| Relations | Name | matrix elements |
|---|---|---|
| $A = A^{T}$ | symmetric | $a_{ij} = a_{ji}$ |
| $A = \\left (A^{T} \\right )^{-1}$ | real orthogonal | $\\sum_k a_{ik} a_{jk} = \\sum_k a_{ki} a_{kj} = \\delta_{ij}$ |
| $A = A^{ * }$ | real matrix | $a_{ij} = a_{ij}^{ * }$ |
| $A = A^{\\dagger}$ | hermitian | $a_{ij} = a_{ji}^{ * }$ |
| $A = \\left (A^{\\dagger} \\right )^{-1}$ | unitary | $\\sum_k a_{ik} a_{jk}^{ * } = \\sum_k a_{ki}^{ * } a_{kj} = \\delta_{ij}$ |
| Discrete PDF | Continuous PDF | |
|---|---|---|
| Domain | $\\left\\{x_1, x_2, x_3, \\dots, x_N\\right\\}$ | $[a,b]$ |
| Probability | $p(x_i)$ | $p(x)dx$ |
| Cumulative | $P_i=\\sum_{l=1}^ip(x_l)$ | $P(x)=\\int_a^xp(t)dt$ |
| Positivity | $ 0\\le p(x_i)\\le 1$ | $ p(x) \\ge 0$ |
| Positivity | $ 0\\le P_i\\le 1$ | $ 0\\le P(x)\\le 1$ |
| Monotonic | $P_i\\ge P_j$ if $x_i\\ge x_j$ | $P(x_i)\\ge P(x_j)$ if $x_i\\ge x_j$ |
| Normalization | $P_N=1$ | $P(b)=1$ |
| $x$-bin | ran0 | ran1 | ran2 | ran3 |
|---|---|---|---|---|
| 0.0-0.1 | 1013 | 991 | 938 | 1047 |
| 0.1-0.2 | 1002 | 1009 | 1040 | 1030 |
| 0.2-0.3 | 989 | 999 | 1030 | 993 |
| 0.3-0.4 | 939 | 960 | 1023 | 937 |
| 0.4-0.5 | 1038 | 1001 | 1002 | 992 |
| 0.5-0.6 | 1037 | 1047 | 1009 | 1009 |
| 0.6-0.7 | 1005 | 989 | 1003 | 989 |
| 0.7-0.8 | 986 | 962 | 985 | 954 |
| 0.8-0.9 | 1000 | 1027 | 1009 | 1023 |
| 0.9-1.0 | 991 | 1015 | 961 | 1026 |
| $\\mu$ | 0.4997 | 0.5018 | 0.4992 | 0.4990 |
| $\\sigma$ | 0.2882 | 0.2892 | 0.2861 | 0.2915 |
| Year | Hares (x1000) | Lynx (x1000) |
|---|---|---|
| 1900 | 30.0 | 4.0 |
| 1901 | 47.2 | 6.1 |
| 1902 | 70.2 | 9.8 |
| 1903 | 77.4 | 35.2 |
| 1904 | 36.3 | 59.4 |
| 1905 | 20.6 | 41.7 |
| 1906 | 18.1 | 19.0 |
| 1907 | 21.4 | 13.0 |
| 1908 | 22.0 | 8.3 |
| 1909 | 25.4 | 9.1 |
| 1910 | 27.1 | 7.4 |
| 1911 | 40.3 | 8.0 |
| 1912 | 57 | 12.3 |
| 1913 | 76.6 | 19.5 |
| 1914 | 52.3 | 45.7 |
| 1915 | 19.5 | 51.1 |
| 1916 | 11.2 | 29.7 |
| 1917 | 7.6 | 15.8 |
| 1918 | 14.6 | 9.7 |
| 1919 | 16.2 | 10.1 |
| 1920 | 24.7 | 8.6 |
![](\"fig/Hudson_Bay_data.png\")
\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "## Why now create a computer model for the hare and lynx populations?\n",
+ "\n",
+ "We see from the plot that there are indeed fluctuations.\n",
+ "We would like to create a mathematical model that explains these\n",
+ "population fluctuations. Ecologists have predicted that in a simple\n",
+ "predator-prey system that a rise in prey population is followed (with\n",
+ "a lag) by a rise in the predator population. When the predator\n",
+ "population is sufficiently high, then the prey population begins\n",
+ "dropping. After the prey population falls, then the predator\n",
+ "population falls, which allows the prey population to recover and\n",
+ "complete one cycle of this interaction. Thus, we see that\n",
+ "qualitatively oscillations occur. Can a mathematical model predict\n",
+ "this? What causes cycles to slow or speed up? What affects the\n",
+ "amplitude of the oscillation or do you expect to see the oscillations\n",
+ "damp to a stable equilibrium? The models tend to ignore factors like\n",
+ "climate and other complicating factors. How significant are these?\n",
+ "\n",
+ " * We see oscillations in the data\n",
+ "\n",
+ " * What causes cycles to slow or speed up?\n",
+ "\n",
+ " * What affects the amplitude of the oscillation or do you expect to see the oscillations damp to a stable equilibrium?\n",
+ "\n",
+ " * With a model we can better *understand the data*\n",
+ "\n",
+ " * More important: we can understand the ecology dynamics of\n",
+ " predator-pray populations\n",
+ "\n",
+ "## The traditional (top-down) approach\n",
+ "\n",
+ "\n",
+ "The classical way (in all books) is to present the Lotka-Volterra equations:"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\begin{align*}\n",
+ "\\frac{dH}{dt} &= H(a - b L)\\\\\n",
+ "\\frac{dL}{dt} &= - L(d - c H)\n",
+ "\\end{align*}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here,\n",
+ "\n",
+ " * $H$ is the number of preys\n",
+ "\n",
+ " * $L$ the number of predators\n",
+ "\n",
+ " * $a$, $b$, $d$, $c$ are parameters\n",
+ "\n",
+ "Most books quickly establish the model and then use considerable space on\n",
+ "discussing the qualitative properties of this *nonlinear system of\n",
+ "ODEs* (which cannot be solved)\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "## Basic mathematics notation\n",
+ "\n",
+ " * Time points: $t_0,t_1,\\ldots,t_m$\n",
+ "\n",
+ " * Uniform distribution of time points: $t_n=n\\Delta t$\n",
+ "\n",
+ " * $H^n$: population of hares at time $t_n$\n",
+ "\n",
+ " * $L^n$: population of lynx at time $t_n$\n",
+ "\n",
+ " * We want to model the changes in populations, $\\Delta H=H^{n+1}-H^n$\n",
+ " and $\\Delta L=L^{n+1}-L^n$ during a general time interval $[t_{n+1},t_n]$\n",
+ " of length $\\Delta t=t_{n+1}-t_n$\n",
+ "\n",
+ "## Basic dynamics of the population of hares\n",
+ "\n",
+ "\n",
+ "The population of hares evolves due to births and deaths exactly as a bacteria population:"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\Delta H = a \\Delta t H^n\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "However, hares have an additional loss in the population because\n",
+ "they are eaten by lynx.\n",
+ "All the hares and lynx can form\n",
+ "$H\\cdot L$ pairs in total. When such pairs meet during a time\n",
+ "interval $\\Delta t$, there is some\n",
+ "small probablity that the lynx will eat the hare.\n",
+ "So in fraction $b\\Delta t HL$, the lynx eat hares. This\n",
+ "loss of hares must be accounted for. Subtracted in the equation for hares:"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\Delta H = a\\Delta t H^n - b \\Delta t H^nL^n\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Basic dynamics of the population of lynx\n",
+ "\n",
+ "\n",
+ "We assume that the primary growth for the lynx population depends on sufficient food for raising lynx kittens, which implies an adequate source of nutrients from predation on hares. Thus, the growth of the lynx population does not only depend of how many lynx there are, but on how many hares they can eat.\n",
+ "In a time interval $\\Delta t HL$ hares and lynx can meet, and in a\n",
+ "fraction $b\\Delta t HL$ the lynx eats the hare. All of this does not\n",
+ "contribute to the growth of lynx, again just a fraction of\n",
+ "$b\\Delta t HL$ that we write as\n",
+ "$d\\Delta t HL$. In addition, lynx die just as in the population\n",
+ "dynamics with one isolated animal population, leading to a loss\n",
+ "$-c\\Delta t L$.\n",
+ "\n",
+ "\n",
+ "\n",
+ "The accounting of lynx then looks like"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\Delta L = d\\Delta t H^nL^n - c\\Delta t L^n\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Evolution equations\n",
+ "\n",
+ "\n",
+ "By writing up the definition of $\\Delta H$ and $\\Delta L$, and putting\n",
+ "all assumed known terms $H^n$ and $L^n$ on the right-hand side, we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "H^{n+1} = H^n + a\\Delta t H^n - b\\Delta t H^n L^n\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "L^{n+1} = L^n + d\\Delta t H^nL^n - c\\Delta t L^n\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Note:\n",
+ "\n",
+ " * These equations are ready to be implemented!\n",
+ "\n",
+ " * But to start, we need $H^0$ and $L^0$ \n",
+ " (which we can get from the data)\n",
+ "\n",
+ " * We also need values for $a$, $b$, $d$, $c$\n",
+ "\n",
+ "## Adapt the model to the Hudson Bay case\n",
+ "\n",
+ "\n",
+ " * As always, models tend to be general - as here, applicable\n",
+ " to \"all\" predator-pray systems\n",
+ "\n",
+ " * The critical issue is whether the *interaction* between hares and lynx\n",
+ " is sufficiently well modeled by $\\hbox{const}HL$\n",
+ "\n",
+ " * The parameters $a$, $b$, $d$, and $c$ must be\n",
+ " estimated from data\n",
+ "\n",
+ " * Measure time in years\n",
+ "\n",
+ " * $t_0=1900$, $t_m=1920$\n",
+ "\n",
+ "## The program"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "\n",
+ "def solver(m, H0, L0, dt, a, b, c, d, t0):\n",
+ " \"\"\"Solve the difference equations for H and L over m years\n",
+ " with time step dt (measured in years.\"\"\"\n",
+ "\n",
+ " num_intervals = int(m/float(dt))\n",
+ " t = np.linspace(t0, t0 + m, num_intervals+1)\n",
+ " H = np.zeros(t.size)\n",
+ " L = np.zeros(t.size)\n",
+ "\n",
+ " print('Init:', H0, L0, dt)\n",
+ " H[0] = H0\n",
+ " L[0] = L0\n",
+ "\n",
+ " for n in range(0, len(t)-1):\n",
+ " H[n+1] = H[n] + a*dt*H[n] - b*dt*H[n]*L[n]\n",
+ " L[n+1] = L[n] + d*dt*H[n]*L[n] - c*dt*L[n]\n",
+ " return H, L, t\n",
+ "\n",
+ "# Load in data file\n",
+ "data = np.loadtxt('src/Hudson_Bay.csv', delimiter=',', skiprows=1)\n",
+ "# Make arrays containing x-axis and hares and lynx populations\n",
+ "t_e = data[:,0]\n",
+ "H_e = data[:,1]\n",
+ "L_e = data[:,2]\n",
+ "\n",
+ "# Simulate using the model\n",
+ "H, L, t = solver(m=20, H0=34.91, L0=3.857, dt=0.1,\n",
+ " a=0.4807, b=0.02482, c=0.9272, d=0.02756,\n",
+ " t0=1900)\n",
+ "\n",
+ "# Visualize simulations and data\n",
+ "plt.plot(t_e, H_e, 'b-+', t_e, L_e, 'r-o', t, H, 'm--', t, L, 'k--')\n",
+ "plt.xlabel('Year')\n",
+ "plt.ylabel('Numbers of hares and lynx')\n",
+ "plt.axis([1900, 1920, 0, 140])\n",
+ "plt.title(r'Population of hares and lynx 1900-1920 (x1000)')\n",
+ "plt.legend(('H_e', 'L_e', 'H', 'L'), loc='upper left')\n",
+ "plt.savefig('Hudson_Bay_sim.pdf')\n",
+ "plt.savefig('Hudson_Bay_sim.png')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## The plot\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "![](\"fig/Hudson_Bay_sim.png\")
\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "If we perform a least-square fitting, we can find optimal values for the parameters $a$, $b$, $d$, $c$. The optimal parameters are $a=0.4807$, $b=0.02482$, $d=0.9272$ and $c=0.02756$. These parameters result in a slightly modified initial conditions, namely $H(0) = 34.91$ and $L(0)=3.857$. With these parameters we are now ready to solve the equations and plot these data together with the experimental values.\n",
+ "\n",
+ "\n",
+ "\n",
+ "## Linear regression in Python"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from IPython.display import display\n",
+ "import sklearn\n",
+ "from sklearn.linear_model import LinearRegression\n",
+ "from sklearn.tree import DecisionTreeRegressor\n",
+ "\n",
+ "\n",
+ "data = np.loadtxt('src/Hudson_Bay.csv', delimiter=',', skiprows=1)\n",
+ "x = data[:,0]\n",
+ "y = data[:,1]\n",
+ "line = np.linspace(1900,1920,1000,endpoint=False).reshape(-1,1)\n",
+ "reg = DecisionTreeRegressor(min_samples_split=3).fit(x.reshape(-1,1),y.reshape(-1,1))\n",
+ "plt.plot(line, reg.predict(line), label=\"decision tree\")\n",
+ "regline = LinearRegression().fit(x.reshape(-1,1),y.reshape(-1,1))\n",
+ "plt.plot(line, regline.predict(line), label= \"Linear Regression\")\n",
+ "plt.plot(x, y, label= \"Linear Regression\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Linear Least squares in R"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ " HudsonBay = read.csv(\"src/Hudson_Bay.csv\",header=T)\n",
+ " fix(HudsonBay)\n",
+ " dim(HudsonBay)\n",
+ " names(HudsonBay)\n",
+ " plot(HudsonBay$Year, HudsonBay$Hares..x1000.)\n",
+ " attach(HudsonBay)\n",
+ " plot(Year, Hares..x1000.)\n",
+ " plot(Year, Hares..x1000., col=\"red\", varwidth=T, xlab=\"Years\", ylab=\"Haresx 1000\")\n",
+ " summary(HudsonBay)\n",
+ " summary(Hares..x1000.)\n",
+ " library(MASS)\n",
+ " library(ISLR)\n",
+ " scatter.smooth(x=Year, y = Hares..x1000.)\n",
+ " linearMod = lm(Hares..x1000. ~ Year)\n",
+ " print(linearMod)\n",
+ " summary(linearMod)\n",
+ " plot(linearMod)\n",
+ " confint(linearMod)\n",
+ " predict(linearMod,data.frame(Year=c(1910,1914,1920)),interval=\"confidence\")\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Non-Linear Least squares in R"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ " set.seed(1485)\n",
+ " len = 24\n",
+ " x = runif(len)\n",
+ " y = x^3+rnorm(len, 0,0.06)\n",
+ " ds = data.frame(x = x, y = y)\n",
+ " str(ds)\n",
+ " plot( y ~ x, main =\"Known cubic with noise\")\n",
+ " s = seq(0,1,length =100)\n",
+ " lines(s, s^3, lty =2, col =\"green\")\n",
+ " m = nls(y ~ I(x^power), data = ds, start = list(power=1), trace = T)\n",
+ " class(m)\n",
+ " summary(m)\n",
+ " power = round(summary(m)$coefficients[1], 3)\n",
+ " power.se = round(summary(m)$coefficients[2], 3)\n",
+ " plot(y ~ x, main = \"Fitted power model\", sub = \"Blue: fit; green: known\")\n",
+ " s = seq(0, 1, length = 100)\n",
+ " lines(s, s^3, lty = 2, col = \"green\")\n",
+ " lines(s, predict(m, list(x = s)), lty = 1, col = \"blue\")\n",
+ " text(0, 0.5, paste(\"y =x^ (\", power, \" +/- \", power.se, \")\", sep = \"\"), pos = 4)\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Important Matrix and vector handling packages\n",
+ "\n",
+ "The Numerical Recipes codes have been rewritten in Fortran 90/95 and\n",
+ "C/C++ by us. The original source codes are taken from the widely used\n",
+ "software package LAPACK, which follows two other popular packages\n",
+ "developed in the 1970s, namely EISPACK and LINPACK.\n",
+ "\n",
+ " * LINPACK: package for linear equations and least square problems.\n",
+ "\n",
+ " * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website | Relations | Name | matrix elements |
|---|---|---|
| $A = A^{T}$ | symmetric | $a_{ij} = a_{ji}$ |
| $A = \\left (A^{T} \\right )^{-1}$ | real orthogonal | $\\sum_k a_{ik} a_{jk} = \\sum_k a_{ki} a_{kj} = \\delta_{ij}$ |
| $A = A^{ * }$ | real matrix | $a_{ij} = a_{ij}^{ * }$ |
| $A = A^{\\dagger}$ | hermitian | $a_{ij} = a_{ji}^{ * }$ |
| $A = \\left (A^{\\dagger} \\right )^{-1}$ | unitary | $\\sum_k a_{ik} a_{jk}^{ * } = \\sum_k a_{ki}^{ * } a_{kj} = \\delta_{ij}$ |
| Discrete PDF | Continuous PDF | |
|---|---|---|
| Domain | $\\left\\{x_1, x_2, x_3, \\dots, x_N\\right\\}$ | $[a,b]$ |
| Probability | $p(x_i)$ | $p(x)dx$ |
| Cumulative | $P_i=\\sum_{l=1}^ip(x_l)$ | $P(x)=\\int_a^xp(t)dt$ |
| Positivity | $ 0\\le p(x_i)\\le 1$ | $ p(x) \\ge 0$ |
| Positivity | $ 0\\le P_i\\le 1$ | $ 0\\le P(x)\\le 1$ |
| Monotonic | $P_i\\ge P_j$ if $x_i\\ge x_j$ | $P(x_i)\\ge P(x_j)$ if $x_i\\ge x_j$ |
| Normalization | $P_N=1$ | $P(b)=1$ |
| $x$-bin | ran0 | ran1 | ran2 | ran3 |
|---|---|---|---|---|
| 0.0-0.1 | 1013 | 991 | 938 | 1047 |
| 0.1-0.2 | 1002 | 1009 | 1040 | 1030 |
| 0.2-0.3 | 989 | 999 | 1030 | 993 |
| 0.3-0.4 | 939 | 960 | 1023 | 937 |
| 0.4-0.5 | 1038 | 1001 | 1002 | 992 |
| 0.5-0.6 | 1037 | 1047 | 1009 | 1009 |
| 0.6-0.7 | 1005 | 989 | 1003 | 989 |
| 0.7-0.8 | 986 | 962 | 985 | 954 |
| 0.8-0.9 | 1000 | 1027 | 1009 | 1023 |
| 0.9-1.0 | 991 | 1015 | 961 | 1026 |
| $\\mu$ | 0.4997 | 0.5018 | 0.4992 | 0.4990 |
| $\\sigma$ | 0.2882 | 0.2892 | 0.2861 | 0.2915 |