How to set the equations to implement a Kalman filter for predicting the foot traffic of a restaurant?

Question

I want to implement a Kalman filter to predict the actual daily number of dine-in customers in a restaurant with the two sets of time series data below.

1. Daily total number of people entering the restaurant: This is not exactly the same as the number of dine-in customers because it also includes the number of staffs and take-away customers entering the restaurant.
2. Daily total number of main dishes sold (dine-in): This is not exactly the same as the number of dine-in customers because some customers may order more than one main dish while some customers may not order any main dish at all.

With the above, how should I set the equations for implementing a Kalman filter to make the best guess of the actual number of dine-in customers?

Answer 1

You must have some relations between the data in order to formulate a Kalman Filtering Problem. The State Equations must be in this format:

State Transition: x_(k+1) = Ak * x_k + Bk * u_k

Measurement: yk = Ck * x_k + Dk * u_k

Your measurements are the datasets you have:

y1: Customers Entered

y2: Dishes Ordered

Let me setup some example equations:

Let the states be:

x1: Number of dine in customers

x2: Number of Staff

x3: Number of takeaway customers

Now y1 = x1 + x2 + x3

y2 = 4 * x1 + 5 * x3

This gives you the C matrix

You must still define an A matrix that defines a relationship between current customers at step k and the customers coming next at step k+1.

Without these relations, you might not have a Kalman Filtering problem.