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I want to perform kernel-density estimate on a grid in parallel. There is some scaling involved and I can't figure out how to obtain the same results when performing the calculation on different threads.
To illustre this, I'm using an simple example from the scipy documentation.
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
def measure(n):
"Measurement model, return two coupled measurements."
m1 = np.random.normal(size=n)
m2 = np.random.normal(scale=0.5, size=n)
return m1+m2, m1-m2
m1, m2 = measure(2000)
xmin = m1.min()
xmax = m1.max()
ymin = m2.min()
ymax = m2.max()
kde_bandwith = 0.5
X, Y = np.mgrid[xmin:xmax:100j, ymin:ymax:100j]
positions = np.vstack([X.ravel(), Y.ravel()])
values = np.vstack([m1, m2])
kernel = stats.gaussian_kde(values, bw_method=kde_bandwith)
Z = np.reshape(kernel.evaluate(positions).T, X.shape)
I'm performing the last two lines on n_split subsets of the data (in practice this would be performed on different thread but here I kept a simple loop for simplicity):
## split
n_split = 10
values_split = np.array_split(values, n_split, axis=1)
results = []
for v in values_split:
kernel_n = stats.gaussian_kde(v, bw_method=kde_bandwith)
Z_n = np.reshape(kernel_n.evaluate(positions).T, X.shape)
results.append(Z_n)
Z2 = np.sum(results, axis=0)
here I'm simply recombining the results from each subset of data by summing the Z_n of each subset together into Z2.
Finally, a simple code to compare the results: left is the original version, middle is the result obtain from combining the subsets, and right the difference between the two.
fig = plt.figure(figsize=(9,2), dpi=200)
ax1 = fig.add_subplot(1,3,1)
ax2 = fig.add_subplot(1,3,2)
ax3 = fig.add_subplot(1,3,3)
c1 = ax1.imshow(np.rot90(Z), cmap=plt.cm.gist_earth_r,
extent=[xmin, xmax, ymin, ymax])
#ax1.plot(m1, m2, 'k.', markersize=2)
ax1.set_xlim([xmin, xmax])
ax1.set_ylim([ymin, ymax])
plt.colorbar(c1, shrink=0.6)
c2 = ax2.imshow(np.rot90(Z2), cmap=plt.cm.gist_earth_r,
extent=[xmin, xmax, ymin, ymax])
#ax2.plot(m1, m2, 'k.', markersize=2)
ax2.set_xlim([xmin, xmax])
ax2.set_ylim([ymin, ymax])
plt.colorbar(c2, shrink=0.6)
c3 = ax3.imshow(np.rot90(Z2-Z), cmap=plt.cm.gist_earth_r,
extent=[xmin, xmax, ymin, ymax])
ax3.plot(m1, m2, 'k.', markersize=0.1)
ax3.set_xlim([xmin, xmax])
ax3.set_ylim([ymin, ymax])
plt.colorbar(c3, shrink=0.6)
plt.show()
Notes: It looks like the magnitude of the KDE is n_split larger, but by simply normalizing with 1/n_split or even len(values_split)/len(values) (which in this case is equal), the results are not exactly the same, shown on the next figure as reference.
I am trying to do a optimization problem which requires the calculation of a new covariance matrix affected by the variable within the implementation.
I am able to do so with scipy optimization Minimize using numpy.cov within my objective function. However, as I need to have integer constraints, I am not able to think of a solution which tackles my issue with cvxpy, gekko since most of the optimization problem online have a fixed covariance matrix.
Below is my code for scipy:
room_revpar = np.array(df.iloc[:,1:10])
nla = np.array([753.2,1077.6, 1278.6,1463.9,1657.0,1990.6,2404.9,2754.6,3464.72])
min_nla = 270517.16
max_nla = 271270.359995
def objective(x, room_revpar,nla,sign = -1.0):
room_revenue = room_revpar * x
avg_revenue = np.mean(room_revenue, axis = 0)
total_revenue = sum(avg_revenue)
cov_matrix = np.cov(room_revenue.T)
total_nla = np.matmul(x.T, nla)
weights = x * nla / total_nla
portfolio_sd = np.sqrt(np.matmul(np.matmul(weights.T, cov_matrix), weights))
adj_risk = total_revenue / portfolio_sd
return sign * adj_risk
def constraint1(x, nla, min_nla):
total_nla = np.matmul(x.T, nla)
return total_nla - min_nla
def constraint2(x, nla, max_nla):
total_nla = np.matmul(x.T, nla)
return max_nla - total_nla
con1 = {'type': 'ineq', 'fun': constraint1, 'args': (nla, min_nla)}
con2 = {'type': 'ineq', 'fun': constraint2, 'args': (nla, max_nla)}
from scipy.optimize import minimize
x = np.ones(9)
sol = minimize(objective,x0 = x, args = (room_revpar, nla), constraints = (con1,con2), options = {'maxiter': 100000})
Would appreciate if anybody has a solution! Thank you.
The covariance of xi and yi is calculated explicitly with np.cov().
import numpy as np
xi = [2.1,2.5,3.6,4.0]
yi = [8,10,12,14]
print(np.cov(xi,yi))
The function np.cov(xi,yi) returns a 2x2 symmetric matrix
[[cov[xi,xi],cov[xi,yi]],
[cov[xi,yi],cov[yi,yi]]]
[[0.80333333 2.26666667]
[2.26666667 6.66666667]]
Gekko needs a symbolic form of the covariance formula for the gradient-based optimizer. Below is a function cov() that creates the symbolic covariance calculation with Gekko variables.
import numpy as np
from gekko import GEKKO
def cov(m,x,y,ddof=1):
''' Calculate the covariance matrix of x, y
Inputs:
m: Gekko model
x: x vector of equal length to y
y: y vector of equal length to x
[ddof=1]: delta degrees of freedom
Returns:
c: covariance as a Gekko variable
'''
nx = len(x); ny = len(y) # length of x, y
if nx!=ny:
print('Error: mismatch of x and y')
xm = m.sum(x)/nx # mean of x
ym = m.sum(y)/ny # mean of y
c = m.Var() # covariance
m.Equation(c==(m.sum([(x[i]-xm)*(y[i]-ym) \
for i in range(nx)]))/(nx-ddof))
return c
m = GEKKO()
n = 4
x = m.Array(m.Param,n)
y = m.Array(m.Param,n)
xi = [2.1,2.5,3.6,4.0]
yi = [8,10,12,14]
for i in range(n):
x[i].value = xi[i]
y[i].value = yi[i]
c0 = cov(m,x,y,ddof=0)
c1 = cov(m,x,y)
m.solve(disp=False)
print('Covariance (Numpy) population cov: ', np.cov(xi,yi,ddof=0)[0,1])
print('Covariance (Numpy) sample cov: ', np.cov(xi,yi)[0,1])
print('Covariance (Gekko) population cov: ', c0.value[0])
print('Covariance (Gekko) sample cov: ', c1.value[0])
Gekko and Numpy produce the same results for the fixed xi and yi values:
Covariance (Numpy) population cov: 1.7
Covariance (Numpy) sample cov: 2.2666666666666666
Covariance (Gekko) population cov: 1.7
Covariance (Gekko) sample cov: 2.2666666667
Now that the cov() function is verified, you can switch x and y to be calculated integer values such as:
x = m.Array(m.Var,n,lb=0,ub=10,integer=True)
y = m.Array(m.Var,n,lb=0,ub=5,integer=True)
To obtain an integer solution, switch to m.options.SOLVER=1 (APOPT) solver before the m.solve() command.
As a Python novice and trying to visualize the curve X2*Y + X*Y2 - X4 - Y4 = 0 with Matplotlib:
from matplotlib.pyplot import *
from sympy import *
from numpy import *
delta = 0.025
p = arange(-0.5, 1.5, delta)
q = arange(-0.5, 1.5, delta)
X, Y = meshgrid(p, q)
Z = X**2*Y + X*Y**2 - X**4 - Y**4
fig, ax = subplots()
CS = ax.contour(X, Y, Z, [0], colors ='k')
ax.set_title('x**2*y + x*y**2 - x**4 - y**4')
show()
the result is that the plot is not connected, whereas mathematically, it should be so. How can the level set be connected?
It's a year later, but for future reference: You just have to choose a smaller stepsize delta. With your delta = 0.025 your get the disconnected picture:
With delta = 0.001 you get a more accurate connected picture:
I am trying to animate a live graph in a wx.Panel. I would like to have the x-axis update like this example. Many of the examples I see are basic and don't take into consideration other controls and functions in the class. Others have so many extras that I get lost in the weeds. I can't get the animation command in the right place or update the x-axis. Here is the code:
import wx
import logging
import numpy as np
import matplotlib
import time
import matplotlib.animation as animation
matplotlib.use('WXAgg')
import matplotlib.pyplot as plt
from matplotlib.backends.backend_wxagg import FigureCanvasWxAgg as FigureCanvas
from matplotlib.backends.backend_wx import NavigationToolbar2Wx
from matplotlib.figure import Figure
fTemp = ""
x = 0
class TempClass(wx.Frame):
def __init__(self):
wx.Frame.__init__(self, None, -1, title="", size=(600,500))
panel = wx.Panel(self)
self.fig = Figure(figsize=(6,4), dpi=75, facecolor='lightskyblue', edgecolor='r')
self.canvas = FigureCanvas(self, -1, self.fig)
self.ax = self.fig.add_subplot(111)
self.ax2 = self.ax.twinx()
self.ax.set_ylim(60,90)
self.ax.set_xlim(0,24)
self.ax2.set_ylim(0,100)
# major ticks every 5, minor ticks every 1
xmajor_ticks = np.arange(0, 24, 5)
xminor_ticks = np.arange(0, 24, 1)
self.ax.set_xticks(xmajor_ticks)
self.ax.set_xticks(xminor_ticks, minor=True)
self.ax.grid()
self.ax.set_xlabel('Hour')
self.ax.set_ylabel('Temp')
self.ax2.set_ylabel('Humidity')
self.ax.set_title('Temperature')
# The graph does not show in the panel when this in uncommented
#self.ani = animation.FuncAnimation(self.fig, self.onPlotTemp, interval=1000)
self.fanSensorTimer = wx.Timer(self)
self.Bind(wx.EVT_TIMER, self.onPlotTemp, self.fanSensorTimer)
self.fanSensorBtn = wx.Button(self, -1, "Start Sensor")
self.Bind(wx.EVT_BUTTON, self.onStartTempPlot, self.fanSensorBtn)
font1 = wx.Font(18, wx.DEFAULT,wx.NORMAL,wx.BOLD)
self.displayTemp = wx.StaticText(self, -1, "Current Tempurature")
self.curTempTxt = wx.TextCtrl(self, -1, "0",size=(100,40), style=wx.TE_READONLY|wx.TE_CENTRE|wx.BORDER_NONE)
self.curTempTxt.SetFont(font1)
self.displayHum = wx.StaticText(self, -1, "Current Humidity")
self.curHumTxt = wx.TextCtrl(self, -1,"0", size=(100,40), style=wx.TE_READONLY|wx.TE_CENTRE|wx.BORDER_NONE)
self.curHumTxt.SetFont(font1)
self.displayBox = wx.GridBagSizer(hgap=5,vgap=5)
self.displayBox.Add(self.displayTemp, pos=(0,0), flag=wx.TOP|wx.LEFT, border=5)
self.displayBox.Add(self.displayHum, pos=(0,1), flag=wx.TOP, border=5)
self.displayBox.Add(self.curTempTxt, pos=(1,0), flag=wx.ALL, border=5)
self.displayBox.Add(self.curHumTxt, pos=(1,1), flag=wx.ALL, border=5)
#---------
self.vbox = wx.BoxSizer(wx.VERTICAL)
self.vbox.Add(self.canvas, wx.ALIGN_CENTER|wx.ALL, 1)
self.vbox.Add(self.fanSensorBtn)
self.vbox.Add(self.displayBox, wx.ALIGN_CENTER|wx.ALL, 1)
self.SetSizer(self.vbox)
self.vbox.Fit(self)
def start(self):
# get temp/humidity reading from node
pass
def readTemp(self, data1, data2):
"Populates Current Temp"
global fTemp
self.curTempTxt.Clear()
a = format(data1, '08b')
b = format(data2, '08b')
x = a+b
y = int(x, base=2)
cTemp = ((175.72 * y)/65536)-46.85
fTemp = cTemp *1.8+32
cel = format(cTemp,'.1f')
far = format(fTemp,'.1f')
self.curTempTxt.WriteText(far + (u'\u00b0')+"F")
def rh1(self, data1, data2):
"Populates Current RH"
global relhum
self.curHumTxt.Clear()
a = format(data1, '08b')
b = format(data2, '08b')
x = a+b
y = int(x, base=2)
rh = ((125 * y)/65536)-6
relhum = format(rh,'.1f')
self.curHumTxt.WriteText(relhum + " %")
def onStartTempPlot(self,event):
#set for a short time period for testing purposes
self.fanSensorTimer.Start(5000)
print "Timer Started"
def onPlotTemp(self,event):
global fTemp, x, relhum
x +=1
y = int(fTemp)
y2 = float(relhum)
self.ax.plot(x,y,'r.')
self.ax2.plot(x,y2,'k.')
self.fig.canvas.draw()
# send message to node for another reading of temp/humidity
if __name__ == "__main__":
app = wx.App(False)
frame = TempClass()
frame.Show()
frame.start()
logging.basicConfig(level=logging.DEBUG)
app.MainLoop()
I would like to see the x axis increment as the data is plotted beyond the 24 hour point on the graph; when data for point 25 appears, the first point is dropped and the x axis shows '25'. The animation is commented out because it causes the graph to disappear until a point is plotted.
Here is a runnable example of what I am trying to achieve with the x axis:
import numpy
from matplotlib.pylab import *
from mpl_toolkits.axes_grid1 import host_subplot
import matplotlib.animation as animation
# Sent for figure
font = {'size' : 9}
matplotlib.rc('font', **font)
# Setup figure and subplots
f0 = figure(num = 0, figsize = (6, 4))#, dpi = 100)
f0.suptitle("Oscillation decay", fontsize=12)
ax01 = subplot2grid((2, 2), (0, 0))
# Set titles of subplots
ax01.set_title('Position vs Time')
# set y-limits
ax01.set_ylim(0,2)
# sex x-limits
ax01.set_xlim(0,1)
# Turn on grids
ax01.grid(True)
# set label names
ax01.set_xlabel("x")
ax01.set_ylabel("py")
# Data Placeholders
yp1=zeros(0)
yv1=zeros(0)
yp2=zeros(0)
yv2=zeros(0)
t=zeros(0)
# set plots
p011, = ax01.plot(t,yp1,'b-', label="yp1")
p012, = ax01.plot(t,yp2,'g-', label="yp2")
# set lagends
ax01.legend([p011,p012], [p011.get_label(),p012.get_label()])
# Data Update
xmin = 0
xmax = 24
x = 0
def updateData(self):
global x
global yp1
global yv1
global yp2
global yv2
global t
tmpp1 = 1 + exp(-x) *sin(2 * pi * x)
tmpv1 = - exp(-x) * sin(2 * pi * x) + exp(-x) * cos(2 * pi * x) * 2 * pi
yp1=append(yp1,tmpp1)
yv1=append(yv1,tmpv1)
yp2=append(yp2,0.5*tmpp1)
yv2=append(yv2,0.5*tmpv1)
t=append(t,x)
x += 1
p011.set_data(t,yp1)
p012.set_data(t,yp2)
if x >= xmax-1:
p011.axes.set_xlim(x-xmax+1,x+1)
return p011
# interval: draw new frame every 'interval' ms
# frames: number of frames to draw
simulation = animation.FuncAnimation(f0, updateData, blit=False, frames=200, interval=20, repeat=False)
plt.show()
You are not incrementing the X axis limit or the ticks.
def onPlotTemp(self,event):
global fTemp, x, relhum
x +=1
y = int(fTemp)
y2 = float(relhum)
if x >= 24-1:
self.ax.set_xlim(x-24+1,x+1)
xmajor_ticks = np.arange(x-24+1,x+5, 5)
xminor_ticks = np.arange(x-24+1, x+1,1)
self.ax.set_xticks(xmajor_ticks)
self.ax.set_xticks(xminor_ticks, minor=True)
self.ax.plot(x,y,'r.')
self.ax2.plot(x,y2,'k.')
self.fig.canvas.draw()
I'm not sure if the above resets the ticks the way you want them but you get the idea. Obviously I have hard-coded 24 as your limit, you may want to create a variable to sort that out.
I'm trying to implement Fabian Timm's eye-tracking algorithm [http://www.inb.uni-luebeck.de/publikationen/pdfs/TiBa11b.pdf] (found here: [http://thume.ca/projects/2012/11/04/simple-accurate-eye-center-tracking-in-opencv/]) in numpy and OpenCV and I've hit a snag. I think I've vectorized my implementation decently enough, but it's still not fast enough to run in real time and it doesn't detect pupils with as much accuracy as I had hoped. This is my first time using numpy, so I'm not sure what I've done wrong.
def find_pupil(eye):
eye_len = np.arange(eye.shape[0])
xx,yy = np.meshgrid(eye_len,eye_len) #coordinates
XX,YY = np.meshgrid(xx.ravel(),yy.ravel()) #all distance vectors
Dx,Dy = [YY-XX, YY-XX] #y2-y1, x2-x1 -- simpler this way because YY = XXT
Dlen = np.sqrt(Dx**2+Dy**2)
Dx,Dy = [Dx/Dlen, Dy/Dlen] #normalized
Gx,Gy = np.gradient(eye)
Gmagn = np.sqrt(Gx**2+Gy**2)
Gx,Gy = [Gx/Gmagn,Gy/Gmagn] #normalized
GX,GY = np.meshgrid(Gx.ravel(),Gy.ravel())
X = (GX*Dx+GY*Dy)**2
eye = cv2.bitwise_not(cv2.GaussianBlur(eye,(5,5),0.005*eye.shape[1])) #inverting and blurring eye for use as w
eyem = np.repeat(eye.ravel()[np.newaxis,:],eye.size,0)
C = (np.nansum(eyem*X, axis=0)/eye.size).reshape(eye.shape)
return np.unravel_index(C.argmax(), C.shape)
and the rest of the code:
def find_eyes(face):
left_x, left_y = [int(floor(0.5 * face.shape[0])), int(floor(0.2 * face.shape[1]))]
right_x, right_y = [int(floor(0.1 * face.shape[0])), int(floor(0.2 * face.shape[1]))]
area = int(floor(0.2 * face.shape[0]))
left_eye = (left_x, left_y, area, area)
right_eye = (right_x, right_y, area, area)
return [left_eye,right_eye]
faceCascade = cv2.CascadeClassifier("haarcascade_frontalface_default.xml")
video_capture = cv2.VideoCapture(0)
while True:
# Capture frame-by-frame
ret, frame = video_capture.read()
gray = cv2.cvtColor(frame, cv2.COLOR_BGR2GRAY)
faces = faceCascade.detectMultiScale(
gray,
scaleFactor=1.1,
minNeighbors=5,
minSize=(30, 30),
flags=cv2.CASCADE_SCALE_IMAGE
)
# Draw a rectangle around the faces
for (x, y, w, h) in faces:
cv2.rectangle(frame, (x, y), (x+w, y+h), (0, 255, 0), 2)
roi_gray = gray[y:y+h, x:x+w]
roi_color = frame[y:y+h, x:x+w]
eyes = find_eyes(roi_gray)
for (ex,ey,ew,eh) in eyes:
eye_gray = roi_gray[ey:ey+eh,ex:ex+ew]
eye_color = roi_color[ey:ey+eh,ex:ex+ew]
cv2.rectangle(roi_color,(ex,ey),(ex+ew,ey+eh),(255,0,0),2)
px,py = find_pupil(eye_gray)
cv2.rectangle(eye_color,(px,py),(px+1,py+1),(255,0,0),2)
# Display the resulting frame
cv2.imshow('Video', frame)
if cv2.waitKey(1) & 0xFF == ord('q'):
break
# When everything is done, release the capture
video_capture.release()
cv2.destroyAllWindows()
You can perform many of those operations that save replicated elements and then perform some mathematical opertaions by directly performing the mathematical operatrions after creating singleton dimensions that would allow NumPy broadcasting. Thus, there would be two benefits - On the fly operations to save workspace memory and performance boost. Also, at the end, we can replace the nansum calculation with a simplified version. Thus, with all of that philosophy in mind, here's one modified approach -
def find_pupil_v2(face, x, y, w, h):
eye = face[x:x+w,y:y+h]
eye_len = np.arange(eye.shape[0])
N = eye_len.size**2
eye_len_diff = eye_len[:,None] - eye_len
Dlen = np.sqrt(2*((eye_len_diff)**2))
Dxy0 = eye_len_diff/Dlen
Gx0,Gy0 = np.gradient(eye)
Gmagn = np.sqrt(Gx0**2+Gy0**2)
Gx,Gy = [Gx0/Gmagn,Gy0/Gmagn] #normalized
B0 = Gy[:,:,None]*Dxy0[:,None,:]
C0 = Gx[:,None,:]*Dxy0
X = ((C0.transpose(1,0,2)[:,None,:,:]+B0[:,:,None,:]).reshape(N,N))**2
eye1 = cv2.bitwise_not(cv2.GaussianBlur(eye,(5,5),0.005*eye.shape[1]))
C = (np.nansum(X,0)*eye1.ravel()/eye1.size).reshape(eye1.shape)
return np.unravel_index(C.argmax(), C.shape)
There's one repeat still left in it at Dxy. It might be possible to avoid that step and Dxy0 could be fed directly into the step that uses Dxy to give us X, but I haven't worked through it. Everything's converted to broadcasting based!
Runtime test and output verification -
In [539]: # Inputs with random elements
...: face = np.random.randint(0,10,(256,256)).astype('uint8')
...: x = 40
...: y = 60
...: w = 64
...: h = 64
...:
In [540]: find_pupil(face,x,y,w,h)
Out[540]: (32, 63)
In [541]: find_pupil_v2(face,x,y,w,h)
Out[541]: (32, 63)
In [542]: %timeit find_pupil(face,x,y,w,h)
1 loops, best of 3: 4.15 s per loop
In [543]: %timeit find_pupil_v2(face,x,y,w,h)
1 loops, best of 3: 529 ms per loop
It seems we are getting close to 8x speedup!