data driven model

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Tutorial on Data-Driven Modeling in Water Resourceand Environmental Engineering Using Matlab

Feb 2014

Waqar S. Qureshi

Teaching AssociateAsian Institute of Technology

February 11, 2014

Waqar Qureshi (AIT) Modeling for WREE February 11, 2014 1 / 33
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Outline

1 Major challenges for water engineers2 System modeling for water engineering

What is modelingTypes of modelingData-driven modelsStatistical modelsSoft computing model

Spatio-temporal complexityType of dataGeneral approach to develop a data-driven model

3 Regression-based models

IntroductionRegression model applicationLinear regressionLinear-regression-example-1Linear-regression-example-2

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Todays major challenges forwater engineers include:

Securing water resources for people

Protecting vital echosystems

Dealing with variability and uncertainty of water in space and time

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System modeling for water engineeringWhat is modeling

The term model refers to tools, softwares, and programs used to

represent real-world systems.

Modeling of a system is used to predict the system behavior and responseto the changing factors.

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System modeling for water engineeringTypes of modeling

Physical model is rescaled copy of the actual system, example, DAM

models.

Mathematical model is baed on mathematical logic, knowledge, andequations.

Figure: classification of models

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System modeling for water engineeringApplications and complexity

As water and environmental engineers, system modeling can be applied inmany applications such as:

Simulation of natural phenomenon

Synthetic data generation

Forecasting and warning of extreme events

Developing decision making rules

Modeling a system in the field of water engineering is difficult:

Physicalcomplexity of natural phenomenon.Time consumingprocess of analyzing different components of thesystem.

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3 Regression-based modelsIntroductionRegression model applicationLinear regressionLinear-regression-example-1Linear-regression-example-2

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System modeling for water engineeringData-driven models

Models that can simulate a system by the experimental data of thatsystem is known as data-driven models.

Data-driven models enable us to mapcausal factorsandconsequentoutcomesfrom the observed patterns (experimental data), without deepunderstanding of the complex physical process.

The purpose of data-driven modeling in water engineering can include thefollowing:

Data classification and clustering.

Extreme value predition with ephasis on floods and droughts.

Water quality simulation and prediction.Extending the length of hydroclimatological data from the historicalones.

Modeling water balance concerning different components of ahydrological system.

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System modeling for water engineeringdata-driven models

For a complex system, data-driven models are inexpensive, accurate,precise, andflexiblein contrast to their counter physical models oranalytical models.

Data-driven models can be used for problems where we have lessinformation about the intrinsic complexityof the phenomenon, in contrastto analytical modeling.

Two groups of Data-driven models are:

Statistical modelingSoft computing (Artificial intelligence)

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System modeling for water engineeringStatistical model

A statistical model is comprised of random and deterministic variables.Deterministic variables are defined by mathematical model and use a set ofequations to generate data, while random variable is represented by aprobabilistic models for example a probability density function to generatedata.

The probabilistic models can be parametric and non parametric.

Parametric model can be described by its mean, variance, etc.

Non-parametric models can be described by loosely confined assumptionssuch as nearest neighbor.

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System modeling for water engineeringSoft computing model

In soft computing the system is modeled using fuzzy logic,neuro-computing, and genetic algorithms.

It is tolerant of imprecision, uncertainty, partial truth, and approximation.

The role model of soft computing is human mind.

Example: A suitable temperature of a room to make people feelcomfortable!


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System modeling for water engineeringSpatio-temporal complexity

The model complexity can also be classified in spacial and temporalmanner.

The spacial and temporal characteristics of a model is essential to study

the effects due to the dynamic change of natural phenomenon on thesystem.

The spacial complexity of a model can be characterized as lumped,semi-distributed, and distributed models.

Let us take an example of rainoff modeling to understand spacialcomplexity of models.


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System modeling for water engineeringLumped models

Lumped modeling methods were used due to complex data collection

methods and software limitations.Lumped models are still useful for producing flood guidance. They requireless data input and less computational power than more modern methods.

Figure: Spacial complexity for runoff model


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System modeling for water engineeringSemi-distributed models

Semi-distributed modeling is a variation of the lumped method and is

sometimes referred to as a pseudo-distributed approach. Using thisapproach, a basin is broken down into smaller sub-basins. Runoff amountsfrom methods such as unit hydrograph are used to estimate stream flowfrom each of these sub-basins


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System modeling for water engineeringSemi-distributed models

A trulydistributed modelingis one that represents processes in a gridded

manner.

Each cell has its parameters allowing for its own stream flow estimates.

If these data in each cell are not available, they must somehow beestimated, introducing an uncertainty factor.


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System modeling for water engineeringTemporal complexity

Data-driven models can be static or dynamic.

A rainoff isdynamicif its parameters changes as it receives newinformation, and is consideredstaticif the model relies only on the

historical data.In summary, thepurposeof modeling is an essential criteria to select amodel and determines itscomplexity,developing time,runtime, itsaccuracy, andprecision.

Modeling also depends upon the type of data that is available, and thetime required to acquire it.


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System modeling for water engineeringType of data

Figure: Types of Data, (a) discrete data, (b) continuous data, (c) spacial data,(d) temporal data


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System modeling for water engineeringGeneral approach to develop a data-driven model

Figure: General approach to develop a data-driven model


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3 Regression-based modelsIntroduction

Regression model applicationLinear regressionLinear-regression-example-1Linear-regression-example-2


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Regression-based models
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Regression based modelsIntroduction

Theregression-basedmodels are data-driven models that are easy to use

and popular.

They ranges from linear to nonlinear and parametric to nonparametricmodels.

Following are the application areas of regression based models

Prediction, forecasting, and estimation of missing data.

Interpolation and extrapolation of data.

They are segregated as

Multiple linear regression model.Conventional non linear regression method.

KNN non parametric model.

logistic regression model.


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gRegression model application

Figure: A summary on the application of regression models in water resources andenvironmental engineering


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gLinear regression

Linear regression is used to model the linear relationship between the

continuous dependent variable (y) and an independent variable (x).The regression model aim to identify what variables are associated with y,to predict the future observations of y.


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gLinear regression

Let x and y two variables, then a plot between x and y shows if y is

positive, negative linear or non-linear function.

Figure: Different type of correlation between Y and X


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Linear regression

The strength of linear relationship between two variables is measured by

simplecorrelation coefficient.The Correlation coefficient between n observations f X and Y is calculatedas

Figure: Correlation coefficient

Figure: sample script for corrcoef(x)


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Linear regression

A simplest regression function is refined as

y=o+1.x, where o and 1 are parameters of the model

Figure: Samples of errors in linear regression fitting

The linear regression modeling tends to fit a line onto the observed datasuch that the sum of absolute errors of fitting for n-observations isminimized.

S2 =n

i=1

2i =n

i=1

(yi o+1.xi)2


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Linear regression

In case of multiple independent variables xi ... xn the model becomes

multiple linear regression and is represented by the equation.

y=o+1.x1+ 2.x2+ 3.x3+ 3.x3+ 4.x4

The dependent variable y can be a deterministic or a probabilistic. In case

it becomes a probabilistic, then the stochastic equation of the form isgiven as

y=o+1.x+e

, where e is the estimation error.The output at any instant of x can be represented by a distributionfunction. The expected value of the estimation is in fact the average valueof this distribution which is given by y=o+1.x.


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Linear regression

Figure: PDF of dependent variable in a linear regression model


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Linear regression-example

Interpolation of water quality values.

Water quality of a river as a function of distane from the upstream of riveris tabulated. Use a linear regression model to interpolate total dissolvedsolution (TDS) at different locations of the river.

Figure: Data presented for Example


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Regression-based modelsLi i l
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Linear regression-example

Solve the above Example in a probabilistic manner and calculate theprobable range of TDS at the distance of 125km from the upstream.


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Matlab Tutorial
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