Regression Analysis Daniel Skoog, Maplesoft Linear regression is one of the fundamental approaches for determining relationships between dependent and exploratory variables.This worksheet covers several introductory examples for regression analysis in Maple. The primary focus of the worksheet is the use of the Statistics:-Fit command to fit a model to data as well as generating reports on the fitted model. Additional examples include fitting a curve using the Optimization package.

The Statistics and CurveFitting packages contain a number of commands for fitting curves to data points including linear, exponential, polynomial, least squares, and spline. Statistics - Regression Curve Fitting ExponentialFitFitLinearFitLogarithmicFitLowessNonlinearFitOneWayANOVAPolynomialFitPowerFitPredictiveLeastSquaresRepeatedMedianEstimatorArrayInterpolationBSplineBSplineCurveInteractiveLeastSquaresLowessPolynomialInterpolationRationalInterpolationSplineThieleInterpolation JSFH

The Statistics:-Fit command is a general purpose command that allows for specification of linear or non-linear models.Syntax:Fit( Model, X::Independent Variable, Y::Dependent Variable, x::Name of independent variable in component function, options );The Fit command fits a model by minimizing the least-squares error.

with(Statistics): Start by specifying some data for an independent variable:X := Vector([1, 2, 3, 4, 5, 6], datatype = float);

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Next add data for a dependent variable:Y := Vector([2, 3, 4, 3.5, 5.8, 7], datatype = float);

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Using the Fit command, you can fit a linear model to data and display summary of the regression model.Fit a least-squares model to the data:Fit(b*t+a, X, Y, t)

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Fit a polynomial of degree 2 to the data:Fit(c*t^2+b*t+a, X, Y, t)

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The 'output' option is useful when you want to return specific values from the regression model. Available outputs are: AtkinsonTstatisticconfidenceintervalsCookDstatisticdegreesoffreedomexternallystandardizedresidualsinternallystandardizedresidualsleastsquaresfunctionleveragesparametervaluesparametervectorresidualsresidualmeansquareresidualstandarddeviationresidualsumofsquaresrsquaredrsquaredadjustedstandarderrorstprobabilitytvaluevariancecovariancematrixOutput the standard errors: Fit(c*t^2+b*t+a, X, Y, t, output=standarderrors)

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Output the r-squared and adjusted r-squared value:Fit(c*t^2+b*t+a, X, Y, t, output=[rsquared,rsquaredadjusted])

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It is often useful to display the results in summary form: Fit(c*t^2+b*t+a, X, Y, t, summarize=true):

Summary: ---------------- Model: 1.9600000+.16500000*t+.11071429*t^2 ---------------- Coefficients: Estimate Std. Error t-value P(>|t|) a 1.9600 1.1720 1.6724 0.1930 b 0.1650 0.7667 0.2152 0.8434 c 0.1107 0.1072 1.0325 0.3778 ---------------- R-squared: 0.9252, Adjusted R-squared: 0.8753

You can also display this summary as an embedded table, complete with more details on residuals.Fit(c*t^2+b*t+a, X, Y, t, summarize=embed): Model:LCgkIikrK2c+ISIoIiIiKiYkIikrK107ISIpRiYlInRHRiZGJiomJCIpSDkyNkYqRiYpRisiIiNGJkYmCoefficientsEstimateStandard Errort-valueP(>|t|)aJCInK2c+ISImJCInKj48IiEiJg==JCInUHM7ISImJCInWEk+ISInbJCInK107ISInJCInW253ISInJCInJT46IyEiJw==JCInOk0lKSEiJw==cJCInOTI2ISInJCInRXM1ISInJCInYEs1ISImJCIncHhQISInR-squared:JCIncF4jKiEiJw==Adjusted R-squared:JCInI0d2KSEiJw==

Residual Sum of SquaresResidual Mean SquareResidual Standard ErrorDegrees of FreedomJCIncihHIiEiJg==JCInUSNIJSEiJw==JCInal5sISInIiIk Five Point Summary MinimumFirst QuartileMedianThird QuartileMaximumJCEnSDkqKSEiJw==JCEnZC5IISInJCInOWQ6ISInJCInJmYhSCEiJw==JCInciZbJiEiJw==

In this example, you will use a random number generator to add noise to a signal, then fit a model to the resulting data. restart; with(Statistics):JSFH Define the noise as a random number generator with mean value 1 and standard deviation 0.5. Noise := Sample(RandomVariable(Normal(1, .5))): We use a sample size of 200 to create a vector of noise values. PureNoiseVector := Noise(200)

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We can then extract entries from the vector. The vector indices go from 1 to 200. To select the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYnUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUqcmVmZXJlbmNlR1ElOTUwNEYnLyUyZGlzcGxheV9yZWZlcmVuY2VHUSU5NTAzRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvRjhRJ25vcm1hbEYnL0Y7USYxODg4NEYnL0Y+USYxODg4M0YnRitGQEZDRkZGSEZKth entry, use the selection operator: PureNoiseVector[i].

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PureNoiseVector[1]

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Next, we define a signal; the following creates a noisy signal by adding noise to the original signal: signal := x -> x*sin(x): Generate a data set using the seq command.noisyData := [seq([.1*(i-100), signal(.1*(i-100))+PureNoiseVector[i]], i = 1..200)]: Plot the signal and noisyData together.plot([noisyData, signal(x)], style=point, symbol=[cross, circle], symbolsize=15, color=["Black", "Blue"], legend=["Noisy Signal", "Signal"], view=[default,-5..5], axes=boxed)

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Fitting a Model to the data:To fit a functional model to the noisy data, use the Fit command from the Statistics package. The calling sequence is Fit(f, X, Y, v), where f is the function model, X and Y are the x and y data, and v is the name of the independent variable in the component function.Create two lists Xdata and Ydata from the data in the two columns of noisyData.Xdata := [seq(noisyData[i, 1], i = 1 .. numelems(noisyData))]: Ydata := [seq(noisyData[i, 2], i = 1 .. numelems(noisyData))]: Tip: It's often useful to do something for each element of a data structure. Here, we do this by i=1..numelems(noisyData). Recall, numelems is the number of elements.The function model is 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.f := Fit(c*x*sin(a*x)+b, Xdata, Ydata, x)

LCYqKF0zRkVGQzU4RkRGRjNEQjYxIiIiSSJ4RzYiRiUtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2IywkKiZdM0ZFRkY0OUIwMUY5OEEwM0YlRiZGJUYlRiVGJV0zRkYwMjQwQTZDMUE2MEY0RiU=

Plot LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EiZkYnLyUlc2l6ZUdRIzExRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGNS9GOVEnbm9ybWFsRic= and the noisyData together to visualize the fit.plot([noisyData, f(x)], x = -10 .. 10, style=[point, line], symbol=cross, symbolsize=15, legend=["Noisy Signal", "Fitted Model"], axes=boxed, color=["Black","Red"], linestyle=dash, thickness=2)

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Exercise: Repeat the noisy signal example using a random number generator based on a Weibull distribution.

The following example adjusts a fitted model by adding or removing variables in order to find better adjusted r-squared values.restart; with(Statistics): The Import command reads a datafile into a new DataFrame.ExperimentalData := Import(FileTools:-JoinPath(["Excel", "ExperimentalData.xls"], base = datadir));

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The convert command converts the values in the DataFrame to a Matrix of values.ExMat := convert( ExperimentalData, Matrix )[..,2..7];

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We start by fitting a model that includes predicting variables for each of the columns of data:Fit( C + C2*v + C3*w + C4*x + C5*y + C6*z, ExMat[..,2..6], ExMat[..,1], [v,w,x,y,z], summarize=embed ): Model:LC4kIikpKSpvbiMhIikhIiIqJiQiKV07eT0hIiYiIiIlInZHRitGJiomJCIpI1siZVMhIihGKyUid0dGK0YmKiYkIilNOEc+ISInRislInhHRitGKyomJCIpcjV6Y0Y1RislInlHRitGKyomJCIpInokUURGMEYrJSJ6R0YrRis=CoefficientsEstimateStandard Errort-valueP(>|t|)CJCEnIXBuIyEiJw==JCInbUxoISInJCEnR2tWISInJCInRzJvISInC2JCEnPHk9ISIkJCInPFs7ISIkJCEnYVI2ISImJCInM2hJISInC3JCEnOmVTISImJCIneSUqKikhIic=JCEnbjZYISImJCInO0pqISIpC4JCInOEc+ISIlJCInUl5NISIlJCInYSdlJiEiJw==JCInIlwrJyEiJw==C5JCInNnpjISIlJCInOy1OISIlJCInZ0A7ISImJCInOmU7ISInC6JCInUVFEISImJCInKT5sIiEiJg==JCInZE86ISImJCInLV09ISInR-squared:JCInSncpKiEiJw==Adjusted R-squared:JCInaV8oKiEiJw==

Residual Sum of SquaresResidual Mean SquareResidual Standard ErrorDegrees of FreedomJCInRms9ISIpJCInYkdQISIqJCInJTQkPiEiKA==IiIm Five Point Summary MinimumFirst QuartileMedianThird QuartileMaximumJCEnYkI+ISIoJCEneTk1ISIoJCInPF04ISIpJCInVlFwISIpJCInPDZJISIo

From the above, we can observe that both the r-squared and adjusted r-squared are reasonably high, however only one of the coefficient values has a significant p-value (C3).Let's try to fit the data again, this time keeping the two coefficients with the lowest p-values and the intercept.Fit( C + C3*v + C5*w, ExMat[..,[3,5]], ExMat[..,1], [v,w], summarize=embed ): Model:LCgkIilIbSpcJiEiKSIiIiomJCIpanZMTSEiKEYmJSJ2R0YmISIiKiYkIilGdk1uRipGJiUid0dGJkYmCoefficientsEstimateStandard Errort-valueP(>|t|)CJCInbSpcJiEiJw==JCInVHY9ISIoJCInXktIISIlJCIndCEpPiEjOQ==C3JCEnd0xNISImJCInV2UiKSEiJw==JCEnJSkzVSEiJg==JCInYmdIISIpC5JCIndk1uISImJCInJ2UjZiEiJg==JCInXU82ISImJCInVScpRyEiJw==R-squared:JCInRy8pKiEiJw==Adjusted R-squared:JCInTmIoKiEiJw==

Residual Sum of SquaresResidual Mean SquareResidual Standard ErrorDegrees of FreedomJCInIipcSCEiKQ==JCInUShvJCEiKg==JCInRT8+ISIoIiIp Five Point Summary MinimumFirst QuartileMedianThird QuartileMaximumJCEnJFE/IyEiKA==JCEnY2I4ISIoJCEnRiFvIyEiKQ==JCInSCZcKCEiKQ==JCInJSpRUSEiKA==

From the above, we can see that the r-squared value does go down, however the adjusted r-squared goes up! Let's fit the model one last time to see if removing C5 increases or decreases the adjusted r-squared.Fit( C + C3*v, ExMat[..,3], ExMat[..,1], [v], summarize=embed ): Model:LCYkIikjM0xiJiEiKSIiIiomJCIpdW08RCEiKEYmJSJ2R0YmISIiCoefficientsEstimateStandard Errort-valueP(>|t|)CJCInSmBiISInJCInQFc9ISIoJCInQDZJISIlJCInWC1DISM6C3JCEnbjxEISImJCInJSp6NyEiJw==JCEnLW4+ISIlJCInTV41ISM4R-squared:JCInb3MoKiEiJw==Adjusted R-squared:JCInVVooKiEiJw==

Residual Sum of SquaresResidual Mean SquareResidual Standard ErrorDegrees of FreedomJCInPUVNISIpJCInKG8hUSEiKg==JCInN14+ISIoIiIq Five Point Summary MinimumFirst QuartileMedianThird QuartileMaximumJCEna2BBISIoJCEnU2ciKiEiKQ==JCEnTVU/ISIpJCInJjNgIyEiKQ==JCInckhbISIo

We can see that the final adjusted r-squared value is lower than the previous two, so we are probably better to keep the additional C5 coefficient value.

There are more ways to fit curves or surfaces to data. The Optimization package is a collection of commands for numerically solving optimization problems, which involve finding the minimum or maximum of an objective function possibly subject to constraints. The package takes advantage of built-in library routines provided by the Numerical Algorithms Group (NAG).The package solves linear programs (LPs), quadratic programs (QPs), nonlinear programs (NLPs), and both linear and nonlinear least-squares problems. Both constrained and unconstrained problems are accepted.

restart; with(Optimization);

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We can use the least squares command to do curve fitting. For example, if we have the following data points, (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):data := [[1, 1.5], [2, 3.5], [2.5, 1.9], [3.1, 4.5], [4.3, 1.9], [4.7, 2.4], [5.8, 3], [6, 5.7], [6.6, 3.4], [6.7, .5], [7.1, 4], [7.4, 1.7]];

Ny43JCIiIiQiIzohIiI3JCIiIyQiI05GJzckJCIjREYnJCIjPkYnNyQkIiNKRickIiNYRic3JCQiI1ZGJ0YvNyQkIiNaRickIiNDRic3JCQiI2VGJyIiJDckIiInJCIjZEYnNyQkIiNtRickIiNNRic3JCQiI25GJyQiIiZGJzckJCIjckYnIiIlNyQkIiN1RickIiM8Ric=

We want to fit the data with a quintic polynomial as follows:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY6LUkjbWlHRiQ2I1EhRictRiM2PEYrLUYjNjotRiw2OFEicEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUqdW5kZXJsaW5lR0Y+LyUqc3Vic2NyaXB0R0Y+LyUsc3VwZXJzY3JpcHRHRj4vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGPi8lK2V4ZWN1dGFibGVHRj4vJSlyZWFkb25seUdGPi8lKWNvbXBvc2VkR0Y+LyUqY29udmVydGVkR0Y+LyUraW1zZWxlY3RlZEdGPi8lLHBsYWNlaG9sZGVyR0Y+LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y+LyUwZm9udF9zdHlsZV9uYW1lR1EqX3BzdHlsZTU0RicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUqcmVmZXJlbmNlR1EmMTg5MDBGJy8lMmRpc3BsYXlfcmVmZXJlbmNlR1EmMTg4OTlGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Zcb1Enbm9ybWFsRicvJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZqcC1JKG1mZW5jZWRHRiQ2OC1GIzY4LUYsNjhRInhGJ0Y2RjlGPEY/RkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRmhuRltvRl5vRmFvRjZGOUY8L0ZARj5GQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GaG5GaG9GXm9GYW9GNkY5RjxGZXFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GaG5GaG9GXm9GYW9GNkY5RjxGZXFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GaG5GaG9GXm9GYW8tRmVvNkFRIj1GJ0Y2RjlGPEZlcUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZobkZob0Zeb0Zhb0Zqb0ZccEZecEZgcEZicEZkcEZmcC9GaXBRLDAuMjc3Nzc3OGVtRicvRlxxRmpxLUYjNkNGKy1GIzY7LUYsNjhRImFGJ0Y2RjlGPEY/RkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRmhuRltvRl5vRmFvLUZlbzYtUTEmSW52aXNpYmxlVGltZXM7RidGaG9Gam9GXHBGXnBGYHBGYnBGZHBGZnBGaHBGW3EtRiM2OC1JJW1zdXBHRiQ2JUZicS1JI21uR0YkNjhRIjVGJ0Y2RjlGPEZlcUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZobkZob0Zeb0Zhby8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGNkY5RjxGZXFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GaG5GaG9GXm9GYW9GK0Y2RjlGPEZlcUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZobkZob0Zeb0Zhby1GZW82QVEiK0YnRjZGOUY8RmVxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRmhuRmhvRl5vRmFvRmpvRlxwRl5wRmBwRmJwRmRwRmZwL0ZpcFEsMC4yMjIyMjIyZW1GJy9GXHFGZnMtRiM2Oy1GLDY4USJiRidGNkY5RjxGP0ZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZobkZbb0Zeb0Zhb0Zjci1GIzY4LUZpcjYlRmJxLUZcczY4USI0RidGNkY5RjxGZXFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GaG5GaG9GXm9GYW9GX3NGNkY5RjxGZXFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GaG5GaG9GXm9GYW9GK0Y2RjlGPEZlcUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZobkZob0Zeb0Zhb0Zicy1GIzY7LUYsNjhRImNGJ0Y2RjlGPEY/RkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRmhuRltvRl5vRmFvRmNyLUYjNjgtRmlyNiVGYnEtRlxzNjhRIjNGJ0Y2RjlGPEZlcUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZobkZob0Zeb0Zhb0Zfc0Y2RjlGPEZlcUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZobkZob0Zeb0Zhb0YrRjZGOUY8RmVxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRmhuRmhvRl5vRmFvRmJzLUYjNjstRiw2OFEiZEYnRjZGOUY8Rj9GQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GaG5GW29GXm9GYW9GY3ItRiM2OC1GaXI2JUZicS1GXHM2OFEiMkYnRjZGOUY8RmVxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRmhuRmhvRl5vRmFvRl9zRjZGOUY8RmVxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRmhuRmhvRl5vRmFvRitGNkY5RjxGZXFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GaG5GaG9GXm9GYW9GYnMtRiM2Oi1GLDY4USJlRidGNkY5RjxGP0ZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZobkZbb0Zeb0Zhb0ZjckZicUY2RjlGPEZlcUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZobkZob0Zeb0Zhb0Zicy1GLDY4USJmRidGNkY5RjxGP0ZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZobkZbb0Zeb0Zhb0Y2RjlGPEZlcUZCRkRGRkZIRktGTkZQRlJGVEZWRlhGWkZmbkZobkZob0Zeb0Zhb0YrRjZGOUY8RmVxRkJGREZGRkhGS0ZORlBGUkZURlZGWEZaRmZuRmhuRmhvRl5vRmFvRitGNkY5RjxGZXFGQkZERkZGSEZLRk5GUEZSRlRGVkZYRlpGZm5GaG5GaG9GXm9GYW8=.Now we need to find a, b, c, d, e, and, f such 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is a minimum.LSSolve takes as input a list of the residues 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. p := a*x^5+b*x^4+c*x^3+d*x^2+e*x+f;

LC4qJkkiYUc2IiIiIilJInhHRiUiIiZGJkYmKiZJImJHRiVGJilGKCIiJUYmRiYqJkkiY0dGJUYmKUYoIiIkRiZGJiomSSJkR0YlRiYpRigiIiNGJkYmKiZJImVHRiVGJkYoRiZGJkkiZkdGJUYm

residues := map( d-> eval(p, x = d[1])-d[2], data);

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

Now we can call LSSolve with the residues.sol := LSSolve(residues);

NyQkIjMnNCd6SDNsVCp6KiEjPDcoL0kiYUc2Il0zRjc0QzI1RUIyNTE5RjhBL0kiYkdGKV1CRkMzMDkyREVGMEU4QTcxL0kiY0dGKV0zRkY4OTg3RDE5MUUyMEI4L0kiZEdGKV1DMDFDMkRGMjlDQ0MzNjVCL0kiZUdGKV00MDJDN0MzNEM5NDNEMzIyL0kiZkdGKV1DMDFDN0M0RThGRjg3NDM0

Our polynomial becomes:poly := eval(p, sol[2]);

LC4qJl0zRjc0QzI1RUIyNTE5RjhBIiIiKUkieEc2IiIiJkYlRiUqJl0zRkMzMDkyREVGMEU4QTcxRiUpRiciIiVGJSEiIiomXTNGRjg5ODdEMTkxRTIwQjhGJSlGJyIiJEYlRiUqJl00MDFDMkRGMjlDQ0MzNjVCRiUpRiciIiNGJUYuKiZdNDAyQzdDMzRDOTQzRDMyMkYlRidGJUYlXTQwMUM3QzRFOEZGODc0MzRGLg==

We can now plot the data and the curvep1 := Statistics:-ScatterPlot(data, symbolsize=15, color="Black", legend="Data"): p2 := plot(poly, x = .9 .. 8, thickness=4, color="Red", linestyle=dash, legend="Fitted Model"): plots:-display(p1, p2, axes=boxed);

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Additional Example: Bivariate Polynomial Regression

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