2.LinearRegression.ipynb

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      "ans = /Users/dann/Documents/Projects/repos/MachineLearning/2.Exercise1\n"
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    "cd ./2.Exercise1/\n",
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    "## Linear Regression with One Variable"
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    "### Goal of this exercise:\n",
    "> Predict the profit of delivering food to multiple cities.\n",
    "\n",
    "**Data:**\n",
    "* population of the city (in 10,000s)\n",
    "* profit (negative value indicates loss; in $10,000s)"
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   "source": [
    "### Loading data:"
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     "text": [
      "m =  97\n",
      "n =  1\n"
     ]
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   "source": [
    "data = load('ex1data1.txt');\n",
    "x = data(:, 1);\n",
    "y = data(:, 2);\n",
    "[m, n] = size(x);\n",
    "m, n"
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       "\t\t<text><tspan font-family=\"Arial\">20</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
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       "\t\t<text><tspan font-family=\"Arial\">25</tspan></text>\n",
       "\t</g>\n",
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       "\t\t<text><tspan font-family=\"Arial\">Profit in $10,000s</tspan></text>\n",
       "\t</g>\n",
       "</g>\n",
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       "\t\t<text><tspan font-family=\"Arial\">Population of City in 10,000s</tspan></text>\n",
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   "source": [
    "plot(x, y, 'rx', 'MarkerSize', 10);\n",
    "ylabel('Profit in $10,000s');\n",
    "xlabel('Population of City in 10,000s');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gradient Descent:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Cost function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "function [cost] = computeCost(X, y, theta)\n",
    "    m = length(X);\n",
    "    cost = (2 * m) ^ -1 * sum((X * theta - y) .^ 2);\n",
    "endfunction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Gradient descent algorithm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "function [theta] = gradientDescent(x, y, ITERATIONS, ALPHA)\n",
    "\n",
    "    [m, n] = size(x);\n",
    "    \n",
    "    X = [ones(m, 1), x];\n",
    "    theta = zeros(n + 1, 1);\n",
    "    \n",
    "    costs = zeros(1, ITERATIONS + 1);\n",
    "    costs(1) = computeCost(X, y, theta);\n",
    "    \n",
    "    for iter = 1:ITERATIONS\n",
    "        theta = theta - (ALPHA * (m ^ -1)) * sum(((X * theta) - y) .* X)';\n",
    "        costs(iter + 1) = computeCost(X, y, theta);\n",
    "    end;\n",
    "    \n",
    "    plot(0:ITERATIONS, costs);\n",
    "    xlabel('iteration');\n",
    "    ylabel('MSE');\n",
    "    title('Training evolution');\n",
    "endfunction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "  -3.6303\n",
      "   1.1664\n",
      "\n"
     ]
    },
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L153.9,337.4 L154.1,337.4 L154.4,337.4 L154.6,337.5   L154.9,337.5 L155.1,337.5 L155.4,337.5 L155.7,337.5 L155.9,337.5 L156.2,337.5 L156.4,337.5 L156.7,337.5   L156.9,337.5 L157.2,337.5 L157.4,337.6 L157.7,337.6 L157.9,337.6 L158.2,337.6 L158.5,337.6 L158.7,337.6   L159.0,337.6 L159.2,337.6 L159.5,337.6 L159.7,337.6 L160.0,337.6 L160.2,337.6 L160.5,337.7 L160.8,337.7   L161.0,337.7 L161.3,337.7 L161.5,337.7 L161.8,337.7 L162.0,337.7 L162.3,337.7 L162.5,337.7 L162.8,337.7   L163.0,337.7 L163.3,337.8 L163.6,337.8 L163.8,337.8 L164.1,337.8 L164.3,337.8 L164.6,337.8 L164.8,337.8   L165.1,337.8 L165.3,337.8 L165.6,337.8 L165.8,337.8 L166.1,337.8 L166.4,337.8 L166.6,337.9 L166.9,337.9   L167.1,337.9 L167.4,337.9 L167.6,337.9 L167.9,337.9 L168.1,337.9 L168.4,337.9 L168.7,337.9 L168.9,337.9   L169.2,337.9 L169.4,337.9 L169.7,337.9 L169.9,338.0 L170.2,338.0 L170.4,338.0 L170.7,338.0 L170.9,338.0   L171.2,338.0 L171.5,338.0 L171.7,338.0 L172.0,338.0 L172.2,338.0 L172.5,338.0 L172.7,338.0 L173.0,338.0   L173.2,338.1 L173.5,338.1 L173.7,338.1 L174.0,338.1 L174.3,338.1 L174.5,338.1 L174.8,338.1 L175.0,338.1   L175.3,338.1 L175.5,338.1 L175.8,338.1 L176.0,338.1 L176.3,338.1 L176.6,338.1 L176.8,338.1 L177.1,338.2   L177.3,338.2 L177.6,338.2 L177.8,338.2 L178.1,338.2 L178.3,338.2 L178.6,338.2 L178.8,338.2 L179.1,338.2   L179.4,338.2 L179.6,338.2 L179.9,338.2 L180.1,338.2 L180.4,338.2 L180.6,338.2 L180.9,338.3 L181.1,338.3   L181.4,338.3 L181.6,338.3 L181.9,338.3 L182.2,338.3 L182.4,338.3 L182.7,338.3 L182.9,338.3 L183.2,338.3   L183.4,338.3 L183.7,338.3 L183.9,338.3 L184.2,338.3 L184.5,338.3 L184.7,338.4 L185.0,338.4 L185.2,338.4   L185.5,338.4 L185.7,338.4 L186.0,338.4 L186.2,338.4 L186.5,338.4 L186.7,338.4 L187.0,338.4 L187.3,338.4   L187.5,338.4 L187.8,338.4 L188.0,338.4 L188.3,338.4 L188.5,338.4 L188.8,338.4 L189.0,338.5 L189.3,338.5   L189.5,338.5 L189.8,338.5 L190.1,338.5 L190.3,338.5 L190.6,338.5 L190.8,338.5 L191.1,338.5 L191.3,338.5   L191.6,338.5 L191.8,338.5 L192.1,338.5 L192.4,338.5 L192.6,338.5 L192.9,338.5 L193.1,338.5 L193.4,338.5   L193.6,338.6 L193.9,338.6 L194.1,338.6 L194.4,338.6 L194.6,338.6 L194.9,338.6 L195.2,338.6 L195.4,338.6   L195.7,338.6 L195.9,338.6 L196.2,338.6 L196.4,338.6 L196.7,338.6 L196.9,338.6 L197.2,338.6 L197.4,338.6   L197.7,338.6 L198.0,338.6 L198.2,338.6 L198.5,338.6 L198.7,338.7 L199.0,338.7 L199.2,338.7 L199.5,338.7   L199.7,338.7 L200.0,338.7 L200.3,338.7 L200.5,338.7 L200.8,338.7 L201.0,338.7 L201.3,338.7 L201.5,338.7   L201.8,338.7 L202.0,338.7 L202.3,338.7 L202.5,338.7 L202.8,338.7 L203.1,338.7 L203.3,338.7 L203.6,338.7   L203.8,338.8 L204.1,338.8 L204.3,338.8 L204.6,338.8 L204.8,338.8 L205.1,338.8 L205.3,338.8 L205.6,338.8   L205.9,338.8 L206.1,338.8 L206.4,338.8 L206.6,338.8 L206.9,338.8 L207.1,338.8 L207.4,338.8 L207.6,338.8   L207.9,338.8 L208.2,338.8 L208.4,338.8 L208.7,338.8 L208.9,338.8 L209.2,338.8 L209.4,338.9 L209.7,338.9   L209.9,338.9 L210.2,338.9 L210.4,338.9 L210.7,338.9 L211.0,338.9 L211.2,338.9 L211.5,338.9 L211.7,338.9   L212.0,338.9 L212.2,338.9 L212.5,338.9 L212.7,338.9 L213.0,338.9 L213.2,338.9 L213.5,338.9 L213.8,338.9   L214.0,338.9 L214.3,338.9 L214.5,338.9 L214.8,338.9 L215.0,338.9 L215.3,338.9 L215.5,338.9 L215.8,339.0   L216.1,339.0 L216.3,339.0 L216.6,339.0 L216.8,339.0 L217.1,339.0 L217.3,339.0 L217.6,339.0 L217.8,339.0   L218.1,339.0 L218.3,339.0 L218.6,339.0 L218.9,339.0 L219.1,339.0 L219.4,339.0 L219.6,339.0 L219.9,339.0   L220.1,339.0 L220.4,339.0 L220.6,339.0 L220.9,339.0 L221.2,339.0 L221.4,339.0 L221.7,339.0 L221.9,339.0   L222.2,339.0 L222.4,339.1 L222.7,339.1 L222.9,339.1 L223.2,339.1 L223.4,339.1 L223.7,339.1 L224.0,339.1   L224.2,339.1 L224.5,339.1 L224.7,339.1 L225.0,339.1 L225.2,339.1 L225.5,339.1 L225.7,339.1 L226.0,339.1   L226.2,339.1 L226.5,339.1 L226.8,339.1 L227.0,339.1 L227.3,339.1 L227.5,339.1 L227.8,339.1 L228.0,339.1   L228.3,339.1 L228.5,339.1 L228.8,339.1 L229.1,339.1 L229.3,339.1 L229.6,339.1 L229.8,339.1 L230.1,339.2   L230.3,339.2 L230.6,339.2 L230.8,339.2 L231.1,339.2 L231.3,339.2 L231.6,339.2 L231.9,339.2 L232.1,339.2   L232.4,339.2 L232.6,339.2 L232.9,339.2 L233.1,339.2 L233.4,339.2 L233.6,339.2 L233.9,339.2 L234.1,339.2   L234.4,339.2 L234.7,339.2 L234.9,339.2 L235.2,339.2 L235.4,339.2 L235.7,339.2 L235.9,339.2 L236.2,339.2   L236.4,339.2 L236.7,339.2 L237.0,339.2 L237.2,339.2 L237.5,339.2 L237.7,339.2 L238.0,339.2 L238.2,339.3   L238.5,339.3 L238.7,339.3 L239.0,339.3 L239.2,339.3 L239.5,339.3 L239.8,339.3 L240.0,339.3 L240.3,339.3   L240.5,339.3 L240.8,339.3 L241.0,339.3 L241.3,339.3 L241.5,339.3 L241.8,339.3 L242.0,339.3 L242.3,339.3   L242.6,339.3 L242.8,339.3 L243.1,339.3 L243.3,339.3 L243.6,339.3 L243.8,339.3 L244.1,339.3 L244.3,339.3   L244.6,339.3 L244.9,339.3 L245.1,339.3 L245.4,339.3 L245.6,339.3 L245.9,339.3 L246.1,339.3 L246.4,339.3   L246.6,339.3 L246.9,339.3 L247.1,339.3 L247.4,339.3 L247.7,339.3 L247.9,339.4 L248.2,339.4 L248.4,339.4   L248.7,339.4 L248.9,339.4 L249.2,339.4 L249.4,339.4 L249.7,339.4 L249.9,339.4 L250.2,339.4 L250.5,339.4   L250.7,339.4 L251.0,339.4 L251.2,339.4 L251.5,339.4 L251.7,339.4 L252.0,339.4 L252.2,339.4 L252.5,339.4   L252.8,339.4 L253.0,339.4 L253.3,339.4 L253.5,339.4 L253.8,339.4 L254.0,339.4 L254.3,339.4 L254.5,339.4   L254.8,339.4 L255.0,339.4 L255.3,339.4 L255.6,339.4 L255.8,339.4 L256.1,339.4 L256.3,339.4 L256.6,339.4   L256.8,339.4 L257.1,339.4 L257.3,339.4 L257.6,339.4 L257.8,339.4 L258.1,339.4 L258.4,339.4 L258.6,339.4   L258.9,339.5 L259.1,339.5 L259.4,339.5 L259.6,339.5 L259.9,339.5 L260.1,339.5 L260.4,339.5 L260.7,339.5   L260.9,339.5 L261.2,339.5 L261.4,339.5 L261.7,339.5 L261.9,339.5 L262.2,339.5 L262.4,339.5 L262.7,339.5   L262.9,339.5 L263.2,339.5 L263.5,339.5 L263.7,339.5 L264.0,339.5 L264.2,339.5 L264.5,339.5 L264.7,339.5   L265.0,339.5 L265.2,339.5 L265.5,339.5 L265.7,339.5 L266.0,339.5 L266.3,339.5 L266.5,339.5 L266.8,339.5   L267.0,339.5 L267.3,339.5 L267.5,339.5 L267.8,339.5 L268.0,339.5 L268.3,339.5 L268.6,339.5 L268.8,339.5   L269.1,339.5 L269.3,339.5 L269.6,339.5 L269.8,339.5 L270.1,339.5 L270.3,339.5 L270.6,339.5 L270.8,339.5   L271.1,339.5 L271.4,339.5 L271.6,339.5 L271.9,339.6 L272.1,339.6 L272.4,339.6 L272.6,339.6 L272.9,339.6   L273.1,339.6 L273.4,339.6 L273.6,339.6 L273.9,339.6 L274.2,339.6 L274.4,339.6 L274.7,339.6 L274.9,339.6   L275.2,339.6 L275.4,339.6 L275.7,339.6 L275.9,339.6 L276.2,339.6 L276.5,339.6 L276.7,339.6 L277.0,339.6   L277.2,339.6 L277.5,339.6 L277.7,339.6 L278.0,339.6 L278.2,339.6 L278.5,339.6 L278.7,339.6 L279.0,339.6   L279.3,339.6 L279.5,339.6 L279.8,339.6 L280.0,339.6 L280.3,339.6 L280.5,339.6 L280.8,339.6 L281.0,339.6   L281.3,339.6 L281.5,339.6 L281.8,339.6 L282.1,339.6 L282.3,339.6 L282.6,339.6 L282.8,339.6 L283.1,339.6   L283.3,339.6 L283.6,339.6 L283.8,339.6 L284.1,339.6 L284.4,339.6 L284.6,339.6 L284.9,339.6 L285.1,339.6   L285.4,339.6 L285.6,339.6 L285.9,339.6 L286.1,339.6 L286.4,339.6 L286.6,339.6 L286.9,339.6 L287.2,339.6   L287.4,339.6 L287.7,339.6 L287.9,339.7 L288.2,339.7 L288.4,339.7 L288.7,339.7 L288.9,339.7 L289.2,339.7   L289.5,339.7 L289.7,339.7 L290.0,339.7 L290.2,339.7 L290.5,339.7 L290.7,339.7 L291.0,339.7 L291.2,339.7   L291.5,339.7 L291.7,339.7 L292.0,339.7 L292.3,339.7 L292.5,339.7 L292.8,339.7 L293.0,339.7 L293.3,339.7   L293.5,339.7 L293.8,339.7 L294.0,339.7 L294.3,339.7 L294.5,339.7 L294.8,339.7 L295.1,339.7 L295.3,339.7   L295.6,339.7 L295.8,339.7 L296.1,339.7 L296.3,339.7 L296.6,339.7 L296.8,339.7 L297.1,339.7 L297.4,339.7   L297.6,339.7 L297.9,339.7 L298.1,339.7 L298.4,339.7 L298.6,339.7 L298.9,339.7 L299.1,339.7 L299.4,339.7   L299.6,339.7 L299.9,339.7 L300.2,339.7 L300.4,339.7 L300.7,339.7 L300.9,339.7 L301.2,339.7 L301.4,339.7   L301.7,339.7 L301.9,339.7 L302.2,339.7 L302.4,339.7 L302.7,339.7 L303.0,339.7 L303.2,339.7 L303.5,339.7   L303.7,339.7 L304.0,339.7 L304.2,339.7 L304.5,339.7 L304.7,339.7 L305.0,339.7 L305.3,339.7 L305.5,339.7   L305.8,339.7 L306.0,339.7 L306.3,339.7 L306.5,339.7 L306.8,339.7 L307.0,339.7 L307.3,339.7 L307.5,339.7   L307.8,339.7 L308.1,339.7 L308.3,339.7 L308.6,339.8 L308.8,339.8 L309.1,339.8 L309.3,339.8 L309.6,339.8   L309.8,339.8 L310.1,339.8 L310.3,339.8 L310.6,339.8 L310.9,339.8 L311.1,339.8 L311.4,339.8 L311.6,339.8   L311.9,339.8 L312.1,339.8 L312.4,339.8 L312.6,339.8 L312.9,339.8 L313.2,339.8 L313.4,339.8 L313.7,339.8   L313.9,339.8 L314.2,339.8 L314.4,339.8 L314.7,339.8 L314.9,339.8 L315.2,339.8 L315.4,339.8 L315.7,339.8   L316.0,339.8 L316.2,339.8 L316.5,339.8 L316.7,339.8 L317.0,339.8 L317.2,339.8 L317.5,339.8 L317.7,339.8   L318.0,339.8 L318.2,339.8 L318.5,339.8 L318.8,339.8 L319.0,339.8 L319.3,339.8 L319.5,339.8 L319.8,339.8   L320.0,339.8 L320.3,339.8 L320.5,339.8 L320.8,339.8 L321.1,339.8 L321.3,339.8 L321.6,339.8 L321.8,339.8   L322.1,339.8 L322.3,339.8 L322.6,339.8 L322.8,339.8 L323.1,339.8 L323.3,339.8 L323.6,339.8 L323.9,339.8   L324.1,339.8 L324.4,339.8 L324.6,339.8 L324.9,339.8 L325.1,339.8 L325.4,339.8 L325.6,339.8 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    "gradientDescent(x, y, ITERATIONS, ALPHA)"
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    "## Linear Regression with Multiple Variables:"
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    "### Goal of the exercise:\n",
    "\n",
    "> Predict the prices of houses.\n",
    "\n",
    "#### Data:\n",
    "\n",
    "* size of the house (in sq feet)\n",
    "* number of bedrooms\n",
    "* price of the house"
   ]
  },
  {
   "cell_type": "code",
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    "data = load('ex1data2.txt');\n",
    "x = data(:, 1:2);\n",
    "y = data(:, 3);"
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    "figure(1);\n",
    "hist(x(:, 1));\n",
    "title('size in sq feet');\n",
    "\n",
    "figure(2);\n",
    "hist(x(:, 2));\n",
    "title('# of bedrooms');\n",
    "\n",
    "figure(3);\n",
    "hist(y);\n",
    "title('price of houses');"
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    "### Feature regularization:\n",
    "\n",
    "> Using standard deviation"
   ]
  },
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   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
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     "text": [
      "stdevs =\n",
      "\n",
      "   794.70235     0.76098\n",
      "\n",
      "means =\n",
      "\n",
      "   2000.6809      3.1702\n",
      "\n"
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    "stdevs = std(x)\n",
    "means = mean(x)"
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   "source": [
    "x = (x - means) ./ stdevs;"
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   "source": [
    "### Cost function:"
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   "execution_count": 70,
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   "outputs": [],
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    "function [cost] = costFunction(X, y, theta)\n",
    "    m = length(X);\n",
    "    cost = (2 * m) ^ (-1) * (X * theta - y)' * (X * theta - y);\n",
    "endfunction"
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   "source": [
    "### Gradient descent:"
   ]
  },
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   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
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      "theta =\n",
      "\n",
      "   340397.96354\n",
      "   109848.00846\n",
      "    -5866.45408\n",
      "\n"
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       "\t</g>\n",
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       "</g>\n",
       "<g color=\"black\" fill=\"none\" stroke=\"black\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n",
       "\t<path d=\"M50.2,384.2 L544.3,384.2 M50.2,39.0 L544.3,39.0 M50.2,384.2 L50.2,39.0 M544.3,384.2 L544.3,39.0    \" stroke=\"rgb( 38,  38,  38)\"/></g>\n",
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       "</g>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta = gradientDescent(x, y, 1000, 0.01)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "prediction =  181572012.56070\n"
     ]
    }
   ],
   "source": [
    "prediction = [1 1650 3] * theta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Normal equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "function [theta] = normalEquation(x, y)\n",
    "\n",
    "    m = length(x);\n",
    "    X = [ones(m, 1), x];\n",
    "    \n",
    "    theta = pinv(X' * X) * X' * y;\n",
    "endfunction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "theta =\n",
      "\n",
      "   340412.65957\n",
      "   110631.05028\n",
      "    -6649.47427\n",
      "\n"
     ]
    }
   ],
   "source": [
    "theta = normalEquation(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "prediction =  182861697.19686\n"
     ]
    }
   ],
   "source": [
    "prediction = [1 1650 3] * theta"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Octave",
   "language": "octave",
   "name": "octave"
  },
  "language_info": {
   "file_extension": ".m",
   "help_links": [
    {
     "text": "GNU Octave",
     "url": "https://www.gnu.org/software/octave/support.html"
    },
    {
     "text": "Octave Kernel",
     "url": "https://github.com/Calysto/octave_kernel"
    },
    {
     "text": "MetaKernel Magics",
     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "octave",
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