Using O-notation for asymptotic estimation of the number of additions in recursive function

the number of additions that are executed during the calculation is a(n). How can i find an asymptotic estimation for the function mystery(n) with the help of the O-notation and master theorem. Note: here the question is not asked for the value of mystery(n), but rather for the number of additions!

def mystery(n):     if n==0:         return n * n     return 2 * mystery(n/3) + 4 * n 

Monte Carlo errors estimation routine

I would value your opinion on the following piece of code. I am rather new to both Python and Monte Carlo analysis, so I was wondering whether the routine makes sense to more experienced and knowledgeable users.

def MC_analysis_a():     x = spin_lock_durations     y_signal_a = (a_norm1, a_norm2, a_norm3, a_norm4, a_norm5, a_norm6, a_norm7, a_norm8)     x = np.array(x, dtype = float)     y_signal_a = np.array(y_signal_a, dtype = float)          def func(x, a, b):         return a * np.exp(-b * x)      initial_guess = [1.0, 1.0]     fitting_parameters, covariance_matrix = optimize.curve_fit(func, x, y_signal_a, initial_guess)     print(round(fitting_parameters[1], 2))      # ---> PRODUCING PARAMETERS ESTIMATES      total_iterations = 5000     MC_pars = np.array([])      for iTrial in range(total_iterations):         xTrial = x         yTrial = y_signal_a + np.random.normal(loc = y_signal_a, scale = e_signal_a, size = np.size(y_signal_a))         try:             iteration_identifiers, covariance_matrix = optimize.curve_fit(func, xTrial, yTrial, initial_guess)         except:             dumdum = 1             continue      # ---> STACKING RESULTS          if np.size(MC_pars) < 1:             MC_pars = np.copy(iteration_identifiers)         else:             MC_pars = np.vstack((MC_pars, iteration_identifiers))      # ---> SLICING THE ARRAY      print(np.shape(MC_pars))     # print(np.median(aFitpyars[:,1]))     print(np.std(MC_pars[:,1])) 

The output I get is apparently satisfactory and plausible.

Many thanks in advance to any contributor!

estimation of unknown parameter of a distribution

I have a normal distribution with unknown parameter $ \mu$ . I have a prior of $ mean = \mu_0, var=\sigma_0^2$

Say I have $ n$ observations from this distribution:

$ $ A_1, A_2, …..A_n$ $

What is $ $ E(\mu | A_1, A_2, A_3 …. A_n)?$ $

My attempt:

Isn’t this just the average over all the observations? (But this looks like a frequentist approach). If we have prior, does this mean that we are adopting a Bayesian approach?

Efficient algorithm for numerical estimation of 3D rotation matrix

I am writing a computer vision related program and stuck on a problem. I have a system of quadratic equations

rot

const

where M is a constant 3×3 matrix and unit is the 3×3 identity matrix.

The first set of equations contains 6 independent equations which leaves us with 3 degrees of freedom of SO(3) group. The second set of 3 equations just states that the matrix sym is symmetric.

Since this program will repeatedly solve these equations, I am searching for an efficient algorithm and/or a C/C++ implementation which I can run on Raspberry Pi, i.e. not monsters like Matlab, Mathematica or root.

[GET][NULLED] – WP Cost Estimation & Payment Forms Builder v9.668

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[GET][NULLED] – WP Cost Estimation & Payment Forms Builder v9.668

Applying a Sigma-Point Kalman Filter to State of Charge Estimation

I’ve found a research paper that would allow me to implement a Kalman filter to estimate state of charge for a LiPo battery, but cannot make sense of some of the symbols as I’m a first year. Can anybody give a rough description of the process used (for the Sigma Point Kalman Filter detailed) and what some of the symbols in the augmented state covariance mean. We have some understanding but do not fully understand the meaning of some of the positive and negative symbols in the superscript and subscript mean.

Need to get an overview so we can finish develop a working model in Matlab.

Here is a link to the paper: https://www.researchgate.net/publication/308672015_State-of-charge_estimation_based_on_microcontroller-implemented_sigma-point_Kalman_filter_in_a_modular_cell_balancing_system_for_Lithium-Ion_battery_packs

Cheers.

Bayesian parameter estimation

I am generally not that knowledgeable for math, so if my question is too broad or inaccurate, please let me know.

I am currently reading a paragraph of one paper (https://www.fil.ion.ucl.ac.uk/spm/doc/papers/karl_bayes_dyn.pdf), which is about Bayesian parameter estimation of nonlinear dynamical system. Here is the link for screenshoted paragraph (link).

I have been going through more or less well entire paragraphs but there is one particular transition that I could not figure out, which is about Equation (13). How I can get specified equations for ‘parameter covariance’ and ‘conditional mean parameters given a data’ as stated in the paper?

University year 1, chapter 7 (estimation)

A random sample of 25 female students is chosen from students at higher education establishments in a particular area of a country, and it is found that their mean height is 163 centimeters with a sample variance of 64.

a. Assuming that the distribution of the heights of the students may be regarded as normally distributed, calculate a 97% confidence interval for the mean height of female students.

b. You are asked to obtain a 97% confidence interval for that mean height of width 2 centimeters. What sample size would be needed in order to achieve that degree of accuracy?

c. Suppose that a sample of 15 had been obtained from a single student hostel, in which there were a large number of female students (still with a distribution of heights which was well approximated by the normal distribution). The mean height is found to be 155 centimeters. Calculate a 99% confidence interval for the mean height of female students in the hostel. How do their heights compare with the group in 4(a)(i) ? [(159.3, 166.7), 339, (148.9, 161.1)]

Hi! Are you able to solve this qns?

Understanding the solution to this polynomial estimation problem

Show that there exist $ K, N > 0$ such that for $ x ∈ R$ , $ $ x ≥ N \implies \frac{3x^ 2 − 4x + 8}{ 5x + 6} ≥ Kx.$ $

Solution:

For $ x ≥ 4,$ we have $ 4x ≤ x^ 2$ so that $ $ 3x^ 2 − 4x + 8 ≥ 3x^ 2 − x^ 2 = 2x^ 2.$ $

Similarly, for $ x ≥ 1,$ we have $ $ 5x + 6 ≤ 11x.$ $ Therefore, when $ x ≥ 4,$ we have $ $ \frac{3x^ 2 − 4x + 8}{ 5x + 6} ≥\frac{ 2x^ 2}{ 11x} = \frac{2} {11} x,$ $

and so we can take $ N = 4$ and $ K = \frac{2}{ 11} .$

I’m struggling to follow the logical steps in this solution. I’m not sure why the inequality of $ x ≥ 4$ and $ x ≥ 1$ has been chosen. Can someone please break down this solution and show where the steps come from?