Generating multivariate random variables

Monte Carlo Simulations in Python

Izzy Weber

Curriculum Manager, DataCamp

Sampling from multivariate distributions

Multinomial distribution

Variables each follow binomial distribution
Probabilities of these variables sum to one

Example: simulating the results of flipping a biased coin

scipy.stats.multinomial.rvs()

Sampling from multivariate distributions

Multivariate normal distribution

Variables each follow normal distribution
Variables can be correlated with each other or not

Example: simulating price and demand

scipy.stats.multivariate_normal.rvs()

Sampling from multinomial distributions

Simulation:.rvs(n, p, size)

$n$: 50
$p$: [0.2, 0.8]
size: 500

results = st.multinomial.rvs(50,
    [0.2, 0.8], size=500)


df_results=pd.DataFrame(
    {"Head":results[:, 0],
     "Tail":results[:, 1]})
sns.pairplot(df_results)

Pairplot of biased coin flip sampling

Sampling from multivariate normal distributions

Simulation: .rvs(mean, size)

mean: [2, 6]
size: 500

results=st.multivariate_normal.rvs(
    mean=[2, 6], size=500)


df_results=pd.DataFrame(
    {"Price":results[:, 0],
     "Demand":results[:, 1]})
sns.pairplot(df_results)

Pairplot of price and demand sampling results

Covariance matrix

Captures the variance and covariances of variables
Definition using two random variables $x$ and $y$:

covariance matrix structure

Example:

df_historical.cov()

|        | Price     | Demand   |
|--------|-----------|----------|
| Price  | 0.920545  | -0.85578 |
| Demand | -0.855780 | 0.98417  |

Multivariate normal sampling with defined covariance

Simulation: .rvs(mean, size)

mean: [2, 6]
size: 500
cov: np.array([[1, -0.9], [-0.9, 1]])

cov_mat = np.array([[1,-0.9], [-0.9,1]])
results = st.multivariate_normal.rvs(
    mean=[2,6], size=500, cov=cov_mat)


df_results = pd.DataFrame(
    {"Price":results[:,0],
     "Demand":results[:,1]})
sns.pairplot(df_results)

a pairplot of the results

Let's practice!

Monte Carlo Simulations in Python