RegularisedLinearRegression.ipynb

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    "## Prepare Data\n",
    "\n",
    "This is a practice to use the Octave to do the Regularised Linear Regression. The dataset is small, so the regularisation will help.\n",
    "\n",
    "In this notebook, the Normal Euations is used to calculate the best results.\n",
    "https://www.mathworks.com/help/matlab/ref/transpose.html"
   ]
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     "text": [
      "x =\n",
      "\n",
      "  -0.99768\n",
      "  -0.69574\n",
      "  -0.40373\n",
      "  -0.10236\n",
      "   0.22024\n",
      "   0.47742\n",
      "   0.82229\n",
      "\n"
     ]
    }
   ],
   "source": [
    "x=load(\"Linx.dat\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
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     "text": [
      "y =\n",
      "\n",
      "   2.08850\n",
      "   1.16460\n",
      "   0.32870\n",
      "   0.46013\n",
      "   0.44808\n",
      "   0.10013\n",
      "  -0.32952\n",
      "\n"
     ]
    }
   ],
   "source": [
    "y =load(\"Liny.dat\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
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   "source": [
    "figure;\n",
    "scatter(x,y);\n",
    "ylabel('y');\n",
    "xlabel('x');\n",
    "hold on"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
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     "text": [
      "theta = -1.2275\n"
     ]
    }
   ],
   "source": [
    "theta = inverse(x'*x)*x'*y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "m =  7\n"
     ]
    }
   ],
   "source": [
    "m=length(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create X Matrix for input"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X =\n",
      "\n",
      " Columns 1 through 5:\n",
      "\n",
      "   1.000000000  -0.997680000   0.995365382  -0.993056135   0.990752244\n",
      "   1.000000000  -0.695740000   0.484054148  -0.336775833   0.234308418\n",
      "   1.000000000  -0.403730000   0.162997913  -0.065807147   0.026568320\n",
      "   1.000000000  -0.102360000   0.010477570  -0.001072484   0.000109779\n",
      "   1.000000000   0.220240000   0.048505658   0.010682886   0.002352799\n",
      "   1.000000000   0.477420000   0.227929856   0.108818272   0.051952019\n",
      "   1.000000000   0.822290000   0.676160844   0.556000300   0.457193487\n",
      "\n",
      " Column 6:\n",
      "\n",
      "  -0.988453699\n",
      "  -0.163017739\n",
      "  -0.010726428\n",
      "  -0.000011237\n",
      "   0.000518180\n",
      "   0.024802933\n",
      "   0.375945633\n",
      "\n"
     ]
    }
   ],
   "source": [
    "% X = [ones(m, 1) x]\n",
    "% X= [X x.^2]\n",
    "% X=[X x.^3]\n",
    "% X=[X x.^4]\n",
    "% X=[X x.^5]\n",
    "% or use this\n",
    "X = [ones(m, 1), x, x.^2, x.^3, x.^4, x.^5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theta_without_reg =\n",
      "\n",
      "   0.47253\n",
      "   0.68135\n",
      "  -1.38013\n",
      "  -5.97769\n",
      "   2.44173\n",
      "   4.73711\n",
      "\n"
     ]
    }
   ],
   "source": [
    "Theta_without_reg = inverse(X'*X)*X'*y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create Diagonal Matrix\n",
    "https://www.mathworks.com/help/matlab/ref/diag.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tmp_matrix =\n",
      "\n",
      "   0   1   1   1   1   1\n",
      "\n",
      "diagonal_matrix =\n",
      "\n",
      "Diagonal Matrix\n",
      "\n",
      "   0   0   0   0   0   0\n",
      "   0   1   0   0   0   0\n",
      "   0   0   1   0   0   0\n",
      "   0   0   0   1   0   0\n",
      "   0   0   0   0   1   0\n",
      "   0   0   0   0   0   1\n",
      "\n"
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   "source": [
    "tmp_matrix= [0 1 1 1 1 1]\n",
    "diagonal_matrix = diag(tmp_matrix)"
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  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
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     "text": [
      "Theta_lambda_1 =\n",
      "\n",
      "   0.39760\n",
      "  -0.42067\n",
      "   0.12959\n",
      "  -0.39747\n",
      "   0.17526\n",
      "  -0.33939\n",
      "\n"
     ]
    }
   ],
   "source": [
    "Theta_lambda_1 = inverse(X'*X + diagonal_matrix)*X'*y"
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   "cell_type": "code",
   "execution_count": 122,
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     "text": [
      "Theta_lambda_10 =\n",
      "\n",
      "   0.520471\n",
      "  -0.182507\n",
      "   0.060643\n",
      "  -0.148177\n",
      "   0.074330\n",
      "  -0.127957\n",
      "\n"
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    }
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   "source": [
    "Theta_lambda_10 = inverse(X'*X + 10*diagonal_matrix)*X'*y"
   ]
  },
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   "execution_count": 123,
   "metadata": {},
   "outputs": [
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     "text": [
      "Y1 =\n",
      "\n",
      "   1.85009\n",
      "   0.98325\n",
      "   0.62301\n",
      "   0.44246\n",
      "   0.30722\n",
      "   0.18373\n",
      "  -0.12915\n",
      "\n",
      "Y10 =\n",
      "\n",
      "   1.11019\n",
      "   0.76498\n",
      "   0.61714\n",
      "   0.53996\n",
      "   0.48174\n",
      "   0.43172\n",
      "   0.31489\n",
      "\n",
      "Y0 =\n",
      "\n",
      "   2.091938\n",
      "   1.143453\n",
      "   0.379923\n",
      "   0.394951\n",
      "   0.499987\n",
      "   0.077113\n",
      "  -0.326745\n",
      "\n"
     ]
    }
   ],
   "source": [
    "Y1=X*Theta_lambda_1\n",
    "Y10=X*Theta_lambda_10\n",
    "Y0=X*Theta_without_reg"
   ]
  },
  {
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   "execution_count": 124,
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