Finding an Optimum Climb Speed using

OLS Regression

Author: Supervisor:Mikael Persson Boualem Djehichemikaepe@kth.se boualem@kth.se

Bachelor Thesis

Div. of Mathematical Statistics, Dep. of MathematicsKTH Royal Institute of Technology

May 13, 2015

Finding an optimum climb speed usingOLS regression

Mikael Persson

Abstract

The purpose of this project was to find an optimum climb speed fora regional turboprop airplane. Ordinary least squares regression wasused to find a model for the climb performance of a Fokker 50 airplane.Data from 103 flights was collected manually with the assistance of thepilots of a Swedish regional airline and used to confirm the model. Theoptimum airspeed was found to depend on wind component and takeoff mass, where a tailwind suggests lower climb speed and a lower takeoff mass suggest a higher climb speed. Findings from this project couldbe used to make better decisions on what climb speed to use in order tosave fuel, with economical and environmental benefits.

1

Finding an optimum climb speed usingOLS regression

Mikael Persson

Foreword

This work is a bachelors thesis in Engineering Science at KTH Royal In-stitute of Technology. The project comprised 15 academic credits and wasmade at the Division of Mathematical Statistics. I would like to thank Pro-fessor Boualem Djehiche for his support and advice. This project would havebeen impossible without the support from the flight department at AmapolaFlyg AB.Finally I would like to thank all the pilots at Amapola Flyg AB,they voluntarily helped me collecting the data without compensation.

2

Contents

1 Introduction 41.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Statistical methods 52.1 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Choice of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Procedure to collect data . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Linear regression model . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 OLS model assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 OLS estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.5 Coefficient of determination R2 and effect size η2 . . . . . . . . . . . 11

2.3 Problems from a miss-specified model . . . . . . . . . . . . . . . . . . . . . 122.3.1 Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Heteroskedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.3 Endogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 Transformation of variables . . . . . . . . . . . . . . . . . . . . . . . 152.4.2 Selecting the best model . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Climb performance review 173.1 Airspeed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Take Off Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Ambient temperature and pressure . . . . . . . . . . . . . . . . . . . . . . . 193.4 Wind component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Hypotheses 21

5 Implementation 215.1 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Selection of regression model . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Result 226.1 Selected final model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2 Result of the final regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7 Discussion 247.1 Discussion of the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7.1.1 The optimal climb speed . . . . . . . . . . . . . . . . . . . . . . . . . 257.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7.2.1 Feedback from the air crews . . . . . . . . . . . . . . . . . . . . . . . 267.2.2 Other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8 Conclusion 288.1 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.2 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

References 30

3

1 Introduction

1.1 Background

The airline industry is continuously struggling to reduce fuel consumption.Environmental as well as economical considerations require this. While en-vironmental considerations are increasingly used in marketing as an edgeagainst competitors [1], the economical benefits of lower fuel consumptioncould be a question of survival for the individual airline [2].

Modern airplanes provide means for the flight crew to select an opti-mum flight profile by use of advanced flight management systems. However,previous generation of airplanes often lack these systems. A new fleet ofaircraft is generally a too big of an investment for the typical airline so anyimprovement in the fuel efficiency using existing systems is of importance.

Operations manuals describing standard operational procedures has ad-vice as to how to perform the climb [3], but since these stem from themanufacturers original documentation, focus is often not on minimizing fuelconsumption during the climb phase. The flight crew will sometimes adjusttheir climb profiles based on rules of thumb and common knowledge subjectto ambient conditions.

This project is made in cooperation with the flight department of aSwedish regional airline: Amapola Flyg AB. The airline is a typical regionalairline operating the Fokker 50 turboprop aircraft with operations in boththe passenger and cargo segments.

1.2 Purpose

To enable more accurate advice on how to perform the climb safely whileoptimizing the climb performance, more information is needed. The pur-pose of present work is to investigate how ambient conditions and selectionsmade by the flight crew affect the climb performance. The questions to beanswered by present work follows:

Q1 What factors will significantly affect the climb performance?

Q2 Given a set of parameters, known to the crew before the flight, whatis the optimum climb speed?

Q3 Is regression analysis based on manually collected data an acceptablemethod to investigate the climb performance of regional turbopropaircraft?

For this work, a measure on the climb performance of the airplane is needed.Less fuel spent in climb is desired, but spent fuel needs to be scaled bydistance to get comparable values between flights. As a measure on the

4

efficiency, the fuel per distance y defined as

y =f

d, (1.1)

is used. Here f is the mass of the fuel used in kg and d is the ground distancecovered in nautical miles. Note that a low value of y is desired.

1.3 Delimitations

Some limitations to the work was set up. The study was limited to onlyconsider basic aircraft performance theory. Manual data collection was usedand this restricts the amount of data available. Furthermore, only basicstatistical methods were used with the analysis limited to linear regressionby ordinary least squares.

2 Statistical methods

2.1 Data collection

2.1.1 Choice of data

In selecting what data to collect from each flight consideration was taken tolimit the workload for the pilots. At the same time, more data gives greaterchance of a successful analysis. Only data considered relevant for the fuelperformance was of interest. The decision was taken to collect the followingdata for each flight:

1. Date of flight, flight number and crew

2. Outside air temperature in celsius

3. Forecasted wind component at cruise level in knots (nm/hr)

4. Selected cruise level in flight level (feet×102 )

5. Take off mass in metric tonnes (kg ×103)

6. Selected airspeed during climb in knots (nm/hr)

7. Total fuel used in climb in kg

8. Ground distance covered in nautical miles (1.852× 103 m)

In addition to these, the following additional data was collected afterwards:

- Aerodrome elevation in feet (0.305 m)

- Atmospheric pressure in millibars (102 Pa)

5

Item 1 was used for administrative purposes. Items 2 through 6 includingthe additional data was used as predictor variables, covariates. Items 7 and8 was used to form the dependant variable. The units are the common unitsused by the aviation industry.

2.1.2 Procedure to collect data

The data was collected by letting the flight crew manually fill in supplieddata tables. Since flight safety is the primary concern of all airline opera-tions, the pilots were instructed to abort the data collection if flight safetyso required. This would typically be due to high workload in the cockpit forvarious reasons. Data was collected during a period of approximately twomonths giving usable data from 103 flights. The filled in data tables wereentered into digital format. An example of completed data table is given infig 1.

Figure 1: A completed data table. The first five values inthe left column are known to the crew before take off. Theremaining values were entered during the climb.

6

2.2 Statistical analysis

Table 1: Variable names used in describing the statisticalanalysis.

Symbol Description

yi i:th obs. of the response variable yxij i:th obs. of the j:th covariate xjβj j:th slope coefficientεi Residual, an observation of εn Number of observationsk Number of covariates

A description of the statistical methods used in the project follows. Thevariable names used in the following sections is given in table 1.

2.2.1 Linear regression model

The method of Ordinary Least Squares is one of the most common tech-niques for estimation using statistical data. The idea is that y is a functionof the independent variables or covariates, xj .

y = f(x1, x2, . . . , xk)

The OLS regression is used to estimate this function. The model specifica-tion for the multiple linear regression is given by

yi = β0 + x1iβ1 + x2iβ2 + . . .+ xkiβk + εi, i = 1, 2, . . . , n. (2.1)

The data set contains n observations of the random variable, y, and thecovariates x1, . . . , xk. β0 is the intercept while βj is the slope coefficient forthe covariate xj . The interpretation is that one unit of change in xj willlead to βj units of change in the dependent variable y [4]. The residuals, εi,are observations of a random variable present due to the fact that the datadoes not perfectly fit the model. The equations (2.1) are commonly writtenin compact matrix notation as

Y = Xβ + ε, (2.2)

where

Y(n×1) =

y1y2...yn

, X(n×k+1) =

1 x11 . . . x1k1 x21 . . . x2k...

.... . .

...1 xn1 . . . xnk

, ε(n×1) =

ε1ε2...εn

,

andβ(k+1×1) = (β0, β1, . . . , βk)

T .

7

2.2.2 OLS model assumptions

The linear regression model (2.1) comes with several assumptions [4], thatunfortunately not always apply.

- The residuals εi are independentent observations of a stochastic vari-able (or random variable, R.V.) with expected value zero and inde-pendent of the covariates xj :

E(ε) = E(ε|xj) = 0 (2.3)

- The covariates are deterministic, that is fixed in repeated samples

- The residuals are assumed to have equal variances across the sample

E(ε2i ) = σ2. (2.4)

If these assumptions are violated various problems will arise (sec. 2.3).

2.2.3 OLS estimation

In finding the vector β that gives the best fit to available data Y and Xit is natural to minimize |ε| = |Y −Xβ|. Here, β denotes an estimation ofthe true value of β. Referring to elementary linear algebra , this is given byletting ε be orthogonal to Xβ that is

(Xβ)T ε = βTXT ε = 0⇒XT ε = 0. (2.5)

The equations (2.5) are referred to as the normal equations. Using these in(2.2) gives

Y = Xβ + ε⇒XTY = XTXβ + 0⇔

β = (XTX)−1XTY (2.6)

which is the OLS estimate of β. An example of a straight line fitted by OLSto some data is shown in figure 2. Note that this estimator will minimize

|ε| =√ε21 + ε22 + . . .+ ε2n =

√εT ε,

which is equivalent to minimizing the sum of squared residuals

n∑i=1

ε2i = ε21 + ε22 + . . .+ ε2n,

8

hence the term Ordinary Least Squares. The OLS estimate β is an unbiasedestimate of β since

β = (XTX)−1XTY = {(2.2)}= (XTX)−1XT (Xβ + ε)⇒

β − β = (XTX)−1XT ε⇒E(β)− β = (XTX)−1XTE(ε) = 0. (2.7)

Similarly the covariance matrix for β given by

Cov(β)ij = E[(βi − βi)(βj − βj)]

is computed:

Cov(β) = E[(β − β)(β − β)T ]

= E[((XTX)−1XT ε)((XTX)−1XT ε)T ]

= E[(XTX)−1XT εεTX(XTX)−1]

= (XTX)−1XTE(εεT )X(XTX)−1

= (XTX)−1XTσ2IX(XTX)−1

= (XTX)−1σ2, (2.8)

where I is the n × n identity matrix. Since the true variance of ε, σ2, isunknown the estimated covariance matrix is given by

Cov(β) = (XTX)−1s2, (2.9)

where s2 = 1n−k−1ε

T ε is an unbiased estimator of σ2 [4]. As the estimated

variance of βi is given by the appropriate diagonal element of (2.9), theestimated standard error of βi:

SE(βi) =√

(XTX)−1(i+1,i+1)s

2. (2.10)

Note that the OLS estimator (2.6) requires a design matrix X of full columnrank due to the required inversion of XTX, a fact used in the section 2.3.

9

0 2 4 6 8

-2

0

2

4

6

8

x

y

yiβ0 + xβ1εi

Figure 2: An example of a straight line, y = β0 +xβ1, fitted byOLS regression to the data.

2.2.4 Hypothesis testing

Statistical testing is a procedure to either reject or accept a certain condition,called a statistical hypothesis. The null hypothesis H0 is tested against thealternative hypothesis HA. By the nature of random variables, an absolutestatement on which is true is impossible. Therefore a risk level is stated andif the test shows that the deviation from the null hypothesis is unlikely to befrom the random sampling, H0 is rejected [5]. Under H0, the test statisticwill be an observation of a certain random variable.

The F test variable

It is often of interest to test if one of the βs are statistically significant or ifit can be left out of the model. For a null hypothesis

H0 : βj = β0j

the F test variable is given by

F =( βj − β0j

SE(βj)

)2, (2.11)

with F ∈ F (1, n−k−1) under H0. H0 is rejected for F ≥ Fα(1, n−k−1), theα-quantile of an F -distribution with 1 numerator and n−k−1 denominatordegrees of freedom [4]. A common case is to test the hypothesis H0 : βj = 0.The procedure is outlined below when using the F -statistic as test variable.

10

1. State H0, HA and risk level, α

2. Compute the F test variable and possibly the p-value

3. If F ≥ Fα, the α quantile of the stated distribution, reject H0, other-wise accept H0

Note that F ≥ Fα ⇔ p-value < α. This means H0 is regarded unlikely andis rejected, see figure 3. The risk referred to is the risk of a Type I error,that is rejecting a true null hypothesis. α = 5% implies a true null will berejected in 5% of the tests. The error of accepting a false null hypothesis isreferred to as a Type II error.

0 2 4 6 8 10

0.0

0.5

1.0

1.5

F (1,50)density

Fα(1,50)

Figure 3: Example of an F (1,50) probability density functionwith the α = 5% quantile marked. If the computed test variableis greater than Fα, H0 is rejected.

2.2.5 Coefficient of determination R2 and effect size η2

The coefficient of determination and effect size are measures on the degreeof

explanation. The coefficient of determination is defined by

R2 =Var(Xβ)

Var(Y )=|ε0|2 − |ε|2

|ε0|2, (2.12)

where ε0 is from the regression on an intercept only. R2 is a measure on howmuch of the variation in y is explained by the variation in the covariates.

11

0 ≤ R2 ≤ 1 where a low value indicates a poor degree of explanation andvice versa. Similarly the effect size η2j is defined for each covariate xj as

η2j =|ε∗|2 − |ε|2

|ε∗|2, (2.13)

where ε∗ is from a restricted regression, that is with all covariates except xjincluded. η2 is a measure on the explanation of that specific covariate.

2.3 Problems from a miss-specified model

2.3.1 Multicollinearity

Multicollinearity is a term used when the columns in the design matrix Xhas a linear dependence or near linear dependence.

Perfect multicollinearity

Consider a regression of some response variable on gender with a model:

y = β0 + (male)β1 + (female)β2 + ε,

where parentheses indicate dummy variable - one which takes the values 1and 0; 1 if the relevant condition applies and 0 otherwise. It is common touse dummy variables for gender (male and female), season (spring, summer,...) and other conditions that are digital by nature. In the model above therows of X sum to the value 2. As an example we may get something like

X =

1 1 01 0 11 0 1...

......

1 1 0

.

Clearly, Rank(X) < 3 giving a singular matrix XTX. The OLS estimator(2.6) does not exist. The remedy in this case is to remove one of the covari-ates.

Near multicollinearity

More common is near multicollinearity, meaning there is a close to linear de-pendency between some of the columns in X. An example from populationsurveys including age, experience and education:

AGE− EDU− EXP ∼ 6

for many individuals giving a close to linear dependancy of the three covari-ates above and the intercept column. A near singular matrix XTX gives

12

large estimated standard errors, (2.10) for some of the covariates. This in-creases the risk of a type II error since the test variable F ∝ SE(βj)

−2,see equation (2.11). Multicollinearity could make the regression sensitive tosmall changes in data or model specification, essentially in the same way asa small sample size would [5].

Detecting multicollinearity

Large estimated standard errors for some of the covariates could indicate amulticollinearity problem. Another way of detecting multicollinearity is tocompute the variance inflation factors, VIFs. The variance of βj is given bythe appropriate diagonal element of the covariance matrix, which under theassumptions in section 2.2.2 could be written

Var(βj) =σ2

(n− 1)Var(xj)· 1

1−R2j

where the last factor is the variance inflation factor [6].

VIFj =1

1−R2j

. (2.14)

The coefficient of determination appearing in (2.14) is from a regression ofxj on all other covariates appearing in the model. There is no general valueof the VIF considered to invalidate the model. Modern textbooks puts lessemphasis on the problem of multicollinearity [5]. It is clear that for a givendata set, less multicollinearity is desired.

2.3.2 Heteroskedasticity

The assumption (2.4) that all residuals εi have the same standard deviation σis the homoskedastic assumption. If violated, the estimated standard errors(2.10) will be inconsistent [4], which in turn renders the F-test statisticinvalid (section 2.2.4). To overcome this, initially an attempt to reformulatethe model is made. If heteroskedasticity is still suspected, the use of White’sheteroskedasticity-consistent variance estimator should be used [7]. Thecovariance matrix (2.9) is adjusted to

Cov(β)White = (XTX)−1XTD(ε2)X(XTX)−1, (2.15)

where D(ε2) is an n × n diagonal matrix such that D(ε2)(i,i) = ε2i . Thisestimator is asymptotically unbiased as n increases. The covariance matrix(2.15) is usually scaled by n/(n− k − 1):

Cov(β)Scaled =n

n− k − 1(XTX)−1XTD(ε2)X(XTX)−1. (2.16)

13

Note that if the assumption (2.4) holds and (2.16) is used then

n

n− k − 1D(ε2)→ Is2

givingCov(β)Scaled = (XTX)−1s2 = Cov(β).

Therefore, there is no harm in using (2.16) when n is large even underhomoskedasticity, in fact it is recommended to do so [4]. Heteroskedasticityis discovered by inspecting the residuals when plotted against the predictedy-values, an example is shown in figure 4. Normality of the residuals couldbe checked by histograms or a so called PP-plot where the residuals areplotted against the expected distribution.

1.0 1.4 1.8

-0.2

-0.1

0.0

0.1

0.2

y

ε

1.0 1.4 1.8

-0.2

-0.1

0.0

0.1

0.2

y

ε

Figure 4: An example of heteroskedastic residuals is shownin the left plot. The right plot shows the homoskedastic casewhere the standard deviation is independent of y.

2.3.3 Endogeneity

If E(εi) 6= 0 because ε is correlated with some of the covariates, the termendogeneity is used. This is a problem when the model is given a structuralinterpretation as the relevant slope coefficient will be biased and the OLS willgive inconsistent estimates. Selection bias, simultaneity, missing covariatesand measurement error is common reasons for endogeneity. [4]

Two stage least squares

The most common method to deal with endogeneity is to use two-stage least

14

squares, 2SLS. A new instrument variable must be found, well correlatedwith the endogenous covariate and uncorrelated with the residual. Theinstrument variable and the exogenous covariates form the matrix Z. Theendogenous design matrix is then projected onto Z to give the new regressormatrix X:

X = Z(ZTZ)−1ZTX.

The normal equations are now

XT ε = 0 (2.17)

and the 2SLS estimation of β is given by using (2.17) in (2.2):

β = (XTX)−1XY

2.4 Model selection

2.4.1 Transformation of variables

A transformation of variables is sometimes recommended and sometimesnecessary.

Log-linear transformation

Recall that (2.1) gives the interpretation of βj as the impact on y whenincreasing xj by one unit. For a dependent variable that are naturallypositive this could impose problems. An example is hourly wages. A log-linear transformation gives the model

log(yi) = β0 + x1iβ1 + . . .+ xkiβk + εi, (2.18)

with the new interpretation that one unit increase in xj gives βj percentincrease in y. This transformation is sometimes preferred to reduce ho-moskedasticity [4].

Log-log transformation

If the covariates are positive by nature a log-log transformation could beused [5], giving the following model:

log(yi) = β0 + log(x1i)β1 + . . .+ log(xki)βk + εi (2.19)

with the interpretation that one percent increase in xj gives βj percent in-crease in y.

Non-linear dependencies

15

Sometimes, there is an expectation that the influence of a covariate is de-creasing with increasing value of that covariate. Consider the physical con-dition of an individual as a function of weekly exercise. A more or less linearreturn is expected for small values of exercise but a limit is reached. Moreexercise will not increase the performance of that individual. A decreasingreturn to hours of exercise is expected and the following model could beused:

log(cond) = β0 + hoursβ1 + hours2β2 + ε, (2.20)

where the expectation is β1 > 0 and β2 < 0 [5].

Interaction effects

If an interaction effect is suspected, such that one covariate will influencethe effect of another covariate, this could be included as an interaction term[5]. An example with lifetime expectancy on cigarettes per week and alcoholconsumption:

log(years) = β0 + cigβ1 + alcβ2 + cig · alcβ3 + ε. (2.21)

When including interactions and non-linearities the variance inflation factorwill increase significantly since the interaction term can be predicted fromthe other covariates with a non trivial degree of accuracy. This is an exampleof when prior knowledge on what covariates to expect significance from isrequired. Multicollinearity arising from interactions and quadratic terms isof less concern than two independent covariates with multicollinearity [8]

2.4.2 Selecting the best model

Selection of the best regression model is one of the most important aspects ofregression analysis. If k covariates including transformations are available,the total number of possible models is 2k, a severely large number even fora moderate number of possible covariates. To select which one to use fromthese 2k number of subsets several methods have been proposed.

Stepwise procedures

Efroymson introduced stepwise regression [9], which has been extensivelyexplored since. The method starts with an intercept only, then adds andremoves covariates based on F -statistics (2.11) in steps until the final modelis found [6]. Backwards elimination is similar to stepwise regression but itstart from the other end [6]. All covariates are included in the model fromstart, and are removed one by one until the model is finalized. The step-wise procedures has received criticism for overfitting data, selecting complexmodels and misuse of tests [10].

Selections based on Information Criterion

16

The Bayesian Information Criterion and Akaike Information Criterion areused to select the best model while avoiding overfitting [11]. The BIC isdefined as

BIC = n log(|ε|2/n) + k log(n), (2.22)

if the residuals are assumed normal i.i.d. The model that minimizes BIC isselected. The AIC is defined as

AIC = n log(|ε|2) + 2k. (2.23)

The BIC applies a larger penalty for complex models if n is large. Thisselection is preferably used in combination with some a-priori knowledge onreasonable models to use in each specific case.

3 Climb performance review

T

L

Dtot

W

γ

eyex

Figure 5: The forces acting on an airplane in the climb. L isthe lift, T is the thrust, Dtot is the drag and W is the weightof the airplane. The climb angle γ is the angle between thehorizontal plane and the flight path of the airplane.

Several factors affect the amount of fuel used, and the ground distancecovered by an aircraft in the climb. In this section basic aircraft performanceis reviewed to give a foundation for the model selection. The forces acting onan airplane is depicted in figure 5. The thrust is provided by the airplanesengines, the lift is an aerodynamic force produced mainly by the wings.Drag is the resistance to the forward motion of the airplane. In steadyclimb (non-accelerated climb) the sum of all forces are zero [12] , hence weget the following equations:

ex : T −Dtot −W sin(γ) = 0, (3.1)

ey : L−W cos(γ) = 0. (3.2)

17

The effect of certain factors on the climb performance is given in the follow-ing sections.

3.1 Airspeed

The airspeed of an airplane will affect the lift generated by the wings andthe drag of the airplane. The airspeed is defined as the speed of the aircraftrelative to the surrounding air. The standard formula for the lift generatedby the wing is given in the equation

L = 12ρv

2CLSw, (3.3)

where ρ is the air density, Sw is the surface area of the wing, v is the airspeedand CL is the coefficient of lift, which depends mainly on the shape and angleof attack of the wing [12]. Since the airplane will be established in a steadystate climb, the forces (fig. 5) will sum to zero. The lift will have to beconstant, this means given equation (3.3) that a lower speed will have to becompensated by a higher coefficient of lift and vice versa.

The drag of the airplane comes in two parts, parasite drag and induceddrag [12]. The parasite drag, or zero lift drag, D0 is due to skin friction,form drag and interference drag and is independent of the lift created. Theparasite drag will depend on the airspeed, the shape of the airplane, via thecoefficient CD0 , and the drag reference area SD according to

D0 = 12ρv

2CD0SD ∝ v2. (3.4)

The induced drag, DI , is an effect of the wing not being perfect in creatinglift. It is essentially a measure on how hard the wing has to work in creatingthe lift and it is proportional to the square of the coefficient of lift [12],

DI ∝ C2L

(3.3)=⇒ DI ∝

1

v4. (3.5)

Combining (3.4) and (3.5) there must be an airspeed for which the totaldrag Dtot = DI + D0 is minimum, VDmin, given that the airplane needs toproduce a certain amount of lift. The situation is depicted in figure 6. Withthe information above it is natural to expect a non linear dependancy onthe climb performance from the selected airspeed.

18

D

v

D0Dtot

DI

vDmin

Figure 6: The total drag, Dtot versus airspeed. The total dragwill be the sum of a term ∝ v2 and a term ∝ v−4. A very highairspeed would be inefficient due to high zero-lift drag while atoo low airspeed would be inefficient due to high induced drag.

3.2 Take Off Mass

The weight of the airplane in steady climb will depend on the mass, M , andthe gravitational acceleration as

W = Mg (3.6)

where g ≈ 9.81ms−2, which would be almost constant throughout the climb.From equation (3.2) we have that

L = W cos(γ). (3.7)

An increased mass will require more lift to be produced, and by (3.3) withconstant airspeed, this require the CL to be increased giving higher (induced)drag. This would then mean that the thrust need to be increased or the climbangle reduced by equation (3.1). Since the thrust is typically set independentof mass [3], an increased mass would mean lower climb angle.

3.3 Ambient temperature and pressure

The ambient temperature and pressure will affect the density, ρ. The idealgas law is an approximation to the relation between the temperature, T ,pressure, p and volume, V , of the air:

pV = NkBT ⇒ ρ =M

V∝ p

T. (3.8)

19

Both the lift (3.3) and the drag (3.4) are affected by the density ρ. Fur-thermore, the airplane of interest in present work is powered by turbopropengines [3], that is gas turbines providing the torque required to turn thepropellers creating the thrust. The propellers are essentially acting as wingscreating lift in the forward direction, subject to (3.3). As the air density isreduced the amount of oxygen available to the engine is reduced, thereby re-ducing the maximum torque available. On the other hand, as the density isreduced, the fuel required to drive the compressors and propeller is reducedimproving the efficiency [13]. The lower density at high altitude is, besidesless weather factors, the reason airplanes cruise at high level. To summarise,both the temperature and pressure should affect the performance as definedin (1.1). Since pressure is reducing with increased altitude, it is commonto express the ambient pressure at take off by aerodrome pressure altitudeaccording to

pa = elevation + (1013− p) · 30 [ft] (3.9)

where p is given as QNH, ambient pressure reduced to mean sea level al-titude. QNH is the pressure reported to pilots in units of millibars, 1013millibar is the mean sea level pressure.

3.4 Wind component

The effect of the wind is an important factor to consider. In the climb phasethe airplane will typically be flown at constant indicated airspeed [3]. Atailwind would give a higher groundspeed and vice versa. If the componentof the wind in the direction of flight is w and the airspeed vA, the groundspeed is given by

vG = vA + w. (3.10)

Clearly, the wind will affect the climb performance as defined in (1.1), seefigure 7. Due to ground friction, the wind will generally increase with al-titude. Since the covariate for wind component is taken as the forecastedwind at cruise level, this will not be accurate for the wind at lower levelseven if forecasted correctly.

20

Altitude

Distance

w < 0 w = 0 w > 0

Figure 7: The climb profile shown for different wind compo-nents, w. The climb angle is reduced as the altitude increases.This is mainly due to the torque available from the engines isreduced with altitude.

4 Hypotheses

Based on the performance review (section 3) the following were consideredlikely results:

H1 There is an optimum climb speed. Higher speeds will increase fuel perdistance as will lower speeds.

H2 The optimum speed will be affected by wind component. In case of atailwind, it is better to climb steep (low airspeed) to reach the strongertailwind at high altitude.

H3 The climb performance will be affected by wind component, ambienttemperature, pressure and take off mass.

5 Implementation

5.1 Preprocessing

The steps taken to finalise the model included treatment of data, regressionof the data using several possible models and evaluating each model. Forthe purpose of compact notation the nomenclature in table 2 is used in thissection. The implementation was made using the R-scripting [14]. The datawas manually entered to one csv-file per flight. Recall that the fuel perdistance was stated in equation (1.1) as

y =f

d

21

Table 2: Variable names used

Variable Description

y Fuel per distanceias Indicated airspeedwc Wind componentcrz Selected cruise leveltom Take off massoat Ambient temperaturepa Pressure altitude

where f is the total fuel consumed during the climb and d is the ground dis-tance covered. First, a script was built that collected all data and estimatedf and d from the data set of each flight. The estimated values of y togetherwith the available covariates (table 2) for each flight were then saved to acsv-file. Data from a total of 103 flights was used. This file was then loadedinto a script to build the model.

5.2 Selection of regression model

The selection of the regression model was based on a-priori knowledge onwhat covariates to use and expect significant. The insignificant covariateswere removed and the model confirmed by BIC, P-values and the relevantη2 for the covariate when adjusting the model specification.

Several models was tested in selecting the best model. The residualsshowed better normality when applying a log-transformation of the depen-dent variable. Initially models without interactions and quadratic effectswere explored. The final model was built using all reasonable covariatesincluding interactions with airspeed and a quadratic airspeed term. Themodel was then reduced by removing insignificant covariates until the BIC,(2.22), was minimised and all covariates were significant on level α = 5%.

6 Result

6.1 Selected final model

The model is given by

log(y) = β0 + ias · β1 + ias2 · β2 + ias · wc · β3 + ias · tom · β4++ wc · β5 + pa · β6 + log(crz)β7 + log(tom)β8 + ε

22

6.2 Result of the final regression

The coefficient estimates, their estimated standard errors, effects sizes andP-values from the final regression is given in table 3. The regression gave acoefficient of determination:

R2 = 0.93.

The residuals are shown in figure 8.

Table 3: Result of the final regresssion

Coeff. Covariate Estimate SE η2 p-value

β0 intercept 18 3.2 0.169 0.0000β1 ias -5.9×10−2 1.6×10−2 0.092 0.0003β2 ias.sq 1.0×10−4 4.2×10−5 0.045 0.0172β3 iasIwc 1.1×10−4 2.0×10−5 0.188 0.0000β4 iasItom 9.6×10−4 3.5×10−4 0.047 0.0067β5 wc -2.2×10−2 3.4×10−3 0.249 0.0000β6 pa 2.8×10−5 1.2×10−5 0.05 0.0261β7 lcrz -7.3×10−1 3.6×10−2 0.833 0.0000β8 ltom -3.1 1.0 0.053 0.0036

1.4 1.6 1.8 2.0

-0.10

-0.05

0.00

0.05

0.10

ε vs y Histogram

Den

sity

-2 -1 0 1 2

0.0

0.1

0.2

0.3

0.4

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

PP Plot

Exp

ecte

dP

rob

abil

ity

Figure 8: The residuals of from the final regression. Left is theresiduals plotted against the predicted y-values. Center is ahistogram showing standardised residual density and a N(0,1)distribution. Right shows the standardised residuals against aN(0,1) distribution.

23

7 Discussion

7.1 Discussion of the result

The influence of the covariates was mostly as expected. A positive windcomponent (tailwind) lowers the fuel per distance as will a higher selectedcruise level. The fact that ambient temperature was not significant couldbe due to the sampling period being short, extending through February andMarch. During this time of year the temperature is fairly constant whilea study over the whole year could give better significance for temperature.This could also be due to ambient temperature on the ground being a poorpredictor of the temperature throughout the climb, which is what wouldaffect the fuel per distance in the climb.

Intuitively the expectation is that a high take off mass would mean higherfuel per distance. The estimated coefficient for the logarithm of take offmass, however, is negative:

β8 = −3.1 (SE 1.0)

meaning one percent increase in the take off mass will reduce fuel per dis-tance by 3.1 percent. An explanation for this could be that since a highermass means lower climb angle and since less torque is available from theengines at higher altitude, more time will be spent there. Since the fuel con-sumption is lower at higher altitudes, this lowers the fuel used per distancein climb. This is an effect of the study being limited to the climb phaseonly. A study considering all phases of the flight would probably find thata higher take off mass means more fuel used.

The estimated coefficient for logarithm of selected cruise level is negativeas expected.

β7 = −0.73 (SE 0.036)

means that one percent increase in selected cruise level reduces the fuel perdistance by 0.73 percent.

The effect of the aerodrome pressure altitude was not expected to bepositive. The estimated coefficient for pressure altitude is

β6 = 2.8× 10−5 (SE 1.2× 10−5)

means an increase of the aerodrome pressure altitude by 100 feet wouldincrease the fuel per distance by 2.8 · 10−3 percent. This is not a largenumber in absolute values but the effect is statistically significant.

A positive wind component gives lower fuel per distance as expected.

β5 = −0.022 (SE 3.4× 10−3)

24

7.1.1 The optimal climb speed

Consider the equation for the predicted y-values:

log(y) = β0 + ias · β1 + ias2 · β2 + ias · wc · β3 + ias · tom · β4++ wc · β5 + pa · β6 + log(crz)β7 + log(tom)β8 + e (7.1)

where e is a random variable. The sign of β1 and β2 is as expected, that is anincrease from a low airspeed gives lower fuel consumption while an increasefrom a high airspeed gives higher fuel consumption. If (7.1) is taken asthe true expression for the fuel performance, the optimum airspeed couldbe computed given wind component and take off mass. Since log(y) is anincreasing function of y, minimising y is equivalent to minimizing log(y).Setting the derivative of (7.1) with respect to indicated airspeed to zerogives

∂ log(y)

∂ias= β1 + 2β2 · ias + β3 · wc + β4 · tom = 0 =⇒

iasopt = − β1

2β2− β3

2β2· wc− β4

2β2· tom. (7.2)

With the estimated parameter values from table 3 the equation becomes:

iasopt = 290− 0.54 · wc− 4.7 · tom. (7.3)

This suggests that a 10 knots tailwind would lower the optimum climb speedby 5.4 knots. Similarily a headwind increases the optimum climb speed.This is in agreement with the expected effect of optimum airspeed from windcomponent. The take off mass is in the range 14.9 ≤ tom ≤ 20.2. If using thelowest value of 14.9 for the take off mass and zero wind component we get anoptimum climb speed of 220 knots which is highly unlikely. The maximumoperating air speed for the Fokker 50 is 224 knots and 200 knots is considereda very high climb speed. This apparent failure of the model could be due toseveral reasons. One reason could be that some of the estimated coefficientsin (7.2) are endogenous. Since no instrument variables are available for thecommon tests (for example a Wu-Haussman test) for endogeneity, all thatcan be done is to inspect the plot of each suspected endogenous variableagainst the residuals. This show no sign of endogeneity (figure 9). Since theinteraction terms and quadratic terms are collinear the model gets sensitiveto miss specifications and changes in data. More data could be the solution.Another reason could be that the model is plain wrong. The fact that themodel was the best performing of many tested implies this was not the case.Finally this result could be to the fact that the study only focused on theclimb phase. Then it is essentially true that a high speed will minimisefuel per distance in the climb, but the fuel performance of the flight as a

25

whole would gain from a lower speed as the cruise level is reached earlier.This is taken as the reasonable explanation since this would also explain theunexpected positive coefficient for take off mass.

150 160 170 180 190

-0.10

-0.05

0.00

0.05

0.10

ias

resi

d

ε

ias25000 30000 35000

-0.10

-0.05

0.00

0.05

0.10

ias.sq

resi

d

ε

ias.sq

-10000 0 5000 10000

-0.10

-0.05

0.00

0.05

0.10

resi

d

ε

ias*wc2400 2800 3200

-0.10

-0.05

0.00

0.05

0.10

resi

d

ε

ias*tom

Figure 9: The residuals of from the final regression plottedagainst the covariates corresponding to the coefficients in equa-tion (7.2). Endogeneity could be discovered by the residualsbeing correlated with the covariates.

7.2 Future work

7.2.1 Feedback from the air crews

The feedback received from the pilots supplying the data calls for a revisionof the data collecting procedure. Even though consideration was given toease the workload as much as possible some pilots expressed concern thatthe collection interfered with the normal climb procedures. This is a seriousconcern as it might have impaired flight safety. The instructions to thepilots were to abort the data collection if flight safety so required, but is not

26

unlikely that some pilots felt that the data collection was important. Analternative approach is to record just distance from departure and fuel usedonce the top of climb was reached. This would have given less accurate fuelper distance hence greater residuals, but the lower workload for the pilotscould compensate this by enabling data from more flights to be collected.Yet another approach would be to use some kind of semi-automatic collectionof data by GPS or other means.

7.2.2 Other methods

The discussion above implies that the analysis of climb performance shouldconsider the cruise phase as well. Therefore a factor that penalises a latetop of climb should be included. This would essentially be an optimisationproblem: Reach the cruise level early and enjoy lower fuel consumption andhigher air speed in the cruise or cover more distance during climb?

This work investigated if the information known before flight is enoughto take a decision on what climb speed to use. Therefore OLS regression waschosen to investigate what model to use. Another method could be a quasi-analytical one. A deeper analytical review of the climb performance wouldthen give a more precise expression for the fuel per distance in the climb,most likely a non linear model would have to be used. Other statisticalmethods could then be used to fit the model to data.

27

8 Conclusion

8.1 Hypotheses

H1 There is an optimum climb speed. Higher speeds will increase fuel perdistance as will lower speeds.

This was confirmed but value of the optimum airspeed was deemedhighly unlikely.

H2 The optimum speed will be affected by wind component. In case of atailwind, it is better to climb steep (low airspeed) to reach the strongertailwind at high altitude.

This was confirmed as the optimum airspeed was found to increase by5.4 knots for every 10 knots headwind and vice versa.

H3 The climb performance will be affected by wind component, ambienttemperature, pressure and take off mass.

The climb performance was found to be affected by the wind compo-nent, pressure and take off mass while the ambient temperature didnot have a significant impact.

28

8.2 Research questions

Q1 What factors will significantly affect the climb performance?

The factors that affect the climb performance are selected airspeed,wind component, pressure altitude, selected cruise level and take offmass. A regression with coefficient of determination R2 = 0.93 wasmade on a model with a quadratic term for airspeed and interactionterms for airspeed and wind component and airspeed and take off massrespectively. Data from a total of 103 flights was used.

Q2 Given a set of parameters, known to the crew before the flight, what isthe optimum climb speed?

The optimum airspeed was found to be dependent on wind and takeoff mass according to equation (7.3). The optimum climb speed foundwas probably too high.

Q3 Is regression analysis based on manually collected data an acceptablemethod to investigate the climb performance of regional turboprop air-craft?

A deeper investigation of climb performance is required, and a moreadvanced model would require other methods than ordinary least squaresto estimate the model parameters. The manual data collection couldbe sufficient but a revision of the procedure to collect the data is calledfor.

29

References

[1] Hagmann C, Semeijn J, Vellenga DB. Exploring the green image of air-lines: Passenger perceptions and airline choice. Journal of Air TransportManagement. 2015;43:37–45.

[2] IATA. Economic Performance of the Airline In-dustry, end year 2014;. Available from: http:

//www.iata.org/whatwedo/Documents/economics/

IATA-Economic-Performance-of-the-Industry-end-year-2014-report.

pdf [cited 2015-04-10].

[3] Amapola Flyg AB. Operations Manual part B. Sturup, Malmo; 2015.

[4] Lang H. Elements of Regression Analysis; 2014. Course litterature,Applied Mathematical Statistics at KTH.

[5] Wooldridge JM. Introductury Econometrics. Cengage Learning; 2013.

[6] Draper NR, Smith H. Applied Regression Analysis. 3rd ed. John Wiley& Sons, Inc.; 1998.

[7] White H. A Heteroskedasticity-Consistent Covariance Matrix Esti-mator and a Direct Test for Heteroskedasticity. Econometrica. 1980May;48(4):817–138.

[8] Allison P. When Can You Safely Ignore Multicollinearity; 2012. Avail-able from: http://statisticalhorizons.com/multicollinearity

[cited 2015-04-29].

[9] Efroymson MA. Multiple Regression Analysis. Mathematical Methodsfor Digital Computers. 1960;p. 191–203.

[10] Flom PL, Cassel DL. Stopping stepwise: Why stepwise and similarselection methods are bad, and what you should use; 2007.

[11] Burnham KP, Andersson DR. Model Selection and Multimodel Infer-ence. 2nd ed. Springer; 2002.

[12] Boiardt K. Principles of Flight. 2nd ed. Lund University School ofAviation and Nordic Aviation Resources A.S.; 2002.

[13] Soares C. Gas Turbines: A Handbook of Air, Land, and Sea Applica-tions. Elsevier Inc.; 2008.

[14] R Core Team. R: A Language and Environment for Statistical Comput-ing. Vienna, Austria; 2014. Available from: http://www.R-project.

org.

30