five_parameter_logistics_curve.ipynb

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            "source": [
                "#helpful sources in 5PL and 4PL discussions\n",
                "1) https://www.bio-rad.com/webroot/web/pdf/lsr/literature/Bulletin_3022.pdf\n",
                "2) https://www.graphpad.com/guides/prism/latest/curve-fitting/reg_asymmetric_dose_response_ec.htm\n",
                "3) https://www.mathworks.com/matlabcentral/fileexchange/38043-five-parameters-logistic-regression-there-and-back-again\n",
                "4) spiess et al. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2386824/\n",
                "5) https://stackoverflow.com/questions/55725139/fit-sigmoid-function-s-shape-curve-to-data-using-python\n",
                "Autoregressive modeling and diagnostics for qPCR amplification\n",
                "6) https://people.duke.edu/~ccc14/pcfb/analysis.html\n"
            ]
        },
        {
            "cell_type": "code",
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            "source": [
                "import numpy as np\n",
                "import numpy.random as npr\n",
                "import matplotlib.pyplot as plt\n",
                "from scipy import optimize\n",
                "from scipy.misc import derivative\n",
                "import plotly.graph_objects as go\n",
                "from scipy.optimize import curve_fit"
            ]
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                "def logistic5(x, a, b, c, d, g): # spiess et al. 5P logistic curve\n",
                "    # a is ground (bottom) asymptote\n",
                "    # b is slope\n",
                "    # c is inflection point\n",
                "    # d is maximum (top) asymptote\n",
                "    # g is \"...the additional asymmetry parameter. Setting parameter g = 1 yields the four-parameter curves...\"\n",
                "    numerator = a-d\n",
                "    denominator = (1+(x/c)**b)**g\n",
                "    return (d+(numerator/denominator))\n",
                "\n",
                "# provide initial ground_asymptote based on data\n",
                "def ground_asymptote (fluor):\n",
                "    # average first n cycles or so \n",
                "    # I think Anitoa does first 13 cycles\n",
                "    n = 5\n",
                "    return (sum(fluor[0:n]) / len(fluor[0:n]))\n",
                "\n",
                "# provide initial ground_asymptote based on data\n",
                "def max_asymptote (fluor):\n",
                "    # average last n cycles or so \n",
                "    # I think Anitoa does first 13 cycles\n",
                "    m = 5\n",
                "    return (sum(fluor[-m:]) / len(fluor[-m:]))\n",
                "\n",
                "# data for iconic, well-behaved rxn\n",
                "iconic_y = [-23.72, -47.4, -27.93, -4.51, 0.65, 12.79, 29.96, 50.24, 79.85, 129.94, 203.74, 321.5, 488.96, 729.49, 1047.78, 1479.57, 1963.18, 2499.28, 3025.52, 3514.9, 3919.58, 4241.47, 4475.93, 4648.39, 4768.29, 4840.66, 4892.92, 4930.58, 4962.46, 4970.39, 4989.07, 4989.16, 4997.8, 4995.48, 5002.82, 5001.02, 5003.96, 5008.01, 5005.17, 5004.64, 5004.83]\n",
                "cycles = np.arange(1,42)\n",
                "\n",
                "# initial guesses: a, b, c, d, g\n",
                "# p0 = [-7.498, 9.945, 18, 5013, 0.5451] #graphpad values with this dataset\n",
                "# p0 = [4.04930663e+00, 6.34869459e+00, 2.12543050e+01, 5.00090309e+03, 2.31685219e+00] #scipy output using graphpad\n",
                "p0 = (ground_asymptote(iconic_y), 10, 20, max_asymptote(iconic_y), 1) # g = 1 is no asymmetry\n",
                "\n",
                "# parameters (optimal values), parameters covariance\n",
                "popt, pcov = curve_fit(logistic5, cycles, iconic_y, p0) #least sq = dogbox\n",
                "print (popt)\n",
                "# calc opt fit\n",
                "opt_y = logistic5(cycles, *popt)\n",
                "\n",
                "# plot data vs opt fit\n",
                "fig = go.Figure()\n",
                "fig.add_trace(go.Scatter(x=cycles, y=iconic_y, mode='lines+markers', name='data', marker=dict(size=4, color='black'), line=dict(color='black', width=0.5)))\n",
                "fig.add_trace(go.Scatter(x=cycles, y=opt_y, mode='lines+markers', name='fit', marker=dict(size=4, color='red'), line=dict(color='red', width=0.5)))\n",
                "\n",
                "fig.update_layout(\n",
                "    autosize=False,\n",
                "    width=800,\n",
                "    height=500, \n",
                "    title=\"Fitting qPCR Data vs Optimal Fit (Logistic 5PL)\",\n",
                "    xaxis_title=\"Cycle\",\n",
                "    yaxis_title=\"RFU\",\n",
                "    legend_title=\"Source\",)\n",
                "fig.show()\n"
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                "# calc 2nd derivative\n",
                "second = []\n",
                "for each in cycles:\n",
                "    second.append(derivative(logistic5, x0=each, n=2, args=popt, dx=1e-6)) #numerical approx, not symbolic 2nd deriv\n",
                "print (second)\n",
                "\n",
                "\n",
                "\n"
            ]
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                                    "title": {
                                        "text": "Cycle"
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            "source": [
                "# print (popt)\n",
                "from sympy import *\n",
                "x, a, b, c, d, g = symbols('x a b c d g')\n",
                "# eqn = d+((a-d)/((1+(x/c)**b)**g))\n",
                "firstD = diff(d+((a-d)/((1+(x/c)**b)**g)), x) #1st deriv\n",
                "secondD = diff(d+((a-d)/((1+(x/c)**b)**g)), x, 2) #2nd deriv\n",
                "secondD_with_subs = secondD.subs({a: 4.04930663e+00, b: 6.34869459e+00, c: 2.12543050e+01, d: 5.00090309e+03, g:2.31685219e+00 })\n",
                "\n",
                "# secondD_with_subs.evalf(subs={x:20})\n",
                "# secondD.evalf(subs={x: 20, a: 4.04930663e+00, b: 6.34869459e+00, c: 2.12543050e+01, d: 5.00090309e+03, g:2.31685219e+00 }) #-70.9297\n",
                "secondDSubs_lambdified = lambdify(x, secondD_with_subs, \"numpy\")\n",
                "secondDPoints = secondDSubs_lambdified(cycles)\n",
                "# print (secondDPoints)\n",
                "\n",
                "# plot 2nd deriv data\n",
                "deriv = go.Figure()\n",
                "deriv.add_trace(go.Scatter(x=cycles, y=second, mode='lines+markers', name='Taylor Approx', marker=dict(size=4, color='black'), line=dict(color='black', width=0.5)))\n",
                "deriv.add_trace(go.Scatter(x=cycles, y=secondDPoints, mode='lines+markers', name='Symbolic Substitution', marker=dict(size=4, color='red'), line=dict(color='red', width=0.5)))\n",
                "\n",
                "deriv.update_layout(\n",
                "    autosize=False,\n",
                "    width=800,\n",
                "    height=500, \n",
                "    title=\"Second Derivative Plot of Logistic 5PL Using Two Methods\",\n",
                "    xaxis_title=\"Cycle\",\n",
                "    yaxis_title=\"RFU?\",\n",
                "    legend_title=\"Legend Title\",)\n",
                "deriv.show()\n"
            ]
        },
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            "cell_type": "code",
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                    "name": "stdout",
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                    "text": [
                        "                        fun: -84.6195681160942\n",
                        " lowest_optimization_result:       fun: -84.6195681160942\n",
                        " hess_inv: array([[0.0730115]])\n",
                        "      jac: array([7.62939453e-06])\n",
                        "  message: 'Optimization terminated successfully.'\n",
                        "     nfev: 10\n",
                        "      nit: 4\n",
                        "     njev: 5\n",
                        "   status: 0\n",
                        "  success: True\n",
                        "        x: array([21.3321877])\n",
                        "                    message: ['requested number of basinhopping iterations completed successfully']\n",
                        "      minimization_failures: 0\n",
                        "                       nfev: 1136\n",
                        "                        nit: 100\n",
                        "                       njev: 568\n",
                        "                          x: array([21.3321877])\n"
                    ]
                }
            ],
            "source": [
                "secondDSubs_scipyed = lambdify(x, secondD_with_subs, \"scipy\")\n",
                "# print (secondDSubs_scipyed)\n",
                "# localMin=optimize.minimize(secondDSubs_scipyed, x0=10)\n",
                "# print (localMin)\n",
                "# myMin = optimize.minimize_scalar(secondDSubs_scipyed)\n",
                "# find global min\n",
                "globMin = optimize.basinhopping(secondDSubs_scipyed, x0=20, stepsize=1)\n",
                "print (globMin)\n",
                "# grid = (0, 40)\n",
                "# xmin_global = optimize.brute(secondDSubs_scipyed, (grid, ))\n",
                "# print(\"Global minima found %s\" % xmin_global)\n",
                "# root = optimize.root(secondDSubs_scipyed, 18)  # our initial guess is 18; find max/min and find halfway pt\n",
                "# print(\"First root found %s\" % root.x)\n",
                "# root2 = optimize.root(f, -2.5)\n",
                "# print(\"Second root found %s\" % root2.x)\n",
                "\n",
                "# # Constrain optimization\n",
                "# xmin_local = optimize.fminbound(f, 0, 10)\n",
                "# print(\"Local minimum found %s\" % xmin_local)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": [
                "# 2nd derivates are nice but they won't help find the Cq.\n",
                "# back ground reading\n",
                "# \"Automatically finding starting values for sigmoid curve\", https://stats.stackexchange.com/questions/242594/automatically-finding-starting-values-for-a-sigmoid-curve\n",
                "# Calculate Ct/Cq from amplication curves, https://github.com/ewallace/tidyqpcr/issues/19\n",
                "# \"How to smooth a curve in the right way?\", https://stackoverflow.com/questions/20618804/how-to-smooth-a-curve-in-the-right-way\n",
                "# \"Python baseline correction library\", Python baseline correction library\n",
                "\n"
            ]
        }
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