State-space models revisited

Skill positionModule 1 of 3 · Topic 1 of 2

1.1 State-space models revisited

—Lab score— / 100Total XP—Streak—

Tracking something you cannot see

The beta of a stock, its volatility, the fair level of a spread: nobody can read these numbers off a screen. What you see each day is a price or a return, which is the hidden quantity plus noise. A state-space model writes this down with two equations. The state $x_t$ is the hidden quantity and it moves a little every day. The observation $y_t$ is what you actually measure. The simplest case is the local level model: the state does a random walk, and each observation is the state plus measurement noise.

x_t=x_{t-1}+w_t,\ \ w_t\sim N(0,q),\qquad y_t=x_t+v_t,\ \ v_t\sim N(0,r)

(1)

Here $q$ is how much the state wanders in one day, and $r$ is how noisy one observation is. Three questions come up again and again. Prediction asks where the state will be tomorrow before you see tomorrow's data. Filtering asks where the state is now, after seeing today's observation. Smoothing asks where it was in the past, once more data has arrived. A trader who needs a live number can only use filtering. All three keep the belief as a bell curve with a mean and a variance. Predicting makes the variance bigger by $q$ , because the state may have moved. Seeing an observation makes it smaller, because it carries information.

A first look with numbersStart with the belief

x_0\sim N(0,2)

, and let

q=0.5

and

r=1

. After one day of motion, with no data yet, the variance of

x_1

2+0.5=2.5

. The observation adds its own noise, so

y_1

has predictive variance

2.5+1=3.5

and mean

0

. The chance that

y_1

comes out above

2

is the chance that a standard normal is above

2/\sqrt{3.5}=1.07

, about

14\%

. Without data, the variance keeps growing by

0.5

per day: after four days it is

2+4\times0.5=4

State versus observation: the state is hidden and moves slowly; the observation is noisy and arrives every day.
Predict then update: the prediction step adds uncertainty $q$ ; the update step uses the data to remove some of it.
Same idea as a hidden Markov model: there the state has a few values, here it is a number on a line.

Lab · what's graded

One step of motion15 pts
Variance of an observation15 pts
Four days without data15 pts
A large observation10 pts
Which quantities are hidden10 pts
The filter cycle10 pts
Match the vocabulary10 pts
Why filter at all15 pts

0/ 8 answered

1Numeric answer15 ptsNot answered

The belief about $x_0$ is $N(0,2)$ and the state moves with $q=0.5$ . What is the variance of $x_1$ before any data arrives?

P =

accepted: ± 0.01

2Numeric answer15 ptsNot answered

The state has variance $2.5$ and the measurement noise has variance $r=1$ . What is the predictive variance of the observation $y_1$ ?

S =

accepted: ± 0.01

3Numeric answer15 ptsNot answered

Start from variance $2$ and let the state move for four days with $q=0.5$ and no observations. What is the variance of $x_4$ ?

P4 =

accepted: ± 0.01

4Numeric answer10 ptsNot answered

The observation $y_1$ is $N(0,3.5)$ . The standardised value of $2$ is $z=1.07$ . Using that the standard normal has $P(Z>1.07)\approx0.143$ , give the chance that $y_1$ is above $2$ to three decimals.

p =

accepted: ± 0.003

5Choice10 ptsNot answered

Which of these are hidden states, never read directly from the market? Pick all that apply.

6Order the steps10 ptsNot answered

Put the steps of one filter cycle in order.

Drag to reorder, or focus an item, press Space to lift it, the arrow keys to move it and Space to drop it.

1Use the observation to update the belief
2Start with yesterday's belief
3Repeat tomorrow with the new belief
4Observe today's value
5Predict where the state has moved

7Match10 ptsNot answered

Match each term with its meaning.

Pick the partner of each item on the left.

State noise q

Measurement noise r

Filtering

Smoothing

8Explain in words15 ptsNot answered

In one or two sentences, explain why combining a prediction with an observation, as a filter does, gives a better estimate of the hidden state than the latest noisy observation alone.

Graded on:One observation is noisy, so it alone can be far from the true stateThe prediction carries what earlier data said, so mixing it with the new data averages out noise0 / 400 characters

Math modeKeyboard or LaTeX inputTask 1 of 8 · Ctrl ⏎ to check

State-space models revisited

Skill positionModule 1 of 3 · Topic 1 of 2

1.1 State-space models revisited

—Lab score— / 100Total XP—Streak—

Tracking something you cannot see

x_t=x_{t-1}+w_t,\ \ w_t\sim N(0,q),\qquad y_t=x_t+v_t,\ \ v_t\sim N(0,r)

(1)

A first look with numbersStart with the belief

x_0\sim N(0,2)

, and let

q=0.5

and

r=1

. After one day of motion, with no data yet, the variance of

x_1

2+0.5=2.5

. The observation adds its own noise, so

y_1

has predictive variance

2.5+1=3.5

and mean

0

. The chance that

y_1

comes out above

2

is the chance that a standard normal is above

2/\sqrt{3.5}=1.07

, about

14\%

. Without data, the variance keeps growing by

0.5

per day: after four days it is

2+4\times0.5=4

State versus observation: the state is hidden and moves slowly; the observation is noisy and arrives every day.
Predict then update: the prediction step adds uncertainty $q$ ; the update step uses the data to remove some of it.
Same idea as a hidden Markov model: there the state has a few values, here it is a number on a line.

Lab · what's graded

One step of motion15 pts
Variance of an observation15 pts
Four days without data15 pts
A large observation10 pts
Which quantities are hidden10 pts
The filter cycle10 pts
Match the vocabulary10 pts
Why filter at all15 pts

0/ 8 answered

1Numeric answer15 ptsNot answered

The belief about $x_0$ is $N(0,2)$ and the state moves with $q=0.5$ . What is the variance of $x_1$ before any data arrives?

P =

accepted: ± 0.01

2Numeric answer15 ptsNot answered

The state has variance $2.5$ and the measurement noise has variance $r=1$ . What is the predictive variance of the observation $y_1$ ?

S =

accepted: ± 0.01

3Numeric answer15 ptsNot answered

Start from variance $2$ and let the state move for four days with $q=0.5$ and no observations. What is the variance of $x_4$ ?

P4 =

accepted: ± 0.01

4Numeric answer10 ptsNot answered

p =

accepted: ± 0.003

5Choice10 ptsNot answered

Which of these are hidden states, never read directly from the market? Pick all that apply.

6Order the steps10 ptsNot answered

Put the steps of one filter cycle in order.

Drag to reorder, or focus an item, press Space to lift it, the arrow keys to move it and Space to drop it.

1Use the observation to update the belief
2Start with yesterday's belief
3Repeat tomorrow with the new belief
4Observe today's value
5Predict where the state has moved

7Match10 ptsNot answered

Match each term with its meaning.

Pick the partner of each item on the left.

State noise q

Measurement noise r

Filtering

Smoothing

8Explain in words15 ptsNot answered

In one or two sentences, explain why combining a prediction with an observation, as a filter does, gives a better estimate of the hidden state than the latest noisy observation alone.

Graded on:One observation is noisy, so it alone can be far from the true stateThe prediction carries what earlier data said, so mixing it with the new data averages out noise0 / 400 characters

Math modeKeyboard or LaTeX inputTask 1 of 8 · Ctrl ⏎ to check

State-space models revisited

Tracking something you cannot see

Lab · what's graded

Log in to run your code.

State-space models revisited

Tracking something you cannot see

Lab · what's graded

Log in to run your code.