Problem description
I am new to probabilistic programming and working through PyMC3's Gaussian Mixture Model sample notebook:
import arviz as az
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import pymc3 as pm
import theano.tensor as tt
# simulate data from a known mixture distribution
np.random.seed(12345) # set random seed for reproducibility
k = 3
ndata = 500
spread = 5
centers = np.array([-spread, 0, spread])
# simulate data from mixture distribution
v = np.random.randint(0, k, ndata)
data = centers[v] + np.random.randn(ndata)
plt.hist(data);
# setup model
model = pm.Model()
with model:
# cluster sizes
p = pm.Dirichlet("p", a=np.array([1.0, 1.0, 1.0]), shape=k)
# ensure all clusters have some points
p_min_potential = pm.Potential("p_min_potential", tt.switch(tt.min(p) < 0.1, -np.inf, 0))
# cluster centers
means = pm.Normal("means", mu=[0, 0, 0], sigma=15, shape=k)
# break symmetry
order_means_potential = pm.Potential(
"order_means_potential",
tt.switch(means[1] - means[0] < 0, -np.inf, 0)
+ tt.switch(means[2] - means[1] < 0, -np.inf, 0),
)
# measurement error
sd = pm.Uniform("sd", lower=0, upper=20)
# latent cluster of each observation
category = pm.Categorical("category", p=p, shape=ndata)
# likelihood for each observed value
points = pm.Normal("obs", mu=means[category], sigma=sd, observed=data)
# fit model
with model:
step1 = pm.Metropolis(vars=[p, sd, means])
step2 = pm.ElemwiseCategorical(vars=[category], values=[0, 1, 2])
tr = pm.sample(10000, step=[step1, step2], tune=5000)
I am struggling with the following expressions:
# ensure all clusters have some points
p_min_potential = pm.Potential("p_min_potential", tt.switch(tt.min(p) < 0.1, -np.inf, 0))
and
# break symmetry
order_means_potential = pm.Potential(
"order_means_potential",
tt.switch(means[1] - means[0] < 0, -np.inf, 0)
+ tt.switch(means[2] - means[1] < 0, -np.inf, 0),
)
What I researched
From looking at related questions and PyMC3's and Theano's documentation I think I understand that pm.Potential() is a way of setting the log-likelihood of an event in your model during sampling without providing observations to it, and that tt.switch() checks whether a certain condition is met and returns one of two values accordingly.
Thus p_min_potential ensures that all values in p are greater than 0.1 by setting the log-likelihood for an event where one value in p to negative infinite, similarly order_means_potential ensures the values in means differ from on another and their ordering stays the same during sampling.
Questions
Unfortunately neither related questions nor the documentations could answer the following questions:
How are the results of these expressions fed back into the model, as neither p_min_potential nor order_means_potential occur as input to any other expression?
Am I right in how tt.switch() works, thus if the condition tt.min(p) < 0.1 is met -np.inf is returned as that events log-likelihood, and 0 in any other case?
Any help would be greatly appreciated, I would like to understand how this example works to the degree where I'll be able to alter and expand it. Specifically I want to implement a mixture model for two or more beta distributions.
I'm trying to code a demonstration of compressed sensing for my final year project but am getting poor image reconstruction when using the Lasso algorithm. I've relied on the following as a reference: http://www.pyrunner.com/weblog/2016/05/26/compressed-sensing-python/
However my code has some differences:
I use scikit-learn to perform a lasso optimisation (basis pursuit) as opposed to using cvxpy to perform an l_1 minimisation with an equality constraint as in the article.
I construct psi differently/more simply, testing seems to show that it's correct.
I use a different package to read and write the image.
import numpy as np
import scipy.fftpack as spfft
import scipy.ndimage as spimg
import imageio
from sklearn.linear_model import Lasso
x_orig = imageio.imread('gt40.jpg', pilmode='L') # read in grayscale
x = spimg.zoom(x_orig, 0.2) #zoom for speed
ny,nx = x.shape
k = round(nx * ny * 0.5) #50% sample
ri = np.random.choice(nx * ny, k, replace=False)
y = x.T.flat[ri] #y is the measured sample
# y = np.expand_dims(y, axis=1) ---- this doesn't seem to make a difference, was presumably required with cvxpy
psi = spfft.idct(np.identity(nx*ny), norm='ortho', axis=0) #my construction of psi
# psi = np.kron(
# spfft.idct(np.identity(nx), norm='ortho', axis=0),
# spfft.idct(np.identity(ny), norm='ortho', axis=0)
# )
# psi = 2*np.random.random_sample((nx*ny,nx*ny)) - 1
theta = psi[ri,:] #equivalent to phi*psi
lasso = Lasso(alpha=0.001, max_iter=10000)
lasso.fit(theta, y)
s = np.array(lasso.coef_)
x_recovered = psi#s
x_recovered = x_recovered.reshape(nx, ny).T
x_recovered_final = x_recovered.astype('uint8') #recovered image is float64 and has negative values..
imageio.imwrite('gt40_recovered.jpg', x_recovered_final)
Unfortunately I'm not allowed to post images yet so here is a link to the original zoomed image, the image recovered with lasso and the image recovered with cvxpy (described later):
https://imgur.com/a/LROSug6
As you can see not only is the recovery poor but the image completely corrupted - the colours seem to be negative and the detail from the 50% sample lost. I think I've managed to track down the problem to the Lasso regression - it returns a vector that, when inverse transformed, has values that are not necessarily in the 0-255 range as expected for the image. So the conversion to from dtype float64 to uint8 is rather random (e.g. -55 becomes 255-55=200).
Following this I tried swapping out lasso for the same optimisation as in the article (minimising the l_1 norm subject to theta*s=y using cvxpy):
import cvxpy as cvx
x_orig = imageio.imread('gt40.jpg', pilmode='L') # read in grayscale
x = spimg.zoom(x_orig, 0.2)
ny,nx = x.shape
k = round(nx * ny * 0.5)
ri = np.random.choice(nx * ny, k, replace=False)
y = x.T.flat[ri]
psi = spfft.idct(np.identity(nx*ny), norm='ortho', axis=0)
theta = psi[ri,:] #equivalent to phi*psi
#NEW CODE STARTS:
vx = cvx.Variable(nx * ny)
objective = cvx.Minimize(cvx.norm(vx, 1))
constraints = [theta#vx == y]
prob = cvx.Problem(objective, constraints)
result = prob.solve(verbose=True)
s = np.array(vx.value).squeeze()
x_recovered = psi#s
x_recovered = x_recovered.reshape(nx, ny).T
x_recovered_final = x_recovered.astype('uint8')
imageio.imwrite('gt40_recovered_altopt.jpg', x_recovered_final)
This took nearly 6 hours but finally I got a somewhat satisfactory result. However I would like to perform a demonstration of lasso if possible. Any help in getting the lasso to return appropriate values or somehow converting its result appropriately would be very much appreciated.
I am trying to carry out linear regression subject using some constraints to get a certain prediction.
I want to make the model predicting half of the linear prediction, and the last half linear prediction near the last value in the first half using a very narrow range (using constraints) similar to a green line in figure.
The full code:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import train_test_split
pd.options.mode.chained_assignment = None # default='warn'
data = [5.269, 5.346, 5.375, 5.482, 5.519, 5.57, 5.593999999999999, 5.627000000000001, 5.724, 5.818, 5.792999999999999, 5.817, 5.8389999999999995, 5.882000000000001, 5.92, 6.025, 6.064, 6.111000000000001, 6.1160000000000005, 6.138, 6.247000000000001, 6.279, 6.332000000000001, 6.3389999999999995, 6.3420000000000005, 6.412999999999999, 6.442, 6.519, 6.596, 6.603, 6.627999999999999, 6.76, 6.837000000000001, 6.781000000000001, 6.8260000000000005, 6.849, 6.875, 6.982, 7.018, 7.042000000000001, 7.068, 7.091, 7.204, 7.228, 7.261, 7.3420000000000005, 7.414, 7.44, 7.516, 7.542000000000001, 7.627000000000001, 7.667000000000001, 7.821000000000001, 7.792999999999999, 7.756, 7.871, 8.006, 8.078, 7.916, 7.974, 8.074, 8.119, 8.228, 7.976, 8.045, 8.312999999999999, 8.335, 8.388, 8.437999999999999, 8.456, 8.227, 8.266, 8.277999999999999, 8.289, 8.299, 8.318, 8.332, 8.34, 8.349, 8.36, 8.363999999999999, 8.368, 8.282, 8.283999999999999]
time = range(1,85,1)
x=int(0.7*len(data))
df = pd.DataFrame(list(zip(*[time, data])))
df.columns = ['time', 'data']
# print df
x=int(0.7*len(df))
train = df[:x]
valid = df[x:]
models = []
names = []
tr_x_ax = []
va_x_ax = []
pr_x_ax = []
tr_y_ax = []
va_y_ax = []
pr_y_ax = []
time_model = []
models.append(('LR', LinearRegression()))
for name, model in models:
x_train=df.iloc[:, 0][:x].values
y_train=df.iloc[:, 1][:x].values
x_valid=df.iloc[:, 0][x:].values
y_valid=df.iloc[:, 1][x:].values
model = LinearRegression()
# poly = PolynomialFeatures(5)
x_train= x_train.reshape(-1, 1)
y_train= y_train.reshape(-1, 1)
x_valid = x_valid.reshape(-1, 1)
y_valid = y_valid.reshape(-1, 1)
# model.fit(x_train,y_train)
model.fit(x_train,y_train.ravel())
# score = model.score(x_train,y_train.ravel())
# print 'score', score
preds = model.predict(x_valid)
tr_x_ax.extend(train['data'])
va_x_ax.extend(valid['data'])
pr_x_ax.extend(preds)
valid['Predictions'] = preds
valid.index = df[x:].index
train.index = df[:x].index
plt.figure(figsize=(5,5))
# plt.plot(train['data'],label='data')
# plt.plot(valid[['Close', 'Predictions']])
x = valid['data']
# print x
# plt.plot(valid['data'],label='validation')
plt.plot(valid['Predictions'],label='Predictions before',color='orange')
y =range(0,58)
y1 =range(58,84)
for index, item in enumerate(pr_x_ax):
if index >13:
pr_x_ax[index] = pr_x_ax[13]
pr_x_ax = list([float(i) for i in pr_x_ax])
va_x_ax = list([float(i) for i in va_x_ax])
tr_x_ax = list([float(i) for i in tr_x_ax])
plt.plot(y,tr_x_ax, label='train' , color='red', linewidth=2)
plt.plot(y1,va_x_ax, label='validation1' , color='blue', linewidth=2)
plt.plot(y1,pr_x_ax, label='Predictions after' , color='green', linewidth=2)
plt.xlabel("time")
plt.ylabel("data")
plt.xticks(rotation=45)
plt.legend()
plt.show()
If you see this figure:
label: Predictions before, the model predicted it without any constraints (I don't need this result).
label: Predictions after, the model predicted it within a constraint but this is after the model predicted AND the all values are equal to last value at index = 71 , item 8.56.
I used for loop for index, item in enumerate(pr_x_ax): in line:64, and the curve is line straight from time 71 to 85 sec as you see in order to show you how I need the model work.
Could I build the model give the same result instead of for loop???
Please your suggestions
I expect that in your question by drawing green line you really expect trained model to predict linear horizontal turn to the right. But current trained model draws just straight orange line.
It is true for any trained model of any algorithm and type that in order to learn some unordinary change in behavior model needs to have at least some samples of that unordinary change. Or at least some hidden meaning in observed data should point to having such unordinary change.
In other words for your model to learn that right turn on green line a model should have points with that right turn in the training data set. But you take for training data just first (leftmost) 70% of data by train = df[:int(0.7 * len(df))] and that training data has no such right turns and this training data just looks close to one straight line.
So you need to re-sample your data into training and validation in a different way - take randomly 70% of samples from whole range of X and the rest goes to validation. So that in your training data samples that do right turn also included.
Second thing is that LinearRegression model always models predictions just with one single straight line, and this line can't have right turns. In order to have right turns you need some more complex model.
One way for a model to have a right turn is to be piece-wise-linear, i.e. having several joined straight lines. I didn't find ready-made piecewise linear models inside sklearn, only using other pip models. So I decided to implement my own simple class PieceWiseLinearRegression that uses np.piecewise() and scipy.optimize.curve_fit() in order to model piecewise linear function.
Next picture shows results of applying two mentioned things above, code goes afterwards, re-sampling dataset in a different way and modeling piece-wise-linear function. Your current linear model LR still makes a prediction using just one straight blue line, while my piecewise linear PWLR2, orange line, consists of two segments and correctly predicts right turn:
To see clearly just one PWLR2 graph I did next picture too:
My class PieceWiseLinearRegression on creation of object accepts just one argument n - number of linear segments to be used for prediction. For picture above n = 2 was used.
import sys, numpy as np, pandas as pd
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
np.random.seed(0)
class PieceWiseLinearRegression:
#classmethod
def nargs_func(cls, f, n):
return eval('lambda ' + ', '.join([f'a{i}'for i in range(n)]) + ': f(' + ', '.join([f'a{i}'for i in range(n)]) + ')', locals())
#classmethod
def piecewise_linear(cls, n):
condlist = lambda xs, xa: [(lambda x: (
(xs[i] <= x if i > 0 else np.full_like(x, True, dtype = np.bool_)) &
(x < xs[i + 1] if i < n - 1 else np.full_like(x, True, dtype = np.bool_))
))(xa) for i in range(n)]
funclist = lambda xs, ys: [(lambda i: (
lambda x: (
(x - xs[i]) * (ys[i + 1] - ys[i]) / (
(xs[i + 1] - xs[i]) if abs(xs[i + 1] - xs[i]) > 10 ** -7 else 10 ** -7 * (-1, 1)[xs[i + 1] - xs[i] >= 0]
) + ys[i]
)
))(j) for j in range(n)]
def f(x, *pargs):
assert len(pargs) == (n + 1) * 2, (n, pargs)
xs, ys = pargs[0::2], pargs[1::2]
xa = x.ravel().astype(np.float64)
ya = np.piecewise(x = xa, condlist = condlist(xs, xa), funclist = funclist(xs, ys)).ravel()
#print('xs', xs, 'ys', ys, 'xa', xa, 'ya', ya)
return ya
return cls.nargs_func(f, 1 + (n + 1) * 2)
def __init__(self, n):
self.n = n
self.f = self.piecewise_linear(self.n)
def fit(self, x, y):
from scipy import optimize
self.p, self.e = optimize.curve_fit(self.f, x, y, p0 = [j for i in range(self.n + 1) for j in (np.amin(x) + i * (np.amax(x) - np.amin(x)) / self.n, 1)])
#print('p', self.p)
def predict(self, x):
return self.f(x, *self.p)
data = [5.269, 5.346, 5.375, 5.482, 5.519, 5.57, 5.593999999999999, 5.627000000000001, 5.724, 5.818, 5.792999999999999, 5.817, 5.8389999999999995, 5.882000000000001, 5.92, 6.025, 6.064, 6.111000000000001, 6.1160000000000005, 6.138, 6.247000000000001, 6.279, 6.332000000000001, 6.3389999999999995, 6.3420000000000005, 6.412999999999999, 6.442, 6.519, 6.596, 6.603, 6.627999999999999, 6.76, 6.837000000000001, 6.781000000000001, 6.8260000000000005, 6.849, 6.875, 6.982, 7.018, 7.042000000000001, 7.068, 7.091, 7.204, 7.228, 7.261, 7.3420000000000005, 7.414, 7.44, 7.516, 7.542000000000001, 7.627000000000001, 7.667000000000001, 7.821000000000001, 7.792999999999999, 7.756, 7.871, 8.006, 8.078, 7.916, 7.974, 8.074, 8.119, 8.228, 7.976, 8.045, 8.312999999999999, 8.335, 8.388, 8.437999999999999, 8.456, 8.227, 8.266, 8.277999999999999, 8.289, 8.299, 8.318, 8.332, 8.34, 8.349, 8.36, 8.363999999999999, 8.368, 8.282, 8.283999999999999]
time = list(range(1, 85))
df = pd.DataFrame(list(zip(time, data)), columns = ['time', 'data'])
choose_train = np.random.uniform(size = (len(df),)) < 0.8
choose_valid = ~choose_train
x_all = df.iloc[:, 0].values
y_all = df.iloc[:, 1].values
x_train = df.iloc[:, 0][choose_train].values
y_train = df.iloc[:, 1][choose_train].values
x_valid = df.iloc[:, 0][choose_valid].values
y_valid = df.iloc[:, 1][choose_valid].values
x_all_lin = np.linspace(np.amin(x_all), np.amax(x_all), 500)
models = []
models.append(('LR', LinearRegression()))
models.append(('PWLR2', PieceWiseLinearRegression(2)))
for imodel, (name, model) in enumerate(models):
model.fit(x_train[:, None], y_train)
x_all_lin_pred = model.predict(x_all_lin[:, None])
plt.plot(x_all_lin, x_all_lin_pred, label = f'pred {name}')
plt.plot(x_train, y_train, label='train')
plt.plot(x_valid, y_valid, label='valid')
plt.xlabel('time')
plt.ylabel('data')
plt.legend()
plt.show()
enter image description here
ratios = [(100,2), (100, 20), (100, 40), (100, 80)]
c_list = [0.001, 1, 100] #list of different values regularizier
for j,i in enumerate(ratios):
plt.figure(figsize=(18,14))
X_p=np.random.normal(0,0.05,size=(i[0],2))
X_n=np.random.normal(0.13,0.02,size=(i[1],2))
y_p=np.array([1]*i[0]).reshape(-1,1)
y_n=np.array([0]*i[1]).reshape(-1,1)
X=np.vstack((X_p,X_n))
y=np.vstack((y_p,y_n))
for a in range(len(c_list)):
plt.subplot(4,3,a+1)
clf = SVC(C=c_list[a],kernel='linear', degree=3,probability=False, tol=0.001, cache_size = 200, class_weight=None, verbose=2, max_iter=1000, decision_function_shape='ovr', random_state=15)
clf.fit(X=X,y=y)
coef = clf.coef_
intercept = clf.intercept_
y_max = np.amax(y)
y_min = np.amin(y)
X_pos = np.empty((i[0],2))
X_neg = np.empty((i[1],2))
l= 0
m =0
for r in range(len(X)):
if y[r]==1:
X_pos[l] = X[r]
l +=1
else:
X_neg[m]= X[r]
m +=1
plt.scatter(X_pos[:, 0], X_pos[:, 1],color = 'blue')
plt.scatter(X_neg[:, 0], X_neg[:, 1],color = 'red')
hyper_plane = draw_hyper_plane(coef,intercept,y_max,y_min)
plt.show()
def draw_hyper_plane(coef,intercept,y_max,y_min):
points=np.array([[((-coef[0][1]*y_min - intercept)/coef[0][0]), y_min],[((-coef[0][1]*y_max - intercept)/coef[0][0]), y_max]])
plt.plot(points[:,0], points[:,1])
i want to know what factors effect the position of hyper plane ?
how can improve the position of hyper plane?
The C parameter is the regularization parameter, and it has a lot of influence here. The imbalanced classes too. When svm is iteratively moving the hyperplane to find the optimal position, the regularization term penalizes the coefficients when it misclassifies an example. It does so by moving a great step back in the right direction, but with a large regularization value (100), it can bounce way too far (as you can see in your first plot).
This is exacerbated with the imbalanced classes, because the majority class will have more weight. As you can see, as the regularization parameter decreases, it becomes more stable.
Moral of the story: start with default regularization values, and move from there. Btw, this is a post better suited for https://stats.stackexchange.com.
I am trying to set up a hierarchical linear regression model using PYMC3. In my particular case, I want to see whether postal codes provide a meaningful structure for other features. Suppose I use the following mock data:
import pandas as pd
import numpy as np
import pymc3 as pm
data = pd.DataFrame({"postalcode": np.floor(np.random.uniform(low=10, high=99, size=1000)),
"x": np.random.normal(size=1000),
"y": np.random.normal(size=1000)})
data["postalcode"] = data["postalcode"].astype(int)
I generate postal codes from 10 to 99, as well as a normally distributed feature x and a target value y. Now I set up my indices for postal code level 1 and level 2:
def create_pc_index(level):
pc = data["postalcode"].astype(str).str[0:level]
unique_pc = pc.unique()
pc_dict = dict(zip(unique_pc, range(0, len(unique_pc))))
return pc_dict, pc.apply(lambda x: pc_dict[x]).values
pc1_dict, pc1_index = create_pc_index(1)
pc2_dict, pc2_index = create_pc_index(2)
Using the first digit of the postal code as hierarchical attribute works fine:
number_of_samples = 1000
x = data["x"]
y = data["y"]
with pm.Model() as model:
sigma = pm.HalfCauchy('sigma', beta=10, testval=0.5, shape=1)
mu_i = pm.Normal("mu_i", 5, sd=25, shape=1)
intercept = pm.Normal('Intercept', mu_i, sd=1, shape=len(pc1_dict))
mu_s = pm.Normal("mu_x", 0, sd=3, shape=1)
x_coeffs = pm.Normal("x", mu_s, 1, shape=len(pc1_dict))
mean = intercept[pc1_index] + x_coeffs[pc1_index] * x
likelihood_mean = pm.Deterministic("mean", mean)
likelihood = pm.Normal('y', mu=likelihood_mean, sd=sigma, observed=y)
trace = pm.sample(number_of_samples)
burned_trace = trace[number_of_samples/2:]
However, if I want to add a second level to my hierarchy (in this case only on the intercept, ignoring x for the moment), I run into shape problems
with pm.Model() as model:
sigma = pm.HalfCauchy('sigma', beta=10, testval=0.5, shape=1)
mu_i_level_1 = pm.Normal("mu_i", 0, sd=25, shape=1)
mu_i_level_2 = pm.Normal("mu_i_level_2", mu_i_level_1, sd=1, shape=len(pc1_dict))
intercept = pm.Normal('Intercept', mu_i_level_2[pc1_index], sd=1, shape=len(pc2_dict))
mu_s = pm.Normal("mu_x", 0, sd=3, shape=1)
x_coeffs = pm.Normal("x", mu_s, 1, shape=len(pc1_dict))
mean = intercept[pc2_index] + x_coeffs[pc1_index] * x
likelihood_mean = pm.Deterministic("mean", mean)
likelihood = pm.Normal('y', mu=likelihood_mean, sd=sigma, observed=y)
trace = pm.sample(number_of_samples)
burned_trace = trace[number_of_samples/2:]
The error message is:
operands could not be broadcast together with shapes (89,) (1000,)
How do I model multiple levels in my regression correctly? Is this just an issue with the correct shape size or is there a more fundamental error on my part?
Thanks in advance!
I don't think intercept can have a shape of len(pc2_dict) but a mu of len(pc1_dict). The contradiction is here:
intercept = pm.Normal('Intercept', mu_i_level_2[pc1_index], sd=1, shape=len(pc2_dict))