12.1: Review of Dots and Boxes Model - Mathematics

Let’s start with a quick review of place value, different bases, and our “Dots & Boxes” model for thinking about these ideas.

The 1←2 Rule

Whenever there are two dots in single box, they “explode,” disappear, and become one dot in the box to the left.

Example: Nine dots 1←2 in the system

We start by placing nine dots in the rightmost box.

Two dots in that box explode and become one dot in the box to the left.

Once again, two dots in that box explode and become one dot in the box to the left.

We do it again!

Hey, now we have more than two dots in the second box, so those can explode and move!

And the rightmost box still has more than two dots.

Keep going, until no box has two dots.

After all this, reading from left to right we are left with one dot, followed by zero dots, zero dots, and one final dot.


The 2←1 code for nine dots is: 1001.

The 1←3 Rule

Whenever there are three dots in single box, they “explode,” disappear, and become one dot in the box to the left.

Example: Fifteen dots in the 1←3 system

Here’s what happens with fifteen dots:


The 1←3 code for fifteen dots is: 120.


Recall that numbers written in the 1←2 system are called binary or base two numbers.

Numbers written in the 1←3 system are called base three numbers.

Numbers written in the 1←4 system are called base four numbers.

Numbers written in the 1←10 system are called base ten numbers.

In general, numbers written in the 1←b system are called base b numbers.

In a base b number system, each place represents a power of b, which means (b^{n}) for some whole number n. Remember this means b multiplied by itself n times:

  • The right-most place is the units or ones place. (Why is this a power of b?)
  • The second spot is the “b” place. (In base ten, it’s the tens place.)
  • The third spot is the “(b^{2})” place. (In base ten, that’s the hundreds place. Note that (10^{2} = 100).)
  • The fourth spot is the “(b^{3})” place. (In base ten, that’s the thousands place, since (10^{3} = 1000).)
  • And so on.


Whenever we’re dealing with numbers written in different bases, we use a subscript to indicate the base so that there can be no confusion. So:

  • (102_{three}) is a base three number (read it as “one-zero-two base three”). This is the base three code for the number eleven.
  • (222_{four}) is a base four number (read it as “two-two-two base four”). This is the base four code for the number forty-two.
  • (54321_{ten}) is a base ten number. (It’s ok to say “fifty-four thousand three hundred and twenty-one.” Why?)

If the base is not written, we assume it’s base ten.

Remember: when you see the subscript, you are seeing the code for some number of dots.

Think / Pair / Share

Work through the two examples above carefully to be sure you remember and understand how the “Dots & Boxes” model works. Then answer these questions:

  • When we write 9 in base 2, why do we write (1001_{two}) instead of just (11_{two})?
  • When we write 15 in base 3, why do we write (120_{three}) instead of just (12_{three})?
  • How many different digits do you need in a base 7 system? In a base 12 system? In a base b system? How do you know?

On Your Own

Work on the following exercises on your own or with a partner.

  1. In base 4, four dots in one box are worth one dot in the box one place to the left.
    1. What is the value of each box?
    2. How do you write (29_{ten}) in base 4?
    3. How do you write (132_{four}) in base 10?
  2. In our familiar base ten system, ten dots in one box are worth one dot in the box one place to the left.
    1. What is the value of each box?
    2. When we write the base ten number 7842:
      • What quantity does the “7” represent?
      • The “4” is four groups of what value?
      • The “8” is eight groups of what value?
      • The “2” is two groups of what value?
  3. Write the following numbers of dots in base two, base three, base five, and base eight. Draw the “Dots & Boxes” model if it helps you remember how to do this! (Note: these numbers are all written in base ten. When we don’t say otherwise, you should assume base ten.) $$(a); 2 qquad (b); 17 qquad (c); 27 qquad (d); 63 ldotp$$
  4. Convert these numbers to our more familiar base ten system. Draw out dots and boxes and “unexplode” the dots if it helps you remember. $$(a); 1101_{two} qquad (b); 102_{three} qquad (c); 24_{five} qquad (d); 24_{nine} ldotp$$

Think / Pair / Share

Quickly compute each of the following. Write your answer in the same base as the problem.

  • (131_{ten}) times ten.
  • (263207_{eight}) times eight.
  • (563872_{nine}) times nine.
  • Use the 1←10 system to explain why multiplying a whole number in base ten by ten results in simply appending a zero to the right end of the number.
  • Suppose you have a whole number written in base b. What is the effect of multiplying that number by b? Justify what you say.

The importance of number sense to mathematics achievement in first and third grades ☆

Children's symbolic number sense was examined at the beginning of first grade with a short screen of competencies related to counting, number knowledge, and arithmetic operations. Conventional mathematics achievement was then assessed at the end of both first and third grades. Controlling for age and cognitive abilities (i.e., language, spatial, and memory), number sense made a unique and meaningful contribution to the variance in mathematics achievement at both first and third grades. Furthermore, the strength of the predictions did not weaken over time. Number sense was most strongly related to the ability to solve applied mathematics problems presented in various contexts. The number sense screen taps important intermediate skills that should be considered in the development of early mathematics assessments and interventions.

Plain Language Summary

Dominant modes (i.e., coherent spatio-temporal patterns of variability) of the climate system, such as the Madden-Julian Oscillation (MJO), influence a wide range of weather and climate phenomena worldwide. The ability of state-of-the-art climate models to accurately simulate these modes is crucial for advancing our understanding of the climate system and reliably predicting its future trends. The Coupled Model Intercomparison Project Phase 6 (CMIP6) will be the foundation for the upcoming Intergovernmental Panel on Climate Change (IPCC) Sixth Assessment Report (AR6). Here, we use a wavelet-based spectral principal component analysis (wsPCA) to quantitatively assess how well historical simulations from 20 CMIP6 models capture MJO as compared to the observations. We first show that the MJO magnitude is not reproduced well in most of CMIP6 models. We then reveal that MJO-related precipitation variability in the Amazonia, Southwest Africa, and Maritime Continent is significantly underestimated in many CMIP6 models. Our results highlight the need to better simulate the coupled ocean-atmosphere dynamics in order to improve the representation of MJO in climate models and increase confidence in projected states of MJO for assessing future tropical and extratropical impacts.

Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures, 4th Edition

Quantum Wells, Wires and Dots provides all the essential information, both theoretical and computational, to develop an understanding of the electronic, optical and transport properties of these semiconductor nanostructures. The book will lead the reader through comprehensive explanations and mathematical derivations to the point where they can design semiconductor nanostructures with the required electronic and optical properties for exploitation in these technologies.

This fully revised and updated 4th edition features new sections that incorporate modern techniques and extensive new material including:

  • Properties of non-parabolic energy bands
  • Matrix solutions of the Poisson and Schrödinger equations
  • Critical thickness of strained materials
  • Carrier scattering by interface roughness, alloy disorder and impurities
  • Density matrix transport modelling
  • Thermal modelling

Written by well-known authors in the field of semiconductor nanostructures and quantum optoelectronics, this user-friendly guide is presented in a lucid style with easy to follow steps, illustrative examples and questions and computational problems in each chapter to help the reader build solid foundations of understanding to a level where they can initiate their own theoretical investigations. Suitable for postgraduate students of semiconductor and condensed matter physics, the book is essential to all those researching in academic and industrial laboratories worldwide.

2 Answers 2

Let me fill in a few details of Noah's excellent answer. The empty theory $T$ in the language $mathcal=<=>$ is just that – the theory that contains no $mathcal$ -sentences. Any set is a model of $T$ . You ask if $T$ implies everything this is not the case. Indeed, the sentences derivable from $T$ are precisely the $mathcal$ -sentences that hold in every set. These are also called the tautologies of the predicate calculus. For instance, the following is a theorem of $T$ : $forall vforall w(v=wvee v eq w),$ since it is true in every set (ie, true in every model of $T$ ). On the other hand, consider the $mathcal$ -sentence $phi:equivexists vexists w(v eq w)$ . $phi$ is not a theorem of $T$ , since there are models of $T$ in which it does not hold. For example, consider a model $$ consisting of a single element. Note that $ egphi$ is not a theorem of $T$ either, since $phi$ holds in the model $$ consisting of $2$ elements. So, in other words, both $Tcup$ and $Tcup< egphi>$ are consistent. Hopefully this clarifies the question a bit.

Now, recall that $S_0(T)$ is the set of all complete (and consistent) $mathcal$ -theories that contain $T$ . Since every $mathcal$ -theory contains $T$ (because the empty set is a subset of any set), this means $S_0(T)$ is just the set of all complete $mathcal$ -theories. Topologically, a basis of clopen sets for $S_0(T)$ is parametrized by the set of all $mathcal$ -sentences: for any $mathcal$ -sentence $phi$ , we define $[phi]subseteq S_0(T)$ to be the set of all elements of $S_0(T)$ that contain $phi$ . So, if $phiequivforall vforall w(v=wwedge v eq w)$ , then $[phi]=S_0(T)$ , since $phi$ holds in any model of $T$ . On the other hand, $[ egphi]=emptyset$ . (Why?)

For a less vacuous example, consider the sentence $phiequivexists vexists w(v eq w)$ . Then both $[phi]$ and $[ egphi]$ are non-empty. Indeed, if $M$ is a model of $phi$ (say $M=$ ), then the set of all $mathcal$ -sentences that hold in $M$ , denoted $operatorname_mathcal M$ , is a complete and consistent $mathcal$ -theory containing $phi$ . So $operatorname_mathcalMin [phi]$ . Likewise, if $N$ is a model of $ egphi$ (say $N=$ ), then $operatorname_mathcal Nin[ egphi]$ .

So, this is the topology we're working with. Recall that a point $x$ in an arbitrary topological space $X$ is "isolated" if $$ is an open subset of $X$ . In particular, a complete $mathcal$ -theory $S$ is isolated in $S_0(T)$ if $$ is open. As an exercise, show that this holds if and only if there exists an $mathcal$ -sentence $phi$ such that $[phi]=$ . Furthermore, can you show that, for such $phi$ , we have $Tmodelsphi opsi$ for every $psiin S$ ? We then say that $S$ is isolated by $phi$ . As a final hint, to complement Noah's answer, consider the sentence $phi_nequivexists v_1dotsexists v_nigwedge_v_i eq v_jwedgeforall wigvee_^n w=v_i.$ Up to isomorphism, what does a model of $phi_n$ look like? (Can there be two non-isomorphic models of $phi_n$ ?) In particular, what can you conclude about the open set $[phi_n]subseteq S_0(T)$ ? (Remember that two isomorphic $mathcal$ -structures $Mcong N$ are elementarily equivalent: $operatorname_mathcalM=operatorname_mathcalN$ .) I'll leave my comments at this hopefully you can now work out the details yourself.

Edit: Okay, here's a complete solution to the question. At each stage, try to read a small bit at a time and work out the rest for yourself! For each $ninmathbb$ , let $phi_n$ be the sentence given above, which says that there exist precisely (and no more than) $n$ elements. Now, because $mathcal$ consists of only equality, any function is an $mathcal$ -embedding, and a function between $mathcal$ -structures is an isomorphism if and only if it is a bijection. In particular, suppose we have models $Mmodelsphi_n$ and $Nmodelsphi_n$ for some $ninmathbb$ . Then $|M|=|N|=n$ (why?) and so we can find a bijection – ie an $mathcal$ -isomorphism – $M o N$ . In particular, since isomorphic structures are elementarily equivalent, this means $operatorname_mathcalM=operatorname_mathcalN$ , and so (since $M$ and $N$ were arbitrary) this shows that any two models of the sentence $phi_n$ have the same $mathcal$ -theory. Denoting this theory by $T_n$ , we then have $[phi_n]=$ , and so in particular $T_n$ is isolated.

Now, there are $mathcal$ -structures in which $phi_n$ does not hold for any $ninmathbb$ – just consider any infinite set. To deal with these models, first fix any infinite cardinal $kappageqslantaleph_0$ , and suppose that $M$ and $N$ are two models of $T$ of size $kappa$ . (Ie, two sets of size $kappa$ .) Then there exists a bijection $M o N$ , which is an $mathcal$ -isomorphism by the comments above, and so $operatorname_mathcalM=operatorname_mathcalN$ . Thus any two models of cardinality $kappa$ have the same complete $mathcal$ -theory, for any $kappageqslantaleph_0$ . In fact, this means that any two infinite models have the same complete $mathcal$ -theory! Indeed, suppose $|M|=kappa$ and $|N|=lambda$ for $kappa,lambdageqslantaleph_0$ . Without loss of generality assume $kappageqslantlambda$ . Since $mathcal$ is finite, the Löwenheim-Skolem theorem tells us that there exists an elementary substructure $M'preccurlyeq M$ with $|M'|=lambda$ . Then $operatorname_mathcalM=operatorname_mathcalM'$ , and, by the argument above, since $|M'|=lambda=|N|$ , we also have $operatorname_mathcalM'=operatorname_mathcalN$ , so $operatorname_mathcalM=operatorname_mathcalN$ as desired. Thus any two infinite models of $T$ have the same complete $mathcal$ -theory call this theory $T_infty$ .

Now, we first claim that $S_0(T)=_<>>cup$ . Indeed, let $M$ be any model of $T$ (ie any set). If $M$ is finite, then $operatorname_mathcalM=T_<|M|>$ , and if $M$ is infinite, then $operatorname_mathcalM=T_infty$ , by the arguments above. In particular, every complete $mathcal$ -theory appears in $_<>>cup$ , as desired. By the argument above we've shown that each $T_n$ is isolated, by the formula $phi_n$ , so it remains only to show that the sequence $(T_n)_<>>$ converges to $T_infty$ . This amounts to the following statement: for any $mathcal$ -sentence $phi$ with $T_inftyin[phi]$ , there exists some $kinmathbb$ such that $T_nin[phi]$ for any $ngeqslant k$ .

To see this, suppose $phi$ is an $mathcal$ -sentence such that $T_n otin[phi]$ for arbitrarily large values of $n$ . This means that, for every $kinmathbb$ , we can find $ngeqslant k$ such that $phi$ does not hold in any $mathcal$ -structure of size $k$ . By the compactness theorem, this means that the following family of sentences is consistent: $Sigma=v_i eq v_j>_<>>cup< egphi>.$ Indeed, for any finite subset $DeltasubsetSigma$ , there will be a largest value of $k$ such that the sentence $exists v_1dotsexists v_kigwedge_v_i eq v_j$ appears in $Delta$ . We will then have $MmodelsDelta$ for any set $M$ such that $kleqslant|M|inmathbb$ and $T_<|M|> otin[phi]$ , meaning that $Delta$ is consistent. So, by compactness, $Sigma$ is consistent, and hence we can find a model $MmodelsSigma$ . Then $M$ is infinite (why?), so $operatorname_mathcalM=T_infty$ (why?), but $Mmodels egphi$ , whence $T_infty otin[phi]$ . This proves the desired result, and so we are done.

Seeing quadratics in a new light: secondary mathematics pre-service teachers’ creation of figural growing patterns

Teachers whose mathematical meanings support understanding across different contexts are likely to convey them in productive ways for coherent student learning. This exploratory study sought to elicit 67 secondary mathematics pre-service teachers’ (PSTs) meanings for quadratics with growing pattern creation and multiple translation tasks. A framework for categorizing their patterns, in terms of structural correspondence to quadratic equation forms, is shared. PSTs’ translations and explanations were analyzed using an existing framework for representational fluency. Evidence was found for some PSTs applying prior productive meanings to an unfamiliar context and others discovering new meanings. It was found that some PSTs were not able to draw on their prior knowledge of quadratic functions successfully, and possible reasons for this are explored. Successful figural pattern creation was found to be associated with higher representational fluency. Seven PSTs were also interviewed to probe their cognitive and affective experiences with a “new” type of quadratics task several described “aha” moments and used emotional languages like “horror” and “delight”. Implications for learning to teach quadratic functions, creative visualization tasks and future research are discussed.

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Upper and Lower Deviations of the Speed

This section is devoted to the proof of Lemma 3.2 and Corollary 3.3. We will prove Lemma 3.2 for (v_+) only but exactly the same proof, with symmetric arguments, holds for (v_-) .

Proof of corollaries

We start by showing how Lemma 3.2 implies Corollary 3.3.

Proof of Corollary 3.3

First note that, by the definition of (v_+( ho )) and (v_-( ho )) , we have that, for any (epsilon >0) ,

Note also that, for any (v_1,v_2in mathbb ) such that (v_1<v_2) and any (H>0) ,

We start by showing that either (v_+( ho )ge 0) or (v_-( ho ) le 0) . Indeed, assume that (v_+( ho )<0) and (v_-( ho )>0) and fix any (epsilon in (0,v_-( ho )/4)) . Lemma 3.2 implies that, for (H in mathbb ) large enough,

Since (epsilon <v_-( ho )-epsilon ) , we obtain from (5.1) and (5.3) that (p_(epsilon , ho ) + ilde

_(v_-( ho ) - epsilon , ho ) < 1) , as soon as i is large enough. This contradicts (5.2).

Now consider the case (v_+( ho )ge 0) . Assume, by contradiction, (v_-( ho )>v_+( ho )) . Fix (epsilon in (0,(v_-( ho )-v_+( ho ))/4)) so that (v_+( ho )+epsilon <v_-( ho )-epsilon ) . By Lemma 3.2, for any (Hin mathbb ) large enough, (p_(v_+( ho )+epsilon , ho )<1/2) . Thus, as soon as i is large enough, we obtain from (5.2) that (p_(v_+( ho ) + epsilon , ho ) + ilde

_(v_-( ho ) - epsilon , ho ) < 1) , which contradicts (5.2) once more. Thus, (v_+( ho )ge 0) implies (v_-( ho )le v_+( ho )) .

By a symmetric argument, (v_-( ho )le 0) implies (v_-( ho )le v_+( ho )) . This completes the proof that (v_-( ho )le v_+( ho )) . (square )

Now, we prove that Theorems 3.4 and 3.5 imply Theorem 2.2, stating that the random walk on the Exclusion process with density 1/2 has zero speed when (p_circ =1-p_ullet ) .

Proof of Theorem 2.2

Note that the law of the exclusion process with ( ho =1/2) is invariant under flipping colors (ullet leftrightarrow circ ) . Thus, for any (p, q in [0,1]) we have (mathbb

^<1/2>_ = mathbb

^<1/2>_) which implies

Furthermore, in the particular case (q = 1-p) we have, for any ( ho ge 0) , any (yin mathbb ) and any Borel set (A in mathbb ) ,

$egin mathbb

^< ho >_ ig [ X^y_H - pi _1(y) in Aig ] = mathbb

^< ho >_ ig [- ig (X^y_H - pi _1(y)ig ) in A ig ]. end$

Volume 3 Ordinary Residuals versus Predicted Residuals

This scatterplot has the predicted residuals ei,−i on the horizontal axis and the ordinary residuals ei on the vertical axis 7 ( Figure 23 ). It is expected that the residuals and the predicted residuals should be of about the same size. Hence, the values are expected to fall randomly at the center along the diagonal line of the plot. A y-outlier will be badly fitted. Hence, it will have both large ei and large ei,−i (point A). A bad leverage point will be fitted much better when it is included in the model than when it is excluded hence its ei,−i will be much larger than its ei (point B).

Figure 23 . Ordinary residuals ei versus predicted residuals ei,−i. Point A is a y-outlier. Point B is a bad leverage point.


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Efficient high-dimensional metamodeling strategy using recursive decomposition coupled with sequential sampling method

Metamodel has been widely used to solve computationally expensive engineering problems, and there have been many studies on how to efficiently and accurately generate metamodels with limited number of samples. However, applications of these methods could be limited in high-dimensional problems since it is still challenging due to curse of dimensionality to generate accurate metamodels in high-dimensional design space. In this paper, recursive decomposition coupled with a sequential sampling method is proposed to identify latent decomposability and efficiently generate high-dimensional metamodels. Whenever a new sample is inserted, variable decomposition is repeatedly performed using interaction estimation from a full-dimension Kriging metamodel. The sampling strategy of the proposed method consists of two units: decomposition unit and accuracy improvement unit. Using the proposed method, latent decomposability of a function can be identified using reasonable number of samples, and a high-dimensional metamodel can be generated very efficiently and accurately using the identified decomposability. Numerical examples using both decomposable and indecomposable problems show that the proposed method shows reasonable decomposition results, and thus improves metamodel accuracy using similar number of samples compared with conventional methods.

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Forecasting: Principles and Practice (2nd ed)

Holt (1957) and Winters (1960) extended Holt’s method to capture seasonality. The Holt-Winters seasonal method comprises the forecast equation and three smoothing equations — one for the level (ell_t) , one for the trend (b_t) , and one for the seasonal component (s_t) , with corresponding smoothing parameters (alpha) , (eta^*) and (gamma) . We use (m) to denote the frequency of the seasonality, i.e., the number of seasons in a year. For example, for quarterly data (m=4) , and for monthly data (m=12) .

There are two variations to this method that differ in the nature of the seasonal component. The additive method is preferred when the seasonal variations are roughly constant through the series, while the multiplicative method is preferred when the seasonal variations are changing proportional to the level of the series. With the additive method, the seasonal component is expressed in absolute terms in the scale of the observed series, and in the level equation the series is seasonally adjusted by subtracting the seasonal component. Within each year, the seasonal component will add up to approximately zero. With the multiplicative method, the seasonal component is expressed in relative terms (percentages), and the series is seasonally adjusted by dividing through by the seasonal component. Within each year, the seasonal component will sum up to approximately (m) .

Holt-Winters’ additive method

The component form for the additive method is: [egin hat_ &= ell_ + hb_ + s_ ell_ &= alpha(y_ - s_) + (1 - alpha)(ell_ + b_) b_ &= eta^*(ell_ - ell_) + (1 - eta^*)b_ s_ &= gamma (y_-ell_-b_) + (1-gamma)s_, end] where (k) is the integer part of ((h-1)/m) , which ensures that the estimates of the seasonal indices used for forecasting come from the final year of the sample. The level equation shows a weighted average between the seasonally adjusted observation ((y_ - s_)) and the non-seasonal forecast ((ell_+b_)) for time (t) . The trend equation is identical to Holt’s linear method. The seasonal equation shows a weighted average between the current seasonal index, ((y_-ell_-b_)) , and the seasonal index of the same season last year (i.e., (m) time periods ago).

The equation for the seasonal component is often expressed as [ s_ = gamma^* (y_-ell_)+ (1-gamma^*)s_. ] If we substitute (ell_t) from the smoothing equation for the level of the component form above, we get [ s_ = gamma^*(1-alpha) (y_-ell_-b_)+ [1-gamma^*(1-alpha)]s_, ] which is identical to the smoothing equation for the seasonal component we specify here, with (gamma=gamma^*(1-alpha)) . The usual parameter restriction is (0legamma^*le1) , which translates to (0legammale 1-alpha) .

Holt-Winters’ multiplicative method

The component form for the multiplicative method is: [egin hat_ &= (ell_ + hb_)s_ ell_ &= alpha frac<>><>> + (1 - alpha)(ell_ + b_) b_ &= eta^*(ell_-ell_) + (1 - eta^*)b_ s_ &= gamma frac<>> <(ell_+ b_)> + (1 - gamma)s_ end]

Example: International tourist visitor nights in Australia

We apply Holt-Winters’ method with both additive and multiplicative seasonality to forecast quarterly visitor nights in Australia spent by international tourists. Figure 7.6 shows the data from 2005, and the forecasts for 2016–2017. The data show an obvious seasonal pattern, with peaks observed in the March quarter of each year, corresponding to the Australian summer.

Figure 7.6: Forecasting international visitor nights in Australia using the Holt-Winters method with both additive and multiplicative seasonality.

Table 7.3: Applying Holt-Winters’ method with additive seasonality for forecasting international visitor nights in Australia. Notice that the additive seasonal component sums to approximately zero. The smoothing parameters and initial estimates for the components have been estimated by minimising RMSE ( (alpha=0.306) , (eta^*=0.0003) , (gamma=0.426) and RMSE (=1.763) ).
(t) (y_t) (ell_t) (b_t) (s_t) (hat_t)
2004 Q1 -3 9.70 2004 Q2 -2 -9.31 2004 Q3 -1 -1.69 2004 Q4 0 32.26 0.70 1.31 2005 Q1 1 42.21 32.82 0.70 9.50 42.66 2005 Q2 2 24.65 33.66 0.70 -9.13 24.21 2005 Q3 3 32.67 34.36 0.70 -1.69 32.67 2005 Q4 4 37.26 35.33 0.70 1.69 36.37 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 2015 Q1 41 73.26 59.96 0.70 12.18 69.05 2015 Q2 42 47.70 60.69 0.70 -13.02 47.59 2015 Q3 43 61.10 61.96 0.70 -1.35 59.24 2015 Q4 44 66.06 63.22 0.70 2.35 64.22 (h) (hat_) 2016 Q1 1 76.10 2016 Q2 2 51.60 2016 Q3 3 63.97 2016 Q4 4 68.37 2017 Q1 5 78.90 2017 Q2 6 54.41 2017 Q3 7 66.77 2017 Q4 8 71.18 Table 7.4: Applying Holt-Winters’ method with multiplicative seasonality for forecasting international visitor nights in Australia. Notice that the multiplicative seasonal component sums to approximately (m=4) . The smoothing parameters and initial estimates for the components have been estimated by minimising RMSE ( (alpha=0.441) , (eta^*=0.030) , (gamma=0.002) and RMSE (=1.576) ).
(t) (y_t) (ell_t) (b_t) (s_t) (hat_t)
2004 Q1 -3 1.24 2004 Q2 -2 0.77 2004 Q3 -1 0.96 2004 Q4 0 32.49 0.70 1.02 2005 Q1 1 42.21 33.51 0.71 1.24 41.29 2005 Q2 2 24.65 33.24 0.68 0.77 26.36 2005 Q3 3 32.67 33.94 0.68 0.96 32.62 2005 Q4 4 37.26 35.40 0.70 1.02 35.44 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 2015 Q1 41 73.26 58.57 0.66 1.24 72.59 2015 Q2 42 47.70 60.42 0.69 0.77 45.62 2015 Q3 43 61.10 62.17 0.72 0.96 58.77 2015 Q4 44 66.06 63.62 0.75 1.02 64.38 (h) (hat_) 2016 Q1 1 80.09 2016 Q2 2 50.15 2016 Q3 3 63.34 2016 Q4 4 68.18 2017 Q1 5 83.80 2017 Q2 6 52.45 2017 Q3 7 66.21 2017 Q4 8 71.23

The applications of both methods (with additive and multiplicative seasonality) are presented in Tables 7.3 and 7.4 respectively. Because both methods have exactly the same number of parameters to estimate, we can compare the training RMSE from both models. In this case, the method with multiplicative seasonality fits the data best. This was to be expected, as the time plot shows that the seasonal variation in the data increases as the level of the series increases. This is also reflected in the two sets of forecasts the forecasts generated by the method with the multiplicative seasonality display larger and increasing seasonal variation as the level of the forecasts increases compared to the forecasts generated by the method with additive seasonality.

The estimated states for both models are plotted in Figure 7.7. The small value of (gamma) for the multiplicative model means that the seasonal component hardly changes over time. The small value of (eta^<*>) for the additive model means the slope component hardly changes over time (check the vertical scale). The increasing size of the seasonal component for the additive model suggests that the model is less appropriate than the multiplicative model.

Figure 7.7: Estimated components for the Holt-Winters method with additive and multiplicative seasonal components.

Holt-Winters’ damped method

Damping is possible with both additive and multiplicative Holt-Winters’ methods. A method that often provides accurate and robust forecasts for seasonal data is the Holt-Winters method with a damped trend and multiplicative seasonality: [egin hat_ &= left[ell_ + (phi+phi^2 + dots + phi^)b_ ight]s_. ell_ &= alpha(y_ / s_) + (1 - alpha)(ell_ + phi b_) b_ &= eta^*(ell_ - ell_) + (1 - eta^*)phi b_ s_ &= gamma frac<>> <(ell_+ phi b_)> + (1 - gamma)s_. end]

Example: Holt-Winters method with daily data

The Holt-Winters method can also be used for daily type of data, where the seasonal period is (m=7) , and the appropriate unit of time for (h) is in days. Here, we generate daily forecasts for the last five weeks for the hyndsight data, which contains the daily pageviews on the Hyndsight blog for one year starting April 30, 2014.

Figure 7.8: Forecasts of daily pageviews on the Hyndsight blog.

Clearly the model has identified the weekly seasonal pattern and the increasing trend at the end of the data, and the forecasts are a close match to the test data.