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btel/ornstein_uhlenbeck_covariance(1).ipynb

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ornstein_uhlenbeck_covariance(1).ipynb
{
"metadata": {
"name": "ornstein_uhlenbeck_covariance"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
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"cells": [
{
"cell_type": "code",
"collapsed": false,
"input": "%config InlineBackend.figure_format = 'svg'",
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 1
},
{
"cell_type": "code",
"collapsed": false,
"input": "import numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy import linalg, special",
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 2
},
{
"cell_type": "code",
"collapsed": false,
"input": "mu, theta, sigma = 1, .3,1",
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 3
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{
"cell_type": "code",
"collapsed": false,
"input": "def cov_matrix(sigma, theta, tmax, dt=1):\n \"\"\"calculate and return covariance matrix of ornstein-uhlenbeck proccess\"\"\"\n time_lag = np.arange(int(tmax/dt))*dt\n column = sigma**2/(2.*theta)*np.exp(-theta*time_lag)\n cov = linalg.toeplitz(column)\n return cov\n\ndef rotate_matrix(vec, mat):\n return np.dot(np.dot(vec, mat), vec.T)\n\ndef gaussian_1d(x, mu, sigma):\n return 1./(np.sqrt(2*np.pi)*sigma)*np.exp(-(x-mu)**2/(2*sigma**2))\n\ndef prob_threshold_crossing(theta, mu, sigma):\n return special.erfc((theta-mu)/(np.sqrt(2)*sigma))/2.\n\ndef calculate_cov_matrix(y, tmax, dt=1.):\n y_nomean = y - np.mean(y)\n n_max = int(tmax/dt)\n n_pts = len(y_nomean)\n cov = [np.mean(y_nomean[i:]*y_nomean[:n_pts-i])\n for i in range(n_max)]\n return linalg.toeplitz(cov)\n ",
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 4
},
{
"cell_type": "code",
"collapsed": false,
"input": "def ornstein_uhlenbeck(mu, sigma, theta, tmax, dt=1.):\n \"\"\"return a realisation of ornstein-uhlenbeck process\"\"\"\n\n diffcoef = sigma**2\n y0 = mu\n tau = 1./theta\n \n n_steps = int(tmax/dt)\n \n A = np.sqrt(diffcoef*tau/2.*(1-np.exp(-2*dt/tau)))\n\n noise = np.random.randn(n_steps)\n\n y = np.zeros(n_steps)\n y[0] = y0\n\n for i in range(n_steps-1):\n y[i+1] = y0 + (y[i]-y0)*np.exp(-dt/tau)+A*noise[i]\n return y",
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 5
},
{
"cell_type": "markdown",
"metadata": {},
"source": "Simulate Ornstein-Uhlenbeck process\n-----------------------------------"
},
{
"cell_type": "code",
"collapsed": false,
"input": "ou_samples = ornstein_uhlenbeck(mu, sigma, theta,100000)\n#ou_samples = np.random.randn(100000)*sigma+mu\nplt.plot(ou_samples[:1000])\nplt.title('sample process')",
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "pyout",
"prompt_number": 6,
"text": "<matplotlib.text.Text at 0x3b6a550>"
},
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"output_type": "display_data",
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"input": "cov_est = calculate_cov_matrix(ou_samples,10)\ncov_theory = cov_matrix(sigma, theta, 10)\n\nplt.plot(cov_theory[:,0], cov_est[:,0], 'o', [0, sigma/2.], [0, sigma/2.])\nplt.xlabel('theory')\nplt.ylabel('estimation')\nplt.title('covariance matrix coefficients')\nplt.xlim([0, sigma/2.])\nplt.ylim([0, sigma/2.])",
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},
{
"cell_type": "code",
"collapsed": false,
"input": "plt.subplot(121)\nplt.imshow(cov_theory, interpolation='nearest')\nplt.subplot(122)\nplt.plot(cov_theory[:,4])",
"language": "python",
"metadata": {},
"outputs": [
{
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"text": "[<matplotlib.lines.Line2D at 0x3dbe890>]"
},
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"source": "Simulate spikes\n---------------"
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"input": "filter_length = 10\ntime = np.arange(-filter_length, 0, 1)\nG_filter = np.exp(time/2.)\n#normalize to unit variance\nG_filter = G_filter/np.sqrt(np.sum(G_filter**2))\nplt.plot(time, G_filter)\nplt.xlabel('time lag')\nplt.title('linear filter')",
"language": "python",
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"outputs": [
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},
{
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"input": "threshold = 3.\nprojected_samples = np.convolve(ou_samples, G_filter)\nspikes = (projected_samples>threshold)\nspike_rate = np.sum(spikes)*1./len(spikes)\nplt.plot(spikes[:1000])\nplt.title('simulated spikes')",
"language": "python",
"metadata": {},
"outputs": [
{
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"text": "<matplotlib.text.Text at 0x403c390>"
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},
{
"cell_type": "markdown",
"metadata": {},
"source": "Estimate firing rate\n--------------------"
},
{
"cell_type": "code",
"collapsed": false,
"input": "cov_theory = cov_matrix(sigma, theta, filter_length)\nsigma_proj = np.sqrt(rotate_matrix(G_filter[::-1], cov_theory))\nmu_proj = np.sum(G_filter)*mu\nprob_theory = prob_threshold_crossing(threshold, mu_proj, sigma_proj)\nprint \"Spike rate (estimated/theory):\", spike_rate,\"/\", prob_theory\nprint \"Projection std. (estimated/theory):\", np.std(projected_samples),\"/\", sigma_proj",
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": "Spike rate (estimated/theory): 0.31606155446 / 0.317601489327\nProjection std. (estimated/theory): 2.09040112349 / 2.09294790519\n"
}
],
"prompt_number": 11
},
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"input": "n, bins = np.histogram(projected_samples,100, density=True)\nplt.bar(bins[:-1],n,width=bins[1]-bins[0], label='simulation',fc='none')\nplt.plot(bins, gaussian_1d(bins,mu_proj, sigma_proj),'r', label='theory')\nplt.title(\"distribution of projection coefficients\")\nplt.axvline(threshold,label='threshold')\nplt.legend(frameon=False,loc='upper left')",
"language": "python",
"metadata": {},
"outputs": [
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"input": "def compare_theory_simulation(mu, theta, sigma, threshold):\n npts = 50000\n ou_samples = ornstein_uhlenbeck(mu, sigma, theta, npts)\n spikes = np.convolve(ou_samples, G_filter)>threshold\n spike_rate = np.sum(spikes)*1./len(spikes)\n \n cov_theory = cov_matrix(sigma, theta, filter_length)\n sigma_proj = np.sqrt(rotate_matrix(G_filter[::-1], cov_theory))\n mu_proj = np.sum(G_filter)*mu\n prob_theory = prob_threshold_crossing(threshold, mu_proj, sigma_proj)\n \n return spike_rate, prob_theory\n\ndef plot_comparison(value_list, title=None, labels=True):\n simulation, theory = np.array(value_list).T\n plt.plot(simulation, theory, 'o')\n xmin, xmax = plt.xlim()\n ymin, ymax = plt.ylim()\n mmax = max(xmax, ymax)\n mmin = min(xmin, ymin)\n plt.plot([mmin, mmax], [mmin, mmax], 'k-')\n plt.xticks([mmin, mmax])\n plt.yticks([mmin, mmax])\n if labels:\n plt.xlabel('spike rate - simulation',labelpad=-8)\n plt.ylabel('spike rate - theory', labelpad=-8)\n \n ax = plt.gca()\n if title:\n plt.text(0.1, 0.8, title, transform=ax.transAxes)\n plt.xlim([mmin, mmax])\n plt.ylim([mmin, mmax])",
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"source": "Compare theoretical and estimated firing rates"
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"input": "vary_theta = [compare_theory_simulation(mu, _, sigma, threshold) for _ in np.linspace(0.5,2,5)]\nvary_sigma = [compare_theory_simulation(mu, theta, _, threshold) for _ in np.linspace(0.5,5,5)]\nvary_mu = [compare_theory_simulation(_, theta, sigma, threshold) for _ in np.linspace(0.5,2,5)]\nvary_threshold = [compare_theory_simulation(mu, theta, sigma, _) for _ in np.linspace(0.5,2,5)]",
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"input": "plt.figure(figsize=(8,5))\nplt.subplot(221)\nplot_comparison(vary_theta,'vary theta')\nplt.subplot(222)\nplot_comparison(vary_sigma,'vary sigma', labels=False)\nplt.subplot(223)\nplot_comparison(vary_mu, 'vary mu', labels=False)\nplt.subplot(224)\nplot_comparison(vary_threshold, 'vary threshold', labels=False)",
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"source": "Recover filters with STA\n------------------------"
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"source": "Compute the spike-triggered average and whiten it using the whitening transformation $z=Vx$, where $V=E D^{-\\frac{1}{2}} E^T$. ($E$ is the matrix of eigenvectors aranged in columns and $D$ is the diagonal matrix of eigenvalues of covariance matrix of $x$)"
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"input": "sta = np.correlate(ou_samples[:]-ou_samples.mean(), spikes[:])[::-1]\nsta = sta/np.sum(spikes)\n\nevals, evecs = np.linalg.eig(cov_theory)\nd = 1./np.sqrt(evals)\nwhitening_filter = np.dot(evecs,np.dot(np.diag(d),evecs.T))\nsta_whitened = np.dot(whitening_filter, sta)\n\n\nplt.plot(time, sta_whitened, label='whitened STA')\nplt.plot(time, G_filter, label='original STA')\nplt.plot(time, sta, label='raw STA')\nplt.title('filter estimates')\nplt.xlabel('time lag')\nplt.legend(loc='upper left', frameon=False)",
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{
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"input": "",
"language": "python",
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}
],
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}
]
}
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