detect highest peaks automatically from noisy data python - python-3.x
Is there any way to detect the highest peaks using a python library without setting any parameter?. I'm developing a user interface and I want the algorithm to be able to detect highest peaks automatically...
I want it to be able to detect these peaks in picture below:
graph here
Data looks like this:
8.60291e-07
-1.5491e-06
5.64568e-07
-9.51195e-07
1.07203e-06
4.6521e-07
6.43967e-07
-9.86092e-07
-9.82323e-07
6.38977e-07
-1.93884e-06
-2.98309e-08
1.33543e-06
1.05064e-06
1.17332e-06
-1.53549e-07
-8.9357e-07
1.59176e-06
-2.17331e-06
1.46756e-06
5.63301e-07
-8.77556e-07
7.47681e-09
-8.30101e-07
-3.6647e-07
5.27046e-07
-1.94983e-06
1.89018e-07
1.22533e-06
8.00735e-07
-8.51166e-07
1.13437e-06
-2.75787e-07
1.79601e-06
-1.67875e-06
1.13529e-06
-1.29865e-06
9.9688e-07
-9.34486e-07
8.89931e-07
-3.88634e-07
1.15124e-06
-4.23569e-07
-1.8029e-07
1.20537e-07
4.10736e-07
-9.99077e-07
-3.62984e-07
2.97916e-06
-1.95828e-06
-1.07398e-06
2.422e-06
-6.33202e-07
-1.36953e-06
1.6694e-06
-4.71764e-07
3.98849e-07
-1.0071e-06
-9.72984e-07
8.13553e-07
2.64193e-06
-3.12365e-06
1.34049e-06
-1.30419e-06
1.48369e-07
1.26033e-06
-2.59872e-07
4.28284e-07
-6.44356e-07
2.99934e-07
8.34335e-07
3.53226e-07
-7.08252e-07
4.1243e-07
2.41525e-06
-8.92159e-07
8.82339e-08
4.31945e-06
3.75152e-06
1.091e-06
3.8204e-06
-1.21356e-06
3.35564e-06
-1.06234e-06
-5.99808e-07
2.18155e-06
5.90652e-07
-1.36728e-06
-4.97017e-07
-7.77283e-08
8.68263e-07
4.37645e-07
-1.26514e-06
2.26413e-06
-8.52966e-07
-7.35596e-07
4.11911e-07
1.7585e-06
-inf
1.10779e-08
-1.49507e-06
9.87305e-07
-3.85296e-06
4.31265e-06
-9.89227e-07
-1.33537e-06
4.1713e-07
1.89362e-07
3.21968e-07
6.80237e-08
2.31636e-07
-2.98523e-07
7.99133e-07
7.36305e-07
6.39862e-07
-1.11932e-06
-1.57262e-06
1.86305e-06
-3.63716e-07
3.83865e-07
-5.23293e-07
1.31812e-06
-1.23608e-06
2.54684e-06
-3.99796e-06
2.90441e-06
-5.20203e-07
1.36295e-06
-1.89317e-06
1.22366e-06
-1.10373e-06
2.71276e-06
9.48181e-07
7.70881e-06
5.17066e-06
6.21254e-06
1.3513e-05
1.47878e-05
8.78543e-06
1.61819e-05
1.68438e-05
1.16082e-05
5.74059e-06
4.92458e-06
1.11884e-06
-1.07419e-06
-1.28517e-06
-2.70949e-06
1.65662e-06
1.42964e-06
3.40604e-06
-5.82825e-07
1.98288e-06
1.42819e-06
1.65517e-06
4.42749e-07
-1.95609e-06
-2.1756e-07
1.69164e-06
8.7204e-08
-5.35324e-07
7.43546e-07
-1.08687e-06
2.07289e-06
2.18529e-06
-2.8161e-06
1.88821e-06
4.07272e-07
1.063e-06
8.47244e-07
1.53879e-06
-9.0799e-07
-1.26709e-07
2.40044e-06
-9.48166e-07
1.41788e-06
3.67615e-07
-1.29199e-06
3.868e-06
9.54654e-06
2.51951e-05
2.2769e-05
7.21716e-06
1.36545e-06
-1.32681e-06
-3.09641e-06
4.90417e-07
2.99335e-06
1.578e-06
6.0025e-07
2.90656e-06
-2.08258e-06
-1.54214e-06
2.19757e-07
3.74982e-06
-1.76944e-06
2.15018e-06
-1.01935e-06
4.37469e-07
1.39078e-06
6.39587e-07
-1.7807e-06
-6.16455e-09
1.61557e-06
1.59644e-06
-2.35217e-06
5.29449e-07
1.9169e-06
-7.54822e-07
2.00342e-06
-3.28452e-06
3.91663e-06
1.66016e-08
-2.65897e-06
-1.4064e-06
4.67987e-07
1.67786e-06
4.69543e-07
-8.90106e-07
-1.4584e-06
1.37915e-06
1.98483e-06
-2.3735e-06
4.45618e-07
1.91504e-06
1.09653e-06
-8.00873e-07
1.32321e-06
2.04846e-06
-1.50656e-06
7.23816e-07
2.06049e-06
-2.43918e-06
1.64417e-06
2.65411e-07
-2.66107e-06
-8.01788e-07
2.05121e-06
-1.74988e-06
1.83594e-06
-8.14026e-07
-2.69342e-06
1.81152e-06
1.11664e-07
-4.21863e-06
-7.20551e-06
-5.92407e-07
-1.44629e-06
-2.08136e-06
2.86105e-06
3.77911e-06
-1.91898e-06
1.41742e-06
2.67914e-07
-8.55835e-07
-9.8584e-07
-2.74115e-06
3.39044e-06
1.39639e-06
-2.4964e-06
8.2486e-07
2.02432e-06
1.65793e-06
-1.43094e-06
-3.36807e-06
-8.96515e-07
5.31323e-06
-8.27209e-07
-1.39221e-06
-3.3754e-06
2.12372e-06
3.08218e-06
-1.42947e-06
-2.36777e-06
3.86218e-06
2.29327e-06
-3.3941e-06
-1.67291e-06
2.63828e-06
2.21008e-07
7.07794e-07
1.8172e-06
-2.00082e-06
1.80664e-06
6.69739e-07
-3.95395e-06
1.92148e-06
-1.07187e-06
-4.04938e-07
-1.76553e-06
2.7099e-06
1.30768e-06
1.41812e-06
-1.55518e-07
-3.78302e-06
4.00137e-06
-8.38623e-07
4.54651e-07
1.00027e-06
1.32196e-06
-2.62717e-06
1.67865e-06
-6.99249e-07
2.8837e-06
-1.00516e-06
-3.68011e-06
1.61847e-06
1.90887e-06
1.59641e-06
4.16779e-07
-1.35245e-06
1.65717e-06
-2.92667e-06
3.6203e-07
2.53528e-06
-2.0578e-07
-3.41919e-07
-1.42154e-06
-2.33322e-06
3.07175e-06
-2.69165e-08
-8.21045e-07
2.3175e-06
-7.22992e-07
1.49069e-06
8.75488e-07
-2.02676e-06
-2.81158e-07
3.6004e-06
-3.94708e-06
4.72983e-06
-1.38873e-06
-6.92139e-08
-1.4678e-06
1.04251e-06
-2.06625e-06
3.10406e-06
-8.13873e-07
7.23694e-07
-9.78912e-07
-8.65967e-07
7.37335e-07
1.52563e-06
-2.33591e-06
1.78265e-06
9.58435e-07
-5.22064e-07
-2.29736e-07
-4.26996e-06
-6.61411e-06
1.14789e-06
-4.32697e-06
-5.32779e-06
2.12241e-06
-1.40726e-06
1.76086e-07
-3.77194e-06
-2.71326e-06
-9.49402e-08
1.70807e-07
-2.495e-06
4.22324e-06
-3.62476e-06
-9.56055e-07
7.16583e-07
3.01447e-06
-1.41229e-06
-1.67694e-06
7.61627e-07
3.55881e-06
2.31015e-06
-9.50378e-07
4.45251e-08
-1.94791e-06
2.27081e-06
-3.34717e-06
3.05688e-06
4.57062e-07
3.87326e-06
-2.39215e-06
-3.52682e-06
-2.05212e-06
5.26495e-06
-3.28613e-07
-5.76569e-07
-7.46338e-07
5.98795e-06
8.80493e-07
-4.82965e-06
2.56839e-06
-1.58792e-06
-2.2294e-06
1.83841e-06
2.65482e-06
-3.10474e-06
-3.46741e-07
2.45557e-06
2.01328e-06
-3.92606e-06
inf
-8.11737e-07
5.72174e-07
1.57245e-06
8.02612e-09
-2.901e-06
1.22079e-06
-6.31714e-07
3.06241e-06
1.20059e-06
-1.80344e-06
4.90784e-07
3.74243e-06
-2.94342e-07
-3.45764e-08
-3.42099e-06
-1.43695e-06
5.91064e-07
3.47308e-06
3.78232e-06
4.01093e-07
-1.58435e-06
-3.47375e-06
1.34943e-06
1.11768e-06
1.95212e-06
-8.28033e-07
1.53705e-06
6.38031e-07
-1.84702e-06
1.34689e-06
-6.98669e-07
1.81653e-06
-2.42355e-06
-1.35257e-06
3.04367e-06
-1.21976e-06
1.61896e-06
-2.69528e-06
1.84601e-06
6.45447e-08
-4.94263e-07
3.47568e-06
-2.00531e-06
3.56693e-06
-3.19446e-06
2.72141e-06
-1.39059e-06
2.20032e-06
-1.76819e-06
2.32727e-07
-3.47382e-07
2.11823e-07
-5.22614e-07
2.69846e-06
-1.47983e-06
2.14554e-06
-6.27594e-07
-8.8501e-10
7.89124e-07
-2.8653e-07
8.30902e-07
-2.12857e-06
-1.90887e-07
1.07593e-06
1.40781e-06
2.41641e-06
-4.52689e-06
2.37207e-06
-2.19479e-06
1.65131e-06
1.2706e-06
-2.18387e-06
-1.72821e-07
5.41687e-07
7.2879e-07
7.56927e-07
1.57739e-06
-3.79395e-07
-1.02887e-06
-1.20987e-06
1.43066e-06
8.96301e-08
5.09766e-07
-2.8812e-06
-2.35944e-06
2.25912e-06
-2.78967e-06
-4.69913e-06
1.60822e-06
6.9342e-07
4.6225e-07
-1.33276e-06
-3.59033e-06
1.11206e-06
1.83521e-06
2.39163e-06
2.3468e-08
5.91431e-07
-8.80249e-07
-2.77405e-08
-1.13184e-06
-1.28036e-06
1.66229e-06
2.81784e-06
-2.97589e-06
8.73413e-08
1.06439e-06
2.39075e-06
-2.76974e-06
1.20862e-06
-5.12817e-07
-5.19104e-07
4.51324e-07
-4.7168e-07
2.35608e-06
5.46906e-07
-1.66748e-06
5.85236e-07
6.42944e-07
2.43164e-07
4.01031e-07
-1.93646e-06
2.07416e-06
-1.16116e-06
4.27155e-07
5.2951e-07
9.09149e-07
-8.71887e-08
-1.5564e-09
1.07266e-06
-9.49402e-08
2.04016e-06
-6.38123e-07
-1.94241e-06
-5.17294e-06
-2.18622e-06
-8.26703e-06
2.54364e-06
4.32614e-06
8.3847e-07
-2.85309e-06
2.72345e-06
-3.42752e-06
-1.36871e-07
2.23346e-06
5.26825e-07
1.3566e-06
-2.17111e-06
2.1463e-07
2.06479e-06
1.76929e-06
-1.2655e-06
-1.3797e-06
3.10706e-06
-4.72189e-06
4.38138e-06
6.41815e-07
-3.25623e-08
-4.93707e-06
5.05743e-06
5.17578e-07
-5.30524e-06
3.62463e-06
5.68909e-07
1.16226e-06
1.10843e-06
-5.00854e-07
9.48761e-07
-2.18701e-06
-3.57635e-07
4.26709e-06
-1.50836e-06
-5.84412e-06
3.5054e-06
3.94019e-06
-4.7623e-06
2.05856e-06
-2.22992e-07
1.64969e-06
2.64694e-06
-8.49487e-07
-3.63562e-06
1.0386e-06
1.69461e-06
-2.05798e-06
3.60349e-06
3.42651e-07
-1.46686e-06
1.19949e-06
-1.60519e-06
2.37793e-07
6.12366e-07
-1.54669e-06
1.43668e-06
1.87009e-06
-2.22626e-06
2.15155e-06
-3.10571e-06
2.05188e-06
-4.40002e-07
2.06683e-06
-1.11362e-06
5.96924e-07
-2.64471e-06
2.4892e-06
1.13083e-06
-3.23181e-07
5.10651e-07
2.73499e-07
-1.24899e-06
1.40564e-06
-9.3158e-07
1.45947e-06
3.70544e-07
-1.62628e-06
-1.70215e-06
1.72098e-06
8.19031e-07
-5.57709e-07
1.10107e-06
-2.81845e-06
1.57654e-07
3.30716e-06
-9.75403e-07
1.73126e-07
1.30447e-06
7.64771e-08
-6.65344e-07
-1.4346e-06
5.03171e-06
-2.84576e-06
2.3212e-06
-2.73373e-06
2.16675e-08
2.24026e-06
-4.11682e-08
-3.36642e-06
1.78775e-06
1.28174e-08
-9.32068e-07
2.97177e-06
-1.05338e-06
9.42505e-07
2.02362e-07
-1.81326e-06
2.16995e-06
2.83722e-07
-1.2648e-06
9.21814e-07
-8.9447e-07
-1.61597e-06
3.5036e-06
-6.79626e-08
1.52823e-06
-2.98682e-06
5.57404e-07
9.5166e-07
7.10419e-07
-1.28528e-06
-3.76038e-07
-1.03845e-06
2.96631e-06
-1.18356e-06
-2.77313e-07
3.24149e-06
-1.85455e-06
-1.27747e-07
3.6264e-07
4.66431e-07
-1.54443e-06
1.38437e-06
-1.53119e-06
7.4231e-07
-1.2388e-06
1.99774e-06
1.15799e-06
1.39478e-06
-2.93527e-06
-2.03012e-06
2.46667e-06
2.16751e-06
-2.50354e-06
3.95905e-07
5.74371e-07
1.33575e-07
-3.98315e-07
4.93927e-07
-5.23987e-07
-1.74713e-07
6.49384e-07
-7.16766e-07
2.35733e-06
-4.91333e-08
-1.88138e-06
1.74722e-06
4.03503e-07
3.5965e-07
1.44836e-07]
The task you are describing could be treated like anomaly/outlier detection.
One possible solution is to use a Z-score transformation and treat every value with a z score above a certain threshold as an outlier. Because there is no clear definition of an outlier it won't be able to detect such peaks without setting any parameters (threshold).
One possible solution could be:
import numpy as np
def detect_outliers(data):
outliers = []
d_mean = np.mean(data)
d_std = np.std(data)
threshold = 3 # this defines what you would consider a peak (outlier)
for point in data:
z_score = (point - d_mean)/d_std
if np.abs(z_score) > threshold:
outliers.append(point)
return outliers
# create normal data
data = np.random.normal(size=100)
# create outliers
outliers = np.random.normal(100, size=3)
# combine normal data and outliers
full_data = data.tolist() + outliers.tolist()
# print outliers
print(detect_outliers(full_data))
If you only want to detect peaks, remove the np.abs function call from the code.
This code snippet is based on a Medium Post, which also provides another way of detecting outliers.
Related
Confusion About Implementing LeafSystem With Vector Output Port Correctly
I'm a student teaching myself Drake, specifically pydrake with Dr. Russ Tedrake's excellent Underactuated Robotics course. I am trying to write a combined energy shaping and lqr controller for keeping a cartpole system balanced upright. I based the diagram on the cartpole example found in Chapter 3 of Underactuated Robotics [http://underactuated.mit.edu/acrobot.html], and the SwingUpAndBalanceController on Chapter 2: [http://underactuated.mit.edu/pend.html].
I have found that due to my use of the cart_pole.sdf model I have to create an abstract input port due receive FramePoseVector from the cart_pole.get_output_port(0). From there I know that I have to create a control signal output of type BasicVector to feed into a Saturation block before feeding into the cartpole's actuation port.
The problem I'm encountering right now is that I'm not sure how to get the system's current state data in the DeclareVectorOutputPort's callback function. I was under the assumption I would use the LeafContext parameter in the callback function, OutputControlSignal, obtaining the BasicVector continuous state vector. However, this resulting vector, x_bar is always NaN. Out of desperation (and testing to make sure the rest of my program worked) I set x_bar to the controller's initialization cart_pole_context and have found that the simulation runs with a control signal of 0.0 (as expected). I can also set output to 100 and the cartpole simulation just flies off into endless space (as expected).
TL;DR: What is the proper way to obtain the continuous state vector in a custom controller extending LeafSystem with a DeclareVectorOutputPort?
Thank you for any help! I really appreciate it :) I've been teaching myself so it's been a little arduous haha.
# Combined Energy Shaping (SwingUp) and LQR (Balance) Controller
# with a simple state machine
class SwingUpAndBalanceController(LeafSystem):
def __init__(self, cart_pole, cart_pole_context, input_i, ouput_i, Q, R, x_star):
LeafSystem.__init__(self)
self.DeclareAbstractInputPort("state_input", AbstractValue.Make(FramePoseVector()))
self.DeclareVectorOutputPort("control_signal", BasicVector(1),
self.OutputControlSignal)
(self.K, self.S) = BalancingLQRCtrlr(cart_pole, cart_pole_context,
input_i, ouput_i, Q, R, x_star).get_LQR_matrices()
(self.A, self.B, self.C, self.D) = BalancingLQRCtrlr(cart_pole, cart_pole_context,
input_i, ouput_i,
Q, R, x_star).get_lin_matrices()
self.energy_shaping = EnergyShapingCtrlr(cart_pole, x_star)
self.energy_shaping_context = self.energy_shaping.CreateDefaultContext()
self.cart_pole_context = cart_pole_context
def OutputControlSignal(self, context, output):
#xbar = copy(self.cart_pole_context.get_continuous_state_vector())
xbar = copy(context.get_continuous_state_vector())
xbar_ = np.array([xbar[0], xbar[1], xbar[2], xbar[3]])
xbar_[1] = wrap_to(xbar_[1], 0, 2.0*np.pi) - np.pi
# If x'Sx <= 2, then use LQR ctrlr. Cost-to-go J_star = x^T * S * x
threshold = np.array([2.0])
if (xbar_.dot(self.S.dot(xbar_)) < 2.0):
#output[:] = -self.K.dot(xbar_) # u = -Kx
output.set_value(-self.K.dot(xbar_))
else:
self.energy_shaping.get_input_port(0).FixValue(self.energy_shaping_context,
self.cart_pole_context.get_continuous_state_vector())
output_val = self.energy_shaping.get_output_port(0).Eval(self.energy_shaping_context)
output.set_value(output_val)
print(output)
Here are two things that might help: If you want to get the state of the cart-pole from MultibodyPlant, you probably want to be connecting to the continuous_state output port, which gives you a normal vector instead of the abstract-type FramePoseVector. In that case, your call to get_input_port().Eval(context) should work just fine. If you do really want to read the FramePoseVector, then you have to evaluate the input port slightly differently. You can find an example of that here.
How does sklearn.linear_model.LinearRegression work with insufficient data?
To solve a 5 parameter model, I need at least 5 data points to get a unique solution. For x and y data below: import numpy as np x = np.array([[-0.24155831, 0.37083184, -1.69002708, 1.4578805 , 0.91790011, 0.31648635, -0.15957368], [-0.37541846, -0.14572825, -2.19695883, 1.01136142, 0.57288752, 0.32080956, -0.82986857], [ 0.33815532, 3.1123936 , -0.29317028, 3.01493602, 1.64978158, 0.56301755, 1.3958912 ], [ 0.84486735, 4.74567324, 0.7982888 , 3.56604097, 1.47633894, 1.38743513, 3.0679506 ], [-0.2752026 , 2.9110031 , 0.19218081, 2.0691105 , 0.49240373, 1.63213241, 2.4235483 ], [ 0.89942508, 5.09052174, 1.26048572, 3.73477373, 1.4302902 , 1.91907482, 3.70126468]]) y = np.array([-0.81388378, -1.59719762, -0.08256274, 0.61297275, 0.99359647, 1.11315445]) I used only 6 data to fit a 8 parameter model (7 slopes and 1 intercept). lr = LinearRegression().fit(x, y) print(lr.coef_) array([-0.83916772, -0.57249998, 0.73025938, -0.02065629, 0.47637768, -0.36962192, 0.99128474]) print(lr.intercept_) 0.2978781587718828 Clearly, it's using some kind of assignment to reduce the degrees of freedom. I tried to look into the source code but couldn't found anything about that. What method do they use to find the parameter of under specified model?
You don't need to reduce the degrees of freedom, it simply finds a solution to the least squares problem min sum_i (dot(beta,x_i)+beta_0-y_i)**2. For example, in the non-sparse case it uses the linalg.lstsq module from scipy. The default solver for this optimization problem is the gelsd LAPACK driver. If A= np.concatenate((ones_v, X), axis=1) is the augmented array with ones as its first column, then your solution is given by x=numpy.linalg.pinv(A.T*A)*A.T*y Where we use the pseudoinverse precisely because the matrix may not be of full rank. Of course, the solver doesn't actually use this formula but uses singular value Decomposition of A to reduce this formula.
PACF function in statsmodels.tsa.stattools gives numbers greater than 1 when using ywunbiased?
I have a dataframe which is of length 177 and I want to calculate and plot the partial auto-correlation function (PACF). I have the data imported etc and I do: from statsmodels.tsa.stattools import pacf ys = pacf(data[key][array].diff(1).dropna(), alpha=0.05, nlags=176, method="ywunbiased") xs = range(lags+1) plt.figure() plt.scatter(xs,ys[0]) plt.grid() plt.vlines(xs, 0, ys[0]) plt.plot(ys[1]) The method used results in numbers greater than 1 for very long lags (90ish) which is incorrect and I get a RuntimeWarning: invalid value encountered in sqrtreturn rho, np.sqrt(sigmasq) but since I can't see their source code I don't know what this means. To be honest, when I search for PACF, all the examples only carry out PACF up to 40 lags or 60 or so and they never have any significant PACF after lag=2 and so I couldn't compare to other examples either. But when I use: method="ols" # or method="ywmle" the numbers are corrected. So it must be the algo they use to solve it. I tried importing inspect and getsource method but its useless it just shows that it uses another package and I can't find that. If you also know where the problem arises from, I would really appreciate the help. For your reference, the values for data[key][array] are: [1131.130005, 1144.939941, 1126.209961, 1107.300049, 1120.680054, 1140.839966, 1101.719971, 1104.23999, 1114.579956, 1130.199951, 1173.819946, 1211.920044, 1181.27002, 1203.599976, 1180.589966, 1156.849976, 1191.5, 1191.329956, 1234.180054, 1220.329956, 1228.810059, 1207.01001, 1249.47998, 1248.290039, 1280.079956, 1280.660034, 1294.869995, 1310.609985, 1270.089966, 1270.199951, 1276.660034, 1303.819946, 1335.849976, 1377.939941, 1400.630005, 1418.300049, 1438.23999, 1406.819946, 1420.859985, 1482.369995, 1530.619995, 1503.349976, 1455.27002, 1473.98999, 1526.75, 1549.380005, 1481.140015, 1468.359985, 1378.550049, 1330.630005, 1322.699951, 1385.589966, 1400.380005, 1280.0, 1267.380005, 1282.829956, 1166.359985, 968.75, 896.23999, 903.25, 825.880005, 735.090027, 797.869995, 872.8099980000001, 919.1400150000001, 919.320007, 987.4799800000001, 1020.6199949999999, 1057.079956, 1036.189941, 1095.630005, 1115.099976, 1073.869995, 1104.48999, 1169.430054, 1186.689941, 1089.410034, 1030.709961, 1101.599976, 1049.329956, 1141.199951, 1183.26001, 1180.550049, 1257.640015, 1286.119995, 1327.219971, 1325.829956, 1363.609985, 1345.199951, 1320.640015, 1292.280029, 1218.890015, 1131.420044, 1253.300049, 1246.959961, 1257.599976, 1312.410034, 1365.680054, 1408.469971, 1397.910034, 1310.329956, 1362.160034, 1379.319946, 1406.579956, 1440.670044, 1412.160034, 1416.180054, 1426.189941, 1498.109985, 1514.680054, 1569.189941, 1597.569946, 1630.73999, 1606.280029, 1685.72998, 1632.969971, 1681.550049, 1756.540039, 1805.810059, 1848.359985, 1782.589966, 1859.449951, 1872.339966, 1883.949951, 1923.569946, 1960.22998, 1930.6700440000002, 2003.369995, 1972.290039, 2018.050049, 2067.560059, 2058.899902, 1994.9899899999998, 2104.5, 2067.889893, 2085.51001, 2107.389893, 2063.110107, 2103.840088, 1972.180054, 1920.030029, 2079.360107, 2080.409912, 2043.939941, 1940.2399899999998, 1932.22998, 2059.73999, 2065.300049, 2096.949951, 2098.860107, 2173.600098, 2170.949951, 2168.27002, 2126.149902, 2198.810059, 2238.830078, 2278.8701170000004, 2363.639893, 2362.719971, 2384.199951, 2411.800049, 2423.409912, 2470.300049, 2471.649902, 2519.360107, 2575.26001, 2584.840088, 2673.610107, 2823.810059, 2713.830078, 2640.8701170000004, 2648.050049, 2705.27002, 2718.3701170000004, 2816.290039, 2901.52002, 2913.97998]
Your time series is pretty clearly not stationary, so that Yule-Walker assumptions are violated. More generally, PACF is usually appropriate with stationary time series. You might difference your data first, before considering the partial autocorrelations.
How to estimate camera pose according to a projective transformation matrix of two consecutive frames?
I'm working on the kitti visual odometry dataset. I use projective transformation to register two 2D consecutive frames(see projective transformation example here ). I want to know how this 3*3 projective transformation matrix is related to the ground truth poses provided by the kitti dataset. This dataset gives the ground truth poses (trajectory) for the sequences, which is described below: Folder 'poses': The folder 'poses' contains the ground truth poses (trajectory) for the first 11 sequences. This information can be used for training/tuning your method. Each file xx.txt contains a N x 12 table, where N is the number of frames of this sequence. Row i represents the i'th pose of the left camera coordinate system (i.e., z pointing forwards) via a 3x4 transformation matrix. The matrices are stored in row aligned order (the first entries correspond to the first row), and take a point in the i'th coordinate system and project it into the first (=0th) coordinate system. Hence, the translational part (3x1 vector of column 4) corresponds to the pose of the left camera coordinate system in the i'th frame with respect to the first (=0th) frame. Your submission results must be provided using the same data format. Some samples of the given groud-truth poses: 1.000000e+00 9.043680e-12 2.326809e-11 5.551115e-17 9.043683e-12 1.000000e+00 2.392370e-10 3.330669e-16 2.326810e-11 2.392370e-10 9.999999e-01 -4.440892e-16 9.999978e-01 5.272628e-04 -2.066935e-03 -4.690294e-02 -5.296506e-04 9.999992e-01 -1.154865e-03 -2.839928e-02 2.066324e-03 1.155958e-03 9.999971e-01 8.586941e-01 9.999910e-01 1.048972e-03 -4.131348e-03 -9.374345e-02 -1.058514e-03 9.999968e-01 -2.308104e-03 -5.676064e-02 4.128913e-03 2.312456e-03 9.999887e-01 1.716275e+00 9.999796e-01 1.566466e-03 -6.198571e-03 -1.406429e-01 -1.587952e-03 9.999927e-01 -3.462706e-03 -8.515762e-02 6.193102e-03 3.472479e-03 9.999747e-01 2.574964e+00 9.999637e-01 2.078471e-03 -8.263498e-03 -1.874858e-01 -2.116664e-03 9.999871e-01 -4.615826e-03 -1.135202e-01 8.253797e-03 4.633149e-03 9.999551e-01 3.432648e+00 9.999433e-01 2.586172e-03 -1.033094e-02 -2.343818e-01 -2.645881e-03 9.999798e-01 -5.770163e-03 -1.419150e-01 1.031581e-02 5.797170e-03 9.999299e-01 4.291335e+00 9.999184e-01 3.088363e-03 -1.239599e-02 -2.812195e-01 -3.174350e-03 9.999710e-01 -6.922975e-03 -1.702743e-01 1.237425e-02 6.961759e-03 9.998991e-01 5.148987e+00 9.998890e-01 3.586305e-03 -1.446384e-02 -3.281178e-01 -3.703403e-03 9.999605e-01 -8.077186e-03 -1.986703e-01 1.443430e-02 8.129853e-03 9.998627e-01 6.007777e+00 9.998551e-01 4.078705e-03 -1.652913e-02 -3.749547e-01 -4.231669e-03 9.999484e-01 -9.229794e-03 -2.270290e-01 1.649063e-02 9.298401e-03 9.998207e-01 6.865477e+00 9.998167e-01 4.566671e-03 -1.859652e-02 -4.218367e-01 -4.760342e-03 9.999347e-01 -1.038342e-02 -2.554151e-01 1.854788e-02 1.047004e-02 9.997731e-01 7.724036e+00 9.997738e-01 5.049868e-03 -2.066463e-02 -4.687329e-01 -5.289072e-03 9.999194e-01 -1.153730e-02 -2.838096e-01 2.060470e-02 1.164399e-02 9.997198e-01 8.582886e+00 9.997264e-01 5.527315e-03 -2.272922e-02 -5.155474e-01 -5.816781e-03 9.999025e-01 -1.268908e-02 -3.121547e-01 2.265686e-02 1.281782e-02 9.996611e-01 9.440275e+00 9.996745e-01 6.000540e-03 -2.479692e-02 -5.624310e-01 -6.345160e-03 9.998840e-01 -1.384246e-02 -3.405416e-01 2.471098e-02 1.399530e-02 9.995966e-01 1.029896e+01 9.996182e-01 6.468772e-03 -2.686440e-02 -6.093087e-01 -6.873365e-03 9.998639e-01 -1.499561e-02 -3.689250e-01 2.676374e-02 1.517453e-02 9.995266e-01 1.115757e+01 9.995562e-01 7.058450e-03 -2.894213e-02 -6.562052e-01 -7.530449e-03 9.998399e-01 -1.623192e-02 -3.973964e-01 2.882292e-02 1.644266e-02 9.994492e-01 1.201541e+01 9.995095e-01 5.595311e-03 -3.081450e-02 -7.018788e-01 -6.093682e-03 9.998517e-01 -1.610315e-02 -4.239119e-01 3.071983e-02 1.628303e-02 9.993953e-01 1.286965e+01
The common name for your "projective transformation" is homography. In a calibrated setup (i.e. if you know your camera's field of view or, equivalently, its focal length) a homography can be decomposed into 3D rotation and translation, the latter only up to scale. The decomposition algorithm additionally produces the normal to the 3D plane inducting the homography. The algorithm has up to 4 solutions, of which only one is feasible when you apply additional constraints, such as that the matched image points triangulate in front of the camera, and that the general direction of the translation match a known prior. More information about the method is in a well-known paper by Malis and Vargas. There is an implementation in OpenCV, under the name decomposeHomographyMat.
scikit-learn roc_curve: why does it return a threshold value = 2 some time?
Correct me if I'm wrong: the "thresholds" returned by scikit-learn's roc_curve should be an array of numbers that are in [0,1]. However, it sometimes gives me an array with the first number close to "2". Is it a bug or I did sth wrong? Thanks. In [1]: import numpy as np In [2]: from sklearn.metrics import roc_curve In [3]: np.random.seed(11) In [4]: aa = np.random.choice([True, False],100) In [5]: bb = np.random.uniform(0,1,100) In [6]: fpr,tpr,thresholds = roc_curve(aa,bb) In [7]: thresholds Out[7]: array([ 1.97396826, 0.97396826, 0.9711752 , 0.95996265, 0.95744405, 0.94983331, 0.93290463, 0.93241372, 0.93214862, 0.93076592, 0.92960511, 0.92245024, 0.91179548, 0.91112166, 0.87529458, 0.84493853, 0.84068543, 0.83303741, 0.82565223, 0.81096657, 0.80656679, 0.79387241, 0.77054807, 0.76763223, 0.7644911 , 0.75964947, 0.73995152, 0.73825262, 0.73466772, 0.73421299, 0.73282534, 0.72391126, 0.71296292, 0.70930102, 0.70116428, 0.69606617, 0.65869235, 0.65670881, 0.65261474, 0.6487222 , 0.64805644, 0.64221486, 0.62699782, 0.62522484, 0.62283401, 0.61601839, 0.611632 , 0.59548669, 0.57555854, 0.56828967, 0.55652111, 0.55063947, 0.53885029, 0.53369398, 0.52157349, 0.51900774, 0.50547317, 0.49749635, 0.493913 , 0.46154029, 0.45275916, 0.44777116, 0.43822067, 0.43795921, 0.43624093, 0.42039077, 0.41866343, 0.41550367, 0.40032843, 0.36761763, 0.36642721, 0.36567017, 0.36148354, 0.35843793, 0.34371331, 0.33436415, 0.33408289, 0.33387442, 0.31887024, 0.31818719, 0.31367915, 0.30216469, 0.30097917, 0.29995201, 0.28604467, 0.26930354, 0.2383461 , 0.22803687, 0.21800338, 0.19301808, 0.16902881, 0.1688173 , 0.14491946, 0.13648451, 0.12704826, 0.09141459, 0.08569481, 0.07500199, 0.06288762, 0.02073298, 0.01934336])
Most of the time these thresholds are not used, for example in calculating the area under the curve, or plotting the False Positive Rate against the True Positive Rate. Yet to plot what looks like a reasonable curve, one needs to have a threshold that incorporates 0 data points. Since Scikit-Learn's ROC curve function need not have normalised probabilities for thresholds (any score is fine), setting this point's threshold to 1 isn't sufficient; setting it to inf is sensible but coders often expect finite data (and it's possible the implementation also works for integer thresholds). Instead the implementation uses max(score) + epsilon where epsilon = 1. This may be cosmetically deficient, but you haven't given any reason why it's a problem!
From the documentation: thresholds : array, shape = [n_thresholds] Decreasing thresholds on the decision function used to compute fpr and tpr. thresholds[0] represents no instances being predicted and is arbitrarily set to max(y_score) + 1. So the first element of thresholds is close to 2 because it is max(y_score) + 1, in your case thresholds[1] + 1.
this seems like a bug to me - in roc_curve(aa,bb), 1 is added to the first threshold. You should create an issue here https://github.com/scikit-learn/scikit-learn/issues