the ideal exponential model is E(Y) = A'B'^X which for comparison can be written as log(E(Y)) = A + XB while log-linear model will be E(log(Y) = A + XB. Note the difference in Expectation.
From the two models we have:
Notice how when we go to higher numbers the log-linear overestimates. While in the lower numbers the exponential overestimates.
Code for image:
from scipy.optimize import least_squares
log_lin = least_squares(lambda b: np.log(puf['val']) -(np.log(b[0]) + b[1] * puf['id']),
[1,1])['x']
expo = least_squares(lambda b: puf['val'] - b[0]*exp(b[1] * puf['id']), [0,1])['x']
exp_fun = lambda x: expo[0] * exp(expo[1]*x)
log_lin_fun = lambda x:log_lin[0] * exp(log_lin[1]*x)
plt.plot(puf.id, puf.val, label = 'original')
plt.plot(puf.id, exp_fun(puf.id), label='exponential')
plt.plot(puf.id, log_lin_fun(puf.id), label='log-linear')
plt.legend()

Your getting that error because np.polyfit(puf['id'],puf['log_val'],1) returns two values array([0.55110679, 6.39614819]) which isn't the shape of your dataframe.
This is what you want
y = a* exp (b*x) -> ln(y)=ln(a)+bx
f = np.polyfit(df['id'], np.log(df['val']), 1)
where
a = np.exp(f[1]) -> 599.5313046712091
b = f[0] -> 0.5511067934637022
Giving
puf['fit'] = a * np.exp(b * puf['id'])
id val fit
0 1 850 1040.290193
1 2 1889 1805.082864
2 3 3289 3132.130026
3 4 6083 5434.785677
4 5 10349 9430.290286
5 6 17860 16363.179739
6 7 28180 28392.938399
7 8 41236 49266.644002

Calculating Batch normalization

Part 1
Im going through this article and wanted to try and calculate a forward and backward pass with batch normalization.
When doing the steps after the first layer I get a batch norm output that are equal for all features.
Here is the code (I have on purpose done it in very small steps):
w = np.array([[0.3, 0.4],[0.5,0.1],[0.2,0.3]])
X = np.array([[0.7,0.1],[0.3,0.8],[0.4,0.6]])
def mu(x,axis=0):
return np.mean(x,axis=axis)
def sigma(z, mu):
Ai = np.sum(z,axis=0)
return np.sqrt((1/len(Ai)) * (Ai-mu)**2)
def Ai(z):
return np.sum(z,axis=0)
def norm(Ai,mu,sigma):
return (Ai-mu)/sigma
z1 = np.dot(w1,X.T)
mu1 = mu(z1)
A1 = Ai(z1)
sigma1 = sigma(z1,mu1)
gamma1 = np.ones(len(A1))
beta1 = np.zeros(len(A1))
Ahat = norm(A1,mu1,sigma1) #since gamma is just ones it does change anything here
The output I get from this is:
[1.73205081 1.73205081 1.73205081]
Part 2
In this image:
Should the sigma_mov and mu_mov be set to zero for the first layer?
EDIT: I think I found what I did wrong. In the normalization step I used A1 and not z1. Also I think I found that its normal to use initlize moving average with zeros for mean and ones for variance. Nice if anyone can confirm this.

How to vectorize indexing and computation when indexed tensors are different dimensions?

I'm trying to vectorize the following for-loop in Pytorch. I'd be happy with just vectorizing the inner for-loop, but doing the whole batch would also be awesome.
# B: the batch size
# N: the number of training examples
# dim: the dimension of each feature vector
# K: the number of discrete labels. each vector has a single label
# delta: margin for hinge loss
batch_data = torch.tensor(...) # Tensor of shape [B x N x d]
batch_labels = torch.tensor(...) # Tensor of shape [B x N x 1], each element is one of K labels (ints)
batch_losses = [] # Ultimately should be [B x 1]
batch_centroids = [] # Ultimately should be [B x K_i x dim]
for i in range(B):
centroids = [] # Keep track of the means for each class.
classes = torch.unique(labels) # Get the unique labels for the classes.
# NOTE: The number of classes K for each item in the batch might actually
# be different. This may complicate batch-level operations.
total_loss = 0
# For each class independently. This is the part I want to vectorize.
for cl in classes:
# Take the subset of training examples with that label.
subset = data[torch.where(labels == cl)]
# Find the centroid of that subset.
centroid = subset.mean(dim=0)
centroids.append(centroid)
# Get the distance between each point in the subset and the centroid.
dists = subset - centroid
norm = torch.linalg.norm(dists, dim=1)
# The loss is the mean of the hinge loss across the subset.
margin = norm - delta
hinge = torch.clamp(margin, min=0.0) ** 2
total_loss += hinge.mean()
# Keep track of everything. If it's too hard to keep track of centroids, that's also OK.
loss = total_loss.mean()
batch_losses.append(loss)
batch_centroids.append(centroids)
I've been scratching my head on how to deal with the irregularly sized tensors. The number of classes in each batch K_i is different, and the size of each subset is different.

It turns out it actually is possible to vectorize across ragged arrays. I'll use numpy, but code should be directly translatable to torch. The key technique is to:
Sort by ragged array membership
Perform an accumulation
Find boundary indices, compute adjacent differences
For a single (non-batch) input of an n x d matrix X and an n-length array label, the following returns the k x d centroids and n-length distances to respective centroids:
def vcentroids(X, label):
"""
Vectorized version of centroids.
"""
# order points by cluster label
ix = np.argsort(label)
label = label[ix]
Xz = X[ix]
# compute pos where pos[i]:pos[i+1] is span of cluster i
d = np.diff(label, prepend=0) # binary mask where labels change
pos = np.flatnonzero(d) # indices where labels change
pos = np.repeat(pos, d[pos]) # repeat for 0-length clusters
pos = np.append(np.insert(pos, 0, 0), len(X))
Xz = np.concatenate((np.zeros_like(Xz[0:1]), Xz), axis=0)
Xsums = np.cumsum(Xz, axis=0)
Xsums = np.diff(Xsums[pos], axis=0)
counts = np.diff(pos)
c = Xsums / np.maximum(counts, 1)[:, np.newaxis]
repeated_centroids = np.repeat(c, counts, axis=0)
aligned_centroids = repeated_centroids[inverse_permutation(ix)]
dist = np.sum((X - aligned_centroids) ** 2, axis=1)
return c, dist
Batching requires little special handling. For an input B x n x d array batch_X, with B x n batch labels batch_labels, create unique labels for each batch:
batch_k = batch_labels.max(axis=1) + 1
batch_k[1:] = batch_k[:-1]
batch_k[0] = 0
base = np.cumsum(batch_k)
batch_labels += base.expand_dims(1)
So now each batch element has a unique contiguous range of labels. I.e., the first batch element will have n labels in some range [0, k0) where k0 = batch_k[0], the second will have range [k0, k0 + k1) where k1 = batch_k[1], etc.
Then just flatten the n x B x d input to n*B x d and call the same vectorized method. Your loss function is derivable using the final distances and same position-array based reduction technique.
For a detailed explanation of how the vectorization works, see my blog post.

You can vectorize the whole thing if you use a one-hot encoding for your classes and a pairwise distance trick for your norms:
import torch
B = 32
N = 1000
dim = 50
K = 25
batch_data = torch.randn((B, N, dim))
batch_labels = torch.randint(0, K, size=(B, N))
batch_one_hot = torch.nn.functional.one_hot(batch_labels)
centroids = torch.matmul(
batch_one_hot.transpose(-1, 1).type(batch_data.dtype),
batch_data
) / batch_one_hot.sum(1)[..., None]
norms = torch.linalg.norm(batch_data[:, :, None] - centroids[:, None], axis=-1)
# Compute the rest of your loss
# ...
A couple things to watch out for:
You'll get a divide by zero for any batches that have a missing class. You can handle this by first computing the class sums (with matmul) and counts (summing the one-hot tensor along axis 1) separately. Then, mask the sums with count == 0 and divide the rest of them by their class counts.
If you have a large number of classes, this will cause memory problems because the one-hot tensor will be too big. In that case, the answer from #VF1 probably makes more sense.

Why does my sigmoid function return values not in the interval ]0,1[?

I am implementing logistic regression in Python with numpy. I have generated the following data set:
# class 0:
# covariance matrix and mean
cov0 = np.array([[5,-4],[-4,4]])
mean0 = np.array([2.,3])
# number of data points
m0 = 1000
# class 1
# covariance matrix
cov1 = np.array([[5,-3],[-3,3]])
mean1 = np.array([1.,1])
# number of data points
m1 = 1000
# generate m gaussian distributed data points with
# mean and cov.
r0 = np.random.multivariate_normal(mean0, cov0, m0)
r1 = np.random.multivariate_normal(mean1, cov1, m1)
X = np.concatenate((r0,r1))
Now I have implemented the sigmoid function with the aid of the following methods:
def logistic_function(x):
""" Applies the logistic function to x, element-wise. """
return 1.0 / (1 + np.exp(-x))
def logistic_hypothesis(theta):
return lambda x : logistic_function(np.dot(generateNewX(x), theta.T))
def generateNewX(x):
x = np.insert(x, 0, 1, axis=1)
return x
After applying logistic regression, I found out that the best thetas are:
best_thetas = [-0.9673200946417307, -1.955812236119612, -5.060885703369424]
However, when I apply the logistic function with these thetas, then the output is numbers that are not inside the interval [0,1]
Example:
data = logistic_hypothesis(np.asarray(best_thetas))(X)
print(data
This gives the following result:
[2.67871968e-11 3.19858822e-09 3.77845881e-09 ... 5.61325410e-03
2.19767618e-01 6.23288747e-01]
Can someone help me understand what has gone wrong with my implementation? I cannot understand why I am getting such big values. Isnt the sigmoid function supposed to only give results in the [0,1] interval?

It does, it's just in scientific notation.
'e' Exponent notation. Prints the number in scientific notation using
the letter ‘e’ to indicate the exponent.
>>> a = [2.67871968e-11, 3.19858822e-09, 3.77845881e-09, 5.61325410e-03]
>>> [0 <= i <= 1 for i in a]
[True, True, True, True]

How to make scipy.optimize.curve_fit result in a better sine regression fit?

I have a problem where I am using scipy.optimize.curve_fit to do a regression fit to a sine/cosine function but the fit does not seem as optimized as I want it to be. How can I change my code to make the fitting better?
I have already tried changing how parameters are tried for the dataset and there is always seemingly a difference in phase-offset of my generated fit or the fitting function is not fitting to the proper minima/maxima.
Here is the code I am using to generate the regression fit. The output (fitfunc) can be plotted to show the result.
def sin_regress(data_x, data_y):
"""Function regression fits data to SIN function; does not need guess of freq.
Parameters
----------
data_x :
Data for X values, most likely a set of voltages.
data_y :
Data for Y values, most likely the resulting powers from voltages.
Returns
-------
__ :
Dictionary containing values for amplitude, angular frequency, phase, offset, frequency, period, fit function, max covariance, initial guess.
"""
data_x = np.array(data_x)
data_y = np.array(data_y)
freqz = np.fft.rfftfreq(len(data_x), (data_x[1] - data_x[0])) # uniform spacing
freq_y = abs(np.fft.rfft(data_y))
guess_freq = abs(freqz[np.argmax(freq_y[1:])+1]) # exclude offset peak
guess_amp = np.std(data_y) * 2.**0.5
guess_offset = np.mean(data_y)
guess = np.array([guess_amp, 2.*np.pi*guess_freq, 0., guess_offset])
def sinfunc(t, A, w, p, c):
"""Raw function to be used to fit data.
Parameters
----------
t :
Voltage array
A :
Amplitude
w :
Angular frequency
p :
Phase
c :
Constant value
Returns
-------
__ :
Formed fit function with provided values.
"""
return A * np.sin(w*t + p) + c
popt, pcov = scipy.optimize.curve_fit(sinfunc, data_x, data_y, p0=guess)
A, w, p, c = popt
f = w/(2.*np.pi)
fitfunc = lambda t: A * np.sin(w*t + p) + c
return {"amp": A, "omega": w, "phase": p, "offset": c, "freq": f, "period": 1./f, "fitfunc": fitfunc, "maxcov": np.max(pcov), "rawres": (guess,popt,pcov)}
With my trial dataset being:
x = np.linspace(3.5,9.5,(9.5-3.5)/0.00625 + 1)
pow1 = [1.8262110863, 1.80944546009, 1.7970185646900003, 1.77120336754, 1.7458101235699999, 1.73597098224, 1.7122529922799998, 1.70015674142, 1.68968617429, 1.6989396515, 1.69760676076, 1.6946375613599998, 1.6895321899, 1.68145658386, 1.68581793183, 1.6920468775900002, 1.6865452951599997, 1.68570953338, 1.6922784791700003, 1.70958957412, 1.71683408637, 1.70360183933, 1.6919669752199997, 1.6669487117300001, 1.6351298032300001, 1.6061729066600001, 1.57344333403, 1.54723708217, 1.5277773737599998, 1.5122628414300001, 1.4962354965200002, 1.4873367459, 1.47567715522, 1.4696584634, 1.46159565032, 1.45320592315, 1.4487225244200002, 1.44572887186, 1.44089260198, 1.4367157657399998, 1.4349226211, 1.43614316806, 1.4381950627400002, 1.43947658627, 1.4483572314200002, 1.4504305909200002, 1.44436990692, 1.43367609757, 1.42637295252, 1.41197427963, 1.4067529511399999, 1.39714414185, 1.38309980493, 1.3730701362500004, 1.3693239836499997, 1.3729558979599998, 1.38291189477, 1.3988274622900003, 1.42112832324, 1.44217266068, 1.4578792438300001, 1.46478639274, 1.46676801398, 1.4646383458800003, 1.45918801344, 1.44561402809, 1.4212145146499997, 1.4012453921299999, 1.38070199226, 1.36215759642, 1.3540496661500003, 1.35470913884, 1.3481165993199997, 1.34059081754, 1.332964567, 1.33426054366, 1.34052562222, 1.3343255632100002, 1.3310385903, 1.33044179339, 1.32827462527, 1.3356201140500001, 1.3400144893900001, 1.3157198001600001, 1.27716313727, 1.2517667292400003, 1.2406836620500001, 1.2354036030700002, 1.23110776291, 1.22492582889, 1.22074838719, 1.21816502762, 1.21015135518, 1.20038737012, 1.1920263929700001, 1.18723010357, 1.19656731125, 1.2237068834899998, 1.2373841696199999, 1.2251076648299999, 1.1963014909299998, 1.16152861736, 1.13940556893, 1.12839812676, 1.12368066547, 1.1190219542100002, 1.11384679759, 1.10555781262, 1.0977575386300003, 1.0901734365399998, 1.0824275375699999, 1.07552931443, 1.0696565210100002, 1.06481394254, 1.0578173014299999, 1.05204230102, 1.0482530038799998, 1.04237087457, 1.0361766944300002, 1.0297906393, 1.0240842912299999, 1.01250548183, 0.9964340353700001, 0.9859450307400002, 0.98614987451, 0.9826424718800002, 0.9739505767299999, 0.9578738177999998, 0.9416973908799999, 0.92975112051, 0.9204409049900001, 0.91821299468, 0.9100360995600001, 0.89589154778, 0.8799530701000002, 0.8640439088, 0.8500274234399999, 0.8428500205999999, 0.8358678326, 0.8333072464999999, 0.83420148485, 0.8362578717, 0.83608947323, 0.83035464861, 0.82315039029, 0.81220152235, 0.80169300598, 0.7918658959, 0.7808782388700001, 0.77684747687, 0.7743299962, 0.76797978094, 0.7591097217, 0.7520710688500001, 0.7452609707, 0.73562753255, 0.7256206568399999, 0.71663518742, 0.70951165178, 0.7035884873, 0.6973768853, 0.6900439160299999, 0.68062538021, 0.67096725454, 0.66585371901, 0.6663177033900001, 0.67214877804, 0.6787934074299999, 0.68365489213, 0.68581510712, 0.6820892084400001, 0.67805153237, 0.67540688376, 0.6724865515, 0.6674502035, 0.6593852224500001, 0.6524835227400001, 0.64758563177, 0.6424489126599999, 0.63385426361, 0.6242639699699999, 0.6143974848999999, 0.60705328516, 0.60087306988, 0.5928024247700001, 0.5864009594799999, 0.5786877362899999, 0.57457744302, 0.57012636848, 0.56554310644, 0.5618750202299999, 0.55731189492, 0.55057384756, 0.5419996086800001, 0.52987726408, 0.51025575876, 0.48599474143000004, 0.46231124366000004, 0.44151899608999995, 0.42632008877, 0.42655368254, 0.42784393651999997, 0.42863940533999995, 0.42506971759, 0.41952014686999994, 0.41337420894, 0.40570705996, 0.39706149294, 0.38721395321, 0.3806321949, 0.37313342483999995, 0.36982676447, 0.36704194004, 0.36189430296, 0.3560628963, 0.34954350131, 0.34540695806, 0.34178605934, 0.33629549256, 0.3293877577, 0.32357672213, 0.31864117490000005, 0.31165906503, 0.30439039263000006, 0.29875160317, 0.29294459105000004, 0.28847285244, 0.28509162173, 0.28265949265, 0.28003828154, 0.27814630873999996, 0.27599048828, 0.27524025386, 0.27406833971, 0.27281988259, 0.27155314420999993, 0.26840999947000005, 0.2634181241, 0.25883622926000005, 0.25503165868, 0.25056988104, 0.24466620872, 0.23932761459000002, 0.23422685251999997, 0.22880456697, 0.22310130485000004, 0.21785542557999998, 0.21366651902000006, 0.20966530780999998, 0.20521315906, 0.20012157666000002, 0.19469597081, 0.18957032591999995, 0.18423432945, 0.17946309866000001, 0.17845044232, 0.17746098912000002, 0.17475331315, 0.17039776599, 0.16363173032999997, 0.15716942518, 0.15214176858, 0.14870803788, 0.14515563527000003, 0.14218680693, 0.13893215828, 0.13546723615, 0.13178983356, 0.12747471604, 0.12350983297, 0.12011202021999998, 0.11627787931000003, 0.11218377746, 0.10821276155, 0.10384311280999999, 0.09960625706000001, 0.09615194041000003, 0.09216061199, 0.08847719376999999, 0.08481545522999999, 0.08163922452000001, 0.07851820869000001, 0.07535195845, 0.07259346216999998, 0.06996658694999999, 0.06748611806, 0.06513859836, 0.06343437948, 0.06174502390000001, 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Here are some additional values for the pow dataset:
(Link to pastebin to not exceed post length limit)
https://pastebin.com/5GP8sj4N
The resulting fit that from the trial dataset (x, pow1) I get is shown here (orange) with the original (pow1) data (blue)
As mentioned, there is an issue with how the phase fits the minima and maxima. Unfortunately the application of getting this fit function correct has very little room for error.
Please help out if you have an idea of how to make this fit the data better!
Edit:
I tried what #Joe mentioned in the comments, with first filtering the data. I utilized a Savitzky-Golay filter and recieved the following result, Original data (blue), the filtered data (green), and the fit to the filtered data (orange). Again the same shift in minima and maxima is still present in the fit function to the filtered data.

Here are my results with more aggressive clipping bounds of 0.5 to 1.75 for each data set.
for pow1:
A = 9.6711505138648990E-01
c = 9.7613787086912507E-01
p = 4.0262076448344617E+00
w = 1.2654001570670070E+00
for pow2:
A = 9.4894637490866129E-01
c = 9.6733405789489280E-01
p = 4.0892433833755097E+00
w = 1.2578627414445132E+00
for pow3:
A = 9.8595630272060597E-01
c = 9.6749868212694512E-01
p = 4.0859456191316230E+00
w = 1.2598547148182329E+00
for pow4:
A = -9.4636707498392481E-01
c = 9.5047597808408602E-01
p = -4.2643913461857056E+02
w = 1.2761107231684055E+00

I think I have this figured out - your data is not a mathematically perfect sine wave + noise, so the fitting software can only come close to modeling a sine function to this data. If you must have more accuracy, try splitting the model into different segments and use a piecewise fit. Here is a close-up of the problem area: