# Build Your Own Model¶

One important feature of `orbit`

is to allow developers to build their own models in a relatively flexible manner to serve their own purpose. This tutorial will go over a demo on how to build up a simple Bayesian linear regression model using Pyro API in the backend with orbit interface.

## Orbit Class Design¶

In version `1.1.0`

, the classes within Orbit are re-designed as such:

Forecaster

Model

Estimator

### Forecaster¶

**Forecaster** provides general interface for users to perform `fit`

and `predict`

task. It is further inherited to provide different types of forecasting methodology:

[Stochastic Variational Inference (SVI)]

Full Bayesian

The discrepancy on these three methods mainly lie on the posteriors estimation where **MAP** will yield point posterior estimate and can be extracted through the method `get_point_posterior()`

. Meanwhile, **SVI** and **Full Bayesian** allow posterior sample extraction through the method `get_posteriors()`

. Alternatively, you can also approximate point estimate by passing through additional arg such as `point_method='median'`

in the `.fit()`

process.

To make use of a **Forecaster**, one must provide these two objects:

Model

Estimator

Theses two objects are prototyped as abstract and next subsections will cover how they work.

### Model¶

**Model** is an object defined by a class inherited from `BaseTemplate`

a.k.a **Model Template** in the diagram below. It mainly turns the logic of `fit()`

and `predict()`

concrete by supplying the `fitter`

as a file (**PyStan**) or a callable class (**Pyro**) and the internal `predict()`

method. This object defines the overall inputs, model structure, parameters and likelihoods.

### Estimator¶

Meanwhile, there are different APIs implement slightly different ways of sampling and optimization (for **MAP**). `orbit`

is designed to support various APIs such as **PyStan** and **Pyro** (hopefully PyMC3, Numpyro in the future!). The logic separating the call of different APIs with different interface is done by the **Estimator** class which is further inherited in `PyroEstimator`

and `StanEstimator`

.

Diagram above shows the interaction across classes under the Orbit package design.

## Creating a Bayesian Linear Regression Model¶

The plan here is to build a classical regression model with the formula below:

where \(\alpha\) is the intercept, \(\beta\) is the coefficients matrix and \(\epsilon\) is the random noise.

To start with let’s load the libraries.

```
[1]:
```

```
import pandas as pd
import numpy as np
import torch
import pyro
import pyro.distributions as dist
from copy import deepcopy
import matplotlib.pyplot as plt
import orbit
from orbit.template.model_template import ModelTemplate
from orbit.forecaster import SVIForecaster
from orbit.estimators.pyro_estimator import PyroEstimatorSVI
from orbit.utils.simulation import make_regression
from orbit.diagnostics.plot import plot_predicted_data
from orbit.utils.plot import get_orbit_style
plt.style.use(get_orbit_style())
%matplotlib inline
```

```
[2]:
```

```
print(orbit.__version__)
```

```
1.1.0
```

Since the **Forecaster** and **Estimator** are already built inside `orbit`

, the rest of the ingredients to construct a new model will be a **Model** object that contains the follow:

a callable class as a fitter

a predict method

### Define a Fitter¶

For **Pyro** users, you should find the code below familiar. All it does is to put a Bayesian linear regression (**BLR**) model code in a callable class. Details of **BLR** will not be covered here. Note that the parameters here need to be consistent .

```
[3]:
```

```
class MyFitter:
max_plate_nesting = 1 # max number of plates nested in model
def __init__(self, data):
for key, value in data.items():
key = key.lower()
if isinstance(value, (list, np.ndarray)):
value = torch.tensor(value, dtype=torch.float)
self.__dict__[key] = value
def __call__(self):
extra_out = {}
p = self.regressor.shape[1]
bias = pyro.sample("bias", dist.Normal(0, 1))
weight = pyro.sample("weight", dist.Normal(0, 1).expand([p]).to_event(1))
yhat = bias + weight @ self.regressor.transpose(-1, -2)
obs_sigma = pyro.sample("obs_sigma", dist.HalfCauchy(self.response_sd))
with pyro.plate("response_plate", self.num_of_obs):
pyro.sample("response", dist.Normal(yhat, obs_sigma), obs=self.response)
return extra_out
```

### Define the Model Class¶

This is the part requires the knowledge of `orbit`

most. First we construct a class by plugging in the `fitter`

callable. Users need to let the `orbit`

estimators know the required input in addition to the defaults (e.g. `response`

, `response_sd`

etc.). In this case, it takes `regressor`

as the matrix input from the data frame. That is why there are lines of code to provide this information in

`_data_input_mapper`

- a list or`Enum`

to let estimator keep tracking required data input`set_dynamic_attributes`

- the logic define the actual inputs i.e.`regressor`

from the data frame. This is a**reserved function**being called inside**Forecaster**.

Finally, we code the logic in `predict()`

to define how we utilize posteriors to perform in-sample / out-of-sample prediction. Note that the output needs to be a dictionary where it supports **components decomposition**.

```
[4]:
```

```
class BayesLinearRegression(ModelTemplate):
_fitter = MyFitter
_data_input_mapper = ['regressor']
_supported_estimator_types = [PyroEstimatorSVI]
def __init__(self, regressor_col, **kwargs):
super().__init__(**kwargs)
self.regressor_col = regressor_col
self.regressor = None
self._model_param_names = ['bias', 'weight', 'obs_sigma']
def set_dynamic_attributes(self, df, training_meta):
self.regressor = df[self.regressor_col].values
def predict(self, posterior_estimates, df, training_meta, prediction_meta, include_error=False, **kwargs):
model = deepcopy(posterior_estimates)
new_regressor = df[self.regressor_col].values.T
bias = np.expand_dims(model.get('bias'),-1)
obs_sigma = np.expand_dims(model.get('obs_sigma'), -1)
weight = model.get('weight')
pred_len = df.shape[0]
batch_size = weight.shape[0]
prediction = bias + np.matmul(weight, new_regressor) + \
np.random.normal(0, obs_sigma, size=(batch_size, pred_len))
return {'prediction': prediction}
```

## Test the New Model with Forecaster¶

Once the model class is defined. User can initialize an object and build a forecaster for fit and predict purpose. Before doing that, the demo provides a simulated dataset here.

### Data Simulation¶

```
[5]:
```

```
x, y, coefs = make_regression(120, [3.0, -1.0], bias=1.0, scale=1.0)
```

```
[6]:
```

```
df = pd.DataFrame(
np.concatenate([y.reshape(-1, 1), x], axis=1), columns=['y', 'x1', 'x2']
)
df['week'] = pd.date_range(start='2016-01-04', periods=len(y), freq='7D')
```

```
[7]:
```

```
df.head(5)
```

```
[7]:
```

y | x1 | x2 | week | |
---|---|---|---|---|

0 | 2.382337 | 0.345584 | 0.000000 | 2016-01-04 |

1 | 2.812929 | 0.330437 | -0.000000 | 2016-01-11 |

2 | 3.600130 | 0.905356 | 0.446375 | 2016-01-18 |

3 | -0.884275 | -0.000000 | 0.581118 | 2016-01-25 |

4 | 2.704941 | 0.364572 | 0.294132 | 2016-02-01 |

```
[8]:
```

```
test_size = 20
train_df = df[:-test_size]
test_df = df[-test_size:]
```

### Create the Forecaster¶

As mentioned previously, model is the inner object to control the math. To use it for fit and predict purpose, we need a **Forecaster**. Since the model is written in **Pyro**, the pick here should be `SVIForecaster`

.

```
[9]:
```

```
model = BayesLinearRegression(
regressor_col=['x1','x2'],
)
```

```
[11]:
```

```
blr = SVIForecaster(
model=model,
response_col='y',
date_col='week',
estimator_type=PyroEstimatorSVI,
verbose=True,
num_steps=501,
seed=2021,
)
```

```
INFO:root:Using 501 steps, 100 samples, 0.1 learning rate and 100 particles for SVI.
```

```
[12]:
```

```
blr
```

```
[12]:
```

```
<orbit.forecaster.svi.SVIForecaster at 0x1400328d0>
```

Now, an object `blr`

is instantiated as a `SVIForecaster`

object and is ready for fit and predict.

```
[13]:
```

```
blr.fit(train_df)
```

```
INFO:root:Guessed max_plate_nesting = 2
```

```
step 0 loss = 27333, scale = 0.077552
step 100 loss = 12590, scale = 0.0092793
step 200 loss = 12597, scale = 0.0098217
step 300 loss = 12591, scale = 0.0095262
step 400 loss = 12593, scale = 0.0092962
step 500 loss = 12591, scale = 0.0095438
```

```
[13]:
```

```
<orbit.forecaster.svi.SVIForecaster at 0x1400328d0>
```

### Compare Coefficients with Truth¶

```
[14]:
```

```
estimated_weights = blr.get_posterior_samples()['weight']
```

The code below is to compare the median of coefficients posteriors which is labeled as `weight`

with the truth.

```
[15]:
```

```
print("True Coef: {:.3f}, {:.3f}".format(coefs[0], coefs[1]) )
estimated_coef = np.median(estimated_weights, axis=0)
print("Estimated Coef: {:.3f}, {:.3f}".format(estimated_coef[0], estimated_coef[1]))
```

```
True Coef: 3.000, -1.000
Estimated Coef: 2.951, -0.964
```

### Examine Forecast Accuracy¶

```
[16]:
```

```
predicted_df = blr.predict(df)
```

```
[17]:
```

```
_ = plot_predicted_data(train_df, predicted_df, 'week', 'y', test_actual_df=test_df, prediction_percentiles=[5, 95])
```

### Additional Notes¶

In general, most of the **diagnostic tools** in orbit such as posteriors checking and plotting is applicable in the model created in this style. Also, users can provide `point_method='median'`

in the `fit()`

under the **SVIForecaster** to extract median of posteriors directly.