Examples of how to prepend an n-by-1 column of ones to a matrix of training input data:

Let's consider the following matrix X of training data X (with n=6 observations and d=3 features):

\begin{equation}

X = \left( \begin{array}{ccc}

x_{11} & x_{12} & x_{13} \\

x_{21} & x_{22} & x_{23} \\

x_{31} & x_{32} & x_{33} \\

x_{41} & x_{42} & x_{43} \\

x_{51} & x_{52} & x_{53} \\

x_{61} & x_{62} & x_{63}

\end{array}\right)

\end{equation}

the goal is to prepend an n-by-1 column of ones to the matrix X:

\begin{equation}

X = \left( \begin{array}{ccc}

1 & x_{11} & x_{12} & x_{13} \\

1 & x_{21} & x_{22} & x_{23} \\

1 & x_{31} & x_{32} & x_{33} \\

1 & x_{41} & x_{42} & x_{43} \\

1 & x_{51} & x_{52} & x_{53} \\

1 & x_{61} & x_{62} & x_{63}

\end{array}\right)

\end{equation}

As an example, let's create a matrix X with random element:

`>>> import numpy as np`

`>>> X = np.random.randint(100, size=(6,3))`

`>>> X`

`array([[46, 16, 11],`

`[79, 95, 54],`

`[35, 3, 90],`

`[62, 71, 63],`

`[14, 61, 80],`

`[92, 69, 57]])`

to create a matrix of ones of size (6-by-1) a solution is to use the numpy finction ones():

`>>> b = np.ones(X.shape[0])`

`>>> b.shape`

`(6,)`

than can be concatenated to the matrix X using the numpy function numpy.c_ , illustration:

`>>> X = np.c_[np.ones(X.shape[0]),X]`

`>>> X`

`array([[ 1., 46., 16., 11.],`

`[ 1., 79., 95., 54.],`

`[ 1., 35., 3., 90.],`

`[ 1., 62., 71., 63.],`

`[ 1., 14., 61., 80.],`

`[ 1., 92., 69., 57.]])`

Note: check if d < n else transpose the matrix:

`>>> X`

`array([[46, 79, 35, 62, 14, 92],`

`[16, 95, 3, 71, 61, 69],`

`[11, 54, 90, 63, 80, 57]])`

`>>> if X.shape[0] < X.shape[1]: X = X.T`

`...`

`>>> X`

`array([[46, 16, 11],`

`[79, 95, 54],`

`[35, 3, 90],`

`[62, 71, 63],`

`[14, 61, 80],`

`[92, 69, 57]])`

To make a prediction it is necessary to add a 1 as well, example:

`>>> Y_new = np.array([11., 64., 20.])`

`>>> Y_new`

`array([ 11., 64., 20.])`

A solution is to use the numpy function numpy.concatenate:

`>>> Y_new = np.concatenate([np.ones(1),Y_new])`

`>>> Y_new`

`array([ 1., 11., 64., 20.])`

Links | Site |
---|---|

numpy.c_ | scipy doc |

numpy.concatenate | scipy doc |

numpy.transpose | scipy doc |

ones() | scipy doc |

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