Linalgerror schur decomposition solver error

Hello, I have a problem with time series analysis. I have a dataset with 5 features. Following is the subset of my input dataset: date,price,year,day,totaltx 1/1/2016 0:00,434.46,2016,1,126762 1/2/...

Hello,
I have a problem with time series analysis. I have a dataset with 5 features. Following is the subset of my input dataset:

date,price,year,day,totaltx
1/1/2016 0:00,434.46,2016,1,126762
1/2/2016 0:00,433.59,2016,2,147449
1/3/2016 0:00,430.36,2016,3,148661
1/4/2016 0:00,433.49,2016,4,185279
1/5/2016 0:00,432.25,2016,5,178723
1/6/2016 0:00,429.46,2016,6,184207

My endogenous data is price column and exogenous data is totaltx price.
This is the code I am running and getting an error:

import statsmodels.api as sm
import pandas as pd
import numpy as np
from numpy.linalg import LinAlgError

def arima(filteredData, coinOutput, window, horizon, trainLength):
    start_index = 0
    end_index = 0
    inputNumber = filteredData.shape[0]
    predictions = np.array([], dtype=np.float32)
    prices = np.array([], dtype=np.float32)
    # sliding on time series data with 1 day step
    while ((end_index) < inputNumber - 1):
        end_index = start_index + trainLength
        trainFeatures = filteredData[start_index:end_index]["totaltx"]
        trainOutput = coinOutput[start_index:end_index]["price"]

        arima = sm.tsa.statespace.SARIMAX(endog=trainOutput.values, exog=trainFeatures.values, order=(window, 0, 0))
        arima_fit = arima.fit(disp=0)
        testdata=filteredData[end_index:end_index+1]["totaltx"]
        total_sample = end_index-start_index
        predicted = arima_fit.predict(start=total_sample, end=total_sample, exog=np.array(testdata.values).reshape(-1,1))
        price = coinOutput[end_index:end_index + 1]["price"].values

        predictions = np.append(predictions, predicted)
        prices = np.append(prices, price)

        start_index = start_index + 1
    return predictions, prices

def processCoins(bitcoinPrice, window, horizon):
    output = bitcoinPrice[horizon:][["date", "day", "year", "price"]]
    return output

trainLength=100;
for window in [3,5]:
    for horizon in [1,2,5,7,10]:
        bitcoinPrice = pd.read_csv("..\prices.csv", sep=",")
        coinOutput = processCoins(bitcoinPrice, window, horizon)
        predictions, prices = arima(bitcoinPrice, coinOutput, window, horizon, trainLength)

In this code, I am using rolling window regression technique. I am training arima for start_index:end_index and predicting the test data with end_index:end_index+1

This the error that is thrown from my code:

Traceback (most recent call last):
  File "C:/PycharmProjects/coinLogPrediction/src/arima.py", line 115, in <module>
    predictions, prices = arima(filteredBitcoinPrice, coinOutput, window, horizon, trainLength, outputFile)
  File "C:/PycharmProjects/coinLogPrediction/src/arima.py", line 64, in arima
    arima_fit = arima.fit(disp=0)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelstsastatespacemlemodel.py", line 469, in fit
    skip_hessian=True, **kwargs)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelsbasemodel.py", line 466, in fit
    full_output=full_output)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelsbaseoptimizer.py", line 191, in _fit
    hess=hessian)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelsbaseoptimizer.py", line 410, in _fit_lbfgs
    **extra_kwargs)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesscipyoptimizelbfgsb.py", line 193, in fmin_l_bfgs_b
    **opts)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesscipyoptimizelbfgsb.py", line 328, in _minimize_lbfgsb
    f, g = func_and_grad(x)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesscipyoptimizelbfgsb.py", line 273, in func_and_grad
    f = fun(x, *args)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesscipyoptimizeoptimize.py", line 292, in function_wrapper
    return function(*(wrapper_args + args))
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelsbasemodel.py", line 440, in f
    return -self.loglike(params, *args) / nobs
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelstsastatespacemlemodel.py", line 646, in loglike
    loglike = self.ssm.loglike(complex_step=complex_step, **kwargs)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelstsastatespacekalman_filter.py", line 825, in loglike
    kfilter = self._filter(**kwargs)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelstsastatespacekalman_filter.py", line 747, in _filter
    self._initialize_state(prefix=prefix, complex_step=complex_step)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelstsastatespacerepresentation.py", line 723, in _initialize_state
    self._statespaces[prefix].initialize_stationary(complex_step)
  File "_representation.pyx", line 1351, in statsmodels.tsa.statespace._representation.dStatespace.initialize_stationary
  File "_tools.pyx", line 1151, in statsmodels.tsa.statespace._tools._dsolve_discrete_lyapunov
numpy.linalg.linalg.LinAlgError: LU decomposition error.

I believe that there is a bug in statsmodels if I do not have any error. Can you please help me to solve it?

Источник

I am getting an error while using SARIMA Model for Energy Consumption Project. Below is the code and error. If anyone have any solutions or suggestions, they would be greatly appreciated.

Code:

from statsmodels.tsa.statespace import sarimax
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

data = pd.read_csv("C:\Users\Jaswal\Desktop\Analytics Vidhya\Assignments\energy consumption.csv", index_col=None)
data

DATE    ENERGY_INDEX
0   01/1939 3.3842
1   02/1939 3.4100
2   03/1939 3.4875
3   04/1939 3.5133
4   05/1939 3.5133
... ... ...
964 05/2019 91.9046
965 06/2019 98.4397
966 07/2019 112.9469
967 08/2019 111.6645
968 09/2019 102.2911

model = sarimax.SARIMAX(data['ENERGY_INDEX'], seasonal_order=(1,1,1,7), order=(2,1,2))
fit1 = model.fit()

valid_data['SARIMA'] = fit1.predict(start="01/1939", end="09/2019", dynamic=True)

Error:

LinAlgError                               Traceback (most recent call last)
<ipython-input-55-0d31300a6c27> in <module>
      1 # fit model
      2 model = sarimax.SARIMAX(data['ENERGY_INDEX'], seasonal_order=(1,1,1,7), order=(2,1,2))
----> 3 fit1 = model.fit()
      4 
      5 # make predictions

~Anaconda3libsite-packagesstatsmodelstsastatespacerepresentation.py in _initialize_state(self, prefix, complex_step)
    982                 raise RuntimeError('Initialization is incomplete.')
    983             self._statespaces[prefix].initialize(self.initialization,
--> 984                                                  complex_step=complex_step)
    985         else:
    986             raise RuntimeError('Statespace model not initialized.')

statsmodelstsastatespace_representation.pyx in statsmodels.tsa.statespace._representation.dStatespace.initialize()

statsmodelstsastatespace_representation.pyx in statsmodels.tsa.statespace._representation.dStatespace.initialize()

statsmodelstsastatespace_initialization.pyx in statsmodels.tsa.statespace._initialization.dInitialization.initialize()

statsmodelstsastatespace_initialization.pyx in statsmodels.tsa.statespace._initialization.dInitialization.initialize_stationary_stationary_cov()

statsmodelstsastatespace_tools.pyx in statsmodels.tsa.statespace._tools._dsolve_discrete_lyapunov()

LinAlgError: Schur decomposition solver error.

Python Version = 3.7.6
Statsmodel = 1.2.0

Источник

Hello,
I have a problem with time series analysis. I have a dataset with 5 features. Following is the subset of my input dataset:

date,price,year,day,totaltx
1/1/2016 0:00,434.46,2016,1,126762
1/2/2016 0:00,433.59,2016,2,147449
1/3/2016 0:00,430.36,2016,3,148661
1/4/2016 0:00,433.49,2016,4,185279
1/5/2016 0:00,432.25,2016,5,178723
1/6/2016 0:00,429.46,2016,6,184207

My endogenous data is price column and exogenous data is totaltx price.
This is the code I am running and getting an error:

import statsmodels.api as sm
import pandas as pd
import numpy as np
from numpy.linalg import LinAlgError

def arima(filteredData, coinOutput, window, horizon, trainLength):
    start_index = 0
    end_index = 0
    inputNumber = filteredData.shape[0]
    predictions = np.array([], dtype=np.float32)
    prices = np.array([], dtype=np.float32)
    # sliding on time series data with 1 day step
    while ((end_index) < inputNumber - 1):
        end_index = start_index + trainLength
        trainFeatures = filteredData[start_index:end_index]["totaltx"]
        trainOutput = coinOutput[start_index:end_index]["price"]

        arima = sm.tsa.statespace.SARIMAX(endog=trainOutput.values, exog=trainFeatures.values, order=(window, 0, 0))
        arima_fit = arima.fit(disp=0)
        testdata=filteredData[end_index:end_index+1]["totaltx"]
        total_sample = end_index-start_index
        predicted = arima_fit.predict(start=total_sample, end=total_sample, exog=np.array(testdata.values).reshape(-1,1))
        price = coinOutput[end_index:end_index + 1]["price"].values

        predictions = np.append(predictions, predicted)
        prices = np.append(prices, price)

        start_index = start_index + 1
    return predictions, prices

def processCoins(bitcoinPrice, window, horizon):
    output = bitcoinPrice[horizon:][["date", "day", "year", "price"]]
    return output

trainLength=100;
for window in [3,5]:
    for horizon in [1,2,5,7,10]:
        bitcoinPrice = pd.read_csv("..\prices.csv", sep=",")
        coinOutput = processCoins(bitcoinPrice, window, horizon)
        predictions, prices = arima(bitcoinPrice, coinOutput, window, horizon, trainLength)

In this code, I am using rolling window regression technique. I am training arima for start_index:end_index and predicting the test data with end_index:end_index+1

This the error that is thrown from my code:

Traceback (most recent call last):
  File "C:/PycharmProjects/coinLogPrediction/src/arima.py", line 115, in <module>
    predictions, prices = arima(filteredBitcoinPrice, coinOutput, window, horizon, trainLength, outputFile)
  File "C:/PycharmProjects/coinLogPrediction/src/arima.py", line 64, in arima
    arima_fit = arima.fit(disp=0)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelstsastatespacemlemodel.py", line 469, in fit
    skip_hessian=True, **kwargs)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelsbasemodel.py", line 466, in fit
    full_output=full_output)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelsbaseoptimizer.py", line 191, in _fit
    hess=hessian)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelsbaseoptimizer.py", line 410, in _fit_lbfgs
    **extra_kwargs)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesscipyoptimizelbfgsb.py", line 193, in fmin_l_bfgs_b
    **opts)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesscipyoptimizelbfgsb.py", line 328, in _minimize_lbfgsb
    f, g = func_and_grad(x)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesscipyoptimizelbfgsb.py", line 273, in func_and_grad
    f = fun(x, *args)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesscipyoptimizeoptimize.py", line 292, in function_wrapper
    return function(*(wrapper_args + args))
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelsbasemodel.py", line 440, in f
    return -self.loglike(params, *args) / nobs
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelstsastatespacemlemodel.py", line 646, in loglike
    loglike = self.ssm.loglike(complex_step=complex_step, **kwargs)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelstsastatespacekalman_filter.py", line 825, in loglike
    kfilter = self._filter(**kwargs)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelstsastatespacekalman_filter.py", line 747, in _filter
    self._initialize_state(prefix=prefix, complex_step=complex_step)
  File "C:AppDataLocalContinuumAnaconda3libsite-packagesstatsmodelstsastatespacerepresentation.py", line 723, in _initialize_state
    self._statespaces[prefix].initialize_stationary(complex_step)
  File "_representation.pyx", line 1351, in statsmodels.tsa.statespace._representation.dStatespace.initialize_stationary
  File "_tools.pyx", line 1151, in statsmodels.tsa.statespace._tools._dsolve_discrete_lyapunov
numpy.linalg.linalg.LinAlgError: LU decomposition error.

I believe that there is a bug in statsmodels if I do not have any error. Can you please help me to solve it?

comp-tsa
comp-tsa-statespace

Thanks for the bug report!

This must have something to do with a bad value getting into one of the arrays being decomposed. I can’t replicate any LU decomposition error on my machine, can you try the following and see what you get:

from scipy import linalg
print(linalg.lapack.dgetrf([np.nan]))
print(linalg.lapack.dgetrf([np.inf]))

I’ve got this problem too. I’m seeing numpy.linalg.LinAlgError: LU decomposition error

Replication example

  • dataset: my_data_sample.csv

  • code:

import pandas as pd
import statsmodels.api as sm

endog = pd.read_csv('my_data_sample.csv', index_col=[0])['1b']
order = (25, 1, 21)
seasonal_order = (0, 0, 0, 0)

mod = sm.tsa.SARIMAX(endog, order=order, seasonal_order=seasonal_order)
res = mod.fit(disp=0)
print(res.llf)

As per previous suggestion, 

import numpy as np
from scipy import linalg
print(linalg.lapack.dgetrf([np.nan]))
print(linalg.lapack.dgetrf([np.inf]))

Output:

(array([nan]), array([0], dtype=int32), 0)
(array([inf]), array([0], dtype=int32), 0)

@ChadFulton, @bashtage,
were you able to replicate this error?

Model estimates for me without issue on Windows.

import pandas as pd
import statsmodels.api as sm

endog = pd.read_csv('my_data_sample.csv', index_col=[0])['1b']
order = (25, 1, 21)
seasonal_order = (0, 0, 0, 0)

mod = sm.tsa.SARIMAX(endog, order=order, seasonal_order=seasonal_order)
res = mod.fit(disp=0)
print(res.llf)
c:gitstatsmodelsstatsmodelstsastatespacesarimax.py:959: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
27662.36341690524

This does not look liek a sensible model

ar.L1     0.002454
ar.L2    -0.001238
ar.L3    -0.004040
ar.L4    -0.003733
ar.L5     0.000023
ar.L6     0.003945
ar.L7     0.003790
ar.L8     0.000220
ar.L9    -0.003202
ar.L10   -0.003236
ar.L11   -0.000019
ar.L12    0.001810
ar.L13    0.001179
ar.L14   -0.000020
ar.L15    0.000488
ar.L16    0.001268
ar.L17   -0.000188
ar.L18   -0.002657
ar.L19   -0.002871
ar.L20    0.000396
ar.L21    0.004333
ar.L22    0.003714
ar.L23   -0.000790
ar.L24   -0.004197
ar.L25   -0.003927
ma.L1     0.405848
ma.L2     0.970924
ma.L3     0.619739
ma.L4     0.510109
ma.L5     0.594587
ma.L6     0.456341
ma.L7     0.738567
ma.L8     0.335967
ma.L9     0.230263
ma.L10    0.494425
ma.L11    0.238442
ma.L12    0.554102
ma.L13    0.095503
ma.L14    0.009116
ma.L15   -0.147249
ma.L16   -0.159329
ma.L17    0.155134
ma.L18   -0.055530
ma.L19   -0.081475
ma.L20   -0.112965
ma.L21   -0.336345
sigma2    0.002654

I ran this model fitting on multiple Ubuntu 18.04 systems with Statsmodels 0.11.0.dev0+563.g044456ca7. I just double checked and reran the code, and I’m still having the same error. Here is my full error log:

/home/ubuntu/anaconda3/envs/my_env/bin/python /home/ubuntu/Projects/my_project/my_program.py
/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/statsmodels/tsa/statespace/sarimax.py:959: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
Traceback (most recent call last):
  File "/home/ubuntu/Projects/my_project/my_program .py", line 9, in <module>
    res = mod.fit(disp=0)
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/statsmodels/tsa/statespace/mlemodel.py", line 633, in fit
    skip_hessian=True, **kwargs)
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/statsmodels/base/model.py", line 526, in fit
    full_output=full_output)
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/statsmodels/base/optimizer.py", line 218, in _fit
    hess=hessian)
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/statsmodels/base/optimizer.py", line 440, in _fit_lbfgs
    **extra_kwargs)
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/scipy/optimize/lbfgsb.py", line 199, in fmin_l_bfgs_b
    **opts)
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/scipy/optimize/lbfgsb.py", line 335, in _minimize_lbfgsb
    f, g = func_and_grad(x)
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/scipy/optimize/lbfgsb.py", line 280, in func_and_grad
    f = fun(x, *args)
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/scipy/optimize/optimize.py", line 326, in function_wrapper
    return function(*(wrapper_args + args))
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/statsmodels/base/model.py", line 500, in f
    return -self.loglike(params, *args) / nobs
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/statsmodels/tsa/statespace/mlemodel.py", line 864, in loglike
    loglike = self.ssm.loglike(complex_step=complex_step, **kwargs)
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/statsmodels/tsa/statespace/kalman_filter.py", line 879, in loglike
    kfilter = self._filter(**kwargs)
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/statsmodels/tsa/statespace/kalman_filter.py", line 799, in _filter
    self._initialize_state(prefix=prefix, complex_step=complex_step)
  File "/home/ubuntu/anaconda3/envs/my_env/lib/python3.7/site-packages/statsmodels/tsa/statespace/representation.py", line 760, in _initialize_state
    complex_step=complex_step)
  File "statsmodels/tsa/statespace/_representation.pyx", line 1328, in statsmodels.tsa.statespace._representation.dStatespace.initialize
  File "statsmodels/tsa/statespace/_representation.pyx", line 1321, in statsmodels.tsa.statespace._representation.dStatespace.initialize
  File "statsmodels/tsa/statespace/_initialization.pyx", line 288, in statsmodels.tsa.statespace._initialization.dInitialization.initialize
  File "statsmodels/tsa/statespace/_initialization.pyx", line 406, in statsmodels.tsa.statespace._initialization.dInitialization.initialize_stationary_stationary_cov
  File "statsmodels/tsa/statespace/_tools.pyx", line 1206, in statsmodels.tsa.statespace._tools._dsolve_discrete_lyapunov
numpy.linalg.LinAlgError: LU decomposition error.

(previous comment was submitted too fast)

Ultimately the issue here is the behavior when the parameters imply a non-stationary model, but only barely. I would guess that the symptom of an exception being raised (or not) has to do with the underlying LAPACK library.

Can you check what happens if you run the following?

mod = sm.tsa.SARIMAX(np.zeros(10))
res = mod.filter([1, 1.])
res.filter_results.initial_state_cov

I’m guessing that you will get an exception. When I run this, I get:

[[-4.98960077e+291]]

and of course this is not a valid covariance matrix. That is because there when the process is not covariance stationary then of course there is no stationary distribution, but the default initialization method, «stationary», tries to solve for such a distribution. That’s why @SureshArr was able to solve the problem by using diffuse initialization, which does not assume a stationary distribution.

The problem is not the above example, though, where an error is appropriate because I have specified a non-stationary model. Solving for the stationary covariance requires solving a discrete Lyapunov equation, and the problem is what precision the underlying solver is able to handle.

To see this, just consider using scipy.linalg.solve_discrete_lyapunov. For me, I get:

> print(solve_discrete_lyapunov([[1 - 6e-17]], [[1.]]))
array([[4.50359963e+15]])
> print(solve_discrete_lyapunov([[1 - 5e-17]], [[1.]]))
...
LinAlgError: Matrix is singular.

Both of these models are theoretically stationary, but somewhere between 1-6e-17 and 1-5e-17, we lose enough precision that the model appears non-stationary.

Anyway, we use an internal Cython version of the discrete Lyapunov solver, and when you get the LU error, that’s basically responding to the same numerical problem that causes the LinAlgError: Matrix is singular. above.

Okay, so what should be done?

  1. In the example I gave above, we should probably raise an exception rather than run the Kalman filter with that invalid (negative) covariance matrix.
  2. Rather than the LU decomposition error, we should raise a more information error (‘Implied model is not stationary to numerical precision’ or something).

But I don’t think we can prevent this error condition from developing in all cases. We use a parameter transformation method to enforce parameters implying stationary models, but we can’t (as far as I know) enforce that it will be stationary to whatever precision is required by the discrete Lyapunov solver.

My results are the same as yours:

mod = sm.tsa.SARIMAX(np.zeros(10))
res = mod.filter([1, 1.])

print(res.filter_results.initial_state_cov)
print(scipy.linalg.solve_discrete_lyapunov([[1 - 6e-17]], [[1.]]))
print(scipy.linalg.solve_discrete_lyapunov([[1 - 5e-17]], [[1.]]))

> [[-4.98960077e+291]]
> [[4.50359963e+15]]
> numpy.linalg.LinAlgError: Matrix is singular.

Thanks for checking in. I guess that the linear algebra error doesn’t show up in all non-stationary cases, but I can replicate the LU decomposition error with e.g.:

mod = sarimax.SARIMAX(np.zeros(10), order=(2, 0, 0))
res = mod.filter([1, 2., 1.])


mod = sarimax.SARIMAX(np.zeros(10), order=(2, 0, 0))
res = mod.filter([1, 2., 1.])

I see the LU decomposition error with this example too

mod = sarimax.SARIMAX(np.zeros(10), order=(2, 0, 0))
res = mod.filter([1, 2., 1.])

This throwing an error that suggests the matrix being factorized is exactly singular, and so there is no solution to the set of equations.

Hi folks,
I was having the same error,

Erroneous code:
mod = sm.tsa.SARIMAX(y, order=(0 1,0), seasonal_order=(1,0,0,12))
res = mod.fit()

This gave me error :
LinAlgError: Schur decomposition solver error

I was able to solve this error after doing some modification in code:
mod = sm.tsa.SARIMAX(y, order=(0 1,0), seasonal_order=(1,0,0,12),enforce_stationarity=False)
res = mod.fit()

Hope this helps..🙂

Good morning.
I have the following model:

x8=[11862, 13079.4, 15704.1, 16117.5, 16175.9, 16172.1, 19346.1, 38204.5, 49605.5, 33171, 12728.3, 13951.5, 11343.3, 13157.5, 13839.3, 14497.5, 15427.3, 14420.4, 24202.3, 30710.2, 24675.3, 22392.1, 15182.5, 16155.2, 13123.3, 12768.6, 16292.3, 16617.2, 17164.3, 17145.4, 23225.3,  31324.2, 42991.3, 31658.2, 15718.4, 12853.6, 13397.1, 13681.5,  15182.5, 7623.843544, 4133.86649, 2737.872985]
print(x8)
arimats = sm.tsa.statespace.SARIMAX(x8, order=((1,1,0,0,0,0,1,1,0,0,0,1), 1, (1,1,1,1,0,0,0,1,1,1,1,1)), trend='c')
results = arimats.fit(disp=0)
print(results.summary())

which give me following error:

G:workPythonvenvlibsite-packagesstatsmodelstsastatespacesarimax.py:901: RuntimeWarning: Mean of empty slice.
  params_variance = (residuals[k_params_ma:]**2).mean()
G:workPythonvenvlibsite-packagesnumpycore_methods.py:161: RuntimeWarning: invalid value encountered in double_scalars
  ret = ret.dtype.type(ret / rcount)
G:workPythonvenvlibsite-packagesstatsmodelstsastatespacesarimax.py:963: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
G:workPythonvenvlibsite-packagesstatsmodelstsastatespacesarimax.py:1027: RuntimeWarning: invalid value encountered in less
  params_variance = np.atleast_1d(max(np.array(params_variance), 1e-10))
Traceback (most recent call last):
  File "C:Program FilesJetBrainsPyCharm Community Edition 2019.3.4pluginspython-cehelperspydevpydevd.py", line 1434, in _exec
    pydev_imports.execfile(file, globals, locals)  # execute the script
  File "C:Program FilesJetBrainsPyCharm Community Edition 2019.3.4pluginspython-cehelperspydev_pydev_imps_pydev_execfile.py", line 18, in execfile
    exec(compile(contents+"n", file, 'exec'), glob, loc)
  File "G:/work/Python/work.py", line 272, in <module>
    results = arimats.fit(disp=0)
  File "G:workPythonvenvlibsite-packagesstatsmodelstsastatespacemlemodel.py", line 659, in fit
    skip_hessian=True, **kwargs)
  File "G:workPythonvenvlibsite-packagesstatsmodelsbasemodel.py", line 526, in fit
    full_output=full_output)
  File "G:workPythonvenvlibsite-packagesstatsmodelsbaseoptimizer.py", line 218, in _fit
    hess=hessian)
  File "G:workPythonvenvlibsite-packagesstatsmodelsbaseoptimizer.py", line 440, in _fit_lbfgs
    **extra_kwargs)
  File "G:workPythonvenvlibsite-packagesscipyoptimizelbfgsb.py", line 199, in fmin_l_bfgs_b
    **opts)
  File "G:workPythonvenvlibsite-packagesscipyoptimizelbfgsb.py", line 345, in _minimize_lbfgsb
    f, g = func_and_grad(x)
  File "G:workPythonvenvlibsite-packagesscipyoptimizelbfgsb.py", line 290, in func_and_grad
    f = fun(x, *args)
  File "G:workPythonvenvlibsite-packagesscipyoptimizeoptimize.py", line 327, in function_wrapper
    return function(*(wrapper_args + args))
  File "G:workPythonvenvlibsite-packagesstatsmodelsbasemodel.py", line 500, in f
    return -self.loglike(params, *args) / nobs
  File "G:workPythonvenvlibsite-packagesstatsmodelstsastatespacemlemodel.py", line 889, in loglike
    loglike = self.ssm.loglike(complex_step=complex_step, **kwargs)
  File "G:workPythonvenvlibsite-packagesstatsmodelstsastatespacekalman_filter.py", line 977, in loglike
    kfilter = self._filter(**kwargs)
  File "G:workPythonvenvlibsite-packagesstatsmodelstsastatespacekalman_filter.py", line 897, in _filter
    self._initialize_state(prefix=prefix, complex_step=complex_step)
  File "G:workPythonvenvlibsite-packagesstatsmodelstsastatespacerepresentation.py", line 950, in _initialize_state
    complex_step=complex_step)
  File "statsmodelstsastatespace_representation.pyx", line 1341, in statsmodels.tsa.statespace._representation.dStatespace.initialize
  File "statsmodelstsastatespace_representation.pyx", line 1334, in statsmodels.tsa.statespace._representation.dStatespace.initialize
  File "statsmodelstsastatespace_initialization.pyx", line 288, in statsmodels.tsa.statespace._initialization.dInitialization.initialize
  File "statsmodelstsastatespace_initialization.pyx", line 406, in statsmodels.tsa.statespace._initialization.dInitialization.initialize_stationary_stationary_cov
  File "statsmodelstsastatespace_tools.pyx", line 1284, in statsmodels.tsa.statespace._tools._dsolve_discrete_lyapunov
numpy.linalg.LinAlgError: Schur decomposition solver error.

but if I estimate same model in EViews 8 everything is fine and get following resalts:

Dependent Variable: D(X8_SELH)              
Method: Least Squares                           
Sample (adjusted): 14 42                
Included observations: 29 after adjustments             
Convergence achieved after 27 iterations                
MA Backcast: 2 13               

Variable    Coefficient Std. Error  t-Statistic Prob.  

C   -141.7014351111465  2411.692491347072   -0.05875601289117833    0.9539768830154852
AR(1)   0.4888321752359961  0.3032011216523018  1.612237357738301   0.1292177441320149
AR(2)   -0.269073872605583  0.2759878673985239  -0.9749481929835809 0.3461319285956381
AR(7)   -0.07480108655403679    0.2345011535973987  -0.3189796101492038 0.7544514341212291
AR(8)   -0.09598503257169881    0.3338611796218039  -0.287499830559606  0.7779413864768987
AR(12)  0.5366312922904591  0.2793446293137732  1.921036726600852   0.07532728154533474
MA(1)   -0.4550973692816319 0.3593869062383028  -1.266315943574931  0.2260649585838845
MA(2)   -0.1111162656550283 0.5085687740381292  -0.2184881796275945 0.8302016300068391
MA(3)   -0.1790735773867052 0.1993304358476184  -0.8983754870410313 0.3841760196103828
MA(4)   -0.07394037635414821    0.1858564372478969  -0.3978359719417516 0.6967512681648265
MA(8)   -0.2795817257592794 0.2716593100934508  -1.029163055972951  0.3208564539230987
MA(9)   0.07547670561898145 0.2291571834633819  0.3293665268452831  0.7467544973419736
MA(10)  -0.751443500342628  0.1638699652117488  -4.5856084693228    0.0004239199815789558
MA(11)  0.4445429747690641  0.3591100637960734  1.237901745414478   0.2361110616686708
MA(12)  0.418387381540578   0.2836378559074345  1.475075956282504   0.1623244425028538

R-squared   0.7461266664269699      Mean dependent var      -296.7388625862064
Adjusted R-squared  0.4922533328539399      S.D. dependent var      5860.105173174856
S.E. of regression  4175.696746690447       Akaike info criterion       19.81819434225076
Sum squared resid   244110206.4844966       Schwarz criterion       20.52541632327824
Log likelihood  -272.3638179626359      Hannan-Quinn criter.        20.03968744354025
F-statistic 2.938972187137318       Durbin-Watson stat      2.22774360866717
Prob(F-statistic)   0.02637520109075338         

Inverted AR Roots         .95        .87-.48i      .87+.48i  .50-.86i
     .50+.86i        .06+.95i      .06-.95i -.43+.84i
    -.43-.84i       -.78-.47i     -.78+.47i      -.90
Inverted MA Roots    .96-.06i        .96+.06i      .77+.63i  .77-.63i
     .26-.96i        .26+.96i     -.28+.95i -.28-.95i
         -.49       -.76+.63i     -.76-.63i      -.97

Is there any way to fix it or any workaround for this error?

Thanks in advance.

Hi folks,
I was having the same error,

Erroneous code:
mod = sm.tsa.SARIMAX(y, order=(0 1,0), seasonal_order=(1,0,0,12))
res = mod.fit()

This gave me error :
LinAlgError: Schur decomposition solver error

I was able to solve this error after doing some modification in code:
mod = sm.tsa.SARIMAX(y, order=(0 1,0), seasonal_order=(1,0,0,12),enforce_stationarity=False)
res = mod.fit()

Hope this helps..🙂

This fixed the error, but what was the problem that was fixed by adding _’enforce_stationarity=False’_?
What did this parameter do exactly?

@ChadFulton Using statsmodels==0.12.0, scipy==1.5.2 and numpy==1.19.1 on two different machines, both on Ubuntu 18.04.5, the following script raises (an LU decomposition error) in one case and is being solved in the other:

import numpy as np
from statsmodels.tsa.arima.model import ARIMA

data = np.linspace(0, 2 * np.pi, 1000)
res = ARIMA(endog=data, order=(15, 0, 0)).fit()
res.summary()

The result, on the machine that solves it, is:

                               SARIMAX Results                                
==============================================================================
Dep. Variable:                      y   No. Observations:                 1000
Model:                ARIMA(15, 0, 0)   Log Likelihood                7014.970
Date:                Tue, 23 Mar 2021   AIC                         -13995.940
Time:                        09:30:36   BIC                         -13912.508
Sample:                             0   HQIC                        -13964.230
                               - 1000                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const          3.1416      0.000   2.08e+04      0.000       3.141       3.142
ar.L1         -6.0715      0.001  -5296.783      0.000      -6.074      -6.069
ar.L2        -17.2462      0.002  -9519.194      0.000     -17.250     -17.243
ar.L3        -29.0517      0.001  -3.86e+04      0.000     -29.053     -29.050
ar.L4        -28.2015      0.002  -1.56e+04      0.000     -28.205     -28.198
ar.L5         -5.2284      0.001  -8335.077      0.000      -5.230      -5.227
ar.L6         31.8979      0.002   1.84e+04      0.000      31.895      31.901
ar.L7         59.3725      0.000   1.38e+05      0.000      59.372      59.373
ar.L8         56.9406      0.001   1.06e+05      0.000      56.940      56.942
ar.L9         26.5174      0.001   4.44e+04      0.000      26.516      26.519
ar.L10        -9.6627      0.000  -3.15e+04      0.000      -9.663      -9.662
ar.L11       -29.3404      0.000   -6.8e+04      0.000     -29.341     -29.340
ar.L12       -27.6095      0.002  -1.32e+04      0.000     -27.614     -27.605
ar.L13       -15.4022      0.001  -1.82e+04      0.000     -15.404     -15.401
ar.L14        -5.1188      0.002  -2409.615      0.000      -5.123      -5.115
ar.L15        -0.7954   8.23e-05  -9661.256      0.000      -0.796      -0.795
sigma2      4.724e-08   4.45e-09     10.622      0.000    3.85e-08     5.6e-08
===================================================================================
Ljung-Box (L1) (Q):                 849.71   Jarque-Bera (JB):            689014.32
Prob(Q):                              0.00   Prob(JB):                         0.00
Heteroskedasticity (H):               0.16   Skew:                             9.20
Prob(H) (two-sided):                  0.00   Kurtosis:                       130.27
===================================================================================
Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
[2] Covariance matrix is singular or near-singular, with condition number 8.95e+17. Standard errors may be unstable.

Given your previous comments, I went ahead and tried the following script on both machines:

import numpy as np
from numpy.linalg import LinAlgError
import statsmodels.api as sm
from scipy.linalg import solve_discrete_lyapunov


print('Debub script')
print('')

print('solve_discrete_lyapunov with 1 - 6e-17')
print(solve_discrete_lyapunov([[1 - 6e-17]], [[1.]]))

try:
    print('')
    print('solve_discrete_lyapunov with 1 - 5e-17')
    print(solve_discrete_lyapunov([[1 - 5e-17]], [[1.]]))
except LinAlgError as exc:
    print(str(exc))


mod = sm.tsa.SARIMAX(np.zeros(10))
try:
    print('')
    print('first sarimax')
    res = mod.filter([1, 1.])
    print(res.filter_results.initial_state_cov)
except LinAlgError as exc:
    print(str(exc))


mod = sm.tsa.SARIMAX(np.zeros(10), order=(2, 0, 0))

try:
    print('')
    print('second sarimax with order=(2, 0, 0)')
    res = mod.filter([1, 2., 1.])
    print(res.filter_results.initial_state_cov)
except LinAlgError as exc:
    print(str(exc))

print('')
print('Done with debug script.')
print('')

And both machines give the same output:

Debub script

solve_discrete_lyapunov with 1 - 6e-17
[[4.50359963e+15]]

solve_discrete_lyapunov with 1 - 5e-17
Matrix is singular.

first sarimax
[[-4.98960077e+291]]

second sarimax with order=(2, 0, 0)
LU decomposition error.

Done with debug script.

So I am assuming this has to do with np.linspace(0, 2 * np.pi, 1000) differing to some extent. On inspection they look identical on both machines (could their machine representation might be different?).

I can live with the LU decomposition error, but I am wondering if there is a way to reproduce that behavior consistently.

Источник

Содержание

  1. Statsmodels: SARIMAX python np.linalg.linalg.LinAlgError: LU decomposition error
  2. All 18 comments
  3. Replication example
  4. SARIMAX python np.linalg.linalg.LinAlgError: LU decomposition error #5459
  5. Comments
  6. Replication example
  7. Нежное введение в матричную факторизацию для машинного обучения
  8. Обзор учебника
  9. Что такое матричная декомпозиция?
  10. LU Matrix Разложение
  11. Разложение QR-матрицы
  12. Холесский Разложение
  13. расширения
  14. Дальнейшее чтение
  15. книги
  16. статьи
  17. Резюме

Statsmodels: SARIMAX python np.linalg.linalg.LinAlgError: LU decomposition error

Hello,
I have a problem with time series analysis. I have a dataset with 5 features. Following is the subset of my input dataset:

date,price,year,day,totaltx
1/1/2016 0:00,434.46,2016,1,126762
1/2/2016 0:00,433.59,2016,2,147449
1/3/2016 0:00,430.36,2016,3,148661
1/4/2016 0:00,433.49,2016,4,185279
1/5/2016 0:00,432.25,2016,5,178723
1/6/2016 0:00,429.46,2016,6,184207

My endogenous data is price column and exogenous data is totaltx price.
This is the code I am running and getting an error:

In this code, I am using rolling window regression technique. I am training arima for start_index:end_index and predicting the test data with end_index:end_index+1

This the error that is thrown from my code:

I believe that there is a bug in statsmodels if I do not have any error. Can you please help me to solve it?

Hi folks,
I was having the same error,

Erroneous code:
mod = sm.tsa.SARIMAX(y, order=(0 1,0), seasonal_order=(1,0,0,12))
res = mod.fit()

This gave me error :
LinAlgError: Schur decomposition solver error

I was able to solve this error after doing some modification in code:
mod = sm.tsa.SARIMAX(y, order=(0 1,0), seasonal_order=(1,0,0,12),enforce_stationarity=False)
res = mod.fit()

Hope this helps..🙂

Thanks for the bug report!

This must have something to do with a bad value getting into one of the arrays being decomposed. I can’t replicate any LU decomposition error on my machine, can you try the following and see what you get:

I have run into the same issue on Python 3.7.2, statsmodels 0.9.0. Setting initialization=’approximate_diffuse’ in the SARIMAX call helps to avoid the error.

I’ve got this problem too. I’m seeing numpy.linalg.LinAlgError: LU decomposition error

Replication example

As per previous suggestion,

@ChadFulton, @bashtage,
were you able to replicate this error?

Model estimates for me without issue on Windows.

This does not look liek a sensible model

I ran this model fitting on multiple Ubuntu 18.04 systems with Statsmodels 0.11.0.dev0+563.g044456ca7 . I just double checked and reran the code, and I’m still having the same error. Here is my full error log:

(previous comment was submitted too fast)

Ultimately the issue here is the behavior when the parameters imply a non-stationary model, but only barely. I would guess that the symptom of an exception being raised (or not) has to do with the underlying LAPACK library.

Can you check what happens if you run the following?

I’m guessing that you will get an exception. When I run this, I get:

and of course this is not a valid covariance matrix. That is because there when the process is not covariance stationary then of course there is no stationary distribution, but the default initialization method, «stationary», tries to solve for such a distribution. That’s why @SureshArr was able to solve the problem by using diffuse initialization, which does not assume a stationary distribution.

The problem is not the above example, though, where an error is appropriate because I have specified a non-stationary model. Solving for the stationary covariance requires solving a discrete Lyapunov equation, and the problem is what precision the underlying solver is able to handle.

To see this, just consider using scipy.linalg.solve_discrete_lyapunov . For me, I get:

Both of these models are theoretically stationary, but somewhere between 1-6e-17 and 1-5e-17 , we lose enough precision that the model appears non-stationary.

Anyway, we use an internal Cython version of the discrete Lyapunov solver, and when you get the LU error, that’s basically responding to the same numerical problem that causes the LinAlgError: Matrix is singular. above.

Okay, so what should be done?

  1. In the example I gave above, we should probably raise an exception rather than run the Kalman filter with that invalid (negative) covariance matrix.
  2. Rather than the LU decomposition error, we should raise a more information error (‘Implied model is not stationary to numerical precision’ or something).

But I don’t think we can prevent this error condition from developing in all cases. We use a parameter transformation method to enforce parameters implying stationary models, but we can’t (as far as I know) enforce that it will be stationary to whatever precision is required by the discrete Lyapunov solver.

Источник

SARIMAX python np.linalg.linalg.LinAlgError: LU decomposition error #5459

Hello,
I have a problem with time series analysis. I have a dataset with 5 features. Following is the subset of my input dataset:

date,price,year,day,totaltx
1/1/2016 0:00,434.46,2016,1,126762
1/2/2016 0:00,433.59,2016,2,147449
1/3/2016 0:00,430.36,2016,3,148661
1/4/2016 0:00,433.49,2016,4,185279
1/5/2016 0:00,432.25,2016,5,178723
1/6/2016 0:00,429.46,2016,6,184207

My endogenous data is price column and exogenous data is totaltx price.
This is the code I am running and getting an error:

In this code, I am using rolling window regression technique. I am training arima for start_index:end_index and predicting the test data with end_index:end_index+1

This the error that is thrown from my code:

I believe that there is a bug in statsmodels if I do not have any error. Can you please help me to solve it?

The text was updated successfully, but these errors were encountered:

Thanks for the bug report!

This must have something to do with a bad value getting into one of the arrays being decomposed. I can’t replicate any LU decomposition error on my machine, can you try the following and see what you get:

I have run into the same issue on Python 3.7.2, statsmodels 0.9.0. Setting initialization=’approximate_diffuse’ in the SARIMAX call helps to avoid the error.

I’ve got this problem too. I’m seeing numpy.linalg.LinAlgError: LU decomposition error

Replication example

As per previous suggestion,

@ChadFulton, @bashtage,
were you able to replicate this error?

Model estimates for me without issue on Windows.

This does not look liek a sensible model

I ran this model fitting on multiple Ubuntu 18.04 systems with Statsmodels 0.11.0.dev0+563.g044456ca7 . I just double checked and reran the code, and I’m still having the same error. Here is my full error log:

(previous comment was submitted too fast)

Ultimately the issue here is the behavior when the parameters imply a non-stationary model, but only barely. I would guess that the symptom of an exception being raised (or not) has to do with the underlying LAPACK library.

Can you check what happens if you run the following?

I’m guessing that you will get an exception. When I run this, I get:

and of course this is not a valid covariance matrix. That is because there when the process is not covariance stationary then of course there is no stationary distribution, but the default initialization method, «stationary», tries to solve for such a distribution. That’s why @SureshArr was able to solve the problem by using diffuse initialization, which does not assume a stationary distribution.

The problem is not the above example, though, where an error is appropriate because I have specified a non-stationary model. Solving for the stationary covariance requires solving a discrete Lyapunov equation, and the problem is what precision the underlying solver is able to handle.

To see this, just consider using scipy.linalg.solve_discrete_lyapunov . For me, I get:

Both of these models are theoretically stationary, but somewhere between 1-6e-17 and 1-5e-17 , we lose enough precision that the model appears non-stationary.

Anyway, we use an internal Cython version of the discrete Lyapunov solver, and when you get the LU error, that’s basically responding to the same numerical problem that causes the LinAlgError: Matrix is singular. above.

Okay, so what should be done?

  1. In the example I gave above, we should probably raise an exception rather than run the Kalman filter with that invalid (negative) covariance matrix.
  2. Rather than the LU decomposition error, we should raise a more information error (‘Implied model is not stationary to numerical precision’ or something).

But I don’t think we can prevent this error condition from developing in all cases. We use a parameter transformation method to enforce parameters implying stationary models, but we can’t (as far as I know) enforce that it will be stationary to whatever precision is required by the discrete Lyapunov solver.

Источник

Нежное введение в матричную факторизацию для машинного обучения

Многие сложные матричные операции не могут быть решены эффективно или со стабильностью, используя ограниченную точность компьютеров.

Матричные декомпозиции — это методы, которые сокращают матрицу на составные части, что облегчает вычисление более сложных матричных операций. Методы матричной декомпозиции, также называемые методами матричной факторизации, являются основой линейной алгебры в компьютерах, даже для базовых операций, таких как решение систем линейных уравнений, вычисление обратного и вычисление определителя матрицы.

В этом уроке вы узнаете о разложении матриц и о том, как их вычислять в Python.

После завершения этого урока вы узнаете:

  • Что такое матричная декомпозиция и почему эти типы операций важны.
  • Как рассчитать LU и QR матричные разложения в Python.
  • Как рассчитать разложение матрицы Холецкого в Python.
  • Обновление март / 2018: Исправлена ​​небольшая опечатка в описании QR-разложения.

Обзор учебника

Этот урок разделен на 4 части; они есть:

  1. Что такое матричная декомпозиция?
  2. LU Matrix Разложение
  3. Разложение QR-матрицы
  4. Холесский Разложение

Что такое матричная декомпозиция?

Разложение матрицы — это способ разложения матрицы на составные части.

Это подход, который может упростить более сложные матричные операции, которые могут быть выполнены на разложенной матрице, а не на самой исходной матрице.

Распространенной аналогией для разложения матриц является разложение чисел, такое как разложение 10 на 2 x 5. По этой причине разложение матриц также называется разложением матриц. Подобно факторизации реальных значений, существует много способов разложения матрицы, поэтому существует целый ряд различных методов разложения матрицы.

Два простых и широко используемых метода декомпозиции матрицы — это декомпозиция матрицы LU и декомпозиция матрицы QR.

Далее мы подробнее рассмотрим каждый из этих методов.

LU Matrix Разложение

Разложение LU предназначено для квадратных матриц и разбивает матрицу на компоненты L и U.

Или без точечной записи.

Где A — квадратная матрица, которую мы хотим разложить, L — матрица нижнего треугольника, а U — матрица верхнего треугольника.

Факторы L и U являются треугольными матрицами. Факторизация, возникающая в результате исключения, равна A = LU.

— страница 97, Введение в линейную алгебру, Пятое издание, 2016.

Разложение LU найдено с помощью итерационного числового процесса и может дать сбой для тех матриц, которые не могут быть легко разложены или разложены.

Вариант этого разложения, который численно более устойчив для решения на практике, называется разложением LUP, или разложением LU с частичным поворотом.

Строки родительской матрицы переупорядочены, чтобы упростить процесс декомпозиции, а дополнительная P-матрица указывает способ перестановки результата или возврата результата в исходный порядок. Есть и другие варианты LU.

Разложение LU часто используется для упрощения решения систем линейных уравнений, например, для нахождения коэффициентов в линейной регрессии, а также для вычисления детерминанта и обратной матрицы.

Декомпозиция LU может быть реализована в Python с помощью функции lu (). Более конкретно, эта функция вычисляет разложение LPU.

В приведенном ниже примере сначала определяется квадратная матрица 3 × 3. Рассчитывается разложение LU, а затем исходная матрица восстанавливается по компонентам.

При выполнении примера сначала печатается определенная матрица 3 × 3, затем компоненты разложения P, L и U, а затем, наконец, восстанавливается исходная матрица.

Разложение QR-матрицы

QR-разложение предназначено для m x n матриц (не ограничено квадратными матрицами) и разбивает матрицу на Q и R компоненты.

Или без точечной записи.

Где A — матрица, которую мы хотим разложить, Q — матрица с размером m x m, а R — верхняя треугольная матрица с размером m x n.

QR-разложение находится с использованием итерационного численного метода, который может дать сбой для тех матриц, которые невозможно разложить или легко разложить.

Как и LU-разложение, QR-разложение часто используется для решения систем линейных уравнений, хотя не ограничивается квадратными матрицами.

QR-разложение может быть реализовано в NumPy с помощью функции qr (). По умолчанию функция возвращает матрицы Q и R с меньшими или «уменьшенными» размерами, что является более экономичным. Мы можем изменить это, чтобы вернуть ожидаемые размеры m x m для Q и m x n для R, указав аргумент mode как «complete», хотя это не требуется для большинства приложений.

В приведенном ниже примере определяется матрица 3 × 2, вычисляется QR-разложение, а затем восстанавливается исходная матрица из разложенных элементов.

При выполнении примера сначала печатается определенная матрица 3 × 2, затем элементы Q и R, а затем, наконец, восстановленная матрица, которая соответствует тому, с чего мы начали.

Холесский Разложение

Разложение Холецкого предназначено для квадратных симметричных матриц, где все значения больше нуля, так называемых положительно определенных матриц.

Что касается наших интересов в машинном обучении, мы сосредоточимся на разложении Холецкого для вещественных матриц и проигнорируем случаи при работе с комплексными числами.

Разложение определяется следующим образом:

Или без точечной записи:

Где A — разлагаемая матрица, L — нижняя треугольная матрица, а L ^ T — транспонирование L.

Разложение также может быть записано как произведение верхней треугольной матрицы, например:

Где U — верхняя треугольная матрица.

Разложение Холецкого используется для решения линейных наименьших квадратов для линейной регрессии, а также для методов моделирования и оптимизации.

При разложении симметричных матриц разложение Холецкого почти в два раза эффективнее разложения LU и должно быть предпочтительным в этих случаях.

Хотя симметричные положительно определенные матрицы являются довольно особыми, они встречаются довольно часто в некоторых приложениях, поэтому об их специальной факторизации, называемой разложением Холецкого, полезно знать. Когда вы можете использовать его, разложение Холецкого примерно в два раза быстрее, чем альтернативные методы решения линейных уравнений.

Разложение Cholesky может быть реализовано в NumPy путем вызова функции cholesky (). Функция возвращает только L, так как мы можем легко получить доступ к транспонированию L по мере необходимости.

Приведенный ниже пример определяет симметричную и положительно определенную матрицу 3 × 3 и вычисляет разложение Холецкого, после чего восстанавливается исходная матрица.

При выполнении примера сначала печатается симметричная матрица, затем нижняя треугольная матрица из разложения, за которым следует восстановленная матрица.

расширения

В этом разделе перечислены некоторые идеи по расширению учебника, которые вы, возможно, захотите изучить.

  • Создайте 5 примеров, используя каждую операцию со своими собственными данными.
  • Найдите документы по машинному обучению и найдите 1 пример каждой используемой операции.

Если вы исследуете какое-либо из этих расширений, я хотел бы знать.

Дальнейшее чтение

Этот раздел предоставляет больше ресурсов по теме, если вы хотите углубиться.

книги

  • Раздел 6.6 Матричные разложения. Руководство по линейной алгебре, 2017
  • Лекция 7 QR Факторизация, Численная линейная алгебра, 1997.
  • Раздел 2.3. Разложение ЛУ и его применения, Числовые рецепты: искусство научных вычислений, третье издание, 2007.
  • Раздел 2.10 QR-разложение, Численные рецепты: искусство научных вычислений Третье издание, 2007.
  • Раздел 2.9 Разложение Холецкого, Численные рецепты: искусство научных вычислений Третье издание, 2007.
  • Лекция 23, Разложение Холецкого, Численная линейная алгебра, 1997.

статьи

Резюме

В этом уроке вы обнаружили матричные декомпозиции и способы их вычисления в Python.

В частности, вы узнали:

  • Что такое матричная декомпозиция и почему эти типы операций важны.
  • Как рассчитать LU и QR матричные разложения в Python.
  • Как рассчитать разложение матрицы Холецкого в Python.

У вас есть вопросы?
Задайте свои вопросы в комментариях ниже, и я сделаю все возможное, чтобы ответить.

Источник

Источник

Pranav Jaswal

Guest


  • #1

Pranav Jaswal Asks: Jupyter Notebook Error: Schur decomposition solver error
I am getting an error while using SARIMA Model for Energy Consumption Project. Below is the code and error. If anyone have any solutions or suggestions, they would be greatly appreciated.

Code:

Code: 

from statsmodels.tsa.statespace import sarimax
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

data = pd.read_csv("C:\Users\Jaswal\Desktop\Analytics Vidhya\Assignments\energy consumption.csv", index_col=None)
data

DATE    ENERGY_INDEX
0   01/1939 3.3842
1   02/1939 3.4100
2   03/1939 3.4875
3   04/1939 3.5133
4   05/1939 3.5133
... ... ...
964 05/2019 91.9046
965 06/2019 98.4397
966 07/2019 112.9469
967 08/2019 111.6645
968 09/2019 102.2911

model = sarimax.SARIMAX(data['ENERGY_INDEX'], seasonal_order=(1,1,1,7), order=(2,1,2))
fit1 = model.fit()

valid_data['SARIMA'] = fit1.predict(start="01/1939", end="09/2019", dynamic=True)

Error:

Code: 

LinAlgError                               Traceback (most recent call last)
<ipython-input-55-0d31300a6c27> in <module>
      1 # fit model
      2 model = sarimax.SARIMAX(data['ENERGY_INDEX'], seasonal_order=(1,1,1,7), order=(2,1,2))
----> 3 fit1 = model.fit()
      4 
      5 # make predictions

~Anaconda3libsite-packagesstatsmodelstsastatespacerepresentation.py in _initialize_state(self, prefix, complex_step)
    982                 raise RuntimeError('Initialization is incomplete.')
    983             self._statespaces[prefix].initialize(self.initialization,
--> 984                                                  complex_step=complex_step)
    985         else:
    986             raise RuntimeError('Statespace model not initialized.')

statsmodelstsastatespace_representation.pyx in statsmodels.tsa.statespace._representation.dStatespace.initialize()

statsmodelstsastatespace_representation.pyx in statsmodels.tsa.statespace._representation.dStatespace.initialize()

statsmodelstsastatespace_initialization.pyx in statsmodels.tsa.statespace._initialization.dInitialization.initialize()

statsmodelstsastatespace_initialization.pyx in statsmodels.tsa.statespace._initialization.dInitialization.initialize_stationary_stationary_cov()

statsmodelstsastatespace_tools.pyx in statsmodels.tsa.statespace._tools._dsolve_discrete_lyapunov()

LinAlgError: Schur decomposition solver error.

Python Version = 3.7.6 Statsmodel = 1.2.0

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  • user379612
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user379612 Asks: Calculating inflation for retail data
I am currently using a sales weighted average, on retail price increases, in order to calulate a categories Year-over-Year inflation rate.

I am running into an issue however, when I am using products that previously were subsidized, so their prices were, lets say, $0.10.

This year, they no longer have the subsidy, so they are 10$.

I do not wish to add the subsidy back in, as i would like to know the true inflation numbers.

Now, lets say i have 2 products in this category:

Code: 

Baby Food: $0.10 -> $10  Sales=$100, therefore, Inflation = 1000%, weighted_inflation = 1000%*$100 = 1000

Baby Diapers: 10$ -> $12   Sales=$100, therefore, Inflation = 20%, weighted_inflation = 20%*$100 = 20

Now, if I sum our weighted_inflation numbers, and divide by total category sales, i will get the categories aggregate inflation.

i.e. (1000 + 20) / $200 = 5.1 So, our inflation for this category is 510%!

My question is, what is a better way of getting this aggregate inflation of a category, that doesn’t get skewed by extreme price changes, that will still fall in line with how National Retail inflation is calculated/defined.

p.s. Yes National inflation is based on CPI, specifically basket sizes of customers, however they use the same calculations «weighted sales data» to determine this.

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1129hiki Asks: consistency of OLS estimator
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Источник

Я получаю сообщение об ошибке при использовании модели SARIMA для проекта энергопотребления. Ниже код и ошибка. Если у кого-то есть какие-либо решения или предложения, они будут очень признательны.

Код:

from statsmodels.tsa.statespace import sarimax
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

data = pd.read_csv("C:\Users\Jaswal\Desktop\Analytics Vidhya\Assignments\energy consumption.csv", index_col=None)
data

DATE    ENERGY_INDEX
0   01/1939 3.3842
1   02/1939 3.4100
2   03/1939 3.4875
3   04/1939 3.5133
4   05/1939 3.5133
... ... ...
964 05/2019 91.9046
965 06/2019 98.4397
966 07/2019 112.9469
967 08/2019 111.6645
968 09/2019 102.2911

model = sarimax.SARIMAX(data['ENERGY_INDEX'], seasonal_order=(1,1,1,7), order=(2,1,2))
fit1 = model.fit()

valid_data['SARIMA'] = fit1.predict(start="01/1939", end="09/2019", dynamic=True)

Ошибка:

LinAlgError                               Traceback (most recent call last)
<ipython-input-55-0d31300a6c27> in <module>
      1 # fit model
      2 model = sarimax.SARIMAX(data['ENERGY_INDEX'], seasonal_order=(1,1,1,7), order=(2,1,2))
----> 3 fit1 = model.fit()
      4 
      5 # make predictions

~Anaconda3libsite-packagesstatsmodelstsastatespacerepresentation.py in _initialize_state(self, prefix, complex_step)
    982                 raise RuntimeError('Initialization is incomplete.')
    983             self._statespaces[prefix].initialize(self.initialization,
--> 984                                                  complex_step=complex_step)
    985         else:
    986             raise RuntimeError('Statespace model not initialized.')

statsmodelstsastatespace_representation.pyx in statsmodels.tsa.statespace._representation.dStatespace.initialize()

statsmodelstsastatespace_representation.pyx in statsmodels.tsa.statespace._representation.dStatespace.initialize()

statsmodelstsastatespace_initialization.pyx in statsmodels.tsa.statespace._initialization.dInitialization.initialize()

statsmodelstsastatespace_initialization.pyx in statsmodels.tsa.statespace._initialization.dInitialization.initialize_stationary_stationary_cov()

statsmodelstsastatespace_tools.pyx in statsmodels.tsa.statespace._tools._dsolve_discrete_lyapunov()

LinAlgError: Schur decomposition solver error.

Версия Python = 3.7.6 Статистическая модель = 1.2.0

1 ответ

Проблема должна заключаться в вашем fit1 = model.fit(). Вы оставили скобки пустыми, так что это неприменимо. Чтобы применить его, вы должны использовать x_train и y_train. Поэтому попробуйте в скобках написать так: (x_train, y_train)

Вот полный, но базовый код того, как должно выглядеть ваше решение. Отредактируйте его по своему вкусу с вашими предпочтениями:

from statsmodels.tsa.statespace import sarimax
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

data = pd.read_csv("C:\Users\Jaswal\Desktop\Analytics Vidhya\Assignments\energy consumption.csv", index_col=None)
data

DATE    ENERGY_INDEX
0   01/1939 3.3842
1   02/1939 3.4100
2   03/1939 3.4875
3   04/1939 3.5133
4   05/1939 3.5133
... ... ...
964 05/2019 91.9046
965 06/2019 98.4397
966 07/2019 112.9469
967 08/2019 111.6645
968 09/2019 102.2911

model = sarimax.SARIMAX(data['ENERGY_INDEX'], seasonal_order=(1,1,1,7), order=(2,1,2))
fit1 = model.fit(x_train,y_train) #update

valid_data['SARIMA'] = fit1.predict(start="01/1939", end="09/2019", dynamic=True)


-1

Erling Olsen
22 Дек 2021 в 21:25

Источник

From Wikipedia, the free encyclopedia

In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix.

Statement[edit]

The Schur decomposition reads as follows: if A is an n × n square matrix with complex entries, then A can be expressed as[1][2][3]

A = Q U Q^{-1}

where Q is a unitary matrix (so that its inverse Q−1 is also the conjugate transpose Q* of Q), and U is an upper triangular matrix, which is called a Schur form of A. Since U is similar to A, it has the same spectrum, and since it is triangular, its eigenvalues are the diagonal entries of U.

The Schur decomposition implies that there exists a nested sequence of A-invariant subspaces {0} = V0 ⊂ V1 ⊂ ⋯ ⊂ Vn = Cn, and that there exists an ordered orthonormal basis (for the standard Hermitian form of Cn) such that the first i basis vectors span Vi for each i occurring in the nested sequence. Phrased somewhat differently, the first part says that a linear operator J on a complex finite-dimensional vector space stabilizes a complete flag (V1,…,Vn).

Proof[edit]

A constructive proof for the Schur decomposition is as follows: every operator A on a complex finite-dimensional vector space has an eigenvalue λ, corresponding to some eigenspace Vλ. Let Vλ⊥ be its orthogonal complement. It is clear that, with respect to this orthogonal decomposition, A has matrix representation (one can pick here any orthonormal bases Z1 and Z2 spanning Vλ and Vλ⊥ respectively)

begin{bmatrix} Z_1 & Z_2 end{bmatrix}^{*} A begin{bmatrix}Z_1 & Z_2end{bmatrix} = begin{bmatrix} lambda , I_{lambda} & A_{12} \ 0 & A_{22} end{bmatrix}: begin{matrix} V_{lambda} \ oplus \ V_{lambda}^{perp} end{matrix} rightarrow begin{matrix} V_{lambda} \ oplus \ V_{lambda}^{perp} end{matrix}

where Iλ is the identity operator on Vλ. The above matrix would be upper-triangular except for the A22 block. But exactly the same procedure can be applied to the sub-matrix A22, viewed as an operator on Vλ⊥, and its submatrices. Continue this way until the resulting matrix is upper triangular. Since each conjugation increases the dimension of the upper-triangular block by at least one, this process takes at most n steps. Thus the space Cn will be exhausted and the procedure has yielded the desired result.

The above argument can be slightly restated as follows: let λ be an eigenvalue of A, corresponding to some eigenspace Vλ. A induces an operator T on the quotient space Cn/Vλ. This operator is precisely the A22 submatrix from above. As before, T would have an eigenspace, say Wμ ⊂ Cn modulo Vλ. Notice the preimage of Wμ under the quotient map is an invariant subspace of A that contains Vλ. Continue this way until the resulting quotient space has dimension 0. Then the successive preimages of the eigenspaces found at each step form a flag that A stabilizes.

Notes[edit]

Although every square matrix has a Schur decomposition, in general this decomposition is not unique. For example, the eigenspace Vλ can have dimension > 1, in which case any orthonormal basis for Vλ would lead to the desired result.

Write the triangular matrix U as U = D + N, where D is diagonal and N is strictly upper triangular (and thus a nilpotent matrix). The diagonal matrix D contains the eigenvalues of A in arbitrary order (hence its Frobenius norm, squared, is the sum of the squared moduli of the eigenvalues of A, while
the Frobenius norm of A, squared, is the sum of the squared singular values of A). The nilpotent part N is generally not unique either, but its Frobenius norm is uniquely determined by A (just because the Frobenius norm of A is equal to the Frobenius norm of U = D + N).

It is clear that if A is a normal matrix, then U from its Schur decomposition must be a diagonal matrix and the column vectors of Q are the eigenvectors of A. Therefore, the Schur decomposition extends the spectral decomposition. In particular, if A is positive definite, the Schur decomposition of A, its spectral decomposition, and its singular value decomposition coincide.

A commuting family {Ai} of matrices can be simultaneously triangularized, i.e. there exists a unitary matrix Q such that, for every Ai in the given family, Q Ai Q* is upper triangular. This can be readily deduced from the above proof. Take element A from {Ai} and again consider an eigenspace VA. Then VA is invariant under all matrices in {Ai}. Therefore, all matrices in {Ai} must share one common eigenvector in VA. Induction then proves the claim. As a corollary, we have that every commuting family of normal matrices can be simultaneously diagonalized.

In the infinite dimensional setting, not every bounded operator on a Banach space has an invariant subspace. However, the upper-triangularization of an arbitrary square matrix does generalize to compact operators. Every compact operator on a complex Banach space has a nest of closed invariant subspaces.

Computation[edit]

The Schur decomposition of a given matrix is numerically computed by the QR algorithm or its variants. In other words, the roots of the characteristic polynomial corresponding to the matrix are not necessarily computed ahead in order to obtain its Schur decomposition. Conversely, the QR algorithm can be used to compute the roots of any given characteristic polynomial by finding the Schur decomposition of its companion matrix. Similarly, the QR algorithm is used to compute the eigenvalues of any given matrix, which are the diagonal entries of the upper triangular matrix of the Schur decomposition. Although the QR algorithm is formally an infinite sequence of operations, convergence to machine precision is practically achieved in {mathcal {O}}(n^{3}) operations.[4]
See the Nonsymmetric Eigenproblems section in LAPACK Users’ Guide.[5]

Applications[edit]

Lie theory applications include:

  • Every invertible operator is contained in a Borel group.
  • Every operator fixes a point of the flag manifold.

Generalized Schur decomposition[edit]

Given square matrices A and B, the generalized Schur decomposition factorizes both matrices as A=QSZ^* and B=QTZ^*, where Q and Z are unitary, and S and T are upper triangular. The generalized Schur decomposition is also sometimes called the QZ decomposition.[2]: 375 

The generalized eigenvalues lambda that solve the generalized eigenvalue problem {displaystyle Amathbf {x} =lambda Bmathbf {x} } (where x is an unknown nonzero vector) can be calculated as the ratio of the diagonal elements of S to those of T. That is, using subscripts to denote matrix elements, the ith generalized eigenvalue lambda _{i} satisfies {displaystyle lambda _{i}=T_{ii}/S_{ii}}.

References[edit]

  1. ^ Horn, R.A. & Johnson, C.R. (1985). Matrix Analysis. Cambridge University Press. ISBN 0-521-38632-2.(Section 2.3 and further at p. 79)
  2. ^ a b Golub, G.H. & Van Loan, C.F. (1996). Matrix Computations (3rd ed.). Johns Hopkins University Press. ISBN 0-8018-5414-8.(Section 7.7 at p. 313)
  3. ^ Schott, James R. (2016). Matrix Analysis for Statistics (3rd ed.). New York: John Wiley & Sons. pp. 175–178. ISBN 978-1-119-09247-6.
  4. ^ Trefethen, Lloyd N.; Bau, David (1997). Numerical linear algebra. Philadelphia: Society for Industrial and Applied Mathematics. pp. 193–194. ISBN 0-89871-361-7. OCLC 36084666.{{cite book}}: CS1 maint: date and year (link)
  5. ^ Anderson, E; Bai, Z; Bischof, C; Blackford, S; Demmel, J; Dongarra, J; Du Croz, J; Greenbaum, A; Hammarling, S; McKenny, A; Sorensen, D (1995). LAPACK Users guide. Philadelphia, PA: Society for Industrial and Applied Mathematics. ISBN 0-89871-447-8.
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