https://networks.tir.tw/wiki/ 2026-05-19T06:28:30+00:00 https://networks.tir.tw/wiki/ https://networks.tir.tw/wiki/_media/logo.png text/html 2022-11-27T05:36:18+00:00 Anonymous (anonymous@undisclosed.example.com) https://networks.tir.tw/wiki/technical:finite-entropy-estimate?rev=1669527378&do=diff Estimate entropy of a finite discrete system using limited samples Following idea of: * Ma, “Calculation of Entropy from Data of Motion”, Journal of Statistical Physics 26, 221–240 (1981) URL. zotero Consider a system with a state space of the size $|\mathbb{X}| = \Gamma$\begin{align} H & = - \sum_{x\in\mathbb{X}} \frac{1}{\Gamma} \ln\frac{1}{\Gamma} \\ & = \ln \Gamma . \end{align}$\Gamma$$\mathbb{X}$$\delta_{x_i,x_j}$$N$$\Gamma$$x_i=x_j$$P_\mathrm{c} = \Gamma^{-1}$$N(N-1)/2$$N$\begin{equat… text/html 2022-10-29T15:10:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://networks.tir.tw/wiki/technical:ising1d?rev=1667056210&do=diff One-dimensional Ising model The Hamiltonian is defined as \begin{equation} H = E(\mathbf{s}) \equiv - J\sum_i s_i s_{i+1} - h\sum_i s_i . \end{equation} where the state vector is $\mathbf{s}=(s_0,s_1,\ldots,s_{N-1})$ with $s_i = \pm 1$ for $i = 0\ldots N-1$ belonging to the least residue system modulo $N$, i.e., $\mathbb{Z}_N$, and corresponding to the periodic boundary condition. Partition function The partition function is defined as \begin{equation} Z \equiv \sum_{\mathbf{s}} e^{-\beta E(… text/html 2021-09-08T03:09:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://networks.tir.tw/wiki/technical:losses?rev=1631070598&do=diff Loss functions Cross entropy loss For the results of a classifier, the output is an array of unnormalized scores $x_i$, $i=0\ldots K-1$, for each class. Assuming the target is $i=y$. The cross entropy loss for each sample is given by: \begin{equation} l_\text{CE}(\mathbf{x}) = -\ln\frac{e^{x_y}}{\sum_ie^{x_i}} = -x_y+\sum_ie^{x_i} \end{equation} while \begin{equation} L_\text{CE} = \langle l_\text{CE}\rangle \end{equation} is the average over a given dataset.$x_i$\begin{equation}r = \mathop{\t… text/html 2019-09-25T13:44:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://networks.tir.tw/wiki/technical:names?rev=1569419049&do=diff Naming convention for data octal sequencing For the binary representations in machines, only numbers with fractions denominated by 2's exponents can be precisely represented with finite number of digits. Among these, octal numbers are closest to the familiar decimal numbers. text/html 2022-10-10T06:03:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://networks.tir.tw/wiki/technical:poisson_spike_train?rev=1665381780&do=diff Inhomogeneous Poisson spike train generation def gen_spikes(r,dt,rng): '''Generate spike train from Poisson rate Parameters ---------- r: Array of spike rates dt: Time step rng: Random number generator ''' i = 0 s = 0 spks = [] while True: s += rng.exponential() while s>r[i]*dt: s -= r[i]*dt i += 1 if i>=len(r): break else: spks.append(i*dt+s/r[i]) continue … text/html 2026-01-26T13:45:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://networks.tir.tw/wiki/technical:python?rev=1769435155&do=diff Python with jupyter-lab Add the conda-forge channel by running: conda config --add channels conda-forge Then, install jupyterlab: conda install jupyterlab Useful packages: conda install jupyter-ruff Create environment for different python version text/html 2021-10-12T15:15:04+00:00 Anonymous (anonymous@undisclosed.example.com) https://networks.tir.tw/wiki/technical:sliding_window?rev=1634051704&do=diff Sliding window time histogram Given the spike time, plot the event rate within a sliding window. def sliding_window(ts,wsz=10): '''calculate sliding window time histogram Parameters ---------- ts: Array of time points wsz: Window size Returns ------- tss: Array of time points rates: Firing rates at the time points ''' tw = ts[0]-wsz # initialized with the time when the first event hits the window i0 = 0 i1 = 0 lt = len(ts)… text/html 2024-09-05T05:37:49+00:00 Anonymous (anonymous@undisclosed.example.com) https://networks.tir.tw/wiki/technical:surprise?rev=1725514669&do=diff Surprise in prediction of spin given the effective local field Consider an event with two possible outcome, + and -. If a prediction of $p_+ = p_\mathrm{p}$ is made, how much surprise will be there when the actual outcome is known? It should be $-\log_2 p_\mathrm{p}$ if the outcome is + and $-\log_2 \left(1-p_\mathrm{p}\right)$$h^\mathrm{eff}$$T=\beta^{-1}=1$$s$$s=+1$$$ \frac{e^{h^\mathrm{eff}}}{e^{h^\mathrm{eff}}+e^{-h^\mathrm{eff}}} = \frac{1}{2}\left(1+\tanh h^\mathrm{eff}\right). $$$s=-1$$…