NumPy Basics 2

• Square Root
• Square
• Random Variables
• Array Operations
• Correlation Matrix
• Mean
• Standard Deviation
• Variance
• Bessel's Correction
• Column and Row Means

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Before testing remember to import NumPy!

import numpy as np

Square root

x = [25, 49, 81]
np.sqrt(x)

array([5., 7., 9.])

Square

numpy_x = np.array(x)
numpy_x**2

array([ 625, 2401, 6561])

Generating 50 independent random variables from a $N (0, 1)$ distribution.

$N(0,1)$ means Normal distribution (bell curve) where 0 is the mean and 1 is the standard deviation.

• Random variables without a default_rng() will be different on each run.

X = np.random.normal(size=50)
X

array([-1.40601626, 0.7214838 , -0.09496628, 2.39735926, 1.18898171, 0.76323124, 1.05925105, 1.48671102, 0.30660224, -1.01944698, -1.18494759, 1.37000356, -1.19952148, -0.03482423, 0.60095877, -0.01830071, 1.59107384, 1.12081996, 0.10044048, 1.66091275, 0.06833242, 0.51939868, 0.40795647, -1.64827681, 0.14310729, -1.1423812 , 1.25225891, -0.10193296, 0.37470172, -0.5934665 , -1.80341075, 0.70369082, 0.23672748, 0.14942 , 0.55380708, -0.77558763, -0.65266172, -0.76610702, -0.80075262, 1.15642067, -0.91172091, 0.6456585 , -1.3155079 , 1.42653333, -0.76599893, -1.46345512, -0.30072812, -0.70081465, -1.15067792, -0.3178047 ])

Creating an array y by adding an independent $N(50,1)$ random variable to each element of X.

y = X + np.random.normal(loc=50, scale=1, size=50)
y

• loc is the mean.
• scale is the standard deviation.
• size determines the number of random variables to generate.

array([51.56076075, 52.13431065, 49.29660332, 48.50495515, 48.18185202, 50.00832704, 50.26956058, 50.51491111, 50.28365849, 48.87579011, 50.84213829, 49.22854746, 50.16560162, 49.66720927, 47.69292912, 49.82166456, 51.1084384 , 48.29948421, 49.22668163, 50.41076018, 52.36561453, 49.70280929, 49.08306489, 48.49155704, 48.55588808, 54.293656 , 48.84553873, 48.98669483, 50.07122583, 50.30267524, 48.42505456, 51.10251517, 47.69915875, 46.45402369, 48.58695418, 48.64699787, 48.85627297, 51.31704619, 50.32932662, 50.45887703, 50.84479216, 50.75839768, 52.77570726, 50.92059056, 51.63525329, 48.45404531, 48.14896332, 52.90059218, 50.66841757, 50.14221893])

Correlation matrix

np.corrcoef(X, y)

array([[1. , 0.71800869], [0.71800869, 1. ]])

([ corr(X,X) , corr(X,y) ], [ corr(y,X) , corr(y,y) ])

Computing mean of an array

np.mean(y), X.mean()

(np.float64(49.89946191298883), np.float64(0.036730680978435036))

Standard deviation

np.std(y), X.std()

(np.float64(1.5086562629961138), np.float64(0.923966607462578))

Variance

Variance is the average of the squared differences from the mean. Subtract the mean from every data point and square it, take the sum of all these values and divide it by the total number of values.

np.var(y), X.var()

(np.float64(2.1311897491972185), np.float64(1.0063147032427298))

Applying Bessel's correction (dividing by $n-1$) change the delta degrees of freedom to 1:

np.var(y, ddof=1)

np.float64(2.322493591711632)

Mean of a column or row of data

rng = np.random.default_rng(3)
X = rng.standard_normal((10, 3))
X

array([[ 2.04091912, -2.55566503, 0.41809885], [-0.56776961, -0.45264929, -0.21559716], [-2.01998613, -0.23193238, -0.86521308], [ 3.32299952, 0.22578661, -0.35263079], [-0.28128742, -0.66804635, -1.05515055], [-0.39080098, 0.48194539, -0.23855361], [ 0.9577587 , -0.19980213, 0.02425957], [ 1.54582085, 0.54510552, -0.50522874], [-0.18283897, 0.54052513, 1.93508803], [-0.26962033, -0.24355868, 1.0023136 ]])

Mean of each column:

X.mean(0)

array([ 0.41551948, -0.25582912, 0.01473861])

Mean of each row:

X.mean(1)

array([-0.03221569, -0.41200535, -1.03904386, 1.06538511, -0.66816144, -0.0491364 , 0.26073871, 0.52856588, 0.76425806, 0.16304486])