How to Convert a Time Series to a Supervised Learning Problem in Python

Machine learning methods like deep learning can be used for time series forecasting.

Before machine learning can be used, time series forecasting problems must be re-framed as supervised learning problems. From a sequence to pairs of input and output sequences.

In this tutorial, you will discover how to transform univariate and multivariate time series forecasting problems into supervised learning problems for use with machine learning algorithms.

After completing this tutorial, you will know:

  • How to develop a function to transform a time series dataset into a supervised learning dataset.
  • How to transform univariate time series data for machine learning.
  • How to transform multivariate time series data for machine learning.

Let’s get started.

How to Convert a Time Series to a Supervised Learning Problem in Python

How to Convert a Time Series to a Supervised Learning Problem in Python
Photo by Quim Gil, some rights reserved.

Time Series vs Supervised Learning

Before we get started, let’s take a moment to better understand the form of time series and supervised learning data.

A time series is a sequence of numbers that are ordered by a time index. This can be thought of as a list or column of ordered values.

For example:

A supervised learning problem is comprised of input patterns (X) and output patterns (y), such that an algorithm can learn how to predict the output patterns from the input patterns.

For example:

For more on this topic, see the post:

Pandas shift() Function

A key function to help transform time series data into a supervised learning problem is the Pandas shift() function.

Given a DataFrame, the shift() function can be used to create copies of columns that are pushed forward (rows of NaN values added to the front) or pulled back (rows of NaN values added to the end).

This is the behavior required to create columns of lag observations as well as columns of forecast observations for a time series dataset in a supervised learning format.

Let’s look at some examples of the shift function in action.

We can define a mock time series dataset as a sequence of 10 numbers, in this case a single column in a DataFrame as follows:

Running the example prints the time series data with the row indices for each observation.

We can shift all the observations down by one time step by inserting one new row at the top. Because the new row has no data, we can use NaN to represent “no data”.

The shift function can do this for us and we can insert this shifted column next to our original series.

Running the example gives us two columns in the dataset. The first with the original observations and a new shifted column.

We can see that shifting the series forward one time step gives us a primitive supervised learning problem, although with X and y in the wrong order. Ignore the column of row labels. The first row would have to be discarded because of the NaN value. The second row shows the input value of 0.0 in the second column (input or X) and the value of 1 in the first column (output or y).

We can see that if we can repeat this process with shifts of 2, 3, and more, how we could create long input sequences (X) that can be used to forecast an output value (y).

The shift operator can also accept a negative integer value. This has the effect of pulling the observations up by inserting new rows at the end. Below is an example:

Running the example shows a new column with a NaN value as the last value.

We can see that the forecast column can be taken as an input (X) and the second as an output value (y). That is the input value of 0 can be used to forecast the output value of 1.

Technically, in time series forecasting terminology the current time (t) and future times (t+1, t+n) are forecast times and past observations (t-1, t-n) are used to make forecasts.

We can see how positive and negative shifts can be used to create a new DataFrame from a time series with sequences of input and output patterns for a supervised learning problem.

This permits not only classical X -> y prediction, but also X -> Y where both input and output can be sequences.

Further, the shift function also works on so-called multivariate time series problems. That is where instead of having one set of observations for a time series, we have multiple (e.g. temperature and pressure). All variates in the time series can be shifted forward or backward to create multivariate input and output sequences. We will explore this more later in the tutorial.

The series_to_supervised() Function

We can use the shift() function in Pandas to automatically create new framings of time series problems given the desired length of input and output sequences.

This would be a useful tool as it would allow us to explore different framings of a time series problem with machine learning algorithms to see which might result in better performing models.

In this section, we will define a new Python function named series_to_supervised() that takes a univariate or multivariate time series and frames it as a supervised learning dataset.

The function takes four arguments:

  • data: Sequence of observations as a list or 2D NumPy array. Required.
  • n_in: Number of lag observations as input (X). Values may be between [1..len(data)] Optional. Defaults to 1.
  • n_out: Number of observations as output (y). Values may be between [0..len(data)-1]. Optional. Defaults to 1.
  • dropnan: Boolean whether or not to drop rows with NaN values. Optional. Defaults to True.

The function returns a single value:

  • return: Pandas DataFrame of series framed for supervised learning.

The new dataset is constructed as a DataFrame, with each column suitably named both by variable number and time step. This allows you to design a variety of different time step sequence type forecasting problems from a given univariate or multivariate time series.

Once the DataFrame is returned, you can decide how to split the rows of the returned DataFrame into X and y components for supervised learning any way you wish.

The function is defined with default parameters so that if you call it with just your data, it will construct a DataFrame with t-1 as X and t as y.

The function is confirmed to be compatible with Python 2 and Python 3.

The complete function is listed below, including function comments.

Can you see obvious ways to make the function more robust or more readable?
Please let me know in the comments below.

Now that we have the whole function, we can explore how it may be used.

One-Step Univariate Forecasting

It is standard practice in time series forecasting to use lagged observations (e.g. t-1) as input variables to forecast the current time step (t).

This is called one-step forecasting.

The example below demonstrates a one lag time step (t-1) to predict the current time step (t).

Running the example prints the output of the reframed time series.

We can see that the observations are named “var1” and that the input observation is suitably named (t-1) and the output time step is named (t).

We can also see that rows with NaN values have been automatically removed from the DataFrame.

We can repeat this example with an arbitrary number length input sequence, such as 3. This can be done by specifying the length of the input sequence as an argument; for example:

The complete example is listed below.

Again, running the example prints the reframed series. We can see that the input sequence is in the correct left-to-right order with the output variable to be predicted on the far right.

Multi-Step or Sequence Forecasting

A different type of forecasting problem is using past observations to forecast a sequence of future observations.

This may be called sequence forecasting or multi-step forecasting.

We can frame a time series for sequence forecasting by specifying another argument. For example, we could frame a forecast problem with an input sequence of 2 past observations to forecast 2 future observations as follows:

The complete example is listed below:

Running the example shows the differentiation of input (t-n) and output (t+n) variables with the current observation (t) considered an output.

Multivariate Forecasting

Another important type of time series is called multivariate time series.

This is where we may have observations of multiple different measures and an interest in forecasting one or more of them.

For example, we may have two sets of time series observations obs1 and obs2 and we wish to forecast one or both of these.

We can call series_to_supervised() in exactly the same way.

For example:

Running the example prints the new framing of the data, showing an input pattern with one time step for both variables and an output pattern of one time step for both variables.

Again, depending on the specifics of the problem, the division of columns into X and Y components can be chosen arbitrarily, such as if the current observation of var1 was also provided as input and only var2 was to be predicted.

You can see how this may be easily used for sequence forecasting with multivariate time series by specifying the length of the input and output sequences as above.

For example, below is an example of a reframing with 1 time step as input and 2 time steps as forecast sequence.

Running the example shows the large reframed DataFrame.

Experiment with your own dataset and try multiple different framings to see what works best.

Summary

In this tutorial, you discovered how to reframe time series datasets as supervised learning problems with Python.

Specifically, you learned:

  • About the Pandas shift() function and how it can be used to automatically define supervised learning datasets from time series data.
  • How to reframe a univariate time series into one-step and multi-step supervised learning problems.
  • How to reframe multivariate time series into one-step and multi-step supervised learning problems.

Do you have any questions?
Ask your questions in the comments below and I will do my best to answer.

Want to Develop Time Series Forecasts with Python?

Introduction to Time Series Forecasting With Python

Develop Your Own Forecasts in Minutes

...with just a few lines of python code

Discover how in my new Ebook:
Introduction to Time Series Forecasting With Python

It covers self-study tutorials and end-to-end projects on topics like:
Loading data, visualization, modeling, algorithm tuning, and much more...

Finally Bring Time Series Forecasting to
Your Own Projects

Skip the Academics. Just Results.

Click to learn more.

62 Responses to How to Convert a Time Series to a Supervised Learning Problem in Python

  1. Mikkel May 8, 2017 at 7:07 pm #

    Hi Jason, thanks for your highly relevant article 🙂

    I am having a hard time following the structure of the dataset. I understand the basics of t-n, t-1, t, t+1, t+n and so forth. Although, what exactly are we describing here in the t and t-1 column? Is it the change over time for a specific explanatory variable? In that case, wouldn’t it make more sense to transpose the data, so that the time were described in the rows rather than columns?

    Also, how would you then characterise following data:

    Customer_ID Month Balance
    1 01 1,500
    1 02 1,600
    1 03 1,700
    1 04 1,900
    2 01 1,000
    2 02 900
    2 03 700
    2 04 500
    3 01 3,500
    3 02 1,500
    3 03 2,500
    3 04 4,500

    Let’s say, that we wanna forcast their balance using supervised learning, or classify the customers as “savers” or “spenders”

    • Jason Brownlee May 9, 2017 at 7:40 am #

      Yes, it is transposing each variable, but allowing control over the length of each row back into time.

  2. Daniel May 9, 2017 at 4:56 pm #

    Hey Jason,

    this is an awesome article! I was looking for that the whole time.

    The only thing is I am general programming in R, so I only found something similar like your code, but I am not sure if it is the same. I have got this from https://www.r-bloggers.com/generating-a-laglead-variables/ and it deals with lagged and leaded values. Also the output includes NA values.

    shift1)
    return(sapply(shift_by,shift, x=x))

    out 0 )
    out<-c(tail(x,-abs_shift_by),rep(NA,abs_shift_by))
    else if (shift_by < 0 )
    out<-c(rep(NA,abs_shift_by), head(x,-abs_shift_by))
    else
    out<-x
    out
    }

    Output:
    x df_lead2 df_lag2
    1 1 3 NA
    2 2 4 NA
    3 3 5 1
    4 4 6 2
    5 5 7 3
    6 6 8 4
    7 7 9 5
    8 8 10 6
    9 9 NA 7
    10 10 NA 8

    I also tried to recompile your code in R, but it failed.

    • Jason Brownlee May 10, 2017 at 8:44 am #

      I would recommend contacting the authors of the R code you reference.

      • chris May 12, 2017 at 3:28 am #

        Can you answer this in Go, Java, C# and COBOL as well????? Thanks, I really don’t want to do anything

        • Jason Brownlee May 12, 2017 at 7:45 am #

          I do my best to help, some need more help than others.

  3. Lee May 9, 2017 at 11:40 pm #

    Hi Jason, good article, but could be much better if you illustrated everything with some actual time series data. Also, no need to repeat the function code 5 times 😉 Gripes aside, this was very timely as I’m just about to get into some time series forecasting, so thanks for this article!!!

  4. Christopher May 12, 2017 at 9:16 pm #

    Hi Jason,
    thank you for the good article! I really like the shifting approach for reframing the training data!
    But my question about this topic is: What do you think is the next step for a one-step univariate forecasting? Which machine learning method is the most suitable for that?
    Obviously a regressor is the best choice but how can I determine the size of the sliding window for the training?

    Thanks a lot for your help and work
    ~ Christopher

  5. tom June 8, 2017 at 4:06 pm #

    hi Jason:
    In this post, you create new framings of time series ,such as t-1, t, t+1.But, what’s the use of these time series .Do you mean these time series can make a good effect on model? Maybe
    my question is too simple ,because I am a newer ,please understand! thank you !

    • Jason Brownlee June 9, 2017 at 6:19 am #

      I am providing a technique to help you convert a series into a supervised learning problem.

      This is valuable because you can then transform your time series problems into supervised learning problems and apply a suite of standard classification and regression techniques in order to make forecasts.

      • tom June 9, 2017 at 11:40 am #

        Wow, your answer always makes me learn a lot。Thank you Jason!

  6. Brad Suzon June 23, 2017 at 11:32 pm #

    If there are multiple variables varXi to train and only one variable varY to predict will the same technique be used in the below way:
    varX1(t-1) varX2(t-1) varX1(t) varX2(t) … varY(t-1) varY(t)
    .. .. .. .. .. ..
    and then use linear regression and as Response= varY(t) ?

    Thanks in advance

    • Jason Brownlee June 24, 2017 at 8:03 am #

      Not sure I follow your question Brad, perhaps you can restate it?

    • Brad June 25, 2017 at 4:47 pm #

      In case there are multiple measures and then make the transformation in order to forecast only varXn:

      var1(t-1) var2(t-1) var1(t) var2(t) … varN(t-1) varN(t)

      linear regression should use as the response variable the varN(t) ?

  7. Geoff June 24, 2017 at 8:10 am #

    Hi Jason,
    I’ve found your articles very useful during my capstone at a bootcamp I’m attending. I have two questions that I hope you could advise where to find better info about.
    First, I’ve run into an issue with running PCA on the newly supervised version only the data. Does PCA recognize that the lagged series are actually the same data? If one was to do PCA do they need to perform it before supervising the data?
    Secondly, what do you propose as the best learning algorithms and proper ways to perform train test splits on the data?
    Thanks again,

  8. Kushal July 1, 2017 at 1:31 pm #

    Hi Jason

    Great post.

    Just one question. If the some of the input variables are continuous and some are categorical with one binary, predicting two output variables.

    How does the shift work then?

    Thanks
    Kushal

    • Jason Brownlee July 2, 2017 at 6:26 am #

      The same, but consider encoding your categorical variables first (e.g. number encoding or one hot encoding).

      • Kushal July 15, 2017 at 5:22 pm #

        Thanks

        Should I then use the lagged versions of the predictors?

        Kushal

        • Jason Brownlee July 16, 2017 at 7:57 am #

          Perhaps, I do not follow your question, perhaps you can restate it with more information?

  9. Viorel Emilian Teodorescu July 8, 2017 at 9:45 am #

    great article, Jason!

  10. Chinesh August 10, 2017 at 5:15 pm #

    very helpful article !!

    I am working on developing an algorithm which will predict the future traffic for the restaurant. The features I am using are: Day,whether there was festival,temperature,climatic condition , current rating,whether there was holiday,service rating,number of reviews etc.Can I solve this problem using time series analysis along with these features,If yes how.
    Please guide me

  11. Hossein August 23, 2017 at 1:16 am #

    Great article Jason. Just a naive question: How does this method different from moving average smoothing? I’m a bit confused!
    Thanks

    • Jason Brownlee August 23, 2017 at 6:56 am #

      This post is just about the framing of the problem.

      Moving average is something to do to the data once it is framed.

  12. pkl520 August 26, 2017 at 10:29 pm #

    Hi , Jason! Good article as always~

    I have a question.

    “Running the example shows the differentiation of input (t-n) and output (t+n) variables with the current observation (t) considered an output.”

    values = [x for x in range(10)]
    data = series_to_supervised(values, 2, 2)
    print(data)

    var1(t-2) var1(t-1) var1(t) var1(t+1)
    2 0.0 1.0 2 3.0
    3 1.0 2.0 3 4.0
    4 2.0 3.0 4 5.0
    5 3.0 4.0 5 6.0
    6 4.0 5.0 6 7.0
    7 5.0 6.0 7 8.0
    8 6.0 7.0 8 9.0

    So above example, var1(t-2) var1(t-1) are input , var1(t) var1(t+1) are output, am I right?

    Then,below example.

    raw = DataFrame()
    raw[‘ob1’] = [x for x in range(10)]
    raw[‘ob2’] = [x for x in range(50, 60)]
    values = raw.values
    data = series_to_supervised(values, 1, 2)
    print(data)
    Running the example shows the large reframed DataFrame.

    var1(t-1) var2(t-1) var1(t) var2(t) var1(t+1) var2(t+1)
    1 0.0 50.0 1 51 2.0 52.0
    2 1.0 51.0 2 52 3.0 53.0
    3 2.0 52.0 3 53 4.0 54.0
    4 3.0 53.0 4 54 5.0 55.0
    5 4.0 54.0 5 55 6.0 56.0
    6 5.0 55.0 6 56 7.0 57.0
    7 6.0 56.0 7 57 8.0 58.0
    8 7.0 57.0 8 58 9.0 59.0

    var1(t-1) var2(t-1) are input, var1(t) var2(t) var1(t+1) var2(t+1) are output.

    can u answer my question? I will be very appreciate!

    • Jason Brownlee August 27, 2017 at 5:48 am #

      Yes, or you can interpret and use the columns any way you wish.

  13. Thabet August 30, 2017 at 7:36 am #

    Thank you Jason!!
    You are the best teacher ever

  14. Charles September 29, 2017 at 12:24 am #

    Jason,

    I love your articles! Keep it up! I have a generalization question. In this data set:

    var1(t-1) var2(t-1) var1(t) var2(t)
    1 0.0 50.0 1 51
    2 1.0 51.0 2 52
    3 2.0 52.0 3 53
    4 3.0 53.0 4 54
    5 4.0 54.0 5 55
    6 5.0 55.0 6 56
    7 6.0 56.0 7 57
    8 7.0 57.0 8 58
    9 8.0 58.0 9 59

    If I was trying to predict var2(t) from the other 3 data, would the input data X shape would be (9,1,3) and the target data Y would be (9,1)? To generalize, what if this was just one instance of multiple time series that I wanted to use. Say I have 1000 instances of time series. Would my data X have the shape (1000,9,3)? And the input target set Y would have shape (1000,9)?

    Is my reasoning off? Am I framing my problem the wrong way?

    Thanks!
    Charles

  15. Sean Maloney October 1, 2017 at 5:24 pm #

    Hi Jason!

    I’m really struggling to make a new prediction once the model has been build. Could you give an example? I’ve been trying to write a method that takes the past time data and returns the yhat for the next time.

    Thanks you.

  16. Sean Maloney October 1, 2017 at 5:28 pm #

    P.S. I’m the most stuck at how to scale the new input values.

    • Jason Brownlee October 2, 2017 at 9:38 am #

      Any data transforms performed on training data must be performed on new data for which you want to make a prediction.

  17. Nish October 23, 2017 at 11:42 am #

    Hi Jason,
    This is great, but what if I have around ten features (say 4 categorical and 6 continuous), a couple of thousand data points per day, around 200 days worth of data in my training set? The shift function could work in theory but you’d be adding hundreds of thousands of columns, which would be computationally horrendous.
    In such situations, what is the recommended approach?

  18. Shud November 1, 2017 at 5:37 pm #

    Hey Jason,

    I converted my time series problem into regression problem and i used GradientBoostingRegressor to model the data. I see my adjusted R-squared keep changing everytime i run the model. I believe this is because of the correlation that exists between the independent variable (lag variables). How to handle this scenario? Though the range of fluctuation is small, i am concerned that this might be a bad model

  19. Nitin Gupta November 13, 2017 at 10:10 pm #

    Hey Jason,

    I applied the concept that you have explained to my data and used linear regression. Can I expand this concept to polynomial regression also, by squaring the t-1 terms?

  20. Samuel November 15, 2017 at 9:43 pm #

    Hey Jason,

    thanks a lot for your article! I already read a lot of your articles. These articles are great, they really helped me a lot.

    But I still have a rather general question, that I can’t seem to wrap my head around.

    The question is basically:
    In which case do I treat a supervised learning problem as a time series problem, or vice versa?

    For further insight, this is my problem I am currently struggling with:
    I have data out of a factory (hundreds of features), which I can use as my input.
    Additionally I have the energy demand of the factory as my output.
    So I already have a lot of input-output-pairs.
    The energy demand of the factory is also the quantity I want to predict.
    Each data point has its own timestamp.
    I can transform the timestamp into several features to take trends and seasonality into account.
    Subsequently I can use different regression models to predict the energy demand of the factory.
    This would then be a classical supervised regression problem.

    But as I unterstood it from your time series articles, I could as well treat the same problem as a time series problem.
    I could use the timestamp to extract time values which I can use in multivariate time series forecasting.

    In most examples you gave in your time series articles, you had the output over time.
    https://machinelearningmastery.com/reframe-time-series-forecasting-problem/
    And in this article you shifted the time series to get an input, in order to treat the problem as a supervised learning problem.

    So let’s suppose you have the same number of features in both cases.
    Is it a promising solution to change the supervised learning problem to a time series problem?
    What would be the benefits and drawbacks of doing this?

    As most regression outputs are over time.
    Is there a general rule, when to use which framing(supervised or time series) of the problem?

    I hope, that I could phrase my confusion in an ordered fashion.

    Thanks a lot for your time and help, I really appreciate it!

    Cheers Samuel

    • Jason Brownlee November 16, 2017 at 10:29 am #

      To use supervised learning algorithms you must represent your time series as a supervised learning problem.

      Not sure I follow what you mean by tuning a supervised learning problem into a series?

      • Samuel November 29, 2017 at 10:19 pm #

        Dear Jason,

        thank you for your fast answer.
        I’m sorry that I couldn’t frame my question comprehensibly, I’m still new to ML.
        I’ll try to explain what I mean with an example.

        Let’s suppose you have the following data, I adapted it from your article:
        https://machinelearningmastery.com/time-series-forecasting-supervised-learning/

        input1(time), input2, output
        1, 0.2, 88
        2, 0.5, 89
        3, 0.7, 87
        4, 0.4, 88
        5, 1.0, 90

        This data is, what you would consider a time series. But as you already have 2 inputs and 1 output you could already use the data for supervised machine learning.
        In order to predict future outputs of the data you would have to know input 1 and 2 at timestep 6. Let’s assume you know from your production plan in a factory that the input2 will have a value of 0.8 at timestep 6 (input1). With this data you could gain y_pred from your model. You would have treated the data purely as a supervised machine learning problem.

        input1(time), input2, output
        1, 0.2, 88
        2, 0.5, 89
        3, 0.7, 87
        4, 0.4, 88
        5, 1.0, 90
        6, 0.8, y_pred

        But you could do time-series forecasting with the same data as well, if I understood your articles correctly.

        input1(time), input2, output
        nan, nan, 88
        1, 0.2, 89
        2, 0.5, 87
        3, 0.7, 88
        4, 0.4, 90
        5, 1.0, y_pred

        This leads to my questions:

        In which case do I treat the data as a supervised learning problem and in which case as a time series problem?
        Is it a promising solution to change the supervised learning problem to a time series problem?
        What would be the benefits and drawbacks of doing this?
        As my regression outputs are over time.
        Is there a general rule, when to use which framing (supervised or time series) of the problem?

        I hope, that I stated my questions more clearly.

        Thanks a lot in advance for your help!

        Best regards Samuel

        • Jason Brownlee November 30, 2017 at 8:16 am #

          I follow your first case mostly, but time would not be an input, it would be removed and assumed. I do not follow your second case.

          I believe it would be:

          What is best for your specific data, I have no idea. Try a suite of different framings (including more or less lag obs) and see which models give the best skill on your problem. That is the only trade-off to consider.

  21. MJ November 18, 2017 at 12:46 am #

    Ver helpful, thanks!

  22. Michael November 30, 2017 at 6:47 am #

    Jason:
    Thank you for all the time and effort you have expended to share your knowledge of Deep Learning, Neural Networks, etc. Nice work.

    I have altered your series_to_supervised function in several ways which might be helpfut to other novices:
    (1) the returned column names are based on the original data
    (2) the current period data is always included so that leading and lagging period counts can be 0.
    (3) the selLag and selFut arguments can limit the subset of columns that are shifted.

    There is a simple set of test code at the bottom of this listing:

  23. Maciej December 1, 2017 at 7:11 am #

    When I do forecasting, let’s say only one step ahead, as the first input value I should use any value that belongs i.e. to validation data (in order to set up initial state of forecasting). In second, third and so on prediction step I should use previous output of forecasting as input of NN. Do I understand correctly ?

    • Jason Brownlee December 1, 2017 at 7:46 am #

      I think so.

      • Maciej December 2, 2017 at 4:15 am #

        Ok, so another question. In the blog post here: https://machinelearningmastery.com/time-series-forecasting-long-short-term-memory-network-python/, as an input for NN you use test values. The predictions are only saved to a list and they are not used to predict further values of timeseries.

        My question is. Is it possible to predict a series of values knowing only the first value ?
        For example. I train a network to predict values of sine wave. Is it possible to predict next N values of sine wave starting from value zero and feeding NN with result of prediction to predict t + 1, t + 2 etc ?

        • Maciej December 2, 2017 at 4:18 am #

          If my above understanding is incorrect then it means that if your test values are completely different than those which were used to train network, we will get even worse predictions.

          • Jason Brownlee December 2, 2017 at 9:05 am #

            Yes. Bad predictions in a recursive model will give even worse subsequent predictions.

            Ideally, you want to get ground truth values as inputs.

        • Jason Brownlee December 2, 2017 at 9:04 am #

          Yes, this is called multi-step forecasting. Here is an example:
          https://machinelearningmastery.com/multi-step-time-series-forecasting-long-short-term-memory-networks-python/

          • Maciej December 3, 2017 at 5:34 am #

            Does it mean that using multi-step forecast (let’s say I will predict 4 values) I can predict a timeseries which contains 100 samples providing only initial step (for example providing only first two values of the timeseries) ?

          • Jason Brownlee December 4, 2017 at 7:40 am #

            Yes, but I would expect the skill to be poor – it’s a very hard problem to predict so many time steps from so little information.

Leave a Reply