Dynamic Trading using Active Inference (Part 1)

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Dynamic optimal investment decisions can increase ROI substantially when applied in the right context. This project lays the groundwork for setting up a Dynamic Trading system making use of Active Inference and which is implemented in RxInfer. In this initial part, we present the essentials of a previous project, done in Python using Reinforcement Learning. In the following parts we will identify parts of the Python code that we will reuse, together with an RxInfer implementation.

1 BUSINESS UNDERSTANDING

This project deals with a client need relating to making optimal investment decisions regarding a given portfolio. Although we only provide for two financial instruments (stocks, bonds, funds, etc.) in the present example, the code can easily be expanded to provide for multiple financial instruments. A decision can handle buy, hold, or sell options.

This project builds upon a previous project.

2 DATA UNDERSTANDING

## import pdb
## pdb.set_trace()
from collections import namedtuple, defaultdict
import pandas as pd
import numpy as np
import random
import matplotlib.pyplot as plt
from matplotlib.ticker import FormatStrFormatter
import matplotlib as mpl
from copy import copy
import math
pd.options.display.float_format = '{:,.4f}'.format
pd.set_option('display.max_columns', None)
pd.set_option('display.max_rows', None)
pd.set_option('display.max_colwidth', None)
! python --version

Python 3.10.12

The parameters of the system-under-steer (SUS) are:

# PARAMETERS
SNAMES = [ #state variable names
    'R_t',   #resource
    'R0_t',  #cash
    'p_t',   #price
]
xNAMES = ['x_t'] #decision variable names
eNAMES = ['AAA', 'BBB']
piNAMES = ['X__HighLow'] #policy names
thNAMES = ['thLo', 'thHi'] #theta names

SEED_TRAIN = 77777777
SEED_EVALU = 88888888
N_SAMPLEPATHS = 500
N_TRANSITIONS = 40

x = ['Up', 'Neutral', 'Down']
W_BIAS_CDFS = pd.DataFrame(
  [[.9, 1., 1.], #'Up' cdf
   [.2, .8, 1.], #'Neutral' cdf
   [0., .1, 1.]], #'Down' cdf
  index=x,
  columns=x,
)
W_BIAS_CDFS
INIT_PRICE = {eNAMES[0]: 179.23, eNAMES[1]: 85.4}
INIT_BIAS = {en: 'Neutral' for en in eNAMES}
W_UP_STEP = 1
W_DOWN_STEP = -1
W_VARIANCE = 2

INIT_RESOURCE = {eNAMES[0]: 20, eNAMES[1]: 30}
INIT_CASH = 1_000.00
S_0_INFO = {
  'R_t': {en: INIT_RESOURCE[en] for en in eNAMES},
  'R0_t': INIT_CASH,
  'p_t': {en: INIT_PRICE[en] for en in eNAMES},
}
#        'AAA'                         'BBB'
TH_HI = {eNAMES[0]: (200.0, 200.5, .1), eNAMES[1]: (90.0, 90.5, .1)}
TH_LO = {eNAMES[0]: (168.0, 168.5, .1), eNAMES[1]: (72.0, 72.5, .1)}

class PriceSimulator():
  def __init__(self,
    biasCdfs=W_BIAS_CDFS,
    upStep=W_UP_STEP,
    downStep=W_DOWN_STEP,
    variance=W_VARIANCE,
    seed=None):

    self.biasCdfs = biasCdfs
    self.upStep = upStep
    self.downStep = downStep
    self.variance = variance
    self.prng = np.random.RandomState(seed)
    self.bias = 'Neutral'

  def simulate(self): #assume the change in price is normal with mean bias and variance 2
    phat_tt1_dict = {}
    b_tt1_dict = {}
    b_tt1_val_dict = {}
    for e in eNAMES:
      b_t = self.prng.choice(['Down', 'Neutral', 'Up'])
      biasCdf = self.biasCdfs.loc[[b_t]]
      coin = self.prng.random_sample()
      # if (coin < float(biasCdf['Up'])):
      if (coin < float(biasCdf['Up'].iloc[0])):  
        b_tt1 = 'Up' #new bias
        b_tt1_val = self.upStep #bias
      # elif (coin >= float(biasCdf['Up']) and coin < float(biasCdf['Neutral'])): #.
      elif (coin >= float(biasCdf['Up'].iloc[0]) and coin < float(biasCdf['Neutral'].iloc[0])): #.
        b_tt1 = 'Neutral' #new bias
        b_tt1_val = 0 #bias
      else:
        b_tt1 = 'Down' #new bias
        b_tt1_val = self.downStep #bias
      self.bias = b_tt1
      # phat_tt1 = self.prng.normal(b_tt1_val, self.variance) #change in price
      phat_tt1_dict[e] = self.prng.normal(b_tt1_val, self.variance) #change in price
      b_tt1_dict[e] = b_tt1
      b_tt1_val_dict[e] = b_tt1_val
    W_tt1 = {
        "p_t": {e: phat_tt1_dict[e] for e in eNAMES},
        "b_t": {e: b_tt1_dict[e] for e in eNAMES}, #just for display
        "b_t_val": {e: b_tt1_val_dict[e] for e in eNAMES} #just for display
    }
    return W_tt1

  def plot_output(self, df1):
    n_charts = 3
    ylabelsize = 16
    mpl.rcParams['lines.linewidth'] = 1.2
    default_colors = plt.rcParams['axes.prop_cycle'].by_key()['color']
    fig, axs = plt.subplots(n_charts, sharex=True)
    fig.set_figwidth(13); fig.set_figheight(9)
    fig.suptitle('Price Simulation', fontsize=20)

    xi = 0
    axs[xi].set_title(f'')
    axs[xi].set_ylim(auto=True); axs[xi].spines['top'].set_visible(False); axs[xi].spines['right'].set_visible(True); axs[xi].spines['bottom'].set_visible(False)
    for e in eNAMES:
      axs[xi].step(df1[f'p_t_{e}'], random.choice(default_colors), where='post')
    axs[xi].axhline(y=0, color='k', linestyle=':')
    y1ab = '$p_{t}$'
    axs[xi].set_ylabel(y1ab, rotation=0, ha='right', va='center', fontweight='bold', size=ylabelsize)

    xi = 1
    axs[xi].set_ylim(auto=True); axs[xi].spines['top'].set_visible(False); axs[xi].spines['right'].set_visible(True); axs[xi].spines['bottom'].set_visible(False)
    for e in eNAMES:
      axs[xi].step(df1[f'b_t_val_{e}'], random.choice(default_colors), where='post')
    axs[xi].axhline(y=0, color='k', linestyle=':')
    y1ab = '$b_{t,val}$'
    axs[xi].set_ylabel(y1ab, rotation=0, ha='right', va='center', fontweight='bold', size=ylabelsize)

    xi = 2
    axs[xi].set_ylim(auto=True); axs[xi].spines['top'].set_visible(False); axs[xi].spines['right'].set_visible(True); axs[xi].spines['bottom'].set_visible(False)
    for e in eNAMES:
      axs[xi].step(df1[f'b_t_{e}'], random.choice(default_colors), where='post')
    y1ab = '$b_{t}$'
    axs[xi].set_ylabel(y1ab, rotation=0, ha='right', va='center', fontweight='bold', size=ylabelsize)

    axs[xi].set_xlabel('$t$', rotation=0, ha='center', va='center', fontweight='bold', size=ylabelsize)

W_BIAS_CDFS

	Up	Neutral	Down
Up	0.9000	1.0000	1.0000
Neutral	0.2000	0.8000	1.0000
Down	0.0000	0.1000	1.0000

price_sim = PriceSimulator(seed=SEED_TRAIN)
T__sim = 20
PriceData = []
for i in range(T__sim):
  res = price_sim.simulate()
  entry = [itm[1][e] for itm in list(res.items()) for e in eNAMES]
  PriceData.append(entry)
labels = [f'{itm[0]}_{e}' for itm in list(res.items()) for e in eNAMES]
df = pd.DataFrame.from_records(data=PriceData, columns=labels); df[:10]
# df = pd.DataFrame.from_records(data=PriceData); df[:10]

	p_t_AAA	p_t_BBB	b_t_AAA	b_t_BBB	b_t_val_AAA	b_t_val_BBB
0	0.3365	3.1084	Up	Neutral	1	0
1	2.0958	1.1369	Neutral	Neutral	0	0
2	0.7226	-1.1433	Up	Up	1	1
3	3.9397	-0.4858	Neutral	Down	0	-1
4	2.1811	0.4218	Down	Down	-1	-1
5	-2.3785	-0.0162	Up	Up	1	1
6	-3.9910	-0.3274	Neutral	Down	0	-1
7	-3.5321	3.9864	Down	Up	-1	1
8	-3.1872	1.2238	Down	Up	-1	1
9	-0.1920	0.7166	Up	Up	1	1

price_sim.plot_output(df)

del price_sim

3 DATA PREPARATION

We will use the data provided by the simulator directly. There is no need to perform additional data preparation.

4 MODELING

4.1 Narrative

We are holding assets in a portfolio and we are looking for the best time to buy/sell each asset.

For an asset to be bought, its price has to fall below a lower value (specific to the asset). All the available cash is used to buy shares of such an asset. In the case of more than one asset being below their lower value, the available cash is equally divided between the assets to be purchased.

When one or more assets rise above a higher value (specific to each asset), such assets are sold.

We assume that our decisions in the current project do not affect the price of any of the assets. The prices vary according to a stochastic process (as coded in the above price simulator).

In this project, we use two fictitious asset (ticker) names: ‘AAA’ and ‘BBB’.

4.2 Core Elements

This section attempts to answer three important questions:

What metrics are we going to track?
What decisions do we intend to make?
What are the sources of uncertainty?

For this problem, the only metric we are interested in is the amount of profit we make after each decision window. A single type of decision needs to be made at the start of each window - whether for each asset we want to buy, hold, or sell the asset at its current price. The only source of uncertainty is the prices of the assets.

4.3 Mathematical Model | SUS Design

A Python class is used to implement the model for the SUS (System-Under-Steer):

class Model():
  def __init__(self, S_0_info):
    ...
    ...

4.3.1 State variables

The state variables represent what we need to know.

$R_{t} = (R_{t e})_{e \in E}$
- $R_{t e}$ = Our position (in shares) in a stock $e \in E$ , where $R_{t e}$ can be either
  - positive (for a long position) or
  - negative (for a short position), and where
$R_{t}^{0}$ is the amount in cash
$p_{t} = (p_{t e})_{e \in E}$
- $p_{t e}$ = The price at $t$ for stock $e$

The state is:

$S_{t} = (R_{t}, R_{t}^{0}, p_{t})$

The state variables are represented by the following variables in the Model class:

self.State = namedtuple('State', SNAMES) # 'class'
self.S_t = self.build_state(info) # 'instance'

where

SNAMES = [ #state variable names
    'R_t',   #resource
    'R0_t',  #cash
    'p_t',   #price
]

4.3.2 Decision variables

The decision variables represent what we control.

$x_{t} = (x_{t e})_{e \in E}$
- $x_{t e}$ = number of shares that we trade
  - $x_{t e} > 0$ : number of shares we buy of stock $e$
  - $x_{t e} < 0$ : number of shares we sell of stock $e$
Constraints
- $\sum_{e = 1}^{M} x_{t e} p_{t e} \leq R_{t, 0}$
Decisions are made with a policy (TBD below):
- $X^{π} (S_{t})$

The decision variables are represented by the following variables in the Model class:

self.Decision = namedtuple('Decision', xNAMES) # 'class'

where

xNAMES = ['x_t']
eNAMES = ['AAA', 'BBB']

4.3.3 Exogenous information variables

The exogenous information variables represent what we did not know (when we made a decision). These are the variables that we cannot control directly. The information in these variables become available after we make the decision $x_{t}$ .

When we assume that the price in each time period is revealed, without any model to predict the price based on past prices, we have, using approach 1:

$p_{t + 1} = W_{t + 1}$

Alternatively, when we assume that we observe the change in price ${\hat{p}}_{t + 1} = p_{t + 1} - p_{t}$ , we have, using approach 2:

$\begin{aligned} p_{t + 1} & = p_{t} + W_{t + 1} \\ = p_{t} + {\hat{p}}_{t + 1} \end{aligned}$

We will make use of approach 2 which means that the exogenous information, $W_{t + 1}$ , is the observed change in price of the share.

The exogenous information is obtained by a call to

SIM = PriceSimulator.simulate(...)

where SIM is a global variable.

The latest exogenous information can be accessed by calling the following method from class Model(), which returns a simulated price change for each asset:

def W_fn(self, t):
    W_tt1 = SIM.simulate()
    return W_ttl

4.3.4 Transition function

The transition function describes how the state variables evolve over time. Because we currently have three state variables in the state, $S_{t} = (R_{t}, R_{t}^{0}, p_{t})$ , we have the equations:

For a position in a stock $R_{t e}$ : $\begin{aligned} R_{t + 1, e} & = R_{t e} + x_{t e} (RLSO - eq 13.13) \end{aligned}$

For a position in cash $R_{t, 0}$ : $\begin{aligned} R_{t + 1}^{0} & = R_{t}^{0} - \sum_{e = 1}^{M} x_{t e} p_{t e} (RLSO - eq 13.14) \end{aligned}$

For the price $p_{t}$ : $\begin{aligned} p_{t + 1, e} & = p_{t e} + {\hat{p}}_{t + 1, e} (RLSO - eq 13.15) \end{aligned}$

Collectively, they represent the general transition function:

$S_{t + 1} = S^{M} (S_{t}, X^{π} (S_{t} | θ), W_{t + 1})$

4.3.5 Objective function

The objective function captures the performance metrics of the solution to the problem.

The transaction cost per transition/step is: $\begin{array}{r} C_{t} (S_{t}, x_{t}) = - c^{t r a n s} \sum_{e = 1}^{M} | x_{t e} | p_{t e}, for t = 0, . . ., T - 1 \end{array}$ where

$c^{t r a n s}$ is the transation cost per dollar
$| x_{t e} |$ is the absolute value of $x_{t e}$ , which gives the quantity of the trade, i.e. it does not matter whether we are buying or selling

The risk at the end of the day is: $\begin{array}{r} ρ (R_{t}) = R_{t}^{'} Σ R_{t} \end{array}$ where $Σ$ is the covariance matrix of returns, which we assume we have estimated from historical data in advance.

The final-period contributions function is given by: $\begin{array}{r} C_{T} (S_{T}, x_{T}) = R_{T}^{0} + \sum_{e = 1}^{M} R_{T e} p_{T e} - ρ (R_{T}) \end{array}$

The objective function is: $max_{π} E {\sum_{t = 0}^{T} C_{t} (S_{t}, X_{t}^{π} (S_{t})) | S_{0}}$

In practice, the expectation is approximated by using historical prices, which avoids the need to develop an underlying stochastic model.

4.3.6 Implementation of SUM Model

class Model():
    def __init__(self, S_0_info):
      self.S_0_info = S_0_info
      self.State = namedtuple('State', SNAMES) #. 'class'
      self.S_t = self.build_state(S_0_info) #. 'instance'
      self.Decision = namedtuple('Decision', xNAMES) #. 'class'
      self.Ccum = 0.0 #. cumulative reward

    def build_state(self, info):
      return self.State(*[info[sn] for sn in SNAMES])

    def build_decision(self, info):
        return self.Decision(*[info[xn] for xn in xNAMES])

    # exogenous information, dependent on a random process (the change in price)
    def W_fn(self):
        W_tt1 = SIM.simulate()
        return W_tt1

    def S__M_fn(self, S_t, x_t, W_tt1, theta, piName):
        # print(f'...in S__M_fn()...\n\t{S_t=}\n\t{x_t=}\n\t{W_tt1=}\n\t{theta=}')

        # R_t
        R_tt1 = {}
        for en in eNAMES:
          R_tt1[en] = S_t.R_t[en] + x_t.x_t[en]

        # R0_t
        cost = 0.0
        for en in eNAMES:
          cost += x_t.x_t[en]*S_t.p_t[en]
        R0_tt1 = S_t.R0_t - cost

        # p_t
          # W_tt1['p_t'] has CHANGE in price
          # clipped at a penny, else division by zero in X__HighLow
        p_t = S_t.p_t
        p_tt1 = {}
        for en in eNAMES:
          p_tt1[en] = max(0.01, p_t[en] + W_tt1['p_t'][en])

        S_tt1 = self.build_state({
            'R_t': R_tt1,
            'R0_t': R0_tt1,
            'p_t': p_tt1,
        })
        return S_tt1

    def C_fn(self, S_t, x_t, W_tt1):
        # print(f'...in C_fn()...\n\t{S_t=}\n\t{x_t=}\n\t{W_tt1=}')
        C_t = 0.0
        for en in eNAMES:
          C_t += -S_t.p_t[en]*x_t.x_t[en]
        return C_t

    def step(self, x_t, theta, piName):
        # print(f'...in step()...\n\t{x_t=}\n\t{theta=}')
        W_tt1 = self.W_fn()
        C = self.C_fn(self.S_t, x_t, W_tt1)
        self.Ccum += C
        self.S_t = self.S__M_fn(self.S_t, x_t, W_tt1, theta, piName)
        return (self.S_t, self.Ccum, x_t, W_tt1['b_t_val']) #. for plotting

4.4 Uncertainty Model

We will simulate the share price $p_{t + 1} = p_{t} + {\hat{p}}_{t + 1} = p_{t} + W_{t + 1}$ as described in section 2, for each asset.

4.5 Policy Design

There are two main meta-classes of policy design. Each of these has two subclasses:

Policy Search
- Policy Function Approximations (PFAs)
- Cost Function Approximations (CFAs)
Lookahead
- Value Function Approximations (VFAs)
- Direct Lookaheads (DLAs)

In this project we will make use of a single policy from the PFA class:

X__HighLow

where

$X^{H i g h L o w} (S_{t e} | θ^{H i g h L o w}) = {\begin{cases} - 1 & if p_{t e} < θ_{e}^{l o w} or p_{t e} > θ_{e}^{h i g h} \\ - 1 & if t = T and R_{t e} = 1 \\ 0 & otherwise \end{cases}$ for each asset $e \in E$

4.5.1 Implementation of Policy Design

class Policy():
  def __init__(self, model):
    self.model = model
    self.Policy = namedtuple('Policy', piNAMES) #. 'class'
    self.Theta = namedtuple('Theta', thNAMES) #. 'class'

  def build_policy(self, info):
    return self.Policy(*[info[pin] for pin in piNAMES])

  def build_theta(self, info):
    return self.Theta(*[info[thn] for thn in thNAMES])

  def X__HighLow(self, t, S_t, theta):
    # print(f'...in X__HighLow()...\n\t{t=}\n\t{S_t=}\n\t{theta=}')
    x_t_info = {
        'x_t': {en: 0 for en in eNAMES} #default is hold
    }
    # print(f'\t%%% {S_t.R0_t=}, {S_t.R_t=}, {S_t.p_t=}')
    tickersToSell = []
    tickersToBuy = []

    # sell all at end of horizon
    if (t == T - 1):
      tickersToSell = [en for en in eNAMES]
      for ticker in tickersToSell:
        nShares = S_t.R_t[ticker]
        x_t_info['x_t'][ticker] = -nShares
      return self.model.build_decision(x_t_info)
    # identify buys and sells
    for en in eNAMES:
      if (S_t.p_t[en] < theta.thLo[en]): #buy
        tickersToBuy.append(en)
      elif (S_t.p_t[en] > theta.thHi[en]): #sell
        tickersToSell.append(en)
    totalFunds = S_t.R0_t; #print(f'\t%%% {totalFunds=}')
    # sell
    # print(f'\t%%% {tickersToSell=}')
    if len(tickersToSell) > 0:
      for ticker in tickersToSell:
        nShares = S_t.R_t[ticker]
        x_t_info['x_t'][ticker] = -nShares
        totalFunds += nShares*S_t.p_t[ticker]
      # print(f'\t%%% totalFunds after selling: {totalFunds}')
    # buy
    # print(f'\t%%% {tickersToBuy=}')
    if len(tickersToBuy) > 0:
      availFundsPerTicker = totalFunds/len(tickersToBuy); #print(f'{availFundsPerTicker=}')
      for ticker in tickersToBuy:
        nShares = int(availFundsPerTicker/S_t.p_t[ticker])
        x_t_info['x_t'][ticker] = +nShares
        totalFunds -= nShares*S_t.p_t[ticker]
      # print(f'\t%%% totalFunds after buying: {totalFunds}')
    return self.model.build_decision(x_t_info)

  def grid_search_thetas_4(self, thetasLo, thetasHi):
    thetas = [
    self.build_theta({'thLo': {eNAMES[0]: th0, eNAMES[1]: th1}, 'thHi': {eNAMES[0]: th2, eNAMES[1]: th3}})
    for th0 in thetasLo[eNAMES[0]]
      for th1 in thetasLo[eNAMES[1]]
        for th2 in thetasHi[eNAMES[0]]
          for th3 in thetasHi[eNAMES[1]]
    ]
    return thetas

  def grid_search_thetas_2(self, thetas1, thetas2):
      thetas = [(th1, th2) for th1 in thetas1 for th2 in thetas2]
      return thetas

  def grid_search_thetas_1(self, thetas1):
      thetas = [(th1,) for th1 in thetas1]
      return thetas

  def run_policy_sample_paths(self, theta, piName, record):
      CcumIomega__lI = []
      for l in list(range(L)): #for each sample-path
          # print(f'%%% {l=}')
          M = Model(S_0_INFO)
          P = Policy(M)
          # SIM = PriceSimulator(seed=SEED_TRAIN)
          record_l = [piName, theta, l]
          for t in range(T): #for each transition/step
              # print(f'\t%%% {t=}')
              x_t = getattr(self, piName)(t, M.S_t, theta)

              S_t, Ccum, x_t, b_t_val = M.step(x_t, theta, piName)

              record_t = [t] + \
                [S_t.R_t[en] for en in eNAMES] + [S_t.R0_t] + [S_t.p_t[en] for en in eNAMES] + \
                [Ccum] + \
                [x_t.x_t[en] for en in eNAMES] + \
                [b_t_val[en] for en in eNAMES] #rather than b_t which is text and not ordered
              record.append(record_l + record_t)
          CcumIomega__lI.append(M.Ccum)
      return CcumIomega__lI

  def perform_grid_search_sample_paths(self, piName, thetas):
      Cbarcum = defaultdict(float)
      Ctilcum = defaultdict(float)
      expCbarcum = defaultdict(float)
      expCtilcum = defaultdict(float)
      num_thetas = len(thetas)
      record = []
      print(f'{num_thetas=}'); print(f'{thetas=}')
      nth = 20
      i = 0; print(f'... printing every {nth}th theta if considered ...')
      for theta in thetas:
          # print(f'\n=== {theta=} ===')
          if i%nth == 0: print(f'=== ({i:,} / {num_thetas:,}), theta={self.round_theta(theta)} ===')

          CcumIomega__lI = self.run_policy_sample_paths(theta, piName, record)

          Cbarcum_tmp = np.array(CcumIomega__lI).mean()
          Ctilcum_tmp = np.sum(np.square(np.array(CcumIomega__lI) - Cbarcum_tmp))/(L - 1)

          # theta=({'AAA': 168.0, 'BBB': 72.0}, {'AAA': 200.0, 'BBB': 90.0})
          # a dict cannot be used as a key, so we define
          # theta_key = ((168.0, 72.0), (200.0, 90.0)):
          theta_key = tuple(tuple(itm.values()) for itm in theta)

          Cbarcum[theta_key] = Cbarcum_tmp
          Ctilcum[theta_key] = np.sqrt(Ctilcum_tmp/L)
          best_theta = max(Cbarcum, key=Cbarcum.get)
          worst_theta = min(Cbarcum, key=Cbarcum.get)

          expCbarcum_tmp = pd.Series(CcumIomega__lI).expanding().mean()
          expCbarcum[theta_key] = expCbarcum_tmp

          expCtilcum_tmp = pd.Series(CcumIomega__lI).expanding().std()
          expCtilcum[theta_key] = expCtilcum_tmp
          i += 1
      best_Cbarcum = Cbarcum[best_theta]
      best_Ctilcum = Ctilcum[best_theta]

      worst_Cbarcum = Cbarcum[worst_theta]
      worst_Ctilcum = Ctilcum[worst_theta]

      thetaStar_expCbarcum = expCbarcum[best_theta]
      thetaStar_expCtilcum = expCtilcum[best_theta]
      thetaStar_expCtilcum[0] = 0 #set NaN to 0

      best_theta_w_enames = tuple(( ({eNAMES[0]: itm[0], eNAMES[1]: itm[1]})) for itm in best_theta)
      best_theta_en = self.build_theta({'thLo': best_theta_w_enames[0], 'thHi': best_theta_w_enames[1]})
      print(f'best_theta_en:\n{best_theta_en}\n{best_Cbarcum=:.2f}\n{best_Ctilcum=:.2f}')

      worst_theta_w_enames = tuple(( ({eNAMES[0]: itm[0], eNAMES[1]: itm[1]})) for itm in worst_theta)
      worst_theta_en = self.build_theta({'thLo': worst_theta_w_enames[0], 'thHi': worst_theta_w_enames[1]})
      print(f'worst_theta_en:\n{worst_theta_en}\n{worst_Cbarcum=:.2f}\n{worst_Ctilcum=:.2f}')

      return \
        thetaStar_expCbarcum, thetaStar_expCtilcum, \
        Cbarcum, Ctilcum, \
        best_theta_en, worst_theta_en, \
        best_Cbarcum, worst_Cbarcum, \
        best_Ctilcum, worst_Ctilcum, \
        record

  def round_theta(self, complex_theta):
    thetas_rounded = []
    for theta in complex_theta:
      evalues_rounded = []
      for _, evalue in theta.items():
        evalues_rounded.append(float(f"{evalue:f}"))
      thetas_rounded.append(tuple(evalues_rounded))
    return str(tuple(thetas_rounded))

  def plot_Fhat_map(self, FhatI_theta_I, thetasX, thetasY, labelX, labelY, title):
      Fhat_values = [FhatI_theta_I[(thetaX,thetaY)] for thetaY in thetasY for thetaX in thetasX]
      Fhats = np.array(Fhat_values)
      increment_count = len(thetasX)
      Fhats = np.reshape(Fhats, (-1, increment_count))#.

      fig, ax = plt.subplots()
      im = ax.imshow(Fhats, cmap='hot', origin='lower', aspect='auto')
      # create colorbar
      cbar = ax.figure.colorbar(im, ax=ax)
      # cbar.ax.set_ylabel(cbarlabel, rotation=-90, va="bottom")

      # ax.set_xticks(np.arange(0, len(thetasX), 5))#.
      ax.set_xticks(np.arange(len(thetasX)))

      # ax.set_yticks(np.arange(0, len(thetasY), 5))#.
      ax.set_yticks(np.arange(len(thetasY)))

      # NOTE: round tick labels, else very messy
      # function round() does not work, have to do this way
      thetasX_form = [f'{th:.1f}' for th in thetasX]
      thetasY_form = [f'{th:.1f}' for th in thetasY]

      # ax.set_xticklabels(thetasX[::5])#.
      # ax.set_xticklabels(thetasX)
      ax.set_xticklabels(thetasX_form)

      # ax.set_yticklabels(thetasY[::5])#.
      # ax.set_yticklabels(thetasY)
      ax.set_yticklabels(thetasY_form)

      # rotate the tick labels and set their alignment.
      #plt.setp(ax.get_xticklabels(), rotation=45, ha="right",rotation_mode="anchor")

      ax.set_title(title)
      ax.set_xlabel(labelX)
      ax.set_ylabel(labelY)

      #fig.tight_layout()
      plt.show()
      return True

  def plot_Fhat_map_4(self, FhatI_theta_I, thetasX, thetasY, labelX, labelY, title, thetaFixed1, thetaFixed2):
      # Fhat_values = [FhatI_theta_I[(thetaX,thetaY)] for thetaY in thetasY for thetaX in thetasX]
      Fhat_values = [FhatI_theta_I[((thetaX,thetaY), (thetaFixed1,thetaFixed2))] for thetaY in thetasY for thetaX in thetasX]
      Fhats = np.array(Fhat_values)
      increment_count = len(thetasX)
      Fhats = np.reshape(Fhats, (-1, increment_count))#.

      fig, ax = plt.subplots()
      im = ax.imshow(Fhats, cmap='hot', origin='lower', aspect='auto')
      # create colorbar
      cbar = ax.figure.colorbar(im, ax=ax)
      # cbar.ax.set_ylabel(cbarlabel, rotation=-90, va="bottom")

      # ax.set_xticks(np.arange(0, len(thetasX), 5))#.
      ax.set_xticks(np.arange(len(thetasX)))

      # ax.set_yticks(np.arange(0, len(thetasY), 5))#.
      ax.set_yticks(np.arange(len(thetasY)))

      # NOTE: round tick labels, else very messy
      # function round() does not work, have to do this way
      thetasX_form = [f'{th:.1f}' for th in thetasX]
      thetasY_form = [f'{th:.1f}' for th in thetasY]

      # ax.set_xticklabels(thetasX[::5])#.
      # ax.set_xticklabels(thetasX)
      ax.set_xticklabels(thetasX_form)

      # ax.set_yticklabels(thetasY[::5])#.
      # ax.set_yticklabels(thetasY)
      ax.set_yticklabels(thetasY_form)

      # rotate the tick labels and set their alignment.
      #plt.setp(ax.get_xticklabels(), rotation=45, ha="right",rotation_mode="anchor")

      ax.set_title(title)
      ax.set_xlabel(labelX)
      ax.set_ylabel(labelY)

      #fig.tight_layout()
      plt.show()
      return True

  # color_style examples: 'r-', 'b:', 'g--'
  def plot_Fhat_chart(self, FhatI_theta_I, thetasX, labelX, labelY, title, color_style, thetaStar):
      mpl.rcParams['lines.linewidth'] = 1.2
      xylabelsize = 16
      # plt.figure(figsize=(13, 9))
      plt.figure(figsize=(13, 4))
      plt.title(title, fontsize=20)
      Fhats = FhatI_theta_I.values()
      plt.plot(thetasX, Fhats, color_style)
      plt.axvline(x=thetaStar, color='k', linestyle=':')
      plt.xlabel(labelX, rotation=0, labelpad=10, ha='right', va='center', fontweight='bold', size=xylabelsize)
      plt.ylabel(labelY, rotation=0, labelpad=1, ha='right', va='center', fontweight='normal', size=xylabelsize)
      plt.show()

  # expanding Fhat chart
  def plot_expFhat_chart(self, df, labelX, labelY, title, color_style):
    mpl.rcParams['lines.linewidth'] = 1.2
    xylabelsize = 16
    plt.figure(figsize=(13, 4))
    plt.title(title, fontsize=20)
    plt.plot(df, color_style)
    plt.xlabel(labelX, rotation=0, labelpad=10, ha='right', va='center', fontweight='bold', size=xylabelsize)
    plt.ylabel(labelY, rotation=0, labelpad=1, ha='right', va='center', fontweight='normal', size=xylabelsize)
    plt.show()

  # expanding Fhat charts
  def plot_expFhat_charts(self, means, stdvs, labelX, labelY, suptitle, pars=defaultdict(str)):
    n_charts = 2
    xlabelsize = 14
    ylabelsize = 14
    mpl.rcParams['lines.linewidth'] = 1.2
    default_colors = plt.rcParams['axes.prop_cycle'].by_key()['color']
    fig, axs = plt.subplots(n_charts, sharex=True)
    fig.set_figwidth(13); fig.set_figheight(9)
    fig.suptitle(suptitle, fontsize=18)

    xi = 0
    legendlabels = []
    axs[xi].set_title(r"$exp\bar{C}^{cum}(\theta^*)$", loc='right', fontsize=16)
    for i,itm in enumerate(means.items()):
      axs[xi].spines['top'].set_visible(False); axs[xi].spines['right'].set_visible(True); axs[xi].spines['bottom'].set_visible(False)
      leg = axs[xi].plot(itm[1], color=pars['colors'][i]); #print(f"{pars['colors'][i]}")
      legendlabels.append(itm[0])
    axs[xi].set_ylabel(labelY, rotation=0, ha='right', va='center', fontweight='normal', size=ylabelsize)

    xi = 1
    axs[xi].set_title(r"$exp\tilde{C}^{cum}(\theta^*)$", loc='right', fontsize=16)
    for i,itm in enumerate(stdvs.items()):
      axs[xi].spines['top'].set_visible(False); axs[xi].spines['right'].set_visible(True); axs[xi].spines['bottom'].set_visible(False)
      # leg = axs[xi].plot(itm[1], default_colors[i], linestyle='--')
      leg = axs[xi].plot(itm[1], pars['colors'][i], linestyle='--')
    axs[xi].set_ylabel(labelY, rotation=0, ha='right', va='center', fontweight='normal', size=ylabelsize)

    fig.legend(
          # [leg],
          labels=legendlabels,
          title="Policies",
          loc='upper right',
          fancybox=True,
          shadow=True,
          ncol=1)
    plt.xlabel(labelX, rotation=0, labelpad=10, ha='right', va='center', fontweight='normal', size=xlabelsize)
    plt.show()

  def plot_records(self, df, df_non, pars=defaultdict(str)):
    # legendlabels = []
    n_e = len(eNAMES) #number of entities
    n_charts = 3*n_e + 1 + 1
    ylabelsize = 14
    mpl.rcParams['lines.linewidth'] = 1.2
    mycolors = ['g', 'b']
    fig, axs = plt.subplots(n_charts, sharex=True)
    fig.set_figwidth(13); fig.set_figheight(9)
    fig.suptitle(pars['suptitle'], fontsize=14)

    xi = 0
    for i,e in enumerate(eNAMES):
      axs[xi+i].set_ylim(auto=True); axs[xi+i].spines['top'].set_visible(False); axs[xi+i].spines['right'].set_visible(True); axs[xi+i].spines['bottom'].set_visible(False)
      axs[xi+i].step(df[f'x_t_{e}'], 'm-', where='post')
      if not df_non is None: axs[xi+i].step(df_non[f'x_t_{e}'], 'c-.', where='post')
      axs[xi+i].axhline(y=0, color='k', linestyle=':')
      # y1ab = '$x_{t}$'
      y1ab = '$x_{t,'+f'{e}'+'}$'
      axs[xi+i].set_ylabel(y1ab, rotation=0, ha='right', va='center', fontweight='bold', size=ylabelsize)
      for j in range(df.shape[0]//T): axs[xi+i].axvline(x=(j+1)*T, color='grey', ls=':')

    xi = n_e
    for i,e in enumerate(eNAMES):
      axs[xi+i].set_ylim(auto=True); axs[xi+i].spines['top'].set_visible(False); axs[xi+i].spines['right'].set_visible(True); axs[xi+i].spines['bottom'].set_visible(False)
      axs[xi+i].step(df[f'p_t_{e}'], mycolors[xi%len(mycolors)], where='post')
      if not df_non is None: axs[xi+i].step(df_non[f'p_t_{e}'], 'c-.', where='post')
      # axs[xi+i].axhline(y=0, color='k', linestyle=':')
      if(pars['thetaLo']): axs[xi+i].text(-4, pars['thetaLo'][e], r'$\theta^{Lo}$' + f"={pars['thetaLo'][e]:.1f}", size=10, color='m')
      if(pars['thetaLo']): axs[xi+i].axhline(y=pars['thetaLo'][e], color='m', linestyle=':')
      if(pars['thetaHi']): axs[xi+i].text(-4, pars['thetaHi'][e], r'$\theta^{Hi}$' + f"={pars['thetaHi'][e]:.1f}", size=10, color='m')
      if(pars['thetaHi']): axs[xi+i].axhline(y=pars['thetaHi'][e], color='m', linestyle=':')

      if(pars['thetaLoNon']): axs[xi+i].text(-4, pars['thetaLoNon'][e], r'$\theta^{Lo}$' + f"={pars['thetaLoNon'][e]:.1f}", size=10, color='c')
      if(pars['thetaLoNon']): axs[xi+i].axhline(y=pars['thetaLoNon'][e], color='c', linestyle=':')
      if(pars['thetaHiNon']): axs[xi+i].text(-4, pars['thetaHiNon'][e], r'$\theta^{Hi}$' + f"={pars['thetaHiNon'][e]:.1f}", size=10, color='c')
      if(pars['thetaHiNon']): axs[xi+i].axhline(y=pars['thetaHiNon'][e], color='c', linestyle=':')
      y1ab = '$p_{t,'+f'{e}'+'}$'
      axs[xi+i].set_ylabel(y1ab, rotation=0, ha='right', va='center', fontweight='bold', size=ylabelsize)
      for j in range(df.shape[0]//T): axs[xi+i].axvline(x=(j+1)*T, color='grey', ls=':')

    xi = 2*n_e
    for i,e in enumerate(eNAMES):
      axs[xi+i].set_ylim(auto=True); axs[xi+i].spines['top'].set_visible(False); axs[xi+i].spines['right'].set_visible(True); axs[xi+i].spines['bottom'].set_visible(False)
      axs[xi+i].step(df[f'R_t_{e}'], 'm-', where='post')
      if not df_non is None: axs[xi+i].step(df_non[f'R_t_{e}'], 'c-.', where='post')
      axs[xi+i].axhline(y=0, color='k', linestyle=':')
      y1ab = '$R_{t,'+f'{e}'+'}$'
      axs[xi+i].set_ylabel(y1ab, rotation=0, ha='right', va='center', fontweight='bold', size=ylabelsize)
      for j in range(df.shape[0]//T): axs[xi+i].axvline(x=(j+1)*T, color='grey', ls=':')

    xi = 3*n_e
    axs[xi].set_ylim(auto=True); axs[xi].spines['top'].set_visible(False); axs[xi].spines['right'].set_visible(True); axs[xi].spines['bottom'].set_visible(False)
    axs[xi].step(df['R0_t'], 'm-', where='post')
    if not df_non is None: axs[xi].step(df_non['Ccum'], 'c-.', where='post')
    axs[xi].axhline(y=0, color='k', linestyle=':')
    axs[xi].set_ylabel('$R^{0}_t$'+'\n'+''+'$\mathrm{[\$]}$', rotation=0, ha='right', va='center', fontweight='bold', size=ylabelsize);
    for j in range(df.shape[0]//T): axs[xi].axvline(x=(j+1)*T, color='grey', ls=':')

    xi = 3*n_e + 1
    axs[xi].set_ylim(auto=True); axs[xi].spines['top'].set_visible(False); axs[xi].spines['right'].set_visible(True); axs[xi].spines['bottom'].set_visible(False)
    axs[xi].step(df['Ccum'], 'm-', where='post')
    if not df_non is None: axs[xi].step(df_non['Ccum'], 'c-.', where='post')
    axs[xi].axhline(y=0, color='k', linestyle=':')
    axs[xi].set_ylabel('$C^{cum}$'+'\n'+'$\mathrm{(Profit)}$'+'\n'+''+'$\mathrm{[\$]}$', rotation=0, ha='right', va='center', fontweight='bold', size=ylabelsize);
    for j in range(df.shape[0]//T): axs[xi].axvline(x=(j+1)*T, color='grey', ls=':')

    axs[xi].set_xlabel('$t\ \mathrm{[decision\ windows]}$', rotation=0, ha='right', va='center', fontweight='bold', size=ylabelsize);
    # fig.legend(labels=legendlabels, loc='lower left', fontsize=16)

4.6 Policy Evaluation

N_SAMPLEPATHS, N_TRANSITIONS

(500, 40)

4.6.1 Training/Tuning

# setup labels to plot info
R_t_labels = ['R_t_'+en for en in eNAMES]
p_t_labels = ['p_t_'+en for en in eNAMES]
b_t_labels = ['b_t']
x_t_labels = ['x_t_'+en for en in eNAMES]
b_t_val_labels = ['b_t_val_'+en for en in eNAMES]
labels = ['piName', 'theta', 'l'] + \
  ['t'] + R_t_labels + ['R0_t'] + p_t_labels + \
  ['Ccum'] + x_t_labels + b_t_val_labels
labels

['piName',
 'theta',
 'l',
 't',
 'R_t_AAA',
 'R_t_BBB',
 'R0_t',
 'p_t_AAA',
 'p_t_BBB',
 'Ccum',
 'x_t_AAA',
 'x_t_BBB',
 'b_t_val_AAA',
 'b_t_val_BBB']

4.6.1.1 X__HighLow

N_SAMPLEPATHS, N_TRANSITIONS

(500, 40)

## LOWER COMPARED TO ORIGINAL PROJECT TO MAKING TRAINING QUICKER
## ORIGINAL PROJECT training time: 
##   Wall time: 5h 3min 47s
N_SAMPLEPATHS = 50 #10 #50 #100
N_TRANSITIONS = 20 #20 #200 #200 #40

%%time
L = N_SAMPLEPATHS
T = N_TRANSITIONS
first_n_t = 4*T
last_n_t = 4*T

M = Model(S_0_INFO)
P = Policy(M)
SIM = PriceSimulator(seed=SEED_TRAIN)

th_lo = {en: np.arange(TH_LO[en][0], TH_LO[en][1], TH_LO[en][2]) for en in eNAMES}
th_hi = {en: np.arange(TH_HI[en][0], TH_HI[en][1], TH_HI[en][2]) for en in eNAMES}
thetas = P.grid_search_thetas_4(th_lo, th_hi)

thetaStar_expCbarcum_HighLow, thetaStar_expCtilcum_HighLow, \
Cbarcum_HighLow, Ctilcum_HighLow, \
best_theta_HighLow, worst_theta_HighLow, \
best_Cbarcum_HighLow, worst_Cbarcum_HighLow, \
best_Ctilcum_HighLow, worst_Ctilcum_HighLow, \
record_HighLow = \
  P.perform_grid_search_sample_paths("X__HighLow", thetas)
f'{thetaStar_expCbarcum_HighLow.iloc[-1]=:.2f}'
df_first_n_t = pd.DataFrame.from_records(record_HighLow[:first_n_t], columns=labels)
df_last_n_t = pd.DataFrame.from_records(record_HighLow[-last_n_t:], columns=labels)

num_thetas=625
thetas=[Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.0), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': 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Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': 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np.float64(168.29999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.0), 'BBB': 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'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.1)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': 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np.float64(168.29999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': 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Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.0)}, thHi={'AAA': np.float64(200.2), 'BBB': 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np.float64(200.1), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.29999999999998)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.1), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.29999999999998), 'BBB': np.float64(90.39999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.0)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.1)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.19999999999999)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.29999999999998)}), Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.39999999999998), 'BBB': np.float64(90.39999999999998)})]
... printing every 20th theta if considered ...
=== (0 / 625), theta=((168.0, 72.0), (200.0, 90.0)) ===
=== (20 / 625), theta=((168.0, 72.0), (200.4, 90.0)) ===
=== (40 / 625), theta=((168.0, 72.1), (200.3, 90.0)) ===
=== (60 / 625), theta=((168.0, 72.2), (200.2, 90.0)) ===
=== (80 / 625), theta=((168.0, 72.3), (200.1, 90.0)) ===
=== (100 / 625), theta=((168.0, 72.4), (200.0, 90.0)) ===
=== (120 / 625), theta=((168.0, 72.4), (200.4, 90.0)) ===
=== (140 / 625), theta=((168.1, 72.0), (200.3, 90.0)) ===
=== (160 / 625), theta=((168.1, 72.1), (200.2, 90.0)) ===
=== (180 / 625), theta=((168.1, 72.2), (200.1, 90.0)) ===
=== (200 / 625), theta=((168.1, 72.3), (200.0, 90.0)) ===
=== (220 / 625), theta=((168.1, 72.3), (200.4, 90.0)) ===
=== (240 / 625), theta=((168.1, 72.4), (200.3, 90.0)) ===
=== (260 / 625), theta=((168.2, 72.0), (200.2, 90.0)) ===
=== (280 / 625), theta=((168.2, 72.1), (200.1, 90.0)) ===
=== (300 / 625), theta=((168.2, 72.2), (200.0, 90.0)) ===
=== (320 / 625), theta=((168.2, 72.2), (200.4, 90.0)) ===
=== (340 / 625), theta=((168.2, 72.3), (200.3, 90.0)) ===
=== (360 / 625), theta=((168.2, 72.4), (200.2, 90.0)) ===
=== (380 / 625), theta=((168.3, 72.0), (200.1, 90.0)) ===
=== (400 / 625), theta=((168.3, 72.1), (200.0, 90.0)) ===
=== (420 / 625), theta=((168.3, 72.1), (200.4, 90.0)) ===
=== (440 / 625), theta=((168.3, 72.2), (200.3, 90.0)) ===
=== (460 / 625), theta=((168.3, 72.3), (200.2, 90.0)) ===
=== (480 / 625), theta=((168.3, 72.4), (200.1, 90.0)) ===
=== (500 / 625), theta=((168.4, 72.0), (200.0, 90.0)) ===
=== (520 / 625), theta=((168.4, 72.0), (200.4, 90.0)) ===
=== (540 / 625), theta=((168.4, 72.1), (200.3, 90.0)) ===
=== (560 / 625), theta=((168.4, 72.2), (200.2, 90.0)) ===
=== (580 / 625), theta=((168.4, 72.3), (200.1, 90.0)) ===
=== (600 / 625), theta=((168.4, 72.4), (200.0, 90.0)) ===
=== (620 / 625), theta=((168.4, 72.4), (200.4, 90.0)) ===
best_theta_en:
Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)})
best_Cbarcum=6272.52
best_Ctilcum=34.06
worst_theta_en:
Theta(thLo={'AAA': np.float64(168.29999999999998), 'BBB': np.float64(72.39999999999998)}, thHi={'AAA': np.float64(200.0), 'BBB': np.float64(90.1)})
worst_Cbarcum=5976.43
worst_Ctilcum=43.08
CPU times: user 7min 53s, sys: 2.71 s, total: 7min 56s
Wall time: 7min 55s

best_theta_HighLow

Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)})

P.plot_Fhat_map_4(
  FhatI_theta_I=Cbarcum_HighLow,
  thetasX=th_lo[eNAMES[0]],
  thetasY=th_lo[eNAMES[1]],
  labelX=r'$\theta^{Lo}_{AAA}$',
  labelY=r'$\theta^{Lo}_{BBB}$',
  title="Sample mean of the cumulative reward"+f"\n $L={L}, T={T}$, "+ \
    r"$\theta^* =$"+ P.round_theta(best_theta_HighLow)+", " \
    r"$\bar{C}^{cum}(\theta^*) =$"+f"{best_Cbarcum_HighLow:.2f}",
  thetaFixed1=best_theta_HighLow.thHi[eNAMES[0]],
  thetaFixed2=best_theta_HighLow.thHi[eNAMES[1]],
)
print()
P.plot_Fhat_map_4(
  FhatI_theta_I=Ctilcum_HighLow,
  thetasX=th_lo[eNAMES[0]],
  thetasY=th_lo[eNAMES[1]],
  labelX=r'$\theta^{Lo}_{AAA}$',
  labelY=r'$\theta^{Lo}_{BBB}$',
  title="Standard error of the cumulative reward"+f"\n $L={L}, T={T}$, "+ \
    r"$\theta^* =$"+ P.round_theta(best_theta_HighLow)+", " \
    r"$\tilde{C}^{cum}(\theta^*) =$"+f"{best_Ctilcum_HighLow:.2f}",
  thetaFixed1=best_theta_HighLow.thHi[eNAMES[0]],
  thetaFixed2=best_theta_HighLow.thHi[eNAMES[1]],
);

P.plot_expFhat_chart(
  thetaStar_expCbarcum_HighLow,
  r'$\ell$',
  r"$exp\bar{C}^{cum}(\theta^*)$"+"\n(Profit)\n[$]",
  "Expanding sample mean of the cumulative reward"+f"\n $L={L}, T={T}$, "+ \
    r"$\theta^* =$"+ P.round_theta(best_theta_HighLow)+", " \
    r"$\bar{C}^{cum}(\theta^*) =$"+f"{best_Cbarcum_HighLow:.2f}",
  'b-'
)
print()
P.plot_expFhat_chart(
  thetaStar_expCtilcum_HighLow,
  r'$\ell$',
  r"$exp\tilde{C}^{cum}(\theta^*)$"+"\n(Profit)\n[$]",
  "Expanding standard error of the cumulative reward"+f"\n $L={L}, T={T}$, "+ \
    r"$\theta^* =$"+ P.round_theta(best_theta_HighLow)+", " \
    r"$\tilde{C}^{cum}(\theta^*) =$"+f"{best_Ctilcum_HighLow:.2f}",
  'b--'
);

f'{len(record_HighLow):,}', L, T

('625,000', 50, 20)

best_theta_HighLow

Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)})

P.plot_records(
  df=df_first_n_t,
  df_non=None,
  pars=defaultdict(str, {
    'policy': 'X__HighLow',
    'thetaLo': {en: best_theta_HighLow.thLo[en] for en in eNAMES},
    'thetaHi': {en: best_theta_HighLow.thHi[en] for en in eNAMES},
    'suptitle': f'TRAINING OF X__HighLow POLICY'+'\n'+f'(first {first_n_t} records)'+'\n'+ \
    f'L = {L}, T = {T}, '+ \
    r'$\theta^*=$'+f'{P.round_theta(best_theta_HighLow)}',
  }),
)

P.plot_records(
  df=df_last_n_t,
  df_non=None,
  pars=defaultdict(str, {
    'policy': 'X__HighLow',
    'thetaLo': {en: best_theta_HighLow.thLo[en] for en in eNAMES},
    'thetaHi': {en: best_theta_HighLow.thHi[en] for en in eNAMES},
    'suptitle': f'TRAINING OF X__HighLow POLICY'+'\n'+f'(last {last_n_t} records)'+'\n'+ \
    f'L = {L}, T = {T}, '+ \
    r'$\theta^*=$'+f'{P.round_theta(best_theta_HighLow)}',
  }),
)

4.6.2 Evaluation

4.6.2.1 X__HighLow

L = 100 #N_SAMPLEPATHS
T = 200 #N_TRANSITIONS
# first_n_t = int(.01*L*T)
first_n_t = int(.05*L*T)

M = Model(S_0_INFO)
P = Policy(M)
SIM = PriceSimulator(seed=SEED_EVALU)
thetasOpt = []; thetasOpt.append(best_theta_HighLow)
thetaStar_expCbarcum_HighLow_evalu, thetaStar_expCtilcum_HighLow_evalu, \
Cbarcum_HighLow_evalu, Ctilcum_HighLow_evalu, \
best_theta_HighLow_evalu, worst_theta_HighLow_evalu, \
best_Cbarcum_HighLow_evalu, worst_Cbarcum_HighLow_evalu, \
best_Ctilcum_HighLow_evalu, worst_Ctilcum_HighLow_evalu, \
record_HighLow_evalu = \
  P.perform_grid_search_sample_paths("X__HighLow", thetasOpt)
df_X__HighLow_evalu = pd.DataFrame.from_records(record_HighLow_evalu[:first_n_t], columns=labels)

M = Model(S_0_INFO)
P = Policy(M)
SIM = PriceSimulator(seed=SEED_EVALU)
# thetasNon = []; thetasNon.append(worst_theta_HighLow)
thetasNon = []; thetasNon.append(
    P.build_theta(
        {'thLo': {'AAA': 320, 'BBB': 400}, 'thHi': {'AAA': 350, 'BBB': 430}}
))
thetaStar_expCbarcum_HighLow_evalu_non, thetaStar_expCtilcum_HighLow_evalu_non, \
Cbarcum_HighLow_evalu_non, Ctilcum_HighLow_evalu_non, \
best_theta_HighLow_evalu_non, worst_theta_HighLow_evalu_non, \
best_Cbarcum_HighLow_evalu_non, worst_Cbarcum_HighLow_evalu_non, \
best_Ctilcum_HighLow_evalu_non, worst_Ctilcum_HighLow_evalu_non, \
record_HighLow_evalu_non = \
  P.perform_grid_search_sample_paths("X__HighLow", thetasNon)
print(
  f'{thetaStar_expCbarcum_HighLow_evalu.iloc[-1]=:.2f}, \
    {thetaStar_expCbarcum_HighLow_evalu_non.iloc[-1]=:.2f}')
df_X__HighLow_evalu_non = pd.DataFrame.from_records(record_HighLow_evalu_non[:first_n_t], columns=labels)

num_thetas=1
thetas=[Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)})]
... printing every 20th theta if considered ...
=== (0 / 1), theta=((168.4, 72.2), (200.2, 90.2)) ===
best_theta_en:
Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)})
best_Cbarcum=6195.52
best_Ctilcum=166.76
worst_theta_en:
Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)})
worst_Cbarcum=6195.52
worst_Ctilcum=166.76
num_thetas=1
thetas=[Theta(thLo={'AAA': 320, 'BBB': 400}, thHi={'AAA': 350, 'BBB': 430})]
... printing every 20th theta if considered ...
=== (0 / 1), theta=((320.0, 400.0), (350.0, 430.0)) ===
best_theta_en:
Theta(thLo={'AAA': 320, 'BBB': 400}, thHi={'AAA': 350, 'BBB': 430})
best_Cbarcum=6130.99
best_Ctilcum=120.31
worst_theta_en:
Theta(thLo={'AAA': 320, 'BBB': 400}, thHi={'AAA': 350, 'BBB': 430})
worst_Cbarcum=6130.99
worst_Ctilcum=120.31
thetaStar_expCbarcum_HighLow_evalu.iloc[-1]=6195.52,     thetaStar_expCbarcum_HighLow_evalu_non.iloc[-1]=6130.99

P.plot_records(
  df=df_X__HighLow_evalu,
  df_non=df_X__HighLow_evalu_non,
  pars=defaultdict(str, {
    'policy': 'X__HighLow',
    'thetaLo': {en: thetasOpt[0].thLo[en] for en in eNAMES},
    'thetaHi': {en: thetasOpt[0].thHi[en] for en in eNAMES},
    'thetaLoNon': {en: thetasNon[0].thLo[en] for en in eNAMES},
    'thetaHiNon': {en: thetasNon[0].thHi[en] for en in eNAMES},
    'suptitle': f'EVALUATION OF X__HighLow POLICY'+'\n'+f'(first {first_n_t} records)'+'\n'+ \
    f'L = {L}, T = {T}, '+ \
    r'$\theta^*=$'+f'{P.round_theta(best_theta_HighLow_evalu)}',
  }),
)

last_n_l = int(1.*L)
# last_n_l = int(.95*L)
# last_n_l = int(.90*L)
P.plot_expFhat_charts(
  means={
      'HighLow optimal': thetaStar_expCbarcum_HighLow_evalu[-last_n_l:],
      'HighLow non-optimal': thetaStar_expCbarcum_HighLow_evalu_non[-last_n_l:],
  },
  stdvs={
      'HighLow optimal': thetaStar_expCtilcum_HighLow_evalu[-last_n_l:],
      'HighLow non-optimal': thetaStar_expCtilcum_HighLow_evalu_non[-last_n_l:],
  },
  labelX='Sample paths, ' + r'$\ell$',
  labelY='Profit\n[$]',
  suptitle=f"Comparison of Optimal/Non-optimal Policies after Evaluation\n \
    L = {L}, T = {T}\n \
    last {last_n_l} records\n \
    ('exp' refers to expanding)",
  pars=defaultdict(str, {
    'colors': ['m', 'c']
  }),
)

Next, we evaluate with a single, very long sample-path:

# SEED_EVALU = 555
L = 1 #N_SAMPLEPATHS
T = 50_000 #5000 #N_TRANSITIONS
first_n_t = int(1*L*T)

M = Model(S_0_INFO)
P = Policy(M)
SIM = PriceSimulator(seed=SEED_EVALU)
thetasOpt = []; thetasOpt.append(best_theta_HighLow)
thetaStar_expCbarcum_HighLow_evalu, thetaStar_expCtilcum_HighLow_evalu, \
Cbarcum_HighLow_evalu, Ctilcum_HighLow_evalu, \
best_theta_HighLow_evalu, worst_theta_HighLow_evalu, \
best_Cbarcum_HighLow_evalu, worst_Cbarcum_HighLow_evalu, \
best_Ctilcum_HighLow_evalu, worst_Ctilcum_HighLow_evalu, \
record_HighLow_evalu = \
  P.perform_grid_search_sample_paths("X__HighLow", thetasOpt)
df_X__HighLow_evalu = pd.DataFrame.from_records(record_HighLow_evalu[:first_n_t], columns=labels)

M = Model(S_0_INFO)
P = Policy(M)
SIM = PriceSimulator(seed=SEED_EVALU)
thetasNon = []; thetasNon.append(
    P.build_theta(
        {'thLo': {'AAA': 320, 'BBB': 400}, 'thHi': {'AAA': 350, 'BBB': 430}}
))
thetaStar_expCbarcum_HighLow_evalu_non, thetaStar_expCtilcum_HighLow_evalu_non, \
Cbarcum_HighLow_evalu_non, Ctilcum_HighLow_evalu_non, \
best_theta_HighLow_evalu_non, worst_theta_HighLow_evalu_non, \
best_Cbarcum_HighLow_evalu_non, worst_Cbarcum_HighLow_evalu_non, \
best_Ctilcum_HighLow_evalu_non, worst_Ctilcum_HighLow_evalu_non, \
record_HighLow_evalu_non = \
  P.perform_grid_search_sample_paths("X__HighLow", thetasNon)
print(
  f'{thetaStar_expCbarcum_HighLow_evalu.iloc[-1]=:.2f}, \
    {thetaStar_expCbarcum_HighLow_evalu_non.iloc[-1]=:.2f}')
df_X__HighLow_evalu_non = pd.DataFrame.from_records(record_HighLow_evalu_non[:first_n_t], columns=labels)

num_thetas=1
thetas=[Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)})]
... printing every 20th theta if considered ...
=== (0 / 1), theta=((168.4, 72.2), (200.2, 90.2)) ===
best_theta_en:
Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)})
best_Cbarcum=16104665.55
best_Ctilcum=nan
worst_theta_en:
Theta(thLo={'AAA': np.float64(168.39999999999998), 'BBB': np.float64(72.19999999999999)}, thHi={'AAA': np.float64(200.2), 'BBB': np.float64(90.19999999999999)})
worst_Cbarcum=16104665.55
worst_Ctilcum=nan
num_thetas=1
thetas=[Theta(thLo={'AAA': 320, 'BBB': 400}, thHi={'AAA': 350, 'BBB': 430})]
... printing every 20th theta if considered ...
=== (0 / 1), theta=((320.0, 400.0), (350.0, 430.0)) ===
best_theta_en:
Theta(thLo={'AAA': 320, 'BBB': 400}, thHi={'AAA': 350, 'BBB': 430})
best_Cbarcum=54325.45
best_Ctilcum=nan
worst_theta_en:
Theta(thLo={'AAA': 320, 'BBB': 400}, thHi={'AAA': 350, 'BBB': 430})
worst_Cbarcum=54325.45
worst_Ctilcum=nan
thetaStar_expCbarcum_HighLow_evalu.iloc[-1]=16104665.55,     thetaStar_expCbarcum_HighLow_evalu_non.iloc[-1]=54325.45

/tmp/ipykernel_959/2683048994.py:113: RuntimeWarning: invalid value encountered in scalar divide
  Ctilcum_tmp = np.sum(np.square(np.array(CcumIomega__lI) - Cbarcum_tmp))/(L - 1)
/tmp/ipykernel_959/2683048994.py:113: RuntimeWarning: invalid value encountered in scalar divide
  Ctilcum_tmp = np.sum(np.square(np.array(CcumIomega__lI) - Cbarcum_tmp))/(L - 1)

P.plot_records(
  df=df_X__HighLow_evalu,
  df_non=df_X__HighLow_evalu_non,
  pars=defaultdict(str, {
    'policy': 'X__HighLow',
    'thetaLo': {en: thetasOpt[0].thLo[en] for en in eNAMES},
    'thetaHi': {en: thetasOpt[0].thHi[en] for en in eNAMES},
    'thetaLoNon': {en: thetasNon[0].thLo[en] for en in eNAMES},
    'thetaHiNon': {en: thetasNon[0].thHi[en] for en in eNAMES},
    'suptitle': f'EVALUATION OF X__HighLow POLICY'+'\n'+f'(first {first_n_t:,} records)'+'\n'+ \
    f'L = {L}, T = {T:,}, '+ \
    r'$\theta^*=$'+f'{P.round_theta(best_theta_HighLow_evalu_non)}',
  }),
)

df_X__HighLow_evalu[df_X__HighLow_evalu['t']==T-1][['Ccum']]

	Ccum
49999	16,104,665.5534

df_X__HighLow_evalu_non[df_X__HighLow_evalu_non['t']==T-1][['Ccum']]

	Ccum
49999	54,325.4473